Properties

Label 1932.2.a.c.1.1
Level $1932$
Weight $2$
Character 1932.1
Self dual yes
Analytic conductor $15.427$
Analytic rank $0$
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] = [1932,2,Mod(1,1932)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1932, 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("1932.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.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: \(15.4270976705\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 1932.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.30278 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.30278 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.60555 q^{11} -1.30278 q^{13} +4.30278 q^{15} -7.60555 q^{17} -7.60555 q^{19} +1.00000 q^{21} -1.00000 q^{23} +13.5139 q^{25} -1.00000 q^{27} +1.60555 q^{29} +3.60555 q^{31} +1.60555 q^{33} +4.30278 q^{35} -1.60555 q^{37} +1.30278 q^{39} -3.60555 q^{41} +0.697224 q^{43} -4.30278 q^{45} +2.60555 q^{47} +1.00000 q^{49} +7.60555 q^{51} -4.90833 q^{53} +6.90833 q^{55} +7.60555 q^{57} -6.90833 q^{59} -5.90833 q^{61} -1.00000 q^{63} +5.60555 q^{65} +9.69722 q^{67} +1.00000 q^{69} +2.09167 q^{71} +11.0000 q^{73} -13.5139 q^{75} +1.60555 q^{77} -15.4222 q^{79} +1.00000 q^{81} +1.78890 q^{83} +32.7250 q^{85} -1.60555 q^{87} +2.51388 q^{89} +1.30278 q^{91} -3.60555 q^{93} +32.7250 q^{95} +18.2111 q^{97} -1.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} + q^{13} + 5 q^{15} - 8 q^{17} - 8 q^{19} + 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{33} + 5 q^{35} + 4 q^{37} - q^{39} + 5 q^{43} - 5 q^{45} - 2 q^{47} + 2 q^{49} + 8 q^{51} + q^{53} + 3 q^{55} + 8 q^{57} - 3 q^{59} - q^{61} - 2 q^{63} + 4 q^{65} + 23 q^{67} + 2 q^{69} + 15 q^{71} + 22 q^{73} - 9 q^{75} - 4 q^{77} - 2 q^{79} + 2 q^{81} + 18 q^{83} + 33 q^{85} + 4 q^{87} - 13 q^{89} - q^{91} + 33 q^{95} + 22 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(5\) −4.30278 −1.92426 −0.962130 0.272591i \(-0.912119\pi\)
−0.962130 + 0.272591i \(0.912119\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.60555 −0.484092 −0.242046 0.970265i \(-0.577818\pi\)
−0.242046 + 0.970265i \(0.577818\pi\)
\(12\) 0 0
\(13\) −1.30278 −0.361325 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 0 0
\(15\) 4.30278 1.11097
\(16\) 0 0
\(17\) −7.60555 −1.84462 −0.922309 0.386454i \(-0.873700\pi\)
−0.922309 + 0.386454i \(0.873700\pi\)
\(18\) 0 0
\(19\) −7.60555 −1.74483 −0.872417 0.488763i \(-0.837448\pi\)
−0.872417 + 0.488763i \(0.837448\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 13.5139 2.70278
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.60555 0.298143 0.149072 0.988826i \(-0.452371\pi\)
0.149072 + 0.988826i \(0.452371\pi\)
\(30\) 0 0
\(31\) 3.60555 0.647576 0.323788 0.946130i \(-0.395044\pi\)
0.323788 + 0.946130i \(0.395044\pi\)
\(32\) 0 0
\(33\) 1.60555 0.279491
\(34\) 0 0
\(35\) 4.30278 0.727302
\(36\) 0 0
\(37\) −1.60555 −0.263951 −0.131976 0.991253i \(-0.542132\pi\)
−0.131976 + 0.991253i \(0.542132\pi\)
\(38\) 0 0
\(39\) 1.30278 0.208611
\(40\) 0 0
\(41\) −3.60555 −0.563093 −0.281546 0.959548i \(-0.590847\pi\)
−0.281546 + 0.959548i \(0.590847\pi\)
\(42\) 0 0
\(43\) 0.697224 0.106326 0.0531629 0.998586i \(-0.483070\pi\)
0.0531629 + 0.998586i \(0.483070\pi\)
\(44\) 0 0
\(45\) −4.30278 −0.641420
\(46\) 0 0
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 7.60555 1.06499
\(52\) 0 0
\(53\) −4.90833 −0.674211 −0.337105 0.941467i \(-0.609448\pi\)
−0.337105 + 0.941467i \(0.609448\pi\)
\(54\) 0 0
\(55\) 6.90833 0.931519
\(56\) 0 0
\(57\) 7.60555 1.00738
\(58\) 0 0
\(59\) −6.90833 −0.899388 −0.449694 0.893183i \(-0.648467\pi\)
