Properties

Label 3800.2.a.z.1.5
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $6$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.253565184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.59277\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.59277 q^{3} -1.66290 q^{7} -0.463073 q^{9} +5.56511 q^{11} -6.31979 q^{13} -4.12142 q^{17} +1.00000 q^{19} -2.64862 q^{21} +1.82315 q^{23} -5.51589 q^{27} -4.08942 q^{29} +6.61202 q^{31} +8.86396 q^{33} -9.66787 q^{37} -10.0660 q^{39} -4.61202 q^{41} +3.75231 q^{43} +3.85299 q^{47} -4.23477 q^{49} -6.56449 q^{51} -5.24594 q^{53} +1.59277 q^{57} -11.5291 q^{59} +8.02587 q^{61} +0.770043 q^{63} -0.155284 q^{67} +2.90387 q^{69} -12.1645 q^{71} -0.795627 q^{73} -9.25422 q^{77} -7.18555 q^{79} -7.39634 q^{81} -5.88500 q^{83} -6.51351 q^{87} +18.5476 q^{89} +10.5092 q^{91} +10.5315 q^{93} -2.77634 q^{97} -2.57705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.59277 0.919588 0.459794 0.888026i \(-0.347923\pi\)
0.459794 + 0.888026i \(0.347923\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.66290 −0.628516 −0.314258 0.949338i \(-0.601756\pi\)
−0.314258 + 0.949338i \(0.601756\pi\)
\(8\) 0 0
\(9\) −0.463073 −0.154358
\(10\) 0 0
\(11\) 5.56511 1.67794 0.838972 0.544174i \(-0.183157\pi\)
0.838972 + 0.544174i \(0.183157\pi\)
\(12\) 0 0
\(13\) −6.31979 −1.75280 −0.876398 0.481588i \(-0.840060\pi\)
−0.876398 + 0.481588i \(0.840060\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.12142 −0.999592 −0.499796 0.866143i \(-0.666592\pi\)
−0.499796 + 0.866143i \(0.666592\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.64862 −0.577976
\(22\) 0 0
\(23\) 1.82315 0.380154 0.190077 0.981769i \(-0.439126\pi\)
0.190077 + 0.981769i \(0.439126\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.51589 −1.06153
\(28\) 0 0
\(29\) −4.08942 −0.759385 −0.379693 0.925113i \(-0.623970\pi\)
−0.379693 + 0.925113i \(0.623970\pi\)
\(30\) 0 0
\(31\) 6.61202 1.18755 0.593777 0.804630i \(-0.297636\pi\)
0.593777 + 0.804630i \(0.297636\pi\)
\(32\) 0 0
\(33\) 8.86396 1.54302
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.66787 −1.58939 −0.794694 0.607010i \(-0.792369\pi\)
−0.794694 + 0.607010i \(0.792369\pi\)
\(38\) 0 0
\(39\) −10.0660 −1.61185
\(40\) 0 0
\(41\) −4.61202 −0.720277 −0.360138 0.932899i \(-0.617271\pi\)
−0.360138 + 0.932899i \(0.617271\pi\)
\(42\) 0 0
\(43\) 3.75231 0.572222 0.286111 0.958196i \(-0.407637\pi\)
0.286111 + 0.958196i \(0.407637\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.85299 0.562017 0.281008 0.959705i \(-0.409331\pi\)
0.281008 + 0.959705i \(0.409331\pi\)
\(48\) 0 0
\(49\) −4.23477 −0.604967
\(50\) 0 0
\(51\) −6.56449 −0.919213
\(52\) 0 0
\(53\) −5.24594 −0.720585 −0.360293 0.932839i \(-0.617323\pi\)
−0.360293 + 0.932839i \(0.617323\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.59277 0.210968
\(58\) 0 0
\(59\) −11.5291 −1.50097 −0.750483 0.660890i \(-0.770179\pi\)
−0.750483 + 0.660890i \(0.770179\pi\)
\(60\) 0 0
\(61\) 8.02587 1.02761 0.513804 0.857908i \(-0.328236\pi\)
0.513804 + 0.857908i \(0.328236\pi\)
\(62\) 0 0
\(63\) 0.770043 0.0970164
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.155284 −0.0189710 −0.00948548 0.999955i \(-0.503019\pi\)
−0.00948548 + 0.999955i \(0.503019\pi\)
\(68\) 0 0
\(69\) 2.90387 0.349585
\(70\) 0 0
\(71\) −12.1645 −1.44366 −0.721831 0.692069i \(-0.756699\pi\)
−0.721831 + 0.692069i \(0.756699\pi\)
\(72\) 0 0
\(73\) −0.795627 −0.0931211 −0.0465606 0.998915i \(-0.514826\pi\)
−0.0465606 + 0.998915i \(0.514826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.25422 −1.05462
\(78\) 0 0
\(79\) −7.18555 −0.808437 −0.404219 0.914662i \(-0.632456\pi\)
−0.404219 + 0.914662i \(0.632456\pi\)
\(80\) 0 0
\(81\) −7.39634 −0.821816
\(82\) 0 0
\(83\) −5.88500 −0.645963 −0.322981 0.946405i \(-0.604685\pi\)
−0.322981 + 0.946405i \(0.604685\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.51351 −0.698322
\(88\) 0 0
\(89\) 18.5476 1.96604 0.983021 0.183492i \(-0.0587401\pi\)
