Properties

Label 380.3.e.a.341.1
Level $380$
Weight $3$
Character 380.341
Analytic conductor $10.354$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,3,Mod(341,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.341");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 341.1
Root \(-4.83157i\) of defining polynomial
Character \(\chi\) \(=\) 380.341
Dual form 380.3.e.a.341.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.83157i q^{3} +2.23607 q^{5} -3.72856 q^{7} -14.3441 q^{9} +O(q^{10})\) \(q-4.83157i q^{3} +2.23607 q^{5} -3.72856 q^{7} -14.3441 q^{9} -13.8938 q^{11} +4.10699i q^{13} -10.8037i q^{15} -15.3087 q^{17} +(-16.4588 - 9.49250i) q^{19} +18.0148i q^{21} +17.4452 q^{23} +5.00000 q^{25} +25.8203i q^{27} -16.3924i q^{29} -15.9121i q^{31} +67.1291i q^{33} -8.33731 q^{35} +5.14304i q^{37} +19.8432 q^{39} +42.2785i q^{41} +1.56664 q^{43} -32.0743 q^{45} -54.5821 q^{47} -35.0979 q^{49} +73.9649i q^{51} +55.9822i q^{53} -31.0676 q^{55} +(-45.8637 + 79.5219i) q^{57} -101.822i q^{59} +80.5719 q^{61} +53.4827 q^{63} +9.18350i q^{65} +24.8838i q^{67} -84.2876i q^{69} -121.245i q^{71} -33.9798 q^{73} -24.1579i q^{75} +51.8040 q^{77} -102.623i q^{79} -4.34417 q^{81} +80.3324 q^{83} -34.2312 q^{85} -79.2009 q^{87} +12.0884i q^{89} -15.3131i q^{91} -76.8802 q^{93} +(-36.8030 - 21.2259i) q^{95} -161.837i q^{97} +199.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0
\(3\) 4.83157i 1.61052i −0.592919 0.805262i \(-0.702025\pi\)
0.592919 0.805262i \(-0.297975\pi\)
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) −3.72856 −0.532651 −0.266326 0.963883i \(-0.585810\pi\)
−0.266326 + 0.963883i \(0.585810\pi\)
\(8\) 0 0
\(9\) −14.3441 −1.59379
\(10\) 0 0
\(11\) −13.8938 −1.26308 −0.631539 0.775344i \(-0.717576\pi\)
−0.631539 + 0.775344i \(0.717576\pi\)
\(12\) 0 0
\(13\) 4.10699i 0.315922i 0.987445 + 0.157961i \(0.0504920\pi\)
−0.987445 + 0.157961i \(0.949508\pi\)
\(14\) 0 0
\(15\) 10.8037i 0.720248i
\(16\) 0 0
\(17\) −15.3087 −0.900510 −0.450255 0.892900i \(-0.648667\pi\)
−0.450255 + 0.892900i \(0.648667\pi\)
\(18\) 0 0
\(19\) −16.4588 9.49250i −0.866253 0.499605i
\(20\) 0 0
\(21\) 18.0148i 0.857847i
\(22\) 0 0
\(23\) 17.4452 0.758486 0.379243 0.925297i \(-0.376184\pi\)
0.379243 + 0.925297i \(0.376184\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) 25.8203i 0.956307i
\(28\) 0 0
\(29\) 16.3924i 0.565254i −0.959230 0.282627i \(-0.908794\pi\)
0.959230 0.282627i \(-0.0912059\pi\)
\(30\) 0 0
\(31\) 15.9121i 0.513292i −0.966505 0.256646i \(-0.917383\pi\)
0.966505 0.256646i \(-0.0826175\pi\)
\(32\) 0 0
\(33\) 67.1291i 2.03422i
\(34\) 0 0
\(35\) −8.33731 −0.238209
\(36\) 0 0
\(37\) 5.14304i 0.139001i 0.997582 + 0.0695005i \(0.0221405\pi\)
−0.997582 + 0.0695005i \(0.977859\pi\)
\(38\) 0 0
\(39\) 19.8432 0.508800
\(40\) 0 0
\(41\) 42.2785i 1.03118i 0.856835 + 0.515591i \(0.172428\pi\)
−0.856835 + 0.515591i \(0.827572\pi\)
\(42\) 0 0
\(43\) 1.56664 0.0364334 0.0182167 0.999834i \(-0.494201\pi\)
0.0182167 + 0.999834i \(0.494201\pi\)
\(44\) 0 0
\(45\) −32.0743 −0.712763
\(46\) 0 0
\(47\) −54.5821 −1.16132 −0.580660 0.814146i \(-0.697206\pi\)
−0.580660 + 0.814146i \(0.697206\pi\)
\(48\) 0 0
\(49\) −35.0979 −0.716283
\(50\) 0 0
\(51\) 73.9649i 1.45029i
\(52\) 0 0
\(53\) 55.9822i 1.05627i 0.849161 + 0.528134i \(0.177108\pi\)
−0.849161 + 0.528134i \(0.822892\pi\)
\(54\) 0 0
\(55\) −31.0676 −0.564865
\(56\) 0 0
\(57\) −45.8637 + 79.5219i −0.804626 + 1.39512i
\(58\) 0 0
\(59\) 101.822i 1.72580i −0.505376 0.862899i \(-0.668646\pi\)
0.505376 0.862899i \(-0.331354\pi\)
\(60\) 0 0
\(61\) 80.5719 1.32085 0.660426 0.750892i \(-0.270376\pi\)
0.660426 + 0.750892i \(0.270376\pi\)
\(62\) 0 0
\(63\) 53.4827 0.848932
\(64\) 0 0
\(65\) 9.18350i 0.141285i
\(66\) 0 0
\(67\) 24.8838i 0.371400i 0.982606 + 0.185700i \(0.0594552\pi\)
−0.982606 + 0.185700i \(0.940545\pi\)
\(68\) 0 0
\(69\) 84.2876i 1.22156i
\(70\) 0 0
\(71\) 121.245i 1.70767i −0.520545 0.853835i \(-0.674271\pi\)
0.520545 0.853835i \(-0.325729\pi\)
\(72\) 0 0
\(73\) −33.9798 −0.465476 −0.232738 0.972539i \(-0.574769\pi\)
−0.232738 + 0.972539i \(0.574769\pi\)
\(74\) 0 0
\(75\) 24.1579i 0.322105i
\(76\) 0 0
\(77\) 51.8040 0.672779
\(78\) 0 0
\(79\) 102.623i 1.29902i −0.760351 0.649512i \(-0.774973\pi\)
0.760351 0.649512i \(-0.225027\pi\)