−0.449694 + 0.893183i \(0.648467\pi\)
\(60\) 0 0
\(61\) −5.90833 −0.756484 −0.378242 0.925707i \(-0.623471\pi\)
−0.378242 + 0.925707i \(0.623471\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 5.60555 0.695283
\(66\) 0 0
\(67\) 9.69722 1.18470 0.592352 0.805679i \(-0.298199\pi\)
0.592352 + 0.805679i \(0.298199\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 2.09167 0.248236 0.124118 0.992267i \(-0.460390\pi\)
0.124118 + 0.992267i \(0.460390\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −13.5139 −1.56045
\(76\) 0 0
\(77\) 1.60555 0.182970
\(78\) 0 0
\(79\) −15.4222 −1.73513 −0.867567 0.497321i \(-0.834317\pi\)
−0.867567 + 0.497321i \(0.834317\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.78890 0.196357 0.0981785 0.995169i \(-0.468698\pi\)
0.0981785 + 0.995169i \(0.468698\pi\)
\(84\) 0 0
\(85\) 32.7250 3.54952
\(86\) 0 0
\(87\) −1.60555 −0.172133
\(88\) 0 0
\(89\) 2.51388 0.266471 0.133235 0.991084i \(-0.457463\pi\)
0.133235 + 0.991084i \(0.457463\pi\)
\(90\) 0 0
\(91\) 1.30278 0.136568
\(92\) 0 0
\(93\) −3.60555 −0.373878
\(94\) 0 0
\(95\) 32.7250 3.35751
\(96\) 0 0
\(97\) 18.2111 1.84906 0.924529 0.381113i \(-0.124459\pi\)
0.924529 + 0.381113i \(0.124459\pi\)
\(98\) 0 0
\(99\) −1.60555 −0.161364
\(100\) 0 0
\(101\) 5.51388 0.548651 0.274326 0.961637i \(-0.411545\pi\)
0.274326 + 0.961637i \(0.411545\pi\)
\(102\) 0 0
\(103\) 17.2111 1.69586 0.847930 0.530108i \(-0.177849\pi\)
0.847930 + 0.530108i \(0.177849\pi\)
\(104\) 0 0
\(105\) −4.30278 −0.419908
\(106\) 0 0
\(107\) −3.30278 −0.319291 −0.159646 0.987174i \(-0.551035\pi\)
−0.159646 + 0.987174i \(0.551035\pi\)
\(108\) 0 0
\(109\) 13.5139 1.29439 0.647197 0.762322i \(-0.275941\pi\)
0.647197 + 0.762322i \(0.275941\pi\)
\(110\) 0 0
\(111\) 1.60555 0.152392
\(112\) 0 0
\(113\) 1.69722 0.159661 0.0798307 0.996808i \(-0.474562\pi\)
0.0798307 + 0.996808i \(0.474562\pi\)
\(114\) 0 0
\(115\) 4.30278 0.401236
\(116\) 0 0
\(117\) −1.30278 −0.120442
\(118\) 0 0
\(119\) 7.60555 0.697200
\(120\) 0 0
\(121\) −8.42221 −0.765655
\(122\) 0 0
\(123\) 3.60555 0.325102
\(124\) 0 0
\(125\) −36.6333 −3.27658
\(126\) 0 0
\(127\) −7.11943 −0.631747 −0.315874 0.948801i \(-0.602298\pi\)
−0.315874 + 0.948801i \(0.602298\pi\)
\(128\) 0 0
\(129\) −0.697224 −0.0613872
\(130\) 0 0
\(131\) 14.8167 1.29454 0.647269 0.762262i \(-0.275911\pi\)
0.647269 + 0.762262i \(0.275911\pi\)
\(132\) 0 0
\(133\) 7.60555 0.659485
\(134\) 0 0
\(135\) 4.30278 0.370324
\(136\) 0 0
\(137\) 1.60555 0.137172 0.0685858 0.997645i \(-0.478151\pi\)
0.0685858 + 0.997645i \(0.478151\pi\)
\(138\) 0 0
\(139\) 10.1194 0.858319 0.429159 0.903229i \(-0.358810\pi\)
0.429159 + 0.903229i \(0.358810\pi\)
\(140\) 0 0
\(141\) −2.60555 −0.219427
\(142\) 0 0
\(143\) 2.09167 0.174914
\(144\) 0 0
\(145\) −6.90833 −0.573705
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −22.6056 −1.85192 −0.925959 0.377623i \(-0.876742\pi\)
−0.925959 + 0.377623i \(0.876742\pi\)
\(150\) 0 0
\(151\) 1.39445 0.113479 0.0567393 0.998389i \(-0.481930\pi\)
0.0567393 + 0.998389i \(0.481930\pi\)
\(152\) 0 0
\(153\) −7.60555 −0.614872
\(154\) 0 0
\(155\) −15.5139 −1.24610
\(156\) 0 0
\(157\) −9.81665 −0.783454 −0.391727 0.920081i \(-0.628122\pi\)
−0.391727 + 0.920081i \(0.628122\pi\)
\(158\) 0 0
\(159\) 4.90833 0.389256
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 16.1194 1.26257 0.631286 0.775550i \(-0.282528\pi\)
0.631286 + 0.775550i \(0.282528\pi\)
\(164\) 0 0
\(165\) −6.90833 −0.537813
\(166\) 0 0
\(167\) 22.0278 1.70456 0.852279 0.523087i \(-0.175220\pi\)
0.852279 + 0.523087i \(0.175220\pi\)
\(168\) 0 0
\(169\) −11.3028 −0.869444
\(170\) 0 0
\(171\) −7.60555 −0.581611
\(172\) 0 0
