Properties

Label 1104.2.a.m.1.2
Level $1104$
Weight $2$
Character 1104.1
Self dual yes
Analytic conductor $8.815$
Analytic rank $1$
Dimension $2$
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] = [1104,2,Mod(1,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, 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("1104.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(8.81548438315\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.23607 q^{5} -3.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} -4.47214 q^{13} +1.23607 q^{15} -2.76393 q^{17} -7.23607 q^{19} -3.23607 q^{21} -1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +4.47214 q^{29} +6.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} +4.47214 q^{37} -4.47214 q^{39} -10.9443 q^{41} +5.70820 q^{43} +1.23607 q^{45} +4.00000 q^{47} +3.47214 q^{49} -2.76393 q^{51} -5.23607 q^{53} -4.94427 q^{55} -7.23607 q^{57} +4.94427 q^{59} +4.47214 q^{61} -3.23607 q^{63} -5.52786 q^{65} -0.763932 q^{67} -1.00000 q^{69} +8.00000 q^{71} +6.94427 q^{73} -3.47214 q^{75} +12.9443 q^{77} -9.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} -3.41641 q^{85} +4.47214 q^{87} -1.23607 q^{89} +14.4721 q^{91} +6.47214 q^{93} -8.94427 q^{95} +8.47214 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 8 q^{11} - 2 q^{15} - 10 q^{17} - 10 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} - 2 q^{45} + 8 q^{47} - 2 q^{49} - 10 q^{51} - 6 q^{53} + 8 q^{55} - 10 q^{57} - 8 q^{59} - 2 q^{63} - 20 q^{65} - 6 q^{67} - 2 q^{69} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{77} - 6 q^{79} + 2 q^{81} - 8 q^{83} + 20 q^{85} + 2 q^{89} + 20 q^{91} + 4 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) −3.23607 −1.22312 −0.611559 0.791199i \(-0.709457\pi\)
−0.611559 + 0.791199i \(0.709457\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 1.23607 0.319151
\(16\) 0 0
\(17\) −2.76393 −0.670352 −0.335176 0.942156i \(-0.608796\pi\)
−0.335176 + 0.942156i \(0.608796\pi\)
\(18\) 0 0
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 6.47214 1.16243 0.581215 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −10.9443 −1.70921 −0.854604 0.519280i \(-0.826200\pi\)
−0.854604 + 0.519280i \(0.826200\pi\)
\(42\) 0 0
\(43\) 5.70820 0.870493 0.435246 0.900311i \(-0.356661\pi\)
0.435246 + 0.900311i \(0.356661\pi\)
\(44\) 0 0
\(45\) 1.23607 0.184262
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) −2.76393 −0.387028
\(52\) 0 0
\(53\) −5.23607 −0.719229 −0.359615 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(54\) 0 0
\(55\) −4.94427 −0.666685
\(56\) 0 0
\(57\) −7.23607 −0.958441
\(58\) 0 0
\(59\) 4.94427 0.643689 0.321845 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 0 0
\(63\) −3.23607 −0.407706
\(64\) 0 0
\(65\) −5.52786 −0.685647
\(66\) 0 0
\(67\) −0.763932 −0.0933292 −0.0466646 0.998911i \(-0.514859\pi\)
−0.0466646 + 0.998911i \(0.514859\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.94427 0.812766 0.406383 0.913703i \(-0.366790\pi\)
0.406383 + 0.913703i \(0.366790\pi\)
\(74\) 0 0
\(75\) −3.47214 −0.400928
\(76\) 0 0
\(77\) 12.9443 1.47514
\(78\) 0 0
\(79\) −9.70820 −1.09226 −0.546129 0.837701i \(-0.683899\pi\)
−0.546129 + 0.837701i \(0.683899\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −3.41641 −0.370561
\(86\) 0 0
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) −1.23607 −0.131023 −0.0655115 0.997852i \(-0.520868\pi\)
−0.0655115 + 0.997852i \(0.520868\pi\)
\(90\) 0 0
\(91\) 14.4721 1.51709
\(92\) 0 0
\(93\) 6.47214 0.671129
\(94\) 0 0
\(95\) −8.94427 −0.917663
\(96\) 0 0
\(97\) 8.47214 0.860215 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −6.94427 −0.690981 −0.345490 0.938422i \(-0.612287\pi\)
−0.345490 + 0.938422i \(0.612287\pi\)
\(102\) 0 0
\(103\) 11.2361 1.10712 0.553561 0.832808i \(-0.313268\pi\)