\(80\) 0 0
\(81\) −4.34417 −0.0536317
\(82\) 0 0
\(83\) 80.3324 0.967860 0.483930 0.875107i \(-0.339209\pi\)
0.483930 + 0.875107i \(0.339209\pi\)
\(84\) 0 0
\(85\) −34.2312 −0.402720
\(86\) 0 0
\(87\) −79.2009 −0.910355
\(88\) 0 0
\(89\) 12.0884i 0.135824i 0.997691 + 0.0679121i \(0.0216338\pi\)
−0.997691 + 0.0679121i \(0.978366\pi\)
\(90\) 0 0
\(91\) 15.3131i 0.168276i
\(92\) 0 0
\(93\) −76.8802 −0.826669
\(94\) 0 0
\(95\) −36.8030 21.2259i −0.387400 0.223430i
\(96\) 0 0
\(97\) 161.837i 1.66842i −0.551447 0.834210i \(-0.685924\pi\)
0.551447 0.834210i \(-0.314076\pi\)
\(98\) 0 0
\(99\) 199.294 2.01308
\(100\) 0 0
\(101\) −6.07921 −0.0601901 −0.0300951 0.999547i \(-0.509581\pi\)
−0.0300951 + 0.999547i \(0.509581\pi\)
\(102\) 0 0
\(103\) 27.8929i 0.270805i 0.990791 + 0.135402i \(0.0432327\pi\)
−0.990791 + 0.135402i \(0.956767\pi\)
\(104\) 0 0
\(105\) 40.2823i 0.383641i
\(106\) 0 0
\(107\) 61.4652i 0.574441i 0.957865 + 0.287220i \(0.0927312\pi\)
−0.957865 + 0.287220i \(0.907269\pi\)
\(108\) 0 0
\(109\) 30.2310i 0.277348i 0.990338 + 0.138674i \(0.0442841\pi\)
−0.990338 + 0.138674i \(0.955716\pi\)
\(110\) 0 0
\(111\) 24.8489 0.223864
\(112\) 0 0
\(113\) 178.939i 1.58353i −0.610826 0.791765i \(-0.709163\pi\)
0.610826 0.791765i \(-0.290837\pi\)
\(114\) 0 0
\(115\) 39.0086 0.339205
\(116\) 0 0
\(117\) 58.9109i 0.503512i
\(118\) 0 0
\(119\) 57.0793 0.479658
\(120\) 0 0
\(121\) 72.0390 0.595364
\(122\) 0 0
\(123\) 204.272 1.66074
\(124\) 0 0
\(125\) 11.1803 0.0894427
\(126\) 0 0
\(127\) 53.4071i 0.420528i 0.977645 + 0.210264i \(0.0674324\pi\)
−0.977645 + 0.210264i \(0.932568\pi\)
\(128\) 0 0
\(129\) 7.56932i 0.0586769i
\(130\) 0 0
\(131\) −109.035 −0.832327 −0.416163 0.909290i \(-0.636626\pi\)
−0.416163 + 0.909290i \(0.636626\pi\)
\(132\) 0 0
\(133\) 61.3676 + 35.3933i 0.461411 + 0.266115i
\(134\) 0 0
\(135\) 57.7359i 0.427673i
\(136\) 0 0
\(137\) −24.9987 −0.182472 −0.0912362 0.995829i \(-0.529082\pi\)
−0.0912362 + 0.995829i \(0.529082\pi\)
\(138\) 0 0
\(139\) 115.083 0.827936 0.413968 0.910291i \(-0.364143\pi\)
0.413968 + 0.910291i \(0.364143\pi\)
\(140\) 0 0
\(141\) 263.717i 1.87033i
\(142\) 0 0
\(143\) 57.0619i 0.399034i
\(144\) 0 0
\(145\) 36.6545i 0.252789i
\(146\) 0 0
\(147\) 169.578i 1.15359i
\(148\) 0 0
\(149\) −141.364 −0.948750 −0.474375 0.880323i \(-0.657326\pi\)
−0.474375 + 0.880323i \(0.657326\pi\)
\(150\) 0 0
\(151\) 250.107i 1.65634i −0.560478 0.828169i \(-0.689383\pi\)
0.560478 0.828169i \(-0.310617\pi\)
\(152\) 0 0
\(153\) 219.589 1.43522
\(154\) 0 0
\(155\) 35.5804i 0.229551i
\(156\) 0 0
\(157\) −14.5947 −0.0929601 −0.0464801 0.998919i \(-0.514800\pi\)
−0.0464801 + 0.998919i \(0.514800\pi\)
\(158\) 0 0
\(159\) 270.482 1.70115
\(160\) 0 0
\(161\) −65.0454 −0.404009
\(162\) 0 0
\(163\) −9.57924 −0.0587684 −0.0293842 0.999568i \(-0.509355\pi\)
−0.0293842 + 0.999568i \(0.509355\pi\)
\(164\) 0 0
\(165\) 150.105i 0.909729i
\(166\) 0 0
\(167\) 302.815i 1.81326i 0.421923 + 0.906631i \(0.361355\pi\)
−0.421923 + 0.906631i \(0.638645\pi\)
\(168\) 0 0
\(169\) 152.133 0.900193
\(170\) 0 0
\(171\) 236.086 + 136.161i 1.38062 + 0.796264i
\(172\) 0 0
\(173\) 111.084i 0.642102i −0.947062 0.321051i \(-0.895964\pi\)
0.947062 0.321051i \(-0.104036\pi\)
\(174\) 0 0
\(175\) −18.6428 −0.106530
\(176\) 0 0
\(177\) −491.961 −2.77944
\(178\) 0 0
\(179\) 247.724i 1.38393i 0.721929 + 0.691967i \(0.243256\pi\)
−0.721929 + 0.691967i \(0.756744\pi\)
\(180\) 0 0
\(181\) 234.560i 1.29591i −0.761678 0.647956i \(-0.775624\pi\)
0.761678 0.647956i \(-0.224376\pi\)
\(182\) 0 0
\(183\) 389.289i 2.12726i
\(184\) 0 0
\(185\) 11.5002i 0.0621631i
\(186\) 0 0
\(187\) 212.696 1.13741
\(188\) 0 0
\(189\) 96.2724i 0.509378i
\(190\) 0 0
\(191\) −250.384 −1.31091 −0.655455 0.755235i \(-0.727523\pi\)
−0.655455 + 0.755235i \(0.727523\pi\)
\(192\) 0 0
\(193\) 122.208i 0.633204i −0.948558 0.316602i \(-0.897458\pi\)
0.948558 0.316602i \(-0.102542\pi\)
\(194\) 0 0
\(195\) 44.3707 0.227542
\(196\) 0 0
\(197\) −343.177 −1.74201 −0.871007 0.491271i \(-0.836533\pi\)
−0.871007 + 0.491271i \(0.836533\pi\)
\(198\) 0 0
\(199\) 297.583 1.49539 0.747696 0.664042i \(-0.231160\pi\)
0.747696 + 0.664042i \(0.231160\pi\)
\(200\) 0 0
\(201\) 120.228 0.598148
\(202\) 0 0
\(203\) 61.1199i 0.301083i
\(204\) 0 0
\(205\) 94.5376i 0.461159i
\(206\) 0 0