\(173\) −13.6056 −1.03441 −0.517205 0.855861i \(-0.673028\pi\)
−0.517205 + 0.855861i \(0.673028\pi\)
\(174\) 0 0
\(175\) −13.5139 −1.02155
\(176\) 0 0
\(177\) 6.90833 0.519262
\(178\) 0 0
\(179\) −21.9361 −1.63958 −0.819790 0.572664i \(-0.805910\pi\)
−0.819790 + 0.572664i \(0.805910\pi\)
\(180\) 0 0
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) 5.90833 0.436756
\(184\) 0 0
\(185\) 6.90833 0.507910
\(186\) 0 0
\(187\) 12.2111 0.892964
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −1.39445 −0.100899 −0.0504494 0.998727i \(-0.516065\pi\)
−0.0504494 + 0.998727i \(0.516065\pi\)
\(192\) 0 0
\(193\) −15.0278 −1.08172 −0.540861 0.841112i \(-0.681901\pi\)
−0.540861 + 0.841112i \(0.681901\pi\)
\(194\) 0 0
\(195\) −5.60555 −0.401422
\(196\) 0 0
\(197\) 20.9361 1.49163 0.745817 0.666151i \(-0.232059\pi\)
0.745817 + 0.666151i \(0.232059\pi\)
\(198\) 0 0
\(199\) 8.90833 0.631495 0.315747 0.948843i \(-0.397745\pi\)
0.315747 + 0.948843i \(0.397745\pi\)
\(200\) 0 0
\(201\) −9.69722 −0.683989
\(202\) 0 0
\(203\) −1.60555 −0.112688
\(204\) 0 0
\(205\) 15.5139 1.08354
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 12.2111 0.844660
\(210\) 0 0
\(211\) 18.3944 1.26633 0.633163 0.774018i \(-0.281756\pi\)
0.633163 + 0.774018i \(0.281756\pi\)
\(212\) 0 0
\(213\) −2.09167 −0.143319
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) −3.60555 −0.244761
\(218\) 0 0
\(219\) −11.0000 −0.743311
\(220\) 0 0
\(221\) 9.90833 0.666506
\(222\) 0 0
\(223\) −14.7250 −0.986058 −0.493029 0.870013i \(-0.664110\pi\)
−0.493029 + 0.870013i \(0.664110\pi\)
\(224\) 0 0
\(225\) 13.5139 0.900925
\(226\) 0 0
\(227\) 1.09167 0.0724569 0.0362284 0.999344i \(-0.488466\pi\)
0.0362284 + 0.999344i \(0.488466\pi\)
\(228\) 0 0
\(229\) −22.1194 −1.46169 −0.730847 0.682542i \(-0.760874\pi\)
−0.730847 + 0.682542i \(0.760874\pi\)
\(230\) 0 0
\(231\) −1.60555 −0.105638
\(232\) 0 0
\(233\) −18.1194 −1.18704 −0.593522 0.804818i \(-0.702263\pi\)
−0.593522 + 0.804818i \(0.702263\pi\)
\(234\) 0 0
\(235\) −11.2111 −0.731332
\(236\) 0 0
\(237\) 15.4222 1.00178
\(238\) 0 0
\(239\) −4.90833 −0.317493 −0.158747 0.987319i \(-0.550745\pi\)
−0.158747 + 0.987319i \(0.550745\pi\)
\(240\) 0 0
\(241\) −6.21110 −0.400092 −0.200046 0.979786i \(-0.564109\pi\)
−0.200046 + 0.979786i \(0.564109\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.30278 −0.274894
\(246\) 0 0
\(247\) 9.90833 0.630452
\(248\) 0 0
\(249\) −1.78890 −0.113367
\(250\) 0 0
\(251\) 25.8167 1.62953 0.814766 0.579789i \(-0.196865\pi\)
0.814766 + 0.579789i \(0.196865\pi\)
\(252\) 0 0
\(253\) 1.60555 0.100940
\(254\) 0 0
\(255\) −32.7250 −2.04932
\(256\) 0 0
\(257\) 4.18335 0.260950 0.130475 0.991452i \(-0.458350\pi\)
0.130475 + 0.991452i \(0.458350\pi\)
\(258\) 0 0
\(259\) 1.60555 0.0997641
\(260\) 0 0
\(261\) 1.60555 0.0993811
\(262\) 0 0
\(263\) 19.4222 1.19762 0.598812 0.800889i \(-0.295640\pi\)
0.598812 + 0.800889i \(0.295640\pi\)
\(264\) 0 0
\(265\) 21.1194 1.29736
\(266\) 0 0
\(267\) −2.51388 −0.153847
\(268\) 0 0
\(269\) −13.1194 −0.799906 −0.399953 0.916536i \(-0.630974\pi\)
−0.399953 + 0.916536i \(0.630974\pi\)
\(270\) 0 0
\(271\) −29.4222 −1.78727 −0.893636 0.448793i \(-0.851854\pi\)
−0.893636 + 0.448793i \(0.851854\pi\)
\(272\) 0 0
\(273\) −1.30278 −0.0788476
\(274\) 0 0
\(275\) −21.6972 −1.30839
\(276\) 0 0
\(277\) −29.5416 −1.77498 −0.887492 0.460822i \(-0.847555\pi\)
−0.887492 + 0.460822i \(0.847555\pi\)
\(278\) 0 0
\(279\) 3.60555 0.215859
\(280\) 0 0
\(281\) −0.605551 −0.0361242 −0.0180621 0.999837i \(-0.505750\pi\)
−0.0180621 + 0.999837i \(0.505750\pi\)
\(282\) 0 0
\(283\) 21.6972 1.28977 0.644883 0.764281i \(-0.276906\pi\)
0.644883 + 0.764281i \(0.276906\pi\)