0.553561 + 0.832808i \(0.313268\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) −16.9443 −1.63806 −0.819032 0.573747i \(-0.805489\pi\)
−0.819032 + 0.573747i \(0.805489\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 4.47214 0.424476
\(112\) 0 0
\(113\) −6.18034 −0.581397 −0.290699 0.956815i \(-0.593888\pi\)
−0.290699 + 0.956815i \(0.593888\pi\)
\(114\) 0 0
\(115\) −1.23607 −0.115264
\(116\) 0 0
\(117\) −4.47214 −0.413449
\(118\) 0 0
\(119\) 8.94427 0.819920
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −10.9443 −0.986812
\(124\) 0 0
\(125\) −10.4721 −0.936656
\(126\) 0 0
\(127\) −1.52786 −0.135576 −0.0677880 0.997700i \(-0.521594\pi\)
−0.0677880 + 0.997700i \(0.521594\pi\)
\(128\) 0 0
\(129\) 5.70820 0.502579
\(130\) 0 0
\(131\) −16.9443 −1.48043 −0.740214 0.672371i \(-0.765276\pi\)
−0.740214 + 0.672371i \(0.765276\pi\)
\(132\) 0 0
\(133\) 23.4164 2.03046
\(134\) 0 0
\(135\) 1.23607 0.106384
\(136\) 0 0
\(137\) −1.23607 −0.105604 −0.0528022 0.998605i \(-0.516815\pi\)
−0.0528022 + 0.998605i \(0.516815\pi\)
\(138\) 0 0
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 17.8885 1.49592
\(144\) 0 0
\(145\) 5.52786 0.459064
\(146\) 0 0
\(147\) 3.47214 0.286377
\(148\) 0 0
\(149\) −11.7082 −0.959173 −0.479587 0.877494i \(-0.659213\pi\)
−0.479587 + 0.877494i \(0.659213\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −2.76393 −0.223451
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −3.52786 −0.281554 −0.140777 0.990041i \(-0.544960\pi\)
−0.140777 + 0.990041i \(0.544960\pi\)
\(158\) 0 0
\(159\) −5.23607 −0.415247
\(160\) 0 0
\(161\) 3.23607 0.255038
\(162\) 0 0
\(163\) −7.41641 −0.580898 −0.290449 0.956890i \(-0.593805\pi\)
−0.290449 + 0.956890i \(0.593805\pi\)
\(164\) 0 0
\(165\) −4.94427 −0.384911
\(166\) 0 0
\(167\) −12.9443 −1.00166 −0.500829 0.865546i \(-0.666971\pi\)
−0.500829 + 0.865546i \(0.666971\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) −7.23607 −0.553356
\(172\) 0 0
\(173\) 4.47214 0.340010 0.170005 0.985443i \(-0.445622\pi\)
0.170005 + 0.985443i \(0.445622\pi\)
\(174\) 0 0
\(175\) 11.2361 0.849367
\(176\) 0 0
\(177\) 4.94427 0.371634
\(178\) 0 0
\(179\) −20.9443 −1.56545 −0.782724 0.622369i \(-0.786170\pi\)
−0.782724 + 0.622369i \(0.786170\pi\)
\(180\) 0 0
\(181\) −11.8885 −0.883669 −0.441834 0.897097i \(-0.645672\pi\)
−0.441834 + 0.897097i \(0.645672\pi\)
\(182\) 0 0
\(183\) 4.47214 0.330590
\(184\) 0 0
\(185\) 5.52786 0.406417
\(186\) 0 0
\(187\) 11.0557 0.808475
\(188\) 0 0
\(189\) −3.23607 −0.235389
\(190\) 0 0
\(191\) −1.52786 −0.110552 −0.0552762 0.998471i \(-0.517604\pi\)
−0.0552762 + 0.998471i \(0.517604\pi\)
\(192\) 0 0
\(193\) 17.4164 1.25366 0.626830 0.779156i \(-0.284352\pi\)
0.626830 + 0.779156i \(0.284352\pi\)
\(194\) 0 0
\(195\) −5.52786 −0.395859
\(196\) 0 0
\(197\) 17.4164 1.24087 0.620434 0.784259i \(-0.286957\pi\)
0.620434 + 0.784259i \(0.286957\pi\)
\(198\) 0 0
\(199\) −9.70820 −0.688196 −0.344098 0.938934i \(-0.611815\pi\)
−0.344098 + 0.938934i \(0.611815\pi\)
\(200\) 0 0
\(201\) −0.763932 −0.0538836
\(202\) 0 0
\(203\) −14.4721 −1.01574
\(204\) 0 0
\(205\) −13.5279 −0.944827
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 28.9443 2.00212
\(210\) 0 0
\(211\) −13.5279 −0.931297 −0.465648 0.884970i \(-0.654179\pi\)
−0.465648 + 0.884970i \(0.654179\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 7.05573 0.481197
\(216\) 0 0
\(217\) −20.9443 −1.42179
\(218\) 0 0
\(219\) 6.94427 0.469250
\(220\) 0 0
\(221\) 12.3607 0.831469
\(222\) 0 0
\(223\) 25.8885 1.73363 0.866813 0.498634i \(-0.166165\pi\)
0.866813 + 0.498634i \(0.166165\pi\)
\(224\) 0 0
\(225\) −3.47214 −0.231476
\(226\) 0 0
\(227\) 13.5279 0.897876 0.448938 0.893563i \(-0.351802\pi\)
0.448938 + 0.893563i \(0.351802\pi\)
\(228\) 0 0
\(229\) 2.94427 0.194563 0.0972815 0.995257i \(-0.468985\pi\)