\(207\) −250.235 −1.20886
\(208\) 0 0
\(209\) 228.676 + 131.887i 1.09414 + 0.631040i
\(210\) 0 0
\(211\) 237.792i 1.12698i −0.826124 0.563488i \(-0.809459\pi\)
0.826124 0.563488i \(-0.190541\pi\)
\(212\) 0 0
\(213\) −585.801 −2.75024
\(214\) 0 0
\(215\) 3.50311 0.0162935
\(216\) 0 0
\(217\) 59.3290i 0.273406i
\(218\) 0 0
\(219\) 164.176i 0.749661i
\(220\) 0 0
\(221\) 62.8725i 0.284491i
\(222\) 0 0
\(223\) 152.390i 0.683365i −0.939815 0.341682i \(-0.889003\pi\)
0.939815 0.341682i \(-0.110997\pi\)
\(224\) 0 0
\(225\) −71.7204 −0.318757
\(226\) 0 0
\(227\) 205.814i 0.906670i −0.891340 0.453335i \(-0.850234\pi\)
0.891340 0.453335i \(-0.149766\pi\)
\(228\) 0 0
\(229\) 135.060 0.589783 0.294892 0.955531i \(-0.404716\pi\)
0.294892 + 0.955531i \(0.404716\pi\)
\(230\) 0 0
\(231\) 250.295i 1.08353i
\(232\) 0 0
\(233\) 100.593 0.431728 0.215864 0.976423i \(-0.430743\pi\)
0.215864 + 0.976423i \(0.430743\pi\)
\(234\) 0 0
\(235\) −122.049 −0.519358
\(236\) 0 0
\(237\) −495.830 −2.09211
\(238\) 0 0
\(239\) −328.565 −1.37475 −0.687373 0.726304i \(-0.741236\pi\)
−0.687373 + 0.726304i \(0.741236\pi\)
\(240\) 0 0
\(241\) 144.249i 0.598543i 0.954168 + 0.299272i \(0.0967437\pi\)
−0.954168 + 0.299272i \(0.903256\pi\)
\(242\) 0 0
\(243\) 253.372i 1.04268i
\(244\) 0 0
\(245\) −78.4812 −0.320331
\(246\) 0 0
\(247\) 38.9856 67.5961i 0.157836 0.273669i
\(248\) 0 0
\(249\) 388.132i 1.55876i
\(250\) 0 0
\(251\) 66.0565 0.263173 0.131587 0.991305i \(-0.457993\pi\)
0.131587 + 0.991305i \(0.457993\pi\)
\(252\) 0 0
\(253\) −242.381 −0.958027
\(254\) 0 0
\(255\) 165.391i 0.648591i
\(256\) 0 0
\(257\) 268.255i 1.04379i 0.853009 + 0.521896i \(0.174775\pi\)
−0.853009 + 0.521896i \(0.825225\pi\)
\(258\) 0 0
\(259\) 19.1761i 0.0740390i
\(260\) 0 0
\(261\) 235.133i 0.900895i
\(262\) 0 0
\(263\) −241.131 −0.916848 −0.458424 0.888734i \(-0.651586\pi\)
−0.458424 + 0.888734i \(0.651586\pi\)
\(264\) 0 0
\(265\) 125.180i 0.472378i
\(266\) 0 0
\(267\) 58.4058 0.218748
\(268\) 0 0
\(269\) 270.274i 1.00473i 0.864654 + 0.502367i \(0.167538\pi\)
−0.864654 + 0.502367i \(0.832462\pi\)
\(270\) 0 0
\(271\) 159.872 0.589934 0.294967 0.955507i \(-0.404691\pi\)
0.294967 + 0.955507i \(0.404691\pi\)
\(272\) 0 0
\(273\) −73.9865 −0.271013
\(274\) 0 0
\(275\) −69.4692 −0.252615
\(276\) 0 0
\(277\) 480.862 1.73596 0.867981 0.496597i \(-0.165417\pi\)
0.867981 + 0.496597i \(0.165417\pi\)
\(278\) 0 0
\(279\) 228.244i 0.818078i
\(280\) 0 0
\(281\) 125.506i 0.446639i −0.974745 0.223320i \(-0.928311\pi\)
0.974745 0.223320i \(-0.0716894\pi\)
\(282\) 0 0
\(283\) −169.824 −0.600086 −0.300043 0.953926i \(-0.597001\pi\)
−0.300043 + 0.953926i \(0.597001\pi\)
\(284\) 0 0
\(285\) −102.554 + 177.816i −0.359840 + 0.623917i
\(286\) 0 0
\(287\) 157.638i 0.549261i
\(288\) 0 0
\(289\) −54.6445 −0.189081
\(290\) 0 0
\(291\) −781.925 −2.68703
\(292\) 0 0
\(293\) 103.213i 0.352263i 0.984367 + 0.176131i \(0.0563583\pi\)
−0.984367 + 0.176131i \(0.943642\pi\)
\(294\) 0 0
\(295\) 227.681i 0.771801i
\(296\) 0 0
\(297\) 358.743i 1.20789i
\(298\) 0 0
\(299\) 71.6472i 0.239623i
\(300\) 0 0
\(301\) −5.84130 −0.0194063
\(302\) 0 0
\(303\) 29.3721i 0.0969377i
\(304\) 0 0
\(305\) 180.164 0.590703
\(306\) 0 0
\(307\) 470.447i 1.53240i 0.642602 + 0.766200i \(0.277855\pi\)
−0.642602 + 0.766200i \(0.722145\pi\)
\(308\) 0 0
\(309\) 134.767 0.436138
\(310\) 0 0
\(311\) −104.966 −0.337511 −0.168756 0.985658i \(-0.553975\pi\)
−0.168756 + 0.985658i \(0.553975\pi\)
\(312\) 0 0
\(313\) 358.845 1.14647 0.573234 0.819392i \(-0.305689\pi\)
0.573234 + 0.819392i \(0.305689\pi\)
\(314\) 0 0
\(315\) 119.591 0.379654
\(316\) 0 0
\(317\) 367.285i 1.15863i 0.815104 + 0.579314i \(0.196680\pi\)
−0.815104 + 0.579314i \(0.803320\pi\)
\(318\) 0 0
\(319\) 227.753i 0.713960i
\(320\) 0 0
\(321\) 296.973 0.925150
\(322\) 0 0
\(323\) 251.963 + 145.318i 0.780070 + 0.449900i
\(324\) 0 0
\(325\) 20.5349i 0.0631844i
\(326\) 0 0
\(327\) 146.063 0.446676
\(328\) 0 0
\(329\) 203.512 0.618579
\(330\) 0 0
\(331\) 496.428i 1.49978i 0.661561 + 0.749892i \(0.269894\pi\)
−0.661561 + 0.749892i \(0.730106\pi\)
\(332\) 0 0
\(333\) 73.7721i 0.221538i
\(334\) 0 0
\(335\) 55.6419i 0.166095i
\(336\) 0 0
\(337\) 52.5036i 0.155797i 0.996961 + 0.0778986i \(0.0248210\pi\)
−0.996961 + 0.0778986i \(0.975179\pi\)
\(338\) 0 0
\(339\) −864.556 −2.55031