\(284\) 0 0
\(285\) −32.7250 −1.93846
\(286\) 0 0
\(287\) 3.60555 0.212829
\(288\) 0 0
\(289\) 40.8444 2.40261
\(290\) 0 0
\(291\) −18.2111 −1.06755
\(292\) 0 0
\(293\) −33.2111 −1.94021 −0.970107 0.242679i \(-0.921974\pi\)
−0.970107 + 0.242679i \(0.921974\pi\)
\(294\) 0 0
\(295\) 29.7250 1.73066
\(296\) 0 0
\(297\) 1.60555 0.0931635
\(298\) 0 0
\(299\) 1.30278 0.0753415
\(300\) 0 0
\(301\) −0.697224 −0.0401873
\(302\) 0 0
\(303\) −5.51388 −0.316764
\(304\) 0 0
\(305\) 25.4222 1.45567
\(306\) 0 0
\(307\) −18.3944 −1.04983 −0.524913 0.851156i \(-0.675902\pi\)
−0.524913 + 0.851156i \(0.675902\pi\)
\(308\) 0 0
\(309\) −17.2111 −0.979105
\(310\) 0 0
\(311\) −15.1194 −0.857344 −0.428672 0.903460i \(-0.641018\pi\)
−0.428672 + 0.903460i \(0.641018\pi\)
\(312\) 0 0
\(313\) −1.57779 −0.0891822 −0.0445911 0.999005i \(-0.514199\pi\)
−0.0445911 + 0.999005i \(0.514199\pi\)
\(314\) 0 0
\(315\) 4.30278 0.242434
\(316\) 0 0
\(317\) −16.3028 −0.915655 −0.457828 0.889041i \(-0.651372\pi\)
−0.457828 + 0.889041i \(0.651372\pi\)
\(318\) 0 0
\(319\) −2.57779 −0.144329
\(320\) 0 0
\(321\) 3.30278 0.184343
\(322\) 0 0
\(323\) 57.8444 3.21855
\(324\) 0 0
\(325\) −17.6056 −0.976580
\(326\) 0 0
\(327\) −13.5139 −0.747319
\(328\) 0 0
\(329\) −2.60555 −0.143649
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) 0 0
\(333\) −1.60555 −0.0879837
\(334\) 0 0
\(335\) −41.7250 −2.27968
\(336\) 0 0
\(337\) −15.7250 −0.856594 −0.428297 0.903638i \(-0.640886\pi\)
−0.428297 + 0.903638i \(0.640886\pi\)
\(338\) 0 0
\(339\) −1.69722 −0.0921806
\(340\) 0 0
\(341\) −5.78890 −0.313486
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −4.30278 −0.231654
\(346\) 0 0
\(347\) −14.2111 −0.762892 −0.381446 0.924391i \(-0.624574\pi\)
−0.381446 + 0.924391i \(0.624574\pi\)
\(348\) 0 0
\(349\) 4.30278 0.230322 0.115161 0.993347i \(-0.463262\pi\)
0.115161 + 0.993347i \(0.463262\pi\)
\(350\) 0 0
\(351\) 1.30278 0.0695370
\(352\) 0 0
\(353\) 8.39445 0.446791 0.223396 0.974728i \(-0.428286\pi\)
0.223396 + 0.974728i \(0.428286\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) −7.60555 −0.402528
\(358\) 0 0
\(359\) −0.908327 −0.0479397 −0.0239698 0.999713i \(-0.507631\pi\)
−0.0239698 + 0.999713i \(0.507631\pi\)
\(360\) 0 0
\(361\) 38.8444 2.04444
\(362\) 0 0
\(363\) 8.42221 0.442051
\(364\) 0 0
\(365\) −47.3305 −2.47739
\(366\) 0 0
\(367\) 7.48612 0.390772 0.195386 0.980726i \(-0.437404\pi\)
0.195386 + 0.980726i \(0.437404\pi\)
\(368\) 0 0
\(369\) −3.60555 −0.187698
\(370\) 0 0
\(371\) 4.90833 0.254828
\(372\) 0 0
\(373\) 32.8167 1.69918 0.849591 0.527442i \(-0.176849\pi\)
0.849591 + 0.527442i \(0.176849\pi\)
\(374\) 0 0
\(375\) 36.6333 1.89174
\(376\) 0 0
\(377\) −2.09167 −0.107727
\(378\) 0 0
\(379\) 18.4222 0.946285 0.473143 0.880986i \(-0.343120\pi\)
0.473143 + 0.880986i \(0.343120\pi\)
\(380\) 0 0
\(381\) 7.11943 0.364739
\(382\) 0 0
\(383\) 16.6333 0.849922 0.424961 0.905212i \(-0.360288\pi\)
0.424961 + 0.905212i \(0.360288\pi\)
\(384\) 0 0
\(385\) −6.90833 −0.352081
\(386\) 0 0
\(387\) 0.697224 0.0354419
\(388\) 0 0
\(389\) 30.6333 1.55317 0.776585 0.630012i \(-0.216950\pi\)
0.776585 + 0.630012i \(0.216950\pi\)
\(390\) 0 0
\(391\) 7.60555 0.384629
\(392\) 0 0
\(393\) −14.8167 −0.747401
\(394\) 0 0
\(395\) 66.3583 3.33885
\(396\) 0 0
\(397\) −1.39445 −0.0699854 −0.0349927 0.999388i \(-0.511141\pi\)
−0.0349927 + 0.999388i \(0.511141\pi\)
\(398\) 0 0
\(399\) −7.60555 −0.380754
\(400\) 0 0
\(401\) 16.6333 0.830628 0.415314 0.909678i \(-0.363672\pi\)
0.415314 + 0.909678i \(0.363672\pi\)
\(402\) 0 0
\(403\) −4.69722 −0.233985
\(404\) 0 0
\(405\) −4.30278 −0.213807
\(406\) 0 0
\(407\) 2.57779 0.127777
\(408\) 0 0