0.0972815 + 0.995257i \(0.468985\pi\)
\(230\) 0 0
\(231\) 12.9443 0.851671
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) 0 0
\(237\) −9.70820 −0.630616
\(238\) 0 0
\(239\) −4.94427 −0.319818 −0.159909 0.987132i \(-0.551120\pi\)
−0.159909 + 0.987132i \(0.551120\pi\)
\(240\) 0 0
\(241\) 19.5279 1.25790 0.628950 0.777446i \(-0.283485\pi\)
0.628950 + 0.777446i \(0.283485\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 4.29180 0.274193
\(246\) 0 0
\(247\) 32.3607 2.05906
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 10.4721 0.660995 0.330498 0.943807i \(-0.392783\pi\)
0.330498 + 0.943807i \(0.392783\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −3.41641 −0.213944
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) −14.4721 −0.899255
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) 0 0
\(263\) −6.47214 −0.399089 −0.199544 0.979889i \(-0.563946\pi\)
−0.199544 + 0.979889i \(0.563946\pi\)
\(264\) 0 0
\(265\) −6.47214 −0.397580
\(266\) 0 0
\(267\) −1.23607 −0.0756461
\(268\) 0 0
\(269\) −0.472136 −0.0287866 −0.0143933 0.999896i \(-0.504582\pi\)
−0.0143933 + 0.999896i \(0.504582\pi\)
\(270\) 0 0
\(271\) 17.5279 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(272\) 0 0
\(273\) 14.4721 0.875894
\(274\) 0 0
\(275\) 13.8885 0.837511
\(276\) 0 0
\(277\) 15.8885 0.954650 0.477325 0.878727i \(-0.341606\pi\)
0.477325 + 0.878727i \(0.341606\pi\)
\(278\) 0 0
\(279\) 6.47214 0.387477
\(280\) 0 0
\(281\) 24.6525 1.47064 0.735322 0.677718i \(-0.237031\pi\)
0.735322 + 0.677718i \(0.237031\pi\)
\(282\) 0 0
\(283\) −5.70820 −0.339318 −0.169659 0.985503i \(-0.554267\pi\)
−0.169659 + 0.985503i \(0.554267\pi\)
\(284\) 0 0
\(285\) −8.94427 −0.529813
\(286\) 0 0
\(287\) 35.4164 2.09056
\(288\) 0 0
\(289\) −9.36068 −0.550628
\(290\) 0 0
\(291\) 8.47214 0.496645
\(292\) 0 0
\(293\) 7.70820 0.450318 0.225159 0.974322i \(-0.427710\pi\)
0.225159 + 0.974322i \(0.427710\pi\)
\(294\) 0 0
\(295\) 6.11146 0.355823
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) −18.4721 −1.06472
\(302\) 0 0
\(303\) −6.94427 −0.398938
\(304\) 0 0
\(305\) 5.52786 0.316525
\(306\) 0 0
\(307\) −2.47214 −0.141092 −0.0705461 0.997509i \(-0.522474\pi\)
−0.0705461 + 0.997509i \(0.522474\pi\)
\(308\) 0 0
\(309\) 11.2361 0.639198
\(310\) 0 0
\(311\) −7.05573 −0.400094 −0.200047 0.979786i \(-0.564109\pi\)
−0.200047 + 0.979786i \(0.564109\pi\)
\(312\) 0 0
\(313\) 5.05573 0.285767 0.142883 0.989740i \(-0.454363\pi\)
0.142883 + 0.989740i \(0.454363\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 2.58359 0.145109 0.0725545 0.997364i \(-0.476885\pi\)
0.0725545 + 0.997364i \(0.476885\pi\)
\(318\) 0 0
\(319\) −17.8885 −1.00157
\(320\) 0 0
\(321\) −16.9443 −0.945737
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 0 0
\(325\) 15.5279 0.861331
\(326\) 0 0
\(327\) −14.9443 −0.826420
\(328\) 0 0
\(329\) −12.9443 −0.713641
\(330\) 0 0
\(331\) −23.4164 −1.28708 −0.643541 0.765412i \(-0.722535\pi\)
−0.643541 + 0.765412i \(0.722535\pi\)
\(332\) 0 0
\(333\) 4.47214 0.245072
\(334\) 0 0
\(335\) −0.944272 −0.0515911
\(336\) 0 0
\(337\) −14.3607 −0.782276 −0.391138 0.920332i \(-0.627918\pi\)
−0.391138 + 0.920332i \(0.627918\pi\)
\(338\) 0 0
\(339\) −6.18034 −0.335670
\(340\) 0 0
\(341\) −25.8885 −1.40194
\(342\) 0 0
\(343\) 11.4164 0.616428
\(344\) 0 0
\(345\) −1.23607 −0.0665477
\(346\) 0 0
\(347\) 9.88854 0.530845 0.265422 0.964132i \(-0.414489\pi\)
0.265422 + 0.964132i \(0.414489\pi\)
\(348\) 0 0
\(349\) −23.5279 −1.25942 −0.629709 0.776831i \(-0.716826\pi\)
−0.629709 + 0.776831i \(0.716826\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) −17.4164 −0.926982 −0.463491 0.886102i \(-0.653403\pi\)
−0.463491 + 0.886102i \(0.653403\pi\)
\(354\) 0 0
\(355\) 9.88854 0.524829
\(356\) 0 0
\(357\) 8.94427 0.473381
\(358\) 0 0