0.983021 + 0.183492i \(0.0587401\pi\)
\(90\) 0 0
\(91\) 10.5092 1.10166
\(92\) 0 0
\(93\) 10.5315 1.09206
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.77634 −0.281894 −0.140947 0.990017i \(-0.545015\pi\)
−0.140947 + 0.990017i \(0.545015\pi\)
\(98\) 0 0
\(99\) −2.57705 −0.259004
\(100\) 0 0
\(101\) −13.6991 −1.36311 −0.681557 0.731765i \(-0.738697\pi\)
−0.681557 + 0.731765i \(0.738697\pi\)
\(102\) 0 0
\(103\) −7.42279 −0.731389 −0.365695 0.930735i \(-0.619169\pi\)
−0.365695 + 0.930735i \(0.619169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.8010 1.33419 0.667097 0.744971i \(-0.267537\pi\)
0.667097 + 0.744971i \(0.267537\pi\)
\(108\) 0 0
\(109\) −12.7543 −1.22164 −0.610818 0.791771i \(-0.709159\pi\)
−0.610818 + 0.791771i \(0.709159\pi\)
\(110\) 0 0
\(111\) −15.3987 −1.46158
\(112\) 0 0
\(113\) −7.81913 −0.735562 −0.367781 0.929912i \(-0.619882\pi\)
−0.367781 + 0.929912i \(0.619882\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.92653 0.270558
\(118\) 0 0
\(119\) 6.85351 0.628260
\(120\) 0 0
\(121\) 19.9705 1.81550
\(122\) 0 0
\(123\) −7.34591 −0.662358
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.07017 −0.449905 −0.224952 0.974370i \(-0.572223\pi\)
−0.224952 + 0.974370i \(0.572223\pi\)
\(128\) 0 0
\(129\) 5.97658 0.526209
\(130\) 0 0
\(131\) 10.8720 0.949895 0.474947 0.880014i \(-0.342467\pi\)
0.474947 + 0.880014i \(0.342467\pi\)
\(132\) 0 0
\(133\) −1.66290 −0.144192
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.53861 −0.644067 −0.322033 0.946728i \(-0.604366\pi\)
−0.322033 + 0.946728i \(0.604366\pi\)
\(138\) 0 0
\(139\) −0.885954 −0.0751457 −0.0375728 0.999294i \(-0.511963\pi\)
−0.0375728 + 0.999294i \(0.511963\pi\)
\(140\) 0 0
\(141\) 6.13694 0.516824
\(142\) 0 0
\(143\) −35.1704 −2.94109
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −6.74503 −0.556321
\(148\) 0 0
\(149\) 12.2594 1.00433 0.502166 0.864771i \(-0.332537\pi\)
0.502166 + 0.864771i \(0.332537\pi\)
\(150\) 0 0
\(151\) −2.95140 −0.240181 −0.120091 0.992763i \(-0.538319\pi\)
−0.120091 + 0.992763i \(0.538319\pi\)
\(152\) 0 0
\(153\) 1.90852 0.154295
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.7951 1.26058 0.630292 0.776358i \(-0.282935\pi\)
0.630292 + 0.776358i \(0.282935\pi\)
\(158\) 0 0
\(159\) −8.35559 −0.662642
\(160\) 0 0
\(161\) −3.03172 −0.238933
\(162\) 0 0
\(163\) 20.4459 1.60144 0.800721 0.599037i \(-0.204450\pi\)
0.800721 + 0.599037i \(0.204450\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.6083 −1.13043 −0.565213 0.824945i \(-0.691206\pi\)
−0.565213 + 0.824945i \(0.691206\pi\)
\(168\) 0 0
\(169\) 26.9398 2.07229
\(170\) 0 0
\(171\) −0.463073 −0.0354121
\(172\) 0 0
\(173\) −19.4244 −1.47681 −0.738406 0.674356i \(-0.764421\pi\)
−0.738406 + 0.674356i \(0.764421\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −18.3633 −1.38027
\(178\) 0 0
\(179\) −7.21080 −0.538960 −0.269480 0.963006i \(-0.586852\pi\)
−0.269480 + 0.963006i \(0.586852\pi\)
\(180\) 0 0
\(181\) −10.1237 −0.752488 −0.376244 0.926521i \(-0.622785\pi\)
−0.376244 + 0.926521i \(0.622785\pi\)
\(182\) 0 0
\(183\) 12.7834 0.944976
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −22.9362 −1.67726
\(188\) 0 0
\(189\) 9.17236 0.667191
\(190\) 0 0
\(191\) −24.4017 −1.76564 −0.882821 0.469709i \(-0.844359\pi\)
−0.882821 + 0.469709i \(0.844359\pi\)
\(192\) 0 0
\(193\) −19.1845 −1.38093 −0.690465 0.723366i \(-0.742594\pi\)
−0.690465 + 0.723366i \(0.742594\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.9453 −0.851066 −0.425533 0.904943i \(-0.639913\pi\)
−0.425533 + 0.904943i \(0.639913\pi\)
\(198\) 0 0
\(199\) 9.20530 0.652546 0.326273 0.945276i \(-0.394207\pi\)
0.326273 + 0.945276i \(0.394207\pi\)
\(200\) 0 0
\(201\) −0.247332 −0.0174455
\(202\) 0 0
\(203\) 6.80028 0.477286
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.844253 −0.0586796
\(208\) 0 0
\(209\) 5.56511 0.384947
\(210\) 0 0
\(211\) 18.2971 1.25962 0.629811 0.776749i \(-0.283133\pi\)
0.629811 + 0.776749i \(0.283133\pi\)
\(212\) 0 0