\(409\) 15.6056 0.771645 0.385822 0.922573i \(-0.373918\pi\)
0.385822 + 0.922573i \(0.373918\pi\)
\(410\) 0 0
\(411\) −1.60555 −0.0791960
\(412\) 0 0
\(413\) 6.90833 0.339937
\(414\) 0 0
\(415\) −7.69722 −0.377842
\(416\) 0 0
\(417\) −10.1194 −0.495551
\(418\) 0 0
\(419\) −1.09167 −0.0533317 −0.0266659 0.999644i \(-0.508489\pi\)
−0.0266659 + 0.999644i \(0.508489\pi\)
\(420\) 0 0
\(421\) −17.9083 −0.872798 −0.436399 0.899753i \(-0.643746\pi\)
−0.436399 + 0.899753i \(0.643746\pi\)
\(422\) 0 0
\(423\) 2.60555 0.126686
\(424\) 0 0
\(425\) −102.780 −4.98559
\(426\) 0 0
\(427\) 5.90833 0.285924
\(428\) 0 0
\(429\) −2.09167 −0.100987
\(430\) 0 0
\(431\) −0.697224 −0.0335841 −0.0167921 0.999859i \(-0.505345\pi\)
−0.0167921 + 0.999859i \(0.505345\pi\)
\(432\) 0 0
\(433\) 2.18335 0.104925 0.0524625 0.998623i \(-0.483293\pi\)
0.0524625 + 0.998623i \(0.483293\pi\)
\(434\) 0 0
\(435\) 6.90833 0.331229
\(436\) 0 0
\(437\) 7.60555 0.363823
\(438\) 0 0
\(439\) −20.3944 −0.973374 −0.486687 0.873576i \(-0.661795\pi\)
−0.486687 + 0.873576i \(0.661795\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −21.2111 −1.00777 −0.503885 0.863771i \(-0.668096\pi\)
−0.503885 + 0.863771i \(0.668096\pi\)
\(444\) 0 0
\(445\) −10.8167 −0.512759
\(446\) 0 0
\(447\) 22.6056 1.06921
\(448\) 0 0
\(449\) 18.1194 0.855109 0.427554 0.903990i \(-0.359375\pi\)
0.427554 + 0.903990i \(0.359375\pi\)
\(450\) 0 0
\(451\) 5.78890 0.272589
\(452\) 0 0
\(453\) −1.39445 −0.0655169
\(454\) 0 0
\(455\) −5.60555 −0.262792
\(456\) 0 0
\(457\) −24.3028 −1.13684 −0.568418 0.822740i \(-0.692444\pi\)
−0.568418 + 0.822740i \(0.692444\pi\)
\(458\) 0 0
\(459\) 7.60555 0.354997
\(460\) 0 0
\(461\) −36.9083 −1.71899 −0.859496 0.511142i \(-0.829222\pi\)
−0.859496 + 0.511142i \(0.829222\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 0 0
\(465\) 15.5139 0.719439
\(466\) 0 0
\(467\) 33.8444 1.56613 0.783066 0.621938i \(-0.213655\pi\)
0.783066 + 0.621938i \(0.213655\pi\)
\(468\) 0 0
\(469\) −9.69722 −0.447776
\(470\) 0 0
\(471\) 9.81665 0.452328
\(472\) 0 0
\(473\) −1.11943 −0.0514714
\(474\) 0 0
\(475\) −102.780 −4.71589
\(476\) 0 0
\(477\) −4.90833 −0.224737
\(478\) 0 0
\(479\) 23.2389 1.06181 0.530905 0.847431i \(-0.321852\pi\)
0.530905 + 0.847431i \(0.321852\pi\)
\(480\) 0 0
\(481\) 2.09167 0.0953721
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −78.3583 −3.55807
\(486\) 0 0
\(487\) −10.2111 −0.462709 −0.231355 0.972869i \(-0.574316\pi\)
−0.231355 + 0.972869i \(0.574316\pi\)
\(488\) 0 0
\(489\) −16.1194 −0.728946
\(490\) 0 0
\(491\) 24.0917 1.08724 0.543621 0.839331i \(-0.317053\pi\)
0.543621 + 0.839331i \(0.317053\pi\)
\(492\) 0 0
\(493\) −12.2111 −0.549960
\(494\) 0 0
\(495\) 6.90833 0.310506
\(496\) 0 0
\(497\) −2.09167 −0.0938244
\(498\) 0 0
\(499\) 38.1194 1.70646 0.853230 0.521535i \(-0.174640\pi\)
0.853230 + 0.521535i \(0.174640\pi\)
\(500\) 0 0
\(501\) −22.0278 −0.984128
\(502\) 0 0
\(503\) −4.11943 −0.183676 −0.0918381 0.995774i \(-0.529274\pi\)
−0.0918381 + 0.995774i \(0.529274\pi\)
\(504\) 0 0
\(505\) −23.7250 −1.05575
\(506\) 0 0
\(507\) 11.3028 0.501974
\(508\) 0 0
\(509\) −26.6056 −1.17927 −0.589635 0.807670i \(-0.700728\pi\)
−0.589635 + 0.807670i \(0.700728\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) 7.60555 0.335793
\(514\) 0 0
\(515\) −74.0555 −3.26328
\(516\) 0 0
\(517\) −4.18335 −0.183983
\(518\) 0 0
\(519\) 13.6056 0.597217
\(520\) 0 0
\(521\) 19.2111 0.841654 0.420827 0.907141i \(-0.361740\pi\)
0.420827 + 0.907141i \(0.361740\pi\)
\(522\) 0 0
\(523\) 16.7889 0.734127 0.367064 0.930196i \(-0.380363\pi\)
0.367064 + 0.930196i \(0.380363\pi\)
\(524\) 0 0
\(525\) 13.5139 0.589794
\(526\) 0 0
\(527\) −27.4222 −1.19453