\(359\) −6.47214 −0.341586 −0.170793 0.985307i \(-0.554633\pi\)
−0.170793 + 0.985307i \(0.554633\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 8.58359 0.449286
\(366\) 0 0
\(367\) −25.7082 −1.34196 −0.670979 0.741477i \(-0.734126\pi\)
−0.670979 + 0.741477i \(0.734126\pi\)
\(368\) 0 0
\(369\) −10.9443 −0.569736
\(370\) 0 0
\(371\) 16.9443 0.879703
\(372\) 0 0
\(373\) 20.4721 1.06001 0.530004 0.847995i \(-0.322191\pi\)
0.530004 + 0.847995i \(0.322191\pi\)
\(374\) 0 0
\(375\) −10.4721 −0.540779
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −30.0689 −1.54453 −0.772267 0.635298i \(-0.780877\pi\)
−0.772267 + 0.635298i \(0.780877\pi\)
\(380\) 0 0
\(381\) −1.52786 −0.0782748
\(382\) 0 0
\(383\) −25.5279 −1.30441 −0.652206 0.758041i \(-0.726156\pi\)
−0.652206 + 0.758041i \(0.726156\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 5.70820 0.290164
\(388\) 0 0
\(389\) 20.6525 1.04712 0.523561 0.851988i \(-0.324603\pi\)
0.523561 + 0.851988i \(0.324603\pi\)
\(390\) 0 0
\(391\) 2.76393 0.139778
\(392\) 0 0
\(393\) −16.9443 −0.854725
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 23.4164 1.17229
\(400\) 0 0
\(401\) 20.0689 1.00219 0.501096 0.865392i \(-0.332930\pi\)
0.501096 + 0.865392i \(0.332930\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) 1.23607 0.0614207
\(406\) 0 0
\(407\) −17.8885 −0.886702
\(408\) 0 0
\(409\) 17.4164 0.861186 0.430593 0.902546i \(-0.358304\pi\)
0.430593 + 0.902546i \(0.358304\pi\)
\(410\) 0 0
\(411\) −1.23607 −0.0609707
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −4.94427 −0.242705
\(416\) 0 0
\(417\) 8.94427 0.438003
\(418\) 0 0
\(419\) −13.8885 −0.678500 −0.339250 0.940696i \(-0.610173\pi\)
−0.339250 + 0.940696i \(0.610173\pi\)
\(420\) 0 0
\(421\) −30.9443 −1.50813 −0.754066 0.656799i \(-0.771910\pi\)
−0.754066 + 0.656799i \(0.771910\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 9.59675 0.465511
\(426\) 0 0
\(427\) −14.4721 −0.700356
\(428\) 0 0
\(429\) 17.8885 0.863667
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 27.8885 1.34024 0.670119 0.742254i \(-0.266243\pi\)
0.670119 + 0.742254i \(0.266243\pi\)
\(434\) 0 0
\(435\) 5.52786 0.265041
\(436\) 0 0
\(437\) 7.23607 0.346148
\(438\) 0 0
\(439\) 9.88854 0.471954 0.235977 0.971759i \(-0.424171\pi\)
0.235977 + 0.971759i \(0.424171\pi\)
\(440\) 0 0
\(441\) 3.47214 0.165340
\(442\) 0 0
\(443\) 34.8328 1.65496 0.827479 0.561497i \(-0.189775\pi\)
0.827479 + 0.561497i \(0.189775\pi\)
\(444\) 0 0
\(445\) −1.52786 −0.0724277
\(446\) 0 0
\(447\) −11.7082 −0.553779
\(448\) 0 0
\(449\) 14.9443 0.705264 0.352632 0.935762i \(-0.385287\pi\)
0.352632 + 0.935762i \(0.385287\pi\)
\(450\) 0 0
\(451\) 43.7771 2.06138
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) 17.8885 0.838628
\(456\) 0 0
\(457\) 8.47214 0.396310 0.198155 0.980171i \(-0.436505\pi\)
0.198155 + 0.980171i \(0.436505\pi\)
\(458\) 0 0
\(459\) −2.76393 −0.129009
\(460\) 0 0
\(461\) −5.41641 −0.252267 −0.126134 0.992013i \(-0.540257\pi\)
−0.126134 + 0.992013i \(0.540257\pi\)
\(462\) 0 0
\(463\) 12.9443 0.601571 0.300786 0.953692i \(-0.402751\pi\)
0.300786 + 0.953692i \(0.402751\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) −20.3607 −0.942180 −0.471090 0.882085i \(-0.656139\pi\)
−0.471090 + 0.882085i \(0.656139\pi\)
\(468\) 0 0
\(469\) 2.47214 0.114153
\(470\) 0 0
\(471\) −3.52786 −0.162555
\(472\) 0 0
\(473\) −22.8328 −1.04985
\(474\) 0 0
\(475\) 25.1246 1.15280
\(476\) 0 0
\(477\) −5.23607 −0.239743
\(478\) 0 0
\(479\) 33.8885 1.54841 0.774204 0.632937i \(-0.218151\pi\)
0.774204 + 0.632937i \(0.218151\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 3.23607 0.147246
\(484\) 0 0
\(485\) 10.4721 0.475515
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) −7.41641 −0.335382
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) −12.3607 −0.556697
\(494\) 0 0