\(340\) 0 0
\(341\) 221.080i 0.648328i
\(342\) 0 0
\(343\) 313.564 0.914180
\(344\) 0 0
\(345\) 188.473i 0.546298i
\(346\) 0 0
\(347\) 409.556 1.18028 0.590138 0.807302i \(-0.299073\pi\)
0.590138 + 0.807302i \(0.299073\pi\)
\(348\) 0 0
\(349\) 562.063 1.61050 0.805248 0.592938i \(-0.202032\pi\)
0.805248 + 0.592938i \(0.202032\pi\)
\(350\) 0 0
\(351\) −106.044 −0.302118
\(352\) 0 0
\(353\) 252.453 0.715165 0.357583 0.933881i \(-0.383601\pi\)
0.357583 + 0.933881i \(0.383601\pi\)
\(354\) 0 0
\(355\) 271.111i 0.763693i
\(356\) 0 0
\(357\) 275.783i 0.772500i
\(358\) 0 0
\(359\) −119.344 −0.332433 −0.166217 0.986089i \(-0.553155\pi\)
−0.166217 + 0.986089i \(0.553155\pi\)
\(360\) 0 0
\(361\) 180.785 + 312.470i 0.500789 + 0.865569i
\(362\) 0 0
\(363\) 348.062i 0.958848i
\(364\) 0 0
\(365\) −75.9811 −0.208167
\(366\) 0 0
\(367\) −690.635 −1.88184 −0.940920 0.338629i \(-0.890037\pi\)
−0.940920 + 0.338629i \(0.890037\pi\)
\(368\) 0 0
\(369\) 606.446i 1.64349i
\(370\) 0 0
\(371\) 208.733i 0.562623i
\(372\) 0 0
\(373\) 491.052i 1.31649i 0.752803 + 0.658246i \(0.228701\pi\)
−0.752803 + 0.658246i \(0.771299\pi\)
\(374\) 0 0
\(375\) 54.0186i 0.144050i
\(376\) 0 0
\(377\) 67.3233 0.178576
\(378\) 0 0
\(379\) 691.775i 1.82526i −0.408782 0.912632i \(-0.634046\pi\)
0.408782 0.912632i \(-0.365954\pi\)
\(380\) 0 0
\(381\) 258.040 0.677271
\(382\) 0 0
\(383\) 252.889i 0.660285i −0.943931 0.330142i \(-0.892903\pi\)
0.943931 0.330142i \(-0.107097\pi\)
\(384\) 0 0
\(385\) 115.837 0.300876
\(386\) 0 0
\(387\) −22.4720 −0.0580671
\(388\) 0 0
\(389\) −504.764 −1.29759 −0.648797 0.760961i \(-0.724728\pi\)
−0.648797 + 0.760961i \(0.724728\pi\)
\(390\) 0 0
\(391\) −267.063 −0.683025
\(392\) 0 0
\(393\) 526.809i 1.34048i
\(394\) 0 0
\(395\) 229.472i 0.580941i
\(396\) 0 0
\(397\) 313.581 0.789876 0.394938 0.918708i \(-0.370766\pi\)
0.394938 + 0.918708i \(0.370766\pi\)
\(398\) 0 0
\(399\) 171.005 296.502i 0.428585 0.743113i
\(400\) 0 0
\(401\) 561.838i 1.40109i 0.713607 + 0.700546i \(0.247060\pi\)
−0.713607 + 0.700546i \(0.752940\pi\)
\(402\) 0 0
\(403\) 65.3506 0.162160
\(404\) 0 0
\(405\) −9.71385 −0.0239848
\(406\) 0 0
\(407\) 71.4566i 0.175569i
\(408\) 0 0
\(409\) 165.613i 0.404921i −0.979290 0.202460i \(-0.935106\pi\)
0.979290 0.202460i \(-0.0648937\pi\)
\(410\) 0 0
\(411\) 120.783i 0.293876i
\(412\) 0 0
\(413\) 379.650i 0.919249i
\(414\) 0 0
\(415\) 179.629 0.432840
\(416\) 0 0
\(417\) 556.032i 1.33341i
\(418\) 0 0
\(419\) −440.399 −1.05107 −0.525536 0.850771i \(-0.676135\pi\)
−0.525536 + 0.850771i \(0.676135\pi\)
\(420\) 0 0
\(421\) 786.831i 1.86896i −0.356018 0.934479i \(-0.615866\pi\)
0.356018 0.934479i \(-0.384134\pi\)
\(422\) 0 0
\(423\) 782.929 1.85090
\(424\) 0 0
\(425\) −76.5434 −0.180102
\(426\) 0 0
\(427\) −300.417 −0.703553
\(428\) 0 0
\(429\) −275.698 −0.642654
\(430\) 0 0
\(431\) 219.539i 0.509372i −0.967024 0.254686i \(-0.918028\pi\)
0.967024 0.254686i \(-0.0819721\pi\)
\(432\) 0 0
\(433\) 450.900i 1.04134i −0.853758 0.520670i \(-0.825682\pi\)
0.853758 0.520670i \(-0.174318\pi\)
\(434\) 0 0
\(435\) −177.099 −0.407123
\(436\) 0 0
\(437\) −287.127 165.598i −0.657041 0.378944i
\(438\) 0 0
\(439\) 319.069i 0.726808i 0.931632 + 0.363404i \(0.118385\pi\)
−0.931632 + 0.363404i \(0.881615\pi\)
\(440\) 0 0
\(441\) 503.446 1.14160
\(442\) 0 0
\(443\) 780.776 1.76247 0.881237 0.472674i \(-0.156711\pi\)
0.881237 + 0.472674i \(0.156711\pi\)
\(444\) 0 0
\(445\) 27.0304i 0.0607425i
\(446\) 0 0
\(447\) 683.009i 1.52798i
\(448\) 0 0
\(449\) 523.244i 1.16535i −0.812704 0.582677i \(-0.802005\pi\)
0.812704 0.582677i \(-0.197995\pi\)
\(450\) 0 0
\(451\) 587.411i 1.30246i
\(452\) 0 0
\(453\) −1208.41 −2.66757
\(454\) 0 0
\(455\) 34.2412i 0.0752554i
\(456\) 0 0
\(457\) 874.643 1.91388 0.956940 0.290285i \(-0.0937500\pi\)
0.956940 + 0.290285i \(0.0937500\pi\)
\(458\) 0 0
\(459\) 395.274i 0.861164i
\(460\) 0 0
\(461\) 120.368 0.261102 0.130551 0.991442i \(-0.458325\pi\)
0.130551 + 0.991442i \(0.458325\pi\)
\(462\) 0 0
\(463\) 256.911 0.554883 0.277441 0.960743i \(-0.410514\pi\)
0.277441 + 0.960743i \(0.410514\pi\)
\(464\) 0 0
\(465\) −171.909 −0.369698
\(466\) 0 0
\(467\) 490.911 1.05120 0.525601 0.850731i \(-0.323840\pi\)
0.525601 + 0.850731i \(0.323840\pi\)
\(468\) 0 0
\(469\) 92.7807i 0.197827i
\(470\) 0 0