\(213\) −19.3753 −1.32757
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.9951 −0.746397
\(218\) 0 0
\(219\) −1.26725 −0.0856331
\(220\) 0 0
\(221\) 26.0466 1.75208
\(222\) 0 0
\(223\) 1.57405 0.105406 0.0527032 0.998610i \(-0.483216\pi\)
0.0527032 + 0.998610i \(0.483216\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.6034 0.969260 0.484630 0.874719i \(-0.338954\pi\)
0.484630 + 0.874719i \(0.338954\pi\)
\(228\) 0 0
\(229\) 9.33328 0.616761 0.308380 0.951263i \(-0.400213\pi\)
0.308380 + 0.951263i \(0.400213\pi\)
\(230\) 0 0
\(231\) −14.7399 −0.969812
\(232\) 0 0
\(233\) −1.02066 −0.0668658 −0.0334329 0.999441i \(-0.510644\pi\)
−0.0334329 + 0.999441i \(0.510644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.4449 −0.743429
\(238\) 0 0
\(239\) 3.66778 0.237249 0.118624 0.992939i \(-0.462152\pi\)
0.118624 + 0.992939i \(0.462152\pi\)
\(240\) 0 0
\(241\) −23.1700 −1.49251 −0.746254 0.665661i \(-0.768150\pi\)
−0.746254 + 0.665661i \(0.768150\pi\)
\(242\) 0 0
\(243\) 4.76697 0.305801
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.31979 −0.402119
\(248\) 0 0
\(249\) −9.37347 −0.594020
\(250\) 0 0
\(251\) −29.6256 −1.86995 −0.934977 0.354709i \(-0.884580\pi\)
−0.934977 + 0.354709i \(0.884580\pi\)
\(252\) 0 0
\(253\) 10.1461 0.637877
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.9768 0.871848 0.435924 0.899984i \(-0.356422\pi\)
0.435924 + 0.899984i \(0.356422\pi\)
\(258\) 0 0
\(259\) 16.0767 0.998956
\(260\) 0 0
\(261\) 1.89370 0.117217
\(262\) 0 0
\(263\) 11.4699 0.707264 0.353632 0.935385i \(-0.384947\pi\)
0.353632 + 0.935385i \(0.384947\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 29.5421 1.80795
\(268\) 0 0
\(269\) 16.0864 0.980808 0.490404 0.871495i \(-0.336849\pi\)
0.490404 + 0.871495i \(0.336849\pi\)
\(270\) 0 0
\(271\) 15.3369 0.931649 0.465824 0.884877i \(-0.345758\pi\)
0.465824 + 0.884877i \(0.345758\pi\)
\(272\) 0 0
\(273\) 16.7387 1.01307
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.0546196 0.00328177 0.00164089 0.999999i \(-0.499478\pi\)
0.00164089 + 0.999999i \(0.499478\pi\)
\(278\) 0 0
\(279\) −3.06185 −0.183308
\(280\) 0 0
\(281\) 20.3315 1.21288 0.606438 0.795130i \(-0.292598\pi\)
0.606438 + 0.795130i \(0.292598\pi\)
\(282\) 0 0
\(283\) 22.3909 1.33100 0.665501 0.746397i \(-0.268218\pi\)
0.665501 + 0.746397i \(0.268218\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.66932 0.452706
\(288\) 0 0
\(289\) −0.0138666 −0.000815685 0
\(290\) 0 0
\(291\) −4.42208 −0.259227
\(292\) 0 0
\(293\) −13.3983 −0.782736 −0.391368 0.920234i \(-0.627998\pi\)
−0.391368 + 0.920234i \(0.627998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −30.6966 −1.78119
\(298\) 0 0
\(299\) −11.5219 −0.666332
\(300\) 0 0
\(301\) −6.23971 −0.359651
\(302\) 0 0
\(303\) −21.8196 −1.25350
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −7.60404 −0.433985 −0.216993 0.976173i \(-0.569625\pi\)
−0.216993 + 0.976173i \(0.569625\pi\)
\(308\) 0 0
\(309\) −11.8228 −0.672577
\(310\) 0 0
\(311\) 31.7283 1.79915 0.899573 0.436770i \(-0.143877\pi\)
0.899573 + 0.436770i \(0.143877\pi\)
\(312\) 0 0
\(313\) −26.2396 −1.48315 −0.741575 0.670870i \(-0.765921\pi\)
−0.741575 + 0.670870i \(0.765921\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.269210 −0.0151203 −0.00756016 0.999971i \(-0.502406\pi\)
−0.00756016 + 0.999971i \(0.502406\pi\)
\(318\) 0 0
\(319\) −22.7581 −1.27421
\(320\) 0 0
\(321\) 21.9819 1.22691
\(322\) 0 0
\(323\) −4.12142 −0.229322
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −20.3146 −1.12340
\(328\) 0 0
\(329\) −6.40713 −0.353237
\(330\) 0 0
\(331\) −9.69472 −0.532870 −0.266435 0.963853i \(-0.585846\pi\)
−0.266435 + 0.963853i \(0.585846\pi\)
\(332\) 0 0
\(333\) 4.47693 0.245334
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.4219 1.00350 0.501751 0.865012i \(-0.332689\pi\)
0.501751 + 0.865012i \(0.332689\pi\)
\(338\) 0 0
\(339\) −12.4541 −0.676414
\(340\) 0 0
\(341\) 36.7967 1.99265
\(342\) 0 0
\(343\) 18.6823 1.00875