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.90833 −0.299796
\(532\) 0 0
\(533\) 4.69722 0.203459
\(534\) 0 0
\(535\) 14.2111 0.614400
\(536\) 0 0
\(537\) 21.9361 0.946612
\(538\) 0 0
\(539\) −1.60555 −0.0691560
\(540\) 0 0
\(541\) −9.02776 −0.388134 −0.194067 0.980988i \(-0.562168\pi\)
−0.194067 + 0.980988i \(0.562168\pi\)
\(542\) 0 0
\(543\) 19.0000 0.815368
\(544\) 0 0
\(545\) −58.1472 −2.49075
\(546\) 0 0
\(547\) 11.3305 0.484459 0.242229 0.970219i \(-0.422121\pi\)
0.242229 + 0.970219i \(0.422121\pi\)
\(548\) 0 0
\(549\) −5.90833 −0.252161
\(550\) 0 0
\(551\) −12.2111 −0.520210
\(552\) 0 0
\(553\) 15.4222 0.655819
\(554\) 0 0
\(555\) −6.90833 −0.293242
\(556\) 0 0
\(557\) 37.8167 1.60234 0.801172 0.598435i \(-0.204210\pi\)
0.801172 + 0.598435i \(0.204210\pi\)
\(558\) 0 0
\(559\) −0.908327 −0.0384181
\(560\) 0 0
\(561\) −12.2111 −0.515553
\(562\) 0 0
\(563\) 15.5416 0.655002 0.327501 0.944851i \(-0.393794\pi\)
0.327501 + 0.944851i \(0.393794\pi\)
\(564\) 0 0
\(565\) −7.30278 −0.307230
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −29.2111 −1.22459 −0.612297 0.790628i \(-0.709754\pi\)
−0.612297 + 0.790628i \(0.709754\pi\)
\(570\) 0 0
\(571\) −44.2111 −1.85018 −0.925089 0.379752i \(-0.876009\pi\)
−0.925089 + 0.379752i \(0.876009\pi\)
\(572\) 0 0
\(573\) 1.39445 0.0582539
\(574\) 0 0
\(575\) −13.5139 −0.563568
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 15.0278 0.624532
\(580\) 0 0
\(581\) −1.78890 −0.0742160
\(582\) 0 0
\(583\) 7.88057 0.326380
\(584\) 0 0
\(585\) 5.60555 0.231761
\(586\) 0 0
\(587\) −17.4861 −0.721729 −0.360865 0.932618i \(-0.617518\pi\)
−0.360865 + 0.932618i \(0.617518\pi\)
\(588\) 0 0
\(589\) −27.4222 −1.12991
\(590\) 0 0
\(591\) −20.9361 −0.861195
\(592\) 0 0
\(593\) 28.6056 1.17469 0.587345 0.809337i \(-0.300173\pi\)
0.587345 + 0.809337i \(0.300173\pi\)
\(594\) 0 0
\(595\) −32.7250 −1.34159
\(596\) 0 0
\(597\) −8.90833 −0.364594
\(598\) 0 0
\(599\) −27.5416 −1.12532 −0.562660 0.826688i \(-0.690222\pi\)
−0.562660 + 0.826688i \(0.690222\pi\)
\(600\) 0 0
\(601\) 26.5416 1.08266 0.541328 0.840812i \(-0.317922\pi\)
0.541328 + 0.840812i \(0.317922\pi\)
\(602\) 0 0
\(603\) 9.69722 0.394902
\(604\) 0 0
\(605\) 36.2389 1.47332
\(606\) 0 0
\(607\) 33.5416 1.36141 0.680706 0.732556i \(-0.261673\pi\)
0.680706 + 0.732556i \(0.261673\pi\)
\(608\) 0 0
\(609\) 1.60555 0.0650602
\(610\) 0 0
\(611\) −3.39445 −0.137325
\(612\) 0 0
\(613\) 14.6333 0.591034 0.295517 0.955338i \(-0.404508\pi\)
0.295517 + 0.955338i \(0.404508\pi\)
\(614\) 0 0
\(615\) −15.5139 −0.625580
\(616\) 0 0
\(617\) 16.6972 0.672205 0.336102 0.941825i \(-0.390891\pi\)
0.336102 + 0.941825i \(0.390891\pi\)
\(618\) 0 0
\(619\) 36.6972 1.47499 0.737493 0.675355i \(-0.236009\pi\)
0.737493 + 0.675355i \(0.236009\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −2.51388 −0.100716
\(624\) 0 0
\(625\) 90.0555 3.60222
\(626\) 0 0
\(627\) −12.2111 −0.487664
\(628\) 0 0
\(629\) 12.2111 0.486889
\(630\) 0 0
\(631\) 23.6056 0.939722 0.469861 0.882740i \(-0.344304\pi\)
0.469861 + 0.882740i \(0.344304\pi\)
\(632\) 0 0
\(633\) −18.3944 −0.731114
\(634\) 0 0
\(635\) 30.6333 1.21565
\(636\) 0 0
\(637\) −1.30278 −0.0516179
\(638\) 0 0
\(639\) 2.09167 0.0827453
\(640\) 0 0
\(641\) 23.5416 0.929839 0.464919 0.885353i \(-0.346083\pi\)
0.464919 + 0.885353i \(0.346083\pi\)
\(642\) 0 0
\(643\) 29.9083 1.17947 0.589735 0.807597i \(-0.299232\pi\)
0.589735 + 0.807597i \(0.299232\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 1.51388 0.0595167 0.0297583 0.999557i \(-0.490526\pi\)
0.0297583 + 0.999557i \(0.490526\pi\)
\(648\) 0 0
\(649\) 11.0917 0.435386
\(650\) 0 0
\(651\) 3.60555 0.141313