\(471\) 70.5155i 0.149714i
\(472\) 0 0
\(473\) −21.7666 −0.0460182
\(474\) 0 0
\(475\) −82.2941 47.4625i −0.173251 0.0999210i
\(476\) 0 0
\(477\) 803.014i 1.68347i
\(478\) 0 0
\(479\) 545.752 1.13936 0.569679 0.821868i \(-0.307068\pi\)
0.569679 + 0.821868i \(0.307068\pi\)
\(480\) 0 0
\(481\) −21.1224 −0.0439135
\(482\) 0 0
\(483\) 314.271i 0.650665i
\(484\) 0 0
\(485\) 361.878i 0.746140i
\(486\) 0 0
\(487\) 243.032i 0.499039i −0.968370 0.249519i \(-0.919727\pi\)
0.968370 0.249519i \(-0.0802726\pi\)
\(488\) 0 0
\(489\) 46.2828i 0.0946479i
\(490\) 0 0
\(491\) −555.332 −1.13102 −0.565511 0.824740i \(-0.691321\pi\)
−0.565511 + 0.824740i \(0.691321\pi\)
\(492\) 0 0
\(493\) 250.946i 0.509017i
\(494\) 0 0
\(495\) 445.636 0.900275
\(496\) 0 0
\(497\) 452.067i 0.909592i
\(498\) 0 0
\(499\) −231.235 −0.463397 −0.231698 0.972788i \(-0.574428\pi\)
−0.231698 + 0.972788i \(0.574428\pi\)
\(500\) 0 0
\(501\) 1463.07 2.92030
\(502\) 0 0
\(503\) −206.048 −0.409638 −0.204819 0.978800i \(-0.565660\pi\)
−0.204819 + 0.978800i \(0.565660\pi\)
\(504\) 0 0
\(505\) −13.5935 −0.0269179
\(506\) 0 0
\(507\) 735.040i 1.44978i
\(508\) 0 0
\(509\) 398.832i 0.783560i 0.920059 + 0.391780i \(0.128141\pi\)
−0.920059 + 0.391780i \(0.871859\pi\)
\(510\) 0 0
\(511\) 126.696 0.247936
\(512\) 0 0
\(513\) 245.099 424.971i 0.477776 0.828404i
\(514\) 0 0
\(515\) 62.3704i 0.121108i
\(516\) 0 0
\(517\) 758.355 1.46684
\(518\) 0 0
\(519\) −536.708 −1.03412
\(520\) 0 0
\(521\) 262.617i 0.504064i −0.967719 0.252032i \(-0.918901\pi\)
0.967719 0.252032i \(-0.0810988\pi\)
\(522\) 0 0
\(523\) 814.031i 1.55646i 0.627977 + 0.778232i \(0.283883\pi\)
−0.627977 + 0.778232i \(0.716117\pi\)
\(524\) 0 0
\(525\) 90.0740i 0.171569i
\(526\) 0 0
\(527\) 243.592i 0.462225i
\(528\) 0 0
\(529\) −224.666 −0.424699
\(530\) 0 0
\(531\) 1460.54i 2.75055i
\(532\) 0 0
\(533\) −173.637 −0.325773
\(534\) 0 0
\(535\) 137.440i 0.256898i
\(536\) 0 0
\(537\) 1196.90 2.22886
\(538\) 0 0
\(539\) 487.644 0.904720
\(540\) 0 0
\(541\) −196.753 −0.363684 −0.181842 0.983328i \(-0.558206\pi\)
−0.181842 + 0.983328i \(0.558206\pi\)
\(542\) 0 0
\(543\) −1133.29 −2.08710
\(544\) 0 0
\(545\) 67.5985i 0.124034i
\(546\) 0 0
\(547\) 759.052i 1.38766i 0.720137 + 0.693832i \(0.244079\pi\)
−0.720137 + 0.693832i \(0.755921\pi\)
\(548\) 0 0
\(549\) −1155.73 −2.10515
\(550\) 0 0
\(551\) −155.605 + 269.799i −0.282404 + 0.489653i
\(552\) 0 0
\(553\) 382.636i 0.691927i
\(554\) 0 0
\(555\) 55.5639 0.100115
\(556\) 0 0
\(557\) 705.968 1.26745 0.633723 0.773560i \(-0.281526\pi\)
0.633723 + 0.773560i \(0.281526\pi\)
\(558\) 0 0
\(559\) 6.43416i 0.0115101i
\(560\) 0 0
\(561\) 1027.66i 1.83183i
\(562\) 0 0
\(563\) 357.744i 0.635424i 0.948187 + 0.317712i \(0.102914\pi\)
−0.948187 + 0.317712i \(0.897086\pi\)
\(564\) 0 0
\(565\) 400.119i 0.708176i
\(566\) 0 0
\(567\) 16.1975 0.0285670
\(568\) 0 0
\(569\) 448.411i 0.788069i 0.919096 + 0.394034i \(0.128921\pi\)
−0.919096 + 0.394034i \(0.871079\pi\)
\(570\) 0 0
\(571\) 203.672 0.356694 0.178347 0.983968i \(-0.442925\pi\)
0.178347 + 0.983968i \(0.442925\pi\)
\(572\) 0 0
\(573\) 1209.75i 2.11125i
\(574\) 0 0
\(575\) 87.2259 0.151697
\(576\) 0 0
\(577\) 232.447 0.402854 0.201427 0.979504i \(-0.435442\pi\)
0.201427 + 0.979504i \(0.435442\pi\)
\(578\) 0 0
\(579\) −590.458 −1.01979
\(580\) 0 0
\(581\) −299.524 −0.515532
\(582\) 0 0
\(583\) 777.809i 1.33415i
\(584\) 0 0
\(585\) 131.729i 0.225178i
\(586\) 0 0
\(587\) −1101.80 −1.87701 −0.938503 0.345271i \(-0.887787\pi\)
−0.938503 + 0.345271i \(0.887787\pi\)
\(588\) 0 0
\(589\) −151.045 + 261.894i −0.256443 + 0.444641i
\(590\) 0 0
\(591\) 1658.08i 2.80555i
\(592\) 0 0
\(593\) −387.409 −0.653303 −0.326651 0.945145i \(-0.605920\pi\)
−0.326651 + 0.945145i \(0.605920\pi\)
\(594\) 0 0
\(595\) 127.633 0.214509
\(596\) 0 0
\(597\) 1437.79i 2.40836i
\(598\) 0 0
\(599\) 649.288i 1.08395i 0.840393 + 0.541977i \(0.182324\pi\)
−0.840393 + 0.541977i \(0.817676\pi\)
\(600\) 0 0
\(601\) 564.191i 0.938754i −0.882998 0.469377i \(-0.844479\pi\)
0.882998 0.469377i \(-0.155521\pi\)
\(602\) 0 0
\(603\) 356.935i 0.591932i
\(604\) 0 0
\(605\) 161.084 0.266255
\(606\) 0 0
\(607\) 431.505i 0.710882i 0.934699 + 0.355441i \(0.115669\pi\)
−0.934699 + 0.355441i \(0.884331\pi\)
\(608\) 0 0
\(609\) 295.305 0.484902
\(610\) 0 0