\(495\) −4.94427 −0.222228
\(496\) 0 0
\(497\) −25.8885 −1.16126
\(498\) 0 0
\(499\) 12.3607 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(500\) 0 0
\(501\) −12.9443 −0.578307
\(502\) 0 0
\(503\) 19.0557 0.849653 0.424826 0.905275i \(-0.360335\pi\)
0.424826 + 0.905275i \(0.360335\pi\)
\(504\) 0 0
\(505\) −8.58359 −0.381965
\(506\) 0 0
\(507\) 7.00000 0.310881
\(508\) 0 0
\(509\) −16.4721 −0.730115 −0.365057 0.930985i \(-0.618951\pi\)
−0.365057 + 0.930985i \(0.618951\pi\)
\(510\) 0 0
\(511\) −22.4721 −0.994109
\(512\) 0 0
\(513\) −7.23607 −0.319480
\(514\) 0 0
\(515\) 13.8885 0.612002
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 4.47214 0.196305
\(520\) 0 0
\(521\) −33.2361 −1.45610 −0.728049 0.685525i \(-0.759573\pi\)
−0.728049 + 0.685525i \(0.759573\pi\)
\(522\) 0 0
\(523\) −15.5967 −0.681998 −0.340999 0.940064i \(-0.610765\pi\)
−0.340999 + 0.940064i \(0.610765\pi\)
\(524\) 0 0
\(525\) 11.2361 0.490382
\(526\) 0 0
\(527\) −17.8885 −0.779237
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.94427 0.214563
\(532\) 0 0
\(533\) 48.9443 2.12001
\(534\) 0 0
\(535\) −20.9443 −0.905500
\(536\) 0 0
\(537\) −20.9443 −0.903812
\(538\) 0 0
\(539\) −13.8885 −0.598222
\(540\) 0 0
\(541\) −15.5279 −0.667595 −0.333798 0.942645i \(-0.608330\pi\)
−0.333798 + 0.942645i \(0.608330\pi\)
\(542\) 0 0
\(543\) −11.8885 −0.510186
\(544\) 0 0
\(545\) −18.4721 −0.791259
\(546\) 0 0
\(547\) −13.5279 −0.578410 −0.289205 0.957267i \(-0.593391\pi\)
−0.289205 + 0.957267i \(0.593391\pi\)
\(548\) 0 0
\(549\) 4.47214 0.190866
\(550\) 0 0
\(551\) −32.3607 −1.37861
\(552\) 0 0
\(553\) 31.4164 1.33596
\(554\) 0 0
\(555\) 5.52786 0.234645
\(556\) 0 0
\(557\) −42.1803 −1.78724 −0.893619 0.448826i \(-0.851842\pi\)
−0.893619 + 0.448826i \(0.851842\pi\)
\(558\) 0 0
\(559\) −25.5279 −1.07971
\(560\) 0 0
\(561\) 11.0557 0.466773
\(562\) 0 0
\(563\) 7.41641 0.312564 0.156282 0.987712i \(-0.450049\pi\)
0.156282 + 0.987712i \(0.450049\pi\)
\(564\) 0 0
\(565\) −7.63932 −0.321389
\(566\) 0 0
\(567\) −3.23607 −0.135902
\(568\) 0 0
\(569\) −10.7639 −0.451248 −0.225624 0.974215i \(-0.572442\pi\)
−0.225624 + 0.974215i \(0.572442\pi\)
\(570\) 0 0
\(571\) −29.7082 −1.24325 −0.621625 0.783315i \(-0.713527\pi\)
−0.621625 + 0.783315i \(0.713527\pi\)
\(572\) 0 0
\(573\) −1.52786 −0.0638274
\(574\) 0 0
\(575\) 3.47214 0.144798
\(576\) 0 0
\(577\) 7.52786 0.313389 0.156695 0.987647i \(-0.449916\pi\)
0.156695 + 0.987647i \(0.449916\pi\)
\(578\) 0 0
\(579\) 17.4164 0.723801
\(580\) 0 0
\(581\) 12.9443 0.537019
\(582\) 0 0
\(583\) 20.9443 0.867423
\(584\) 0 0
\(585\) −5.52786 −0.228549
\(586\) 0 0
\(587\) 0.944272 0.0389743 0.0194871 0.999810i \(-0.493797\pi\)
0.0194871 + 0.999810i \(0.493797\pi\)
\(588\) 0 0
\(589\) −46.8328 −1.92971
\(590\) 0 0
\(591\) 17.4164 0.716415
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 11.0557 0.453241
\(596\) 0 0
\(597\) −9.70820 −0.397330
\(598\) 0 0
\(599\) 3.05573 0.124854 0.0624268 0.998050i \(-0.480116\pi\)
0.0624268 + 0.998050i \(0.480116\pi\)
\(600\) 0 0
\(601\) −42.3607 −1.72793 −0.863964 0.503553i \(-0.832026\pi\)
−0.863964 + 0.503553i \(0.832026\pi\)
\(602\) 0 0
\(603\) −0.763932 −0.0311097
\(604\) 0 0
\(605\) 6.18034 0.251267
\(606\) 0 0
\(607\) 14.8328 0.602045 0.301023 0.953617i \(-0.402672\pi\)
0.301023 + 0.953617i \(0.402672\pi\)
\(608\) 0 0
\(609\) −14.4721 −0.586441
\(610\) 0 0
\(611\) −17.8885 −0.723693
\(612\) 0 0
\(613\) −40.4721 −1.63465 −0.817327 0.576174i \(-0.804545\pi\)
−0.817327 + 0.576174i \(0.804545\pi\)
\(614\) 0 0
\(615\) −13.5279 −0.545496
\(616\) 0 0
\(617\) −20.2918 −0.816917 −0.408458 0.912777i \(-0.633934\pi\)
−0.408458 + 0.912777i \(0.633934\pi\)
\(618\) 0 0
\(619\) 9.12461 0.366749 0.183375 0.983043i \(-0.441298\pi\)
0.183375 + 0.983043i \(0.441298\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 28.9443 1.15592