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.85913 0.475583 0.237791 0.971316i \(-0.423577\pi\)
0.237791 + 0.971316i \(0.423577\pi\)
\(348\) 0 0
\(349\) −6.36382 −0.340647 −0.170324 0.985388i \(-0.554481\pi\)
−0.170324 + 0.985388i \(0.554481\pi\)
\(350\) 0 0
\(351\) 34.8593 1.86065
\(352\) 0 0
\(353\) 26.4213 1.40627 0.703133 0.711058i \(-0.251784\pi\)
0.703133 + 0.711058i \(0.251784\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.9161 0.577740
\(358\) 0 0
\(359\) 0.904490 0.0477372 0.0238686 0.999715i \(-0.492402\pi\)
0.0238686 + 0.999715i \(0.492402\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 31.8085 1.66951
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −20.6657 −1.07874 −0.539370 0.842069i \(-0.681338\pi\)
−0.539370 + 0.842069i \(0.681338\pi\)
\(368\) 0 0
\(369\) 2.13570 0.111180
\(370\) 0 0
\(371\) 8.72347 0.452900
\(372\) 0 0
\(373\) −23.7919 −1.23190 −0.615949 0.787786i \(-0.711227\pi\)
−0.615949 + 0.787786i \(0.711227\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.8443 1.33105
\(378\) 0 0
\(379\) 27.9667 1.43655 0.718276 0.695758i \(-0.244931\pi\)
0.718276 + 0.695758i \(0.244931\pi\)
\(380\) 0 0
\(381\) −8.07563 −0.413727
\(382\) 0 0
\(383\) −12.1083 −0.618707 −0.309353 0.950947i \(-0.600113\pi\)
−0.309353 + 0.950947i \(0.600113\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.73760 −0.0883269
\(388\) 0 0
\(389\) 0.871634 0.0441936 0.0220968 0.999756i \(-0.492966\pi\)
0.0220968 + 0.999756i \(0.492966\pi\)
\(390\) 0 0
\(391\) −7.51398 −0.379998
\(392\) 0 0
\(393\) 17.3167 0.873512
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.7833 1.04308 0.521542 0.853226i \(-0.325357\pi\)
0.521542 + 0.853226i \(0.325357\pi\)
\(398\) 0 0
\(399\) −2.64862 −0.132597
\(400\) 0 0
\(401\) −16.0237 −0.800184 −0.400092 0.916475i \(-0.631022\pi\)
−0.400092 + 0.916475i \(0.631022\pi\)
\(402\) 0 0
\(403\) −41.7866 −2.08154
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −53.8028 −2.66691
\(408\) 0 0
\(409\) 20.1554 0.996621 0.498311 0.866999i \(-0.333954\pi\)
0.498311 + 0.866999i \(0.333954\pi\)
\(410\) 0 0
\(411\) −12.0073 −0.592276
\(412\) 0 0
\(413\) 19.1718 0.943382
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.41112 −0.0691031
\(418\) 0 0
\(419\) 5.15507 0.251842 0.125921 0.992040i \(-0.459811\pi\)
0.125921 + 0.992040i \(0.459811\pi\)
\(420\) 0 0
\(421\) 26.5417 1.29356 0.646781 0.762676i \(-0.276115\pi\)
0.646781 + 0.762676i \(0.276115\pi\)
\(422\) 0 0
\(423\) −1.78422 −0.0867516
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −13.3462 −0.645868
\(428\) 0 0
\(429\) −56.0184 −2.70460
\(430\) 0 0
\(431\) −21.2788 −1.02496 −0.512481 0.858699i \(-0.671274\pi\)
−0.512481 + 0.858699i \(0.671274\pi\)
\(432\) 0 0
\(433\) 6.44963 0.309949 0.154975 0.987918i \(-0.450470\pi\)
0.154975 + 0.987918i \(0.450470\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.82315 0.0872132
\(438\) 0 0
\(439\) 22.2324 1.06110 0.530548 0.847655i \(-0.321986\pi\)
0.530548 + 0.847655i \(0.321986\pi\)
\(440\) 0 0
\(441\) 1.96101 0.0933813
\(442\) 0 0
\(443\) −7.56611 −0.359477 −0.179738 0.983714i \(-0.557525\pi\)
−0.179738 + 0.983714i \(0.557525\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 19.5265 0.923571
\(448\) 0 0
\(449\) 22.5411 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(450\) 0 0
\(451\) −25.6664 −1.20858
\(452\) 0 0
\(453\) −4.70091 −0.220868
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.9942 −1.40307 −0.701536 0.712634i \(-0.747502\pi\)
−0.701536 + 0.712634i \(0.747502\pi\)
\(458\) 0 0
\(459\) 22.7333 1.06110
\(460\) 0 0
\(461\) −35.7493 −1.66501 −0.832506 0.554016i \(-0.813095\pi\)
−0.832506 + 0.554016i \(0.813095\pi\)
\(462\) 0 0
\(463\) 9.81829 0.456295 0.228147 0.973627i \(-0.426733\pi\)
0.228147 + 0.973627i \(0.426733\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.96046 −0.0907194 −0.0453597 0.998971i \(-0.514443\pi\)
−0.0453597 + 0.998971i \(0.514443\pi\)
\(468\) 0 0
\(469\) 0.258222 0.0119236
\(470\) 0 0
\(471\) 25.1580 1.15922
\(472\) 0 0