\(611\) 224.168i 0.366887i
\(612\) 0 0
\(613\) 360.161 0.587539 0.293769 0.955876i \(-0.405090\pi\)
0.293769 + 0.955876i \(0.405090\pi\)
\(614\) 0 0
\(615\) 456.765 0.742707
\(616\) 0 0
\(617\) −255.900 −0.414749 −0.207375 0.978262i \(-0.566492\pi\)
−0.207375 + 0.978262i \(0.566492\pi\)
\(618\) 0 0
\(619\) −231.220 −0.373538 −0.186769 0.982404i \(-0.559802\pi\)
−0.186769 + 0.982404i \(0.559802\pi\)
\(620\) 0 0
\(621\) 450.440i 0.725346i
\(622\) 0 0
\(623\) 45.0722i 0.0723470i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 637.223 1104.87i 1.01630 1.76215i
\(628\) 0 0
\(629\) 78.7331i 0.125172i
\(630\) 0 0
\(631\) 55.6688 0.0882232 0.0441116 0.999027i \(-0.485954\pi\)
0.0441116 + 0.999027i \(0.485954\pi\)
\(632\) 0 0
\(633\) −1148.91 −1.81502
\(634\) 0 0
\(635\) 119.422i 0.188066i
\(636\) 0 0
\(637\) 144.146i 0.226290i
\(638\) 0 0
\(639\) 1739.14i 2.72166i
\(640\) 0 0
\(641\) 1086.58i 1.69513i −0.530692 0.847565i \(-0.678068\pi\)
0.530692 0.847565i \(-0.321932\pi\)
\(642\) 0 0
\(643\) 132.021 0.205320 0.102660 0.994717i \(-0.467265\pi\)
0.102660 + 0.994717i \(0.467265\pi\)
\(644\) 0 0
\(645\) 16.9255i 0.0262411i
\(646\) 0 0
\(647\) 139.001 0.214840 0.107420 0.994214i \(-0.465741\pi\)
0.107420 + 0.994214i \(0.465741\pi\)
\(648\) 0 0
\(649\) 1414.70i 2.17982i
\(650\) 0 0
\(651\) 286.652 0.440326
\(652\) 0 0
\(653\) −2.61433 −0.00400356 −0.00200178 0.999998i \(-0.500637\pi\)
−0.00200178 + 0.999998i \(0.500637\pi\)
\(654\) 0 0
\(655\) −243.809 −0.372228
\(656\) 0 0
\(657\) 487.408 0.741870
\(658\) 0 0
\(659\) 1115.25i 1.69234i −0.532911 0.846171i \(-0.678902\pi\)
0.532911 0.846171i \(-0.321098\pi\)
\(660\) 0 0
\(661\) 1032.56i 1.56212i −0.624453 0.781062i \(-0.714678\pi\)
0.624453 0.781062i \(-0.285322\pi\)
\(662\) 0 0
\(663\) −303.773 −0.458180
\(664\) 0 0
\(665\) 137.222 + 79.1419i 0.206349 + 0.119010i
\(666\) 0 0
\(667\) 285.968i 0.428738i
\(668\) 0 0
\(669\) −736.285 −1.10058
\(670\) 0 0
\(671\) −1119.45 −1.66834
\(672\) 0 0
\(673\) 923.804i 1.37267i −0.727288 0.686333i \(-0.759219\pi\)
0.727288 0.686333i \(-0.240781\pi\)
\(674\) 0 0
\(675\) 129.101i 0.191261i
\(676\) 0 0
\(677\) 1144.25i 1.69017i 0.534631 + 0.845086i \(0.320451\pi\)
−0.534631 + 0.845086i \(0.679549\pi\)
\(678\) 0 0
\(679\) 603.417i 0.888686i
\(680\) 0 0
\(681\) −994.406 −1.46021
\(682\) 0 0
\(683\) 686.997i 1.00585i −0.864330 0.502926i \(-0.832257\pi\)
0.864330 0.502926i \(-0.167743\pi\)
\(684\) 0 0
\(685\) −55.8988 −0.0816041
\(686\) 0 0
\(687\) 652.554i 0.949860i
\(688\) 0 0
\(689\) −229.918 −0.333699
\(690\) 0 0
\(691\) −1216.77 −1.76088 −0.880442 0.474154i \(-0.842754\pi\)
−0.880442 + 0.474154i \(0.842754\pi\)
\(692\) 0 0
\(693\) −743.081 −1.07227
\(694\) 0 0
\(695\) 257.334 0.370264
\(696\) 0 0
\(697\) 647.228i 0.928591i
\(698\) 0 0
\(699\) 486.020i 0.695308i
\(700\) 0 0
\(701\) −1082.00 −1.54351 −0.771753 0.635922i \(-0.780620\pi\)
−0.771753 + 0.635922i \(0.780620\pi\)
\(702\) 0 0
\(703\) 48.8203 84.6482i 0.0694456 0.120410i
\(704\) 0 0
\(705\) 589.690i 0.836439i
\(706\) 0 0
\(707\) 22.6667 0.0320604
\(708\) 0 0
\(709\) −639.509 −0.901987 −0.450994 0.892527i \(-0.648930\pi\)
−0.450994 + 0.892527i \(0.648930\pi\)
\(710\) 0 0
\(711\) 1472.03i 2.07037i
\(712\) 0 0
\(713\) 277.589i 0.389325i
\(714\) 0 0
\(715\) 127.594i 0.178453i
\(716\) 0 0
\(717\) 1587.48i 2.21406i
\(718\) 0 0
\(719\) −95.9878 −0.133502 −0.0667509 0.997770i \(-0.521263\pi\)
−0.0667509 + 0.997770i \(0.521263\pi\)
\(720\) 0 0
\(721\) 104.000i 0.144245i
\(722\) 0 0
\(723\) 696.949 0.963968
\(724\) 0 0
\(725\) 81.9619i 0.113051i
\(726\) 0 0
\(727\) −368.875 −0.507394 −0.253697 0.967284i \(-0.581647\pi\)
−0.253697 + 0.967284i \(0.581647\pi\)
\(728\) 0 0
\(729\) 1185.09 1.62563
\(730\) 0 0
\(731\) −23.9831 −0.0328087
\(732\) 0 0
\(733\) −865.251 −1.18042 −0.590212 0.807248i \(-0.700956\pi\)
−0.590212 + 0.807248i \(0.700956\pi\)
\(734\) 0 0
\(735\) 379.187i 0.515901i
\(736\) 0 0
\(737\) 345.732i 0.469107i
\(738\) 0 0
\(739\) −934.331 −1.26432 −0.632159 0.774839i \(-0.717831\pi\)
−0.632159 + 0.774839i \(0.717831\pi\)
\(740\) 0 0
\(741\) −326.595 188.362i −0.440750 0.254199i
\(742\) 0 0
\(743\) 235.551i 0.317027i −0.987357 0.158514i \(-0.949330\pi\)
0.987357 0.158514i \(-0.0506702\pi\)
\(744\) 0 0
\(745\) −316.099 −0.424294
\(746\) 0 0
\(747\) −1152.29 −1.54256