\(628\) 0 0
\(629\) −12.3607 −0.492853
\(630\) 0 0
\(631\) 16.1803 0.644129 0.322065 0.946718i \(-0.395623\pi\)
0.322065 + 0.946718i \(0.395623\pi\)
\(632\) 0 0
\(633\) −13.5279 −0.537684
\(634\) 0 0
\(635\) −1.88854 −0.0749446
\(636\) 0 0
\(637\) −15.5279 −0.615236
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 49.0132 1.93590 0.967952 0.251137i \(-0.0808044\pi\)
0.967952 + 0.251137i \(0.0808044\pi\)
\(642\) 0 0
\(643\) −38.0689 −1.50129 −0.750645 0.660706i \(-0.770257\pi\)
−0.750645 + 0.660706i \(0.770257\pi\)
\(644\) 0 0
\(645\) 7.05573 0.277819
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −19.7771 −0.776319
\(650\) 0 0
\(651\) −20.9443 −0.820871
\(652\) 0 0
\(653\) −22.9443 −0.897879 −0.448939 0.893562i \(-0.648198\pi\)
−0.448939 + 0.893562i \(0.648198\pi\)
\(654\) 0 0
\(655\) −20.9443 −0.818360
\(656\) 0 0
\(657\) 6.94427 0.270922
\(658\) 0 0
\(659\) −36.3607 −1.41641 −0.708205 0.706006i \(-0.750495\pi\)
−0.708205 + 0.706006i \(0.750495\pi\)
\(660\) 0 0
\(661\) −11.5279 −0.448382 −0.224191 0.974545i \(-0.571974\pi\)
−0.224191 + 0.974545i \(0.571974\pi\)
\(662\) 0 0
\(663\) 12.3607 0.480049
\(664\) 0 0
\(665\) 28.9443 1.12241
\(666\) 0 0
\(667\) −4.47214 −0.173162
\(668\) 0 0
\(669\) 25.8885 1.00091
\(670\) 0 0
\(671\) −17.8885 −0.690580
\(672\) 0 0
\(673\) 2.58359 0.0995902 0.0497951 0.998759i \(-0.484143\pi\)
0.0497951 + 0.998759i \(0.484143\pi\)
\(674\) 0 0
\(675\) −3.47214 −0.133643
\(676\) 0 0
\(677\) −47.4853 −1.82501 −0.912504 0.409068i \(-0.865854\pi\)
−0.912504 + 0.409068i \(0.865854\pi\)
\(678\) 0 0
\(679\) −27.4164 −1.05215
\(680\) 0 0
\(681\) 13.5279 0.518389
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) −1.52786 −0.0583767
\(686\) 0 0
\(687\) 2.94427 0.112331
\(688\) 0 0
\(689\) 23.4164 0.892094
\(690\) 0 0
\(691\) −4.36068 −0.165888 −0.0829440 0.996554i \(-0.526432\pi\)
−0.0829440 + 0.996554i \(0.526432\pi\)
\(692\) 0 0
\(693\) 12.9443 0.491712
\(694\) 0 0
\(695\) 11.0557 0.419368
\(696\) 0 0
\(697\) 30.2492 1.14577
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 25.2361 0.953153 0.476577 0.879133i \(-0.341878\pi\)
0.476577 + 0.879133i \(0.341878\pi\)
\(702\) 0 0
\(703\) −32.3607 −1.22051
\(704\) 0 0
\(705\) 4.94427 0.186212
\(706\) 0 0
\(707\) 22.4721 0.845152
\(708\) 0 0
\(709\) −32.8328 −1.23306 −0.616531 0.787331i \(-0.711463\pi\)
−0.616531 + 0.787331i \(0.711463\pi\)
\(710\) 0 0
\(711\) −9.70820 −0.364086
\(712\) 0 0
\(713\) −6.47214 −0.242383
\(714\) 0 0
\(715\) 22.1115 0.826922
\(716\) 0 0
\(717\) −4.94427 −0.184647
\(718\) 0 0
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) −36.3607 −1.35414
\(722\) 0 0
\(723\) 19.5279 0.726249
\(724\) 0 0
\(725\) −15.5279 −0.576690
\(726\) 0 0
\(727\) −1.34752 −0.0499769 −0.0249885 0.999688i \(-0.507955\pi\)
−0.0249885 + 0.999688i \(0.507955\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.7771 −0.583537
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 4.29180 0.158305
\(736\) 0 0
\(737\) 3.05573 0.112559
\(738\) 0 0
\(739\) 26.8328 0.987061 0.493531 0.869728i \(-0.335706\pi\)
0.493531 + 0.869728i \(0.335706\pi\)
\(740\) 0 0
\(741\) 32.3607 1.18880
\(742\) 0 0
\(743\) 24.3607 0.893707 0.446853 0.894607i \(-0.352545\pi\)
0.446853 + 0.894607i \(0.352545\pi\)
\(744\) 0 0
\(745\) −14.4721 −0.530218
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 54.8328 2.00355
\(750\) 0 0
\(751\) −50.0689 −1.82704 −0.913520 0.406794i \(-0.866647\pi\)
−0.913520 + 0.406794i \(0.866647\pi\)
\(752\) 0 0
\(753\) 10.4721 0.381626
\(754\) 0 0
\(755\) 19.7771 0.719762
\(756\) 0 0
\(757\) 39.8885 1.44977 0.724887 0.688868i \(-0.241892\pi\)
0.724887 + 0.688868i \(0.241892\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 42.3607 1.53557 0.767787 0.640706i \(-0.221358\pi\)
0.767787 + 0.640706i \(0.221358\pi\)