\(652\) 0 0
\(653\) 7.09167 0.277519 0.138759 0.990326i \(-0.455689\pi\)
0.138759 + 0.990326i \(0.455689\pi\)
\(654\) 0 0
\(655\) −63.7527 −2.49103
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) −34.8167 −1.35626 −0.678132 0.734940i \(-0.737210\pi\)
−0.678132 + 0.734940i \(0.737210\pi\)
\(660\) 0 0
\(661\) 23.4222 0.911018 0.455509 0.890231i \(-0.349457\pi\)
0.455509 + 0.890231i \(0.349457\pi\)
\(662\) 0 0
\(663\) −9.90833 −0.384808
\(664\) 0 0
\(665\) −32.7250 −1.26902
\(666\) 0 0
\(667\) −1.60555 −0.0621672
\(668\) 0 0
\(669\) 14.7250 0.569301
\(670\) 0 0
\(671\) 9.48612 0.366208
\(672\) 0 0
\(673\) 5.00000 0.192736 0.0963679 0.995346i \(-0.469277\pi\)
0.0963679 + 0.995346i \(0.469277\pi\)
\(674\) 0 0
\(675\) −13.5139 −0.520149
\(676\) 0 0
\(677\) −4.88057 −0.187576 −0.0937878 0.995592i \(-0.529898\pi\)
−0.0937878 + 0.995592i \(0.529898\pi\)
\(678\) 0 0
\(679\) −18.2111 −0.698878
\(680\) 0 0
\(681\) −1.09167 −0.0418330
\(682\) 0 0
\(683\) 12.6333 0.483400 0.241700 0.970351i \(-0.422295\pi\)
0.241700 + 0.970351i \(0.422295\pi\)
\(684\) 0 0
\(685\) −6.90833 −0.263954
\(686\) 0 0
\(687\) 22.1194 0.843909
\(688\) 0 0
\(689\) 6.39445 0.243609
\(690\) 0 0
\(691\) −13.9083 −0.529098 −0.264549 0.964372i \(-0.585223\pi\)
−0.264549 + 0.964372i \(0.585223\pi\)
\(692\) 0 0
\(693\) 1.60555 0.0609898
\(694\) 0 0
\(695\) −43.5416 −1.65163
\(696\) 0 0
\(697\) 27.4222 1.03869
\(698\) 0 0
\(699\) 18.1194 0.685340
\(700\) 0 0
\(701\) −27.9361 −1.05513 −0.527566 0.849514i \(-0.676895\pi\)
−0.527566 + 0.849514i \(0.676895\pi\)
\(702\) 0 0
\(703\) 12.2111 0.460550
\(704\) 0 0
\(705\) 11.2111 0.422235
\(706\) 0 0
\(707\) −5.51388 −0.207371
\(708\) 0 0
\(709\) 5.11943 0.192264 0.0961321 0.995369i \(-0.469353\pi\)
0.0961321 + 0.995369i \(0.469353\pi\)
\(710\) 0 0
\(711\) −15.4222 −0.578378
\(712\) 0 0
\(713\) −3.60555 −0.135029
\(714\) 0 0
\(715\) −9.00000 −0.336581
\(716\) 0 0
\(717\) 4.90833 0.183305
\(718\) 0 0
\(719\) −15.8444 −0.590897 −0.295448 0.955359i \(-0.595469\pi\)
−0.295448 + 0.955359i \(0.595469\pi\)
\(720\) 0 0
\(721\) −17.2111 −0.640975
\(722\) 0 0
\(723\) 6.21110 0.230993
\(724\) 0 0
\(725\) 21.6972 0.805815
\(726\) 0 0
\(727\) −13.6056 −0.504602 −0.252301 0.967649i \(-0.581187\pi\)
−0.252301 + 0.967649i \(0.581187\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.30278 −0.196130
\(732\) 0 0
\(733\) −0.972244 −0.0359106 −0.0179553 0.999839i \(-0.505716\pi\)
−0.0179553 + 0.999839i \(0.505716\pi\)
\(734\) 0 0
\(735\) 4.30278 0.158710
\(736\) 0 0
\(737\) −15.5694 −0.573506
\(738\) 0 0
\(739\) −15.1833 −0.558528 −0.279264 0.960214i \(-0.590091\pi\)
−0.279264 + 0.960214i \(0.590091\pi\)
\(740\) 0 0
\(741\) −9.90833 −0.363991
\(742\) 0 0
\(743\) 45.5139 1.66974 0.834871 0.550445i \(-0.185542\pi\)
0.834871 + 0.550445i \(0.185542\pi\)
\(744\) 0 0
\(745\) 97.2666 3.56357
\(746\) 0 0
\(747\) 1.78890 0.0654523
\(748\) 0 0
\(749\) 3.30278 0.120681
\(750\) 0 0
\(751\) 3.93608 0.143630 0.0718149 0.997418i \(-0.477121\pi\)
0.0718149 + 0.997418i \(0.477121\pi\)
\(752\) 0 0
\(753\) −25.8167 −0.940811
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) −0.816654 −0.0296818 −0.0148409 0.999890i \(-0.504724\pi\)
−0.0148409 + 0.999890i \(0.504724\pi\)
\(758\) 0 0
\(759\) −1.60555 −0.0582778
\(760\) 0 0
\(761\) 27.6333 1.00171 0.500853 0.865532i \(-0.333020\pi\)
0.500853 + 0.865532i \(0.333020\pi\)
\(762\) 0 0
\(763\) −13.5139 −0.489235
\(764\) 0 0
\(765\) 32.7250 1.18317
\(766\) 0 0
\(767\) 9.00000 0.324971
\(768\) 0 0
\(769\) −9.81665 −0.353998 −0.176999 0.984211i \(-0.556639\pi\)
−0.176999 + 0.984211i \(0.556639\pi\)
\(770\) 0 0
\(771\) −4.18335 −0.150660
\(772\) 0 0