\(473\) 20.8821 0.960158
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.42925 0.111228
\(478\) 0 0
\(479\) −9.88045 −0.451450 −0.225725 0.974191i \(-0.572475\pi\)
−0.225725 + 0.974191i \(0.572475\pi\)
\(480\) 0 0
\(481\) 61.0989 2.78587
\(482\) 0 0
\(483\) −4.82884 −0.219720
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.03706 −0.409508 −0.204754 0.978813i \(-0.565639\pi\)
−0.204754 + 0.978813i \(0.565639\pi\)
\(488\) 0 0
\(489\) 32.5656 1.47267
\(490\) 0 0
\(491\) −20.3715 −0.919353 −0.459676 0.888086i \(-0.652035\pi\)
−0.459676 + 0.888086i \(0.652035\pi\)
\(492\) 0 0
\(493\) 16.8542 0.759076
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.2283 0.907365
\(498\) 0 0
\(499\) −13.9386 −0.623979 −0.311990 0.950086i \(-0.600995\pi\)
−0.311990 + 0.950086i \(0.600995\pi\)
\(500\) 0 0
\(501\) −23.2678 −1.03953
\(502\) 0 0
\(503\) 39.7299 1.77147 0.885735 0.464192i \(-0.153655\pi\)
0.885735 + 0.464192i \(0.153655\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 42.9090 1.90566
\(508\) 0 0
\(509\) −12.5200 −0.554941 −0.277471 0.960734i \(-0.589496\pi\)
−0.277471 + 0.960734i \(0.589496\pi\)
\(510\) 0 0
\(511\) 1.32305 0.0585282
\(512\) 0 0
\(513\) −5.51589 −0.243533
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 21.4423 0.943033
\(518\) 0 0
\(519\) −30.9387 −1.35806
\(520\) 0 0
\(521\) −0.0426487 −0.00186847 −0.000934237 1.00000i \(-0.500297\pi\)
−0.000934237 1.00000i \(0.500297\pi\)
\(522\) 0 0
\(523\) −29.4459 −1.28758 −0.643789 0.765203i \(-0.722639\pi\)
−0.643789 + 0.765203i \(0.722639\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −27.2509 −1.18707
\(528\) 0 0
\(529\) −19.6761 −0.855483
\(530\) 0 0
\(531\) 5.33883 0.231686
\(532\) 0 0
\(533\) 29.1470 1.26250
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.4852 −0.495621
\(538\) 0 0
\(539\) −23.5670 −1.01510
\(540\) 0 0
\(541\) −20.0558 −0.862264 −0.431132 0.902289i \(-0.641886\pi\)
−0.431132 + 0.902289i \(0.641886\pi\)
\(542\) 0 0
\(543\) −16.1248 −0.691979
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14.8952 −0.636874 −0.318437 0.947944i \(-0.603158\pi\)
−0.318437 + 0.947944i \(0.603158\pi\)
\(548\) 0 0
\(549\) −3.71657 −0.158619
\(550\) 0 0
\(551\) −4.08942 −0.174215
\(552\) 0 0
\(553\) 11.9488 0.508116
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.26068 −0.392388 −0.196194 0.980565i \(-0.562858\pi\)
−0.196194 + 0.980565i \(0.562858\pi\)
\(558\) 0 0
\(559\) −23.7138 −1.00299
\(560\) 0 0
\(561\) −36.5322 −1.54239
\(562\) 0 0
\(563\) −12.9472 −0.545661 −0.272830 0.962062i \(-0.587960\pi\)
−0.272830 + 0.962062i \(0.587960\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.2994 0.516525
\(568\) 0 0
\(569\) 20.0593 0.840929 0.420464 0.907309i \(-0.361867\pi\)
0.420464 + 0.907309i \(0.361867\pi\)
\(570\) 0 0
\(571\) −5.64571 −0.236266 −0.118133 0.992998i \(-0.537691\pi\)
−0.118133 + 0.992998i \(0.537691\pi\)
\(572\) 0 0
\(573\) −38.8663 −1.62366
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −33.5402 −1.39630 −0.698148 0.715953i \(-0.745992\pi\)
−0.698148 + 0.715953i \(0.745992\pi\)
\(578\) 0 0
\(579\) −30.5566 −1.26989
\(580\) 0 0
\(581\) 9.78616 0.405998
\(582\) 0 0
\(583\) −29.1943 −1.20910
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.5517 1.13718 0.568591 0.822621i \(-0.307489\pi\)
0.568591 + 0.822621i \(0.307489\pi\)
\(588\) 0 0
\(589\) 6.61202 0.272444
\(590\) 0 0
\(591\) −19.0261 −0.782630
\(592\) 0 0
\(593\) 24.4353 1.00344 0.501719 0.865031i \(-0.332701\pi\)
0.501719 + 0.865031i \(0.332701\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.6620 0.600074
\(598\) 0 0
\(599\) −3.51535 −0.143633 −0.0718166 0.997418i \(-0.522880\pi\)
−0.0718166 + 0.997418i \(0.522880\pi\)
\(600\) 0 0
\(601\) −1.21824 −0.0496932 −0.0248466 0.999691i \(-0.507910\pi\)
−0.0248466 + 0.999691i \(0.507910\pi\)
\(602\) 0 0
\(603\) 0.0719079 0.00292832
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −13.1872 −0.535251 −0.267625 0.963523i \(-0.586239\pi\)
−0.267625 + 0.963523i \(0.586239\pi\)
\(608\) 0 0