\(762\) 0 0
\(763\) 48.3607 1.75077
\(764\) 0 0
\(765\) −3.41641 −0.123520
\(766\) 0 0
\(767\) −22.1115 −0.798398
\(768\) 0 0
\(769\) 16.8328 0.607007 0.303503 0.952830i \(-0.401844\pi\)
0.303503 + 0.952830i \(0.401844\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 0 0
\(773\) 36.2918 1.30533 0.652663 0.757649i \(-0.273652\pi\)
0.652663 + 0.757649i \(0.273652\pi\)
\(774\) 0 0
\(775\) −22.4721 −0.807223
\(776\) 0 0
\(777\) −14.4721 −0.519185
\(778\) 0 0
\(779\) 79.1935 2.83740
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 4.47214 0.159821
\(784\) 0 0
\(785\) −4.36068 −0.155639
\(786\) 0 0
\(787\) 13.7082 0.488645 0.244322 0.969694i \(-0.421434\pi\)
0.244322 + 0.969694i \(0.421434\pi\)
\(788\) 0 0
\(789\) −6.47214 −0.230414
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) −6.47214 −0.229543
\(796\) 0 0
\(797\) 12.6525 0.448174 0.224087 0.974569i \(-0.428060\pi\)
0.224087 + 0.974569i \(0.428060\pi\)
\(798\) 0 0
\(799\) −11.0557 −0.391124
\(800\) 0 0
\(801\) −1.23607 −0.0436743
\(802\) 0 0
\(803\) −27.7771 −0.980232
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) −0.472136 −0.0166200
\(808\) 0 0
\(809\) −43.3050 −1.52252 −0.761261 0.648446i \(-0.775419\pi\)
−0.761261 + 0.648446i \(0.775419\pi\)
\(810\) 0 0
\(811\) 23.4164 0.822261 0.411131 0.911576i \(-0.365134\pi\)
0.411131 + 0.911576i \(0.365134\pi\)
\(812\) 0 0
\(813\) 17.5279 0.614729
\(814\) 0 0
\(815\) −9.16718 −0.321112
\(816\) 0 0
\(817\) −41.3050 −1.44508
\(818\) 0 0
\(819\) 14.4721 0.505697
\(820\) 0 0
\(821\) 12.1115 0.422693 0.211346 0.977411i \(-0.432215\pi\)
0.211346 + 0.977411i \(0.432215\pi\)
\(822\) 0 0
\(823\) 25.5279 0.889845 0.444923 0.895569i \(-0.353231\pi\)
0.444923 + 0.895569i \(0.353231\pi\)
\(824\) 0 0
\(825\) 13.8885 0.483537
\(826\) 0 0
\(827\) −8.58359 −0.298481 −0.149240 0.988801i \(-0.547683\pi\)
−0.149240 + 0.988801i \(0.547683\pi\)
\(828\) 0 0
\(829\) 10.3607 0.359841 0.179921 0.983681i \(-0.442416\pi\)
0.179921 + 0.983681i \(0.442416\pi\)
\(830\) 0 0
\(831\) 15.8885 0.551167
\(832\) 0 0
\(833\) −9.59675 −0.332508
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 6.47214 0.223710
\(838\) 0 0
\(839\) 12.5836 0.434434 0.217217 0.976123i \(-0.430302\pi\)
0.217217 + 0.976123i \(0.430302\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 24.6525 0.849076
\(844\) 0 0
\(845\) 8.65248 0.297654
\(846\) 0 0
\(847\) −16.1803 −0.555963
\(848\) 0 0
\(849\) −5.70820 −0.195905
\(850\) 0 0
\(851\) −4.47214 −0.153303
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) −8.94427 −0.305888
\(856\) 0 0
\(857\) −42.9443 −1.46695 −0.733474 0.679717i \(-0.762102\pi\)
−0.733474 + 0.679717i \(0.762102\pi\)
\(858\) 0 0
\(859\) 7.05573 0.240738 0.120369 0.992729i \(-0.461592\pi\)
0.120369 + 0.992729i \(0.461592\pi\)
\(860\) 0 0
\(861\) 35.4164 1.20699
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 5.52786 0.187953
\(866\) 0 0
\(867\) −9.36068 −0.317905
\(868\) 0 0
\(869\) 38.8328 1.31731
\(870\) 0 0
\(871\) 3.41641 0.115761
\(872\) 0 0
\(873\) 8.47214 0.286738
\(874\) 0 0
\(875\) 33.8885 1.14564
\(876\) 0 0
\(877\) 33.0557 1.11621 0.558106 0.829769i \(-0.311528\pi\)
0.558106 + 0.829769i \(0.311528\pi\)
\(878\) 0 0
\(879\) 7.70820 0.259991
\(880\) 0 0
\(881\) 26.5410 0.894190 0.447095 0.894487i \(-0.352459\pi\)
0.447095 + 0.894487i \(0.352459\pi\)
\(882\) 0 0
\(883\) 7.41641 0.249582 0.124791 0.992183i \(-0.460174\pi\)
0.124791 + 0.992183i \(0.460174\pi\)
\(884\) 0 0
\(885\) 6.11146 0.205434
\(886\) 0 0
\(887\) −31.0557 −1.04275 −0.521375 0.853328i \(-0.674581\pi\)
−0.521375 + 0.853328i \(0.674581\pi\)
\(888\) 0 0
\(889\) 4.94427 0.165826
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) −28.9443 −0.968583
\(894\) 0 0
\(895\) −25.8885 −0.865359
\(896\) 0 0
\(897\) 4.47214 0.149320
\(898\) 0 0
\(899\) 28.9443 0.965346