\(748\) 0 0
\(749\) 229.176i 0.305976i
\(750\) 0 0
\(751\) 971.042i 1.29300i −0.762915 0.646499i \(-0.776232\pi\)
0.762915 0.646499i \(-0.223768\pi\)
\(752\) 0 0
\(753\) 319.156i 0.423847i
\(754\) 0 0
\(755\) 559.256i 0.740737i
\(756\) 0 0
\(757\) −177.074 −0.233916 −0.116958 0.993137i \(-0.537314\pi\)
−0.116958 + 0.993137i \(0.537314\pi\)
\(758\) 0 0
\(759\) 1171.08i 1.54292i
\(760\) 0 0
\(761\) −159.521 −0.209621 −0.104810 0.994492i \(-0.533424\pi\)
−0.104810 + 0.994492i \(0.533424\pi\)
\(762\) 0 0
\(763\) 112.718i 0.147730i
\(764\) 0 0
\(765\) 491.015 0.641850
\(766\) 0 0
\(767\) 418.182 0.545218
\(768\) 0 0
\(769\) 707.134 0.919551 0.459775 0.888035i \(-0.347930\pi\)
0.459775 + 0.888035i \(0.347930\pi\)
\(770\) 0 0
\(771\) 1296.09 1.68105
\(772\) 0 0
\(773\) 799.572i 1.03438i −0.855872 0.517188i \(-0.826979\pi\)
0.855872 0.517188i \(-0.173021\pi\)
\(774\) 0 0
\(775\) 79.5603i 0.102658i
\(776\) 0 0
\(777\) −92.6507 −0.119242
\(778\) 0 0
\(779\) 401.329 695.854i 0.515184 0.893265i
\(780\) 0 0
\(781\) 1684.55i 2.15692i
\(782\) 0 0
\(783\) 423.256 0.540557
\(784\) 0 0
\(785\) −32.6348 −0.0415730
\(786\) 0 0
\(787\) 1366.97i 1.73693i −0.495749 0.868466i \(-0.665107\pi\)
0.495749 0.868466i \(-0.334893\pi\)
\(788\) 0 0
\(789\) 1165.04i 1.47661i
\(790\) 0 0
\(791\) 667.184i 0.843469i
\(792\) 0 0
\(793\) 330.908i 0.417286i
\(794\) 0 0
\(795\) 604.817 0.760776
\(796\) 0 0
\(797\) 121.258i 0.152143i −0.997102 0.0760717i \(-0.975762\pi\)
0.997102 0.0760717i \(-0.0242378\pi\)
\(798\) 0 0
\(799\) 835.579 1.04578
\(800\) 0 0
\(801\) 173.396i 0.216475i
\(802\) 0 0
\(803\) 472.110 0.587933
\(804\) 0 0
\(805\) −145.446 −0.180678
\(806\) 0 0
\(807\) 1305.85 1.61815
\(808\) 0 0
\(809\) 820.488 1.01420 0.507100 0.861887i \(-0.330717\pi\)
0.507100 + 0.861887i \(0.330717\pi\)
\(810\) 0 0
\(811\) 537.722i 0.663036i 0.943449 + 0.331518i \(0.107561\pi\)
−0.943449 + 0.331518i \(0.892439\pi\)
\(812\) 0 0
\(813\) 772.434i 0.950103i
\(814\) 0 0
\(815\) −21.4198 −0.0262820
\(816\) 0 0
\(817\) −25.7850 14.8713i −0.0315606 0.0182023i
\(818\) 0 0
\(819\) 219.653i 0.268196i
\(820\) 0 0
\(821\) −54.0424 −0.0658250 −0.0329125 0.999458i \(-0.510478\pi\)
−0.0329125 + 0.999458i \(0.510478\pi\)
\(822\) 0 0
\(823\) −599.052 −0.727889 −0.363944 0.931421i \(-0.618570\pi\)
−0.363944 + 0.931421i \(0.618570\pi\)
\(824\) 0 0
\(825\) 335.646i 0.406843i
\(826\) 0 0
\(827\) 620.339i 0.750108i 0.927003 + 0.375054i \(0.122376\pi\)
−0.927003 + 0.375054i \(0.877624\pi\)
\(828\) 0 0
\(829\) 36.8107i 0.0444038i 0.999754 + 0.0222019i \(0.00706766\pi\)
−0.999754 + 0.0222019i \(0.992932\pi\)
\(830\) 0 0
\(831\) 2323.32i 2.79581i
\(832\) 0 0
\(833\) 537.302 0.645020
\(834\) 0 0
\(835\) 677.115i 0.810916i
\(836\) 0 0
\(837\) 410.854 0.490865
\(838\) 0 0
\(839\) 531.549i 0.633551i 0.948501 + 0.316776i \(0.102600\pi\)
−0.948501 + 0.316776i \(0.897400\pi\)
\(840\) 0 0
\(841\) 572.290 0.680488
\(842\) 0 0
\(843\) −606.389 −0.719323
\(844\) 0 0
\(845\) 340.179 0.402579
\(846\) 0 0
\(847\) −268.602 −0.317121
\(848\) 0 0
\(849\) 820.519i 0.966453i
\(850\) 0 0
\(851\) 89.7212i 0.105430i
\(852\) 0 0
\(853\) 16.6518 0.0195214 0.00976072 0.999952i \(-0.496893\pi\)
0.00976072 + 0.999952i \(0.496893\pi\)
\(854\) 0 0
\(855\) 527.905 + 304.466i 0.617433 + 0.356100i
\(856\) 0 0
\(857\) 656.701i 0.766279i 0.923691 + 0.383139i \(0.125157\pi\)
−0.923691 + 0.383139i \(0.874843\pi\)
\(858\) 0 0
\(859\) −7.72122 −0.00898862 −0.00449431 0.999990i \(-0.501431\pi\)
−0.00449431 + 0.999990i \(0.501431\pi\)
\(860\) 0 0
\(861\) −761.638 −0.884597
\(862\) 0 0
\(863\) 1132.71i 1.31253i −0.754530 0.656266i \(-0.772135\pi\)
0.754530 0.656266i \(-0.227865\pi\)
\(864\) 0 0
\(865\) 248.390i 0.287157i
\(866\) 0 0
\(867\) 264.019i 0.304520i
\(868\) 0 0
\(869\) 1425.83i 1.64077i
\(870\) 0 0
\(871\) −102.197 −0.117333
\(872\) 0 0
\(873\) 2321.40i 2.65910i
\(874\) 0 0
\(875\) −41.6865 −0.0476418
\(876\) 0 0
\(877\) 826.040i 0.941893i 0.882162 + 0.470947i \(0.156088\pi\)
−0.882162 + 0.470947i \(0.843912\pi\)
\(878\) 0 0
\(879\) 498.681 0.567327
\(880\) 0 0
\(881\) 691.162 0.784520 0.392260 0.919854i \(-0.371693\pi\)
0.392260 + 0.919854i \(0.371693\pi\)
\(882\) 0 0
\(883\) −1339.08 −1.51651 −0.758255 0.651958i \(-0.773948\pi\)
−0.758255 + 0.651958i \(0.773948\pi\)
\(884\) 0 0