\(609\) 10.8313 0.438907
\(610\) 0 0
\(611\) −24.3501 −0.985100
\(612\) 0 0
\(613\) −9.67880 −0.390923 −0.195462 0.980711i \(-0.562621\pi\)
−0.195462 + 0.980711i \(0.562621\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.4077 0.418999 0.209500 0.977809i \(-0.432816\pi\)
0.209500 + 0.977809i \(0.432816\pi\)
\(618\) 0 0
\(619\) −32.7144 −1.31490 −0.657452 0.753496i \(-0.728366\pi\)
−0.657452 + 0.753496i \(0.728366\pi\)
\(620\) 0 0
\(621\) −10.0563 −0.403546
\(622\) 0 0
\(623\) −30.8428 −1.23569
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.86396 0.353993
\(628\) 0 0
\(629\) 39.8454 1.58874
\(630\) 0 0
\(631\) −37.2305 −1.48212 −0.741061 0.671438i \(-0.765677\pi\)
−0.741061 + 0.671438i \(0.765677\pi\)
\(632\) 0 0
\(633\) 29.1431 1.15833
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 26.7629 1.06038
\(638\) 0 0
\(639\) 5.63306 0.222840
\(640\) 0 0
\(641\) 23.3803 0.923464 0.461732 0.887019i \(-0.347228\pi\)
0.461732 + 0.887019i \(0.347228\pi\)
\(642\) 0 0
\(643\) −24.6705 −0.972911 −0.486455 0.873705i \(-0.661710\pi\)
−0.486455 + 0.873705i \(0.661710\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.2591 1.85795 0.928973 0.370147i \(-0.120693\pi\)
0.928973 + 0.370147i \(0.120693\pi\)
\(648\) 0 0
\(649\) −64.1610 −2.51854
\(650\) 0 0
\(651\) −17.5127 −0.686378
\(652\) 0 0
\(653\) 31.5361 1.23410 0.617051 0.786924i \(-0.288327\pi\)
0.617051 + 0.786924i \(0.288327\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.368434 0.0143740
\(658\) 0 0
\(659\) −21.3511 −0.831722 −0.415861 0.909428i \(-0.636520\pi\)
−0.415861 + 0.909428i \(0.636520\pi\)
\(660\) 0 0
\(661\) −31.2161 −1.21416 −0.607082 0.794639i \(-0.707660\pi\)
−0.607082 + 0.794639i \(0.707660\pi\)
\(662\) 0 0
\(663\) 41.4863 1.61119
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.45563 −0.288683
\(668\) 0 0
\(669\) 2.50711 0.0969304
\(670\) 0 0
\(671\) 44.6649 1.72427
\(672\) 0 0
\(673\) −4.16240 −0.160449 −0.0802243 0.996777i \(-0.525564\pi\)
−0.0802243 + 0.996777i \(0.525564\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 35.5734 1.36720 0.683599 0.729857i \(-0.260414\pi\)
0.683599 + 0.729857i \(0.260414\pi\)
\(678\) 0 0
\(679\) 4.61677 0.177175
\(680\) 0 0
\(681\) 23.2599 0.891320
\(682\) 0 0
\(683\) 26.8628 1.02788 0.513938 0.857827i \(-0.328186\pi\)
0.513938 + 0.857827i \(0.328186\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.8658 0.567166
\(688\) 0 0
\(689\) 33.1533 1.26304
\(690\) 0 0
\(691\) 32.0064 1.21758 0.608791 0.793331i \(-0.291655\pi\)
0.608791 + 0.793331i \(0.291655\pi\)
\(692\) 0 0
\(693\) 4.28538 0.162788
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 19.0081 0.719983
\(698\) 0 0
\(699\) −1.62568 −0.0614890
\(700\) 0 0
\(701\) −0.822942 −0.0310821 −0.0155410 0.999879i \(-0.504947\pi\)
−0.0155410 + 0.999879i \(0.504947\pi\)
\(702\) 0 0
\(703\) −9.66787 −0.364631
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.7802 0.856739
\(708\) 0 0
\(709\) 24.8057 0.931599 0.465800 0.884890i \(-0.345767\pi\)
0.465800 + 0.884890i \(0.345767\pi\)
\(710\) 0 0
\(711\) 3.32743 0.124789
\(712\) 0 0
\(713\) 12.0547 0.451453
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.84194 0.218171
\(718\) 0 0
\(719\) 13.8765 0.517506 0.258753 0.965943i \(-0.416688\pi\)
0.258753 + 0.965943i \(0.416688\pi\)
\(720\) 0 0
\(721\) 12.3433 0.459690
\(722\) 0 0
\(723\) −36.9045 −1.37249
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.7158 1.06501 0.532506 0.846426i \(-0.321251\pi\)
0.532506 + 0.846426i \(0.321251\pi\)
\(728\) 0 0
\(729\) 29.7817 1.10303
\(730\) 0 0
\(731\) −15.4649 −0.571989
\(732\) 0 0
\(733\) 47.1903 1.74301 0.871506 0.490384i \(-0.163144\pi\)
0.871506 + 0.490384i \(0.163144\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.864174 −0.0318322
\(738\) 0 0
\(739\) −7.61271 −0.280038 −0.140019 0.990149i \(-0.544716\pi\)
−0.140019 + 0.990149i \(0.544716\pi\)
\(740\) 0 0
\(741\) −10.0660 −0.369784
\(742\) 0 0
\(743\) −15.8127 −0.580111 −0.290055 0.957010i \(-0.593674\pi\)