\(900\) 0 0
\(901\) 14.4721 0.482137
\(902\) 0 0
\(903\) −18.4721 −0.614714
\(904\) 0 0
\(905\) −14.6950 −0.488480
\(906\) 0 0
\(907\) 1.12461 0.0373421 0.0186711 0.999826i \(-0.494056\pi\)
0.0186711 + 0.999826i \(0.494056\pi\)
\(908\) 0 0
\(909\) −6.94427 −0.230327
\(910\) 0 0
\(911\) 40.7214 1.34916 0.674579 0.738202i \(-0.264325\pi\)
0.674579 + 0.738202i \(0.264325\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) 5.52786 0.182746
\(916\) 0 0
\(917\) 54.8328 1.81074
\(918\) 0 0
\(919\) 23.0132 0.759134 0.379567 0.925164i \(-0.376073\pi\)
0.379567 + 0.925164i \(0.376073\pi\)
\(920\) 0 0
\(921\) −2.47214 −0.0814596
\(922\) 0 0
\(923\) −35.7771 −1.17762
\(924\) 0 0
\(925\) −15.5279 −0.510553
\(926\) 0 0
\(927\) 11.2361 0.369041
\(928\) 0 0
\(929\) −28.8328 −0.945974 −0.472987 0.881069i \(-0.656824\pi\)
−0.472987 + 0.881069i \(0.656824\pi\)
\(930\) 0 0
\(931\) −25.1246 −0.823426
\(932\) 0 0
\(933\) −7.05573 −0.230994
\(934\) 0 0
\(935\) 13.6656 0.446914
\(936\) 0 0
\(937\) −12.8328 −0.419230 −0.209615 0.977784i \(-0.567221\pi\)
−0.209615 + 0.977784i \(0.567221\pi\)
\(938\) 0 0
\(939\) 5.05573 0.164987
\(940\) 0 0
\(941\) −42.5410 −1.38680 −0.693399 0.720554i \(-0.743888\pi\)
−0.693399 + 0.720554i \(0.743888\pi\)
\(942\) 0 0
\(943\) 10.9443 0.356395
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 1.16718 0.0379284 0.0189642 0.999820i \(-0.493963\pi\)
0.0189642 + 0.999820i \(0.493963\pi\)
\(948\) 0 0
\(949\) −31.0557 −1.00811
\(950\) 0 0
\(951\) 2.58359 0.0837787
\(952\) 0 0
\(953\) 12.0689 0.390949 0.195475 0.980709i \(-0.437375\pi\)
0.195475 + 0.980709i \(0.437375\pi\)
\(954\) 0 0
\(955\) −1.88854 −0.0611118
\(956\) 0 0
\(957\) −17.8885 −0.578254
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) −16.9443 −0.546022
\(964\) 0 0
\(965\) 21.5279 0.693006
\(966\) 0 0
\(967\) 7.63932 0.245664 0.122832 0.992427i \(-0.460802\pi\)
0.122832 + 0.992427i \(0.460802\pi\)
\(968\) 0 0
\(969\) 20.0000 0.642493
\(970\) 0 0
\(971\) −17.3050 −0.555342 −0.277671 0.960676i \(-0.589563\pi\)
−0.277671 + 0.960676i \(0.589563\pi\)
\(972\) 0 0
\(973\) −28.9443 −0.927911
\(974\) 0 0
\(975\) 15.5279 0.497290
\(976\) 0 0
\(977\) −32.0689 −1.02597 −0.512987 0.858396i \(-0.671461\pi\)
−0.512987 + 0.858396i \(0.671461\pi\)
\(978\) 0 0
\(979\) 4.94427 0.158020
\(980\) 0 0
\(981\) −14.9443 −0.477134
\(982\) 0 0
\(983\) −22.8328 −0.728254 −0.364127 0.931349i \(-0.618633\pi\)
−0.364127 + 0.931349i \(0.618633\pi\)
\(984\) 0 0
\(985\) 21.5279 0.685935
\(986\) 0 0
\(987\) −12.9443 −0.412021
\(988\) 0 0
\(989\) −5.70820 −0.181510
\(990\) 0 0
\(991\) 1.52786 0.0485342 0.0242671 0.999706i \(-0.492275\pi\)
0.0242671 + 0.999706i \(0.492275\pi\)
\(992\) 0 0
\(993\) −23.4164 −0.743097
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 42.3607 1.34158 0.670788 0.741649i \(-0.265956\pi\)
0.670788 + 0.741649i \(0.265956\pi\)
\(998\) 0 0
\(999\) 4.47214 0.141492
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 1104.2.a.m.1.2 2
3.2 odd 2 3312.2.a.bb.1.1 2
4.3 odd 2 69.2.a.b.1.1 2
8.3 odd 2 4416.2.a.bm.1.1 2
8.5 even 2 4416.2.a.bg.1.1 2
12.11 even 2 207.2.a.c.1.2 2
20.3 even 4 1725.2.b.o.1174.3 4
20.7 even 4 1725.2.b.o.1174.2 4
20.19 odd 2 1725.2.a.ba.1.2 2
28.27 even 2 3381.2.a.t.1.1 2
44.43 even 2 8349.2.a.i.1.2 2
60.59 even 2 5175.2.a.bk.1.1 2
92.91 even 2 1587.2.a.i.1.1 2
276.275 odd 2 4761.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.1 2 4.3 odd 2
207.2.a.c.1.2 2 12.11 even 2
1104.2.a.m.1.2 2 1.1 even 1 trivial
1587.2.a.i.1.1 2 92.91 even 2
1725.2.a.ba.1.2 2 20.19 odd 2
1725.2.b.o.1174.2 4 20.7 even 4
1725.2.b.o.1174.3 4 20.3 even 4
3312.2.a.bb.1.1 2 3.2 odd 2
3381.2.a.t.1.1 2 28.27 even 2
4416.2.a.bg.1.1 2 8.5 even 2
4416.2.a.bm.1.1 2 8.3 odd 2
4761.2.a.v.1.2 2 276.275 odd 2
5175.2.a.bk.1.1 2 60.59 even 2
8349.2.a.i.1.2 2 44.43 even 2