\(885\) −1100.06 −1.24300
\(886\) 0 0
\(887\) 1263.16i 1.42408i −0.702139 0.712040i \(-0.747772\pi\)
0.702139 0.712040i \(-0.252228\pi\)
\(888\) 0 0
\(889\) 199.131i 0.223995i
\(890\) 0 0
\(891\) 60.3572 0.0677409
\(892\) 0 0
\(893\) 898.356 + 518.120i 1.00600 + 0.580202i
\(894\) 0 0
\(895\) 553.928i 0.618914i
\(896\) 0 0
\(897\) 346.168 0.385918
\(898\) 0 0
\(899\) −260.836 −0.290141
\(900\) 0 0
\(901\) 857.014i 0.951181i
\(902\) 0 0
\(903\) 28.2227i 0.0312543i
\(904\) 0 0
\(905\) 524.492i 0.579549i
\(906\) 0 0
\(907\) 531.490i 0.585987i 0.956114 + 0.292994i \(0.0946515\pi\)
−0.956114 + 0.292994i \(0.905349\pi\)
\(908\) 0 0
\(909\) 87.2006 0.0959302
\(910\) 0 0
\(911\) 214.421i 0.235368i −0.993051 0.117684i \(-0.962453\pi\)
0.993051 0.117684i \(-0.0375471\pi\)
\(912\) 0 0
\(913\) −1116.13 −1.22248
\(914\) 0 0
\(915\) 870.476i 0.951340i
\(916\) 0 0
\(917\) 406.542 0.443340
\(918\) 0 0
\(919\) −1523.80 −1.65811 −0.829055 0.559167i \(-0.811121\pi\)
−0.829055 + 0.559167i \(0.811121\pi\)
\(920\) 0 0
\(921\) 2273.00 2.46797
\(922\) 0 0
\(923\) 497.950 0.539490
\(924\) 0 0
\(925\) 25.7152i 0.0278002i
\(926\) 0 0
\(927\) 400.098i 0.431605i
\(928\) 0 0
\(929\) −269.533 −0.290132 −0.145066 0.989422i \(-0.546339\pi\)
−0.145066 + 0.989422i \(0.546339\pi\)
\(930\) 0 0
\(931\) 577.669 + 333.166i 0.620482 + 0.357859i
\(932\) 0 0
\(933\) 507.151i 0.543570i
\(934\) 0 0
\(935\) 475.604 0.508667
\(936\) 0 0
\(937\) −492.223 −0.525318 −0.262659 0.964889i \(-0.584599\pi\)
−0.262659 + 0.964889i \(0.584599\pi\)
\(938\) 0 0
\(939\) 1733.78i 1.84641i
\(940\) 0 0
\(941\) 1364.05i 1.44958i −0.688972 0.724788i \(-0.741938\pi\)
0.688972 0.724788i \(-0.258062\pi\)
\(942\) 0 0
\(943\) 737.556i 0.782138i
\(944\) 0 0
\(945\) 215.272i 0.227801i
\(946\) 0 0
\(947\) −1082.40 −1.14298 −0.571490 0.820609i \(-0.693634\pi\)
−0.571490 + 0.820609i \(0.693634\pi\)
\(948\) 0 0
\(949\) 139.554i 0.147054i
\(950\) 0 0
\(951\) 1774.56 1.86600
\(952\) 0 0
\(953\) 196.828i 0.206535i 0.994654 + 0.103268i \(0.0329298\pi\)
−0.994654 + 0.103268i \(0.967070\pi\)
\(954\) 0 0
\(955\) −559.875 −0.586256
\(956\) 0 0
\(957\) 1100.41 1.14985
\(958\) 0 0
\(959\) 93.2092 0.0971941
\(960\) 0 0
\(961\) 707.806 0.736531
\(962\) 0 0
\(963\) 881.661i 0.915536i
\(964\) 0 0
\(965\) 273.266i 0.283177i
\(966\) 0 0
\(967\) 1381.14 1.42828 0.714138 0.700005i \(-0.246819\pi\)
0.714138 + 0.700005i \(0.246819\pi\)
\(968\) 0 0
\(969\) 702.112 1217.37i 0.724574 1.25632i
\(970\) 0 0
\(971\) 1116.21i 1.14954i 0.818314 + 0.574771i \(0.194909\pi\)
−0.818314 + 0.574771i \(0.805091\pi\)
\(972\) 0 0
\(973\) −429.094 −0.441001
\(974\) 0 0
\(975\) 99.2160 0.101760
\(976\) 0 0
\(977\) 1784.43i 1.82644i −0.407471 0.913218i \(-0.633589\pi\)
0.407471 0.913218i \(-0.366411\pi\)
\(978\) 0 0
\(979\) 167.954i 0.171557i
\(980\) 0 0
\(981\) 433.635i 0.442034i
\(982\) 0 0
\(983\) 600.203i 0.610583i −0.952259 0.305291i \(-0.901246\pi\)
0.952259 0.305291i \(-0.0987538\pi\)
\(984\) 0 0
\(985\) −767.366 −0.779052
\(986\) 0 0
\(987\) 983.285i 0.996236i
\(988\) 0 0
\(989\) 27.3303 0.0276343
\(990\) 0 0
\(991\) 14.0179i 0.0141452i 0.999975 + 0.00707259i \(0.00225129\pi\)
−0.999975 + 0.00707259i \(0.997749\pi\)
\(992\) 0 0
\(993\) 2398.53 2.41544
\(994\) 0 0
\(995\) 665.415 0.668759
\(996\) 0 0
\(997\) 137.727 0.138142 0.0690708 0.997612i \(-0.477997\pi\)
0.0690708 + 0.997612i \(0.477997\pi\)
\(998\) 0 0
\(999\) −132.795 −0.132928
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 380.3.e.a.341.1 12
3.2 odd 2 3420.3.o.a.721.4 12
4.3 odd 2 1520.3.h.b.721.12 12
5.2 odd 4 1900.3.g.c.949.2 24
5.3 odd 4 1900.3.g.c.949.23 24
5.4 even 2 1900.3.e.f.1101.12 12
19.18 odd 2 inner 380.3.e.a.341.12 yes 12
57.56 even 2 3420.3.o.a.721.3 12
76.75 even 2 1520.3.h.b.721.1 12
95.18 even 4 1900.3.g.c.949.1 24
95.37 even 4 1900.3.g.c.949.24 24
95.94 odd 2 1900.3.e.f.1101.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.e.a.341.1 12 1.1 even 1 trivial
380.3.e.a.341.12 yes 12 19.18 odd 2 inner
1520.3.h.b.721.1 12 76.75 even 2
1520.3.h.b.721.12 12 4.3 odd 2
1900.3.e.f.1101.1 12 95.94 odd 2
1900.3.e.f.1101.12 12 5.4 even 2
1900.3.g.c.949.1 24 95.18 even 4
1900.3.g.c.949.2 24 5.2 odd 4
1900.3.g.c.949.23 24 5.3 odd 4
1900.3.g.c.949.24 24 95.37 even 4
3420.3.o.a.721.3 12 57.56 even 2
3420.3.o.a.721.4 12 3.2 odd 2