−0.290055 + 0.957010i \(0.593674\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.72519 0.0997093
\(748\) 0 0
\(749\) −22.9497 −0.838563
\(750\) 0 0
\(751\) 35.7516 1.30459 0.652296 0.757964i \(-0.273806\pi\)
0.652296 + 0.757964i \(0.273806\pi\)
\(752\) 0 0
\(753\) −47.1869 −1.71959
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.8136 0.465717 0.232859 0.972511i \(-0.425192\pi\)
0.232859 + 0.972511i \(0.425192\pi\)
\(758\) 0 0
\(759\) 16.1604 0.586584
\(760\) 0 0
\(761\) 18.8830 0.684508 0.342254 0.939608i \(-0.388810\pi\)
0.342254 + 0.939608i \(0.388810\pi\)
\(762\) 0 0
\(763\) 21.2090 0.767818
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 72.8618 2.63089
\(768\) 0 0
\(769\) −16.0228 −0.577798 −0.288899 0.957360i \(-0.593289\pi\)
−0.288899 + 0.957360i \(0.593289\pi\)
\(770\) 0 0
\(771\) 22.2618 0.801741
\(772\) 0 0
\(773\) −9.91039 −0.356452 −0.178226 0.983990i \(-0.557036\pi\)
−0.178226 + 0.983990i \(0.557036\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 25.6065 0.918628
\(778\) 0 0
\(779\) −4.61202 −0.165243
\(780\) 0 0
\(781\) −67.6969 −2.42238
\(782\) 0 0
\(783\) 22.5568 0.806113
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 30.9671 1.10386 0.551929 0.833891i \(-0.313892\pi\)
0.551929 + 0.833891i \(0.313892\pi\)
\(788\) 0 0
\(789\) 18.2689 0.650392
\(790\) 0 0
\(791\) 13.0024 0.462313
\(792\) 0 0
\(793\) −50.7219 −1.80119
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 54.8059 1.94132 0.970662 0.240448i \(-0.0772945\pi\)
0.970662 + 0.240448i \(0.0772945\pi\)
\(798\) 0 0
\(799\) −15.8798 −0.561787
\(800\) 0 0
\(801\) −8.58890 −0.303474
\(802\) 0 0
\(803\) −4.42776 −0.156252
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 25.6221 0.901939
\(808\) 0 0
\(809\) 25.6862 0.903080 0.451540 0.892251i \(-0.350875\pi\)
0.451540 + 0.892251i \(0.350875\pi\)
\(810\) 0 0
\(811\) 8.01224 0.281348 0.140674 0.990056i \(-0.455073\pi\)
0.140674 + 0.990056i \(0.455073\pi\)
\(812\) 0 0
\(813\) 24.4282 0.856733
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.75231 0.131277
\(818\) 0 0
\(819\) −4.86652 −0.170050
\(820\) 0 0
\(821\) −48.8946 −1.70643 −0.853217 0.521555i \(-0.825352\pi\)
−0.853217 + 0.521555i \(0.825352\pi\)
\(822\) 0 0
\(823\) −39.8314 −1.38843 −0.694217 0.719766i \(-0.744249\pi\)
−0.694217 + 0.719766i \(0.744249\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.8758 −1.00411 −0.502056 0.864835i \(-0.667423\pi\)
−0.502056 + 0.864835i \(0.667423\pi\)
\(828\) 0 0
\(829\) −2.35519 −0.0817991 −0.0408995 0.999163i \(-0.513022\pi\)
−0.0408995 + 0.999163i \(0.513022\pi\)
\(830\) 0 0
\(831\) 0.0869966 0.00301788
\(832\) 0 0
\(833\) 17.4533 0.604720
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −36.4712 −1.26063
\(838\) 0 0
\(839\) 26.0230 0.898415 0.449207 0.893428i \(-0.351706\pi\)
0.449207 + 0.893428i \(0.351706\pi\)
\(840\) 0 0
\(841\) −12.2767 −0.423334
\(842\) 0 0
\(843\) 32.3835 1.11535
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −33.2089 −1.14107
\(848\) 0 0
\(849\) 35.6636 1.22397
\(850\) 0 0
\(851\) −17.6260 −0.604211
\(852\) 0 0
\(853\) −37.7907 −1.29393 −0.646964 0.762520i \(-0.723962\pi\)
−0.646964 + 0.762520i \(0.723962\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.7796 0.675657 0.337828 0.941208i \(-0.390308\pi\)
0.337828 + 0.941208i \(0.390308\pi\)
\(858\) 0 0
\(859\) 22.0641 0.752816 0.376408 0.926454i \(-0.377159\pi\)
0.376408 + 0.926454i \(0.377159\pi\)
\(860\) 0 0
\(861\) 12.2155 0.416303
\(862\) 0 0
\(863\) 32.3505 1.10122 0.550612 0.834761i \(-0.314394\pi\)
0.550612 + 0.834761i \(0.314394\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.0220864 −0.000750094 0
\(868\) 0 0
\(869\) −39.9884 −1.35651
\(870\) 0 0
\(871\) 0.981363 0.0332522
\(872\) 0 0
\(873\) 1.28565 0.0435126
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.5876 0.762728 0.381364 0.924425i \(-0.375454\pi\)
0.381364 + 0.924425i \(0.375454\pi\)
\(878\) 0 0
\(879\) −21.3404 −0.719794
\(880\) 0 0
\(881\) 31.2658 1.05337 0.526686 0.850060i \(-0.323434\pi\)
0.526686 + 0.850060i \(0.323434\pi\)
\(882\) 0 0
\(883\) −22.4584 −0.755786 −0.377893 0.925849i \(-0.623351\pi\)
−0.377893 + 0.925849i \(0.623351\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −14.4883 −0.486470 −0.243235 0.969967i \(-0.578209\pi\)
−0.243235 + 0.969967i \(0.578209\pi\)
\(888\) 0 0
\(889\) 8.43117 0.282772
\(890\) 0 0
\(891\) −41.1615 −1.37896
\(892\) 0 0
\(893\) 3.85299 0.128935
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −18.3519 −0.612751
\(898\) 0 0
\(899\) −27.0393 −0.901811
\(900\) 0 0
\(901\) 21.6207 0.720291
\(902\) 0 0
\(903\) −9.93845 −0.330731
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.58465 0.318253 0.159127 0.987258i \(-0.449132\pi\)
0.159127 + 0.987258i \(0.449132\pi\)
\(908\) 0 0
\(909\) 6.34370 0.210407
\(910\) 0 0
\(911\) 29.8122 0.987721 0.493860 0.869541i \(-0.335585\pi\)
0.493860 + 0.869541i \(0.335585\pi\)
\(912\) 0 0
\(913\) −32.7507 −1.08389
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −18.0791 −0.597025
\(918\) 0 0
\(919\) −22.0075 −0.725961 −0.362981 0.931797i \(-0.618241\pi\)
−0.362981 + 0.931797i \(0.618241\pi\)
\(920\) 0 0
\(921\) −12.1115 −0.399088
\(922\) 0 0
\(923\) 76.8772 2.53044
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.43729 0.112896
\(928\) 0 0
\(929\) −22.7036 −0.744880 −0.372440 0.928056i \(-0.621479\pi\)
−0.372440 + 0.928056i \(0.621479\pi\)
\(930\) 0 0
\(931\) −4.23477 −0.138789
\(932\) 0 0
\(933\) 50.5360 1.65447
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −10.5567 −0.344871 −0.172436 0.985021i \(-0.555164\pi\)
−0.172436 + 0.985021i \(0.555164\pi\)
\(938\) 0 0
\(939\) −41.7937 −1.36389
\(940\) 0 0
\(941\) −24.7961 −0.808331 −0.404166 0.914686i \(-0.632438\pi\)
−0.404166 + 0.914686i \(0.632438\pi\)
\(942\) 0 0
\(943\) −8.40842 −0.273816
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.83000 0.189449 0.0947247 0.995504i \(-0.469803\pi\)
0.0947247 + 0.995504i \(0.469803\pi\)
\(948\) 0 0
\(949\) 5.02820 0.163222
\(950\) 0 0
\(951\) −0.428790 −0.0139045
\(952\) 0 0
\(953\) −7.15932 −0.231913 −0.115957 0.993254i \(-0.536993\pi\)
−0.115957 + 0.993254i \(0.536993\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −36.2484 −1.17175
\(958\) 0 0
\(959\) 12.5359 0.404806
\(960\) 0 0
\(961\) 12.7188 0.410285
\(962\) 0 0
\(963\) −6.39088 −0.205943
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 10.3120 0.331612 0.165806 0.986158i \(-0.446977\pi\)
0.165806 + 0.986158i \(0.446977\pi\)
\(968\) 0 0
\(969\) −6.56449 −0.210882
\(970\) 0 0
\(971\) 38.2430 1.22728 0.613639 0.789587i \(-0.289705\pi\)
0.613639 + 0.789587i \(0.289705\pi\)
\(972\) 0 0
\(973\) 1.47325 0.0472303
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.5114 0.592232 0.296116 0.955152i \(-0.404308\pi\)
0.296116 + 0.955152i \(0.404308\pi\)
\(978\) 0 0
\(979\) 103.220 3.29891
\(980\) 0 0
\(981\) 5.90615 0.188569
\(982\) 0 0
\(983\) 10.9590 0.349539 0.174769 0.984609i \(-0.444082\pi\)
0.174769 + 0.984609i \(0.444082\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −10.2051 −0.324832
\(988\) 0 0
\(989\) 6.84104 0.217532
\(990\) 0 0
\(991\) −36.8766 −1.17142 −0.585712 0.810519i \(-0.699185\pi\)
−0.585712 + 0.810519i \(0.699185\pi\)
\(992\) 0 0
\(993\) −15.4415 −0.490021
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −36.4988 −1.15593 −0.577964 0.816062i \(-0.696153\pi\)
−0.577964 + 0.816062i \(0.696153\pi\)
\(998\) 0 0
\(999\) 53.3269 1.68719
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 3800.2.a.z.1.5 6
4.3 odd 2 7600.2.a.cn.1.2 6
5.2 odd 4 760.2.d.e.609.4 12
5.3 odd 4 760.2.d.e.609.9 yes 12
5.4 even 2 3800.2.a.be.1.2 6
20.3 even 4 1520.2.d.k.609.4 12
20.7 even 4 1520.2.d.k.609.9 12
20.19 odd 2 7600.2.a.cg.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.e.609.4 12 5.2 odd 4
760.2.d.e.609.9 yes 12 5.3 odd 4
1520.2.d.k.609.4 12 20.3 even 4
1520.2.d.k.609.9 12 20.7 even 4
3800.2.a.z.1.5 6 1.1 even 1 trivial
3800.2.a.be.1.2 6 5.4 even 2
7600.2.a.cg.1.5 6 20.19 odd 2
7600.2.a.cn.1.2 6 4.3 odd 2