\(773\) 12.2111 0.439203 0.219601 0.975590i \(-0.429524\pi\)
0.219601 + 0.975590i \(0.429524\pi\)
\(774\) 0 0
\(775\) 48.7250 1.75025
\(776\) 0 0
\(777\) −1.60555 −0.0575988
\(778\) 0 0
\(779\) 27.4222 0.982502
\(780\) 0 0
\(781\) −3.35829 −0.120169
\(782\) 0 0
\(783\) −1.60555 −0.0573777
\(784\) 0 0
\(785\) 42.2389 1.50757
\(786\) 0 0
\(787\) −5.88057 −0.209620 −0.104810 0.994492i \(-0.533423\pi\)
−0.104810 + 0.994492i \(0.533423\pi\)
\(788\) 0 0
\(789\) −19.4222 −0.691449
\(790\) 0 0
\(791\) −1.69722 −0.0603464
\(792\) 0 0
\(793\) 7.69722 0.273336
\(794\) 0 0
\(795\) −21.1194 −0.749029
\(796\) 0 0
\(797\) −28.1833 −0.998305 −0.499153 0.866514i \(-0.666355\pi\)
−0.499153 + 0.866514i \(0.666355\pi\)
\(798\) 0 0
\(799\) −19.8167 −0.701063
\(800\) 0 0
\(801\) 2.51388 0.0888235
\(802\) 0 0
\(803\) −17.6611 −0.623245
\(804\) 0 0
\(805\) −4.30278 −0.151653
\(806\) 0 0
\(807\) 13.1194 0.461826
\(808\) 0 0
\(809\) 7.48612 0.263198 0.131599 0.991303i \(-0.457989\pi\)
0.131599 + 0.991303i \(0.457989\pi\)
\(810\) 0 0
\(811\) −32.6333 −1.14591 −0.572955 0.819587i \(-0.694203\pi\)
−0.572955 + 0.819587i \(0.694203\pi\)
\(812\) 0 0
\(813\) 29.4222 1.03188
\(814\) 0 0
\(815\) −69.3583 −2.42951
\(816\) 0 0
\(817\) −5.30278 −0.185521
\(818\) 0 0
\(819\) 1.30278 0.0455227
\(820\) 0 0
\(821\) 47.4500 1.65602 0.828008 0.560717i \(-0.189474\pi\)
0.828008 + 0.560717i \(0.189474\pi\)
\(822\) 0 0
\(823\) −30.4861 −1.06268 −0.531340 0.847159i \(-0.678311\pi\)
−0.531340 + 0.847159i \(0.678311\pi\)
\(824\) 0 0
\(825\) 21.6972 0.755400
\(826\) 0 0
\(827\) 26.3028 0.914637 0.457319 0.889303i \(-0.348810\pi\)
0.457319 + 0.889303i \(0.348810\pi\)
\(828\) 0 0
\(829\) 22.8167 0.792455 0.396228 0.918152i \(-0.370319\pi\)
0.396228 + 0.918152i \(0.370319\pi\)
\(830\) 0 0
\(831\) 29.5416 1.02479
\(832\) 0 0
\(833\) −7.60555 −0.263517
\(834\) 0 0
\(835\) −94.7805 −3.28001
\(836\) 0 0
\(837\) −3.60555 −0.124626
\(838\) 0 0
\(839\) 25.5416 0.881795 0.440898 0.897557i \(-0.354660\pi\)
0.440898 + 0.897557i \(0.354660\pi\)
\(840\) 0 0
\(841\) −26.4222 −0.911111
\(842\) 0 0
\(843\) 0.605551 0.0208563
\(844\) 0 0
\(845\) 48.6333 1.67304
\(846\) 0 0
\(847\) 8.42221 0.289390
\(848\) 0 0
\(849\) −21.6972 −0.744647
\(850\) 0 0
\(851\) 1.60555 0.0550376
\(852\) 0 0
\(853\) 4.60555 0.157691 0.0788455 0.996887i \(-0.474877\pi\)
0.0788455 + 0.996887i \(0.474877\pi\)
\(854\) 0 0
\(855\) 32.7250 1.11917
\(856\) 0 0
\(857\) −27.0278 −0.923251 −0.461625 0.887075i \(-0.652734\pi\)
−0.461625 + 0.887075i \(0.652734\pi\)
\(858\) 0 0
\(859\) −44.0555 −1.50315 −0.751577 0.659645i \(-0.770707\pi\)
−0.751577 + 0.659645i \(0.770707\pi\)
\(860\) 0 0
\(861\) −3.60555 −0.122877
\(862\) 0 0
\(863\) 9.81665 0.334163 0.167081 0.985943i \(-0.446566\pi\)
0.167081 + 0.985943i \(0.446566\pi\)
\(864\) 0 0
\(865\) 58.5416 1.99048
\(866\) 0 0
\(867\) −40.8444 −1.38715
\(868\) 0 0
\(869\) 24.7611 0.839964
\(870\) 0 0
\(871\) −12.6333 −0.428063
\(872\) 0 0
\(873\) 18.2111 0.616352
\(874\) 0 0
\(875\) 36.6333 1.23843
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) 33.2111 1.12018
\(880\) 0 0
\(881\) 48.2389 1.62521 0.812604 0.582816i \(-0.198049\pi\)
0.812604 + 0.582816i \(0.198049\pi\)
\(882\) 0 0
\(883\) 3.90833 0.131526 0.0657628 0.997835i \(-0.479052\pi\)
0.0657628 + 0.997835i \(0.479052\pi\)
\(884\) 0 0
\(885\) −29.7250 −0.999194
\(886\) 0 0
\(887\) 16.5139 0.554482 0.277241 0.960800i \(-0.410580\pi\)
0.277241 + 0.960800i \(0.410580\pi\)
\(888\) 0 0
\(889\) 7.11943 0.238778
\(890\) 0 0
\(891\) −1.60555 −0.0537880
\(892\) 0 0
\(893\) −19.8167 −0.663139
\(894\) 0 0
\(895\) 94.3860 3.15498
\(896\) 0 0
\(897\) −1.30278 −0.0434984
\(898\) 0 0
\(899\) 5.78890 0.193071
\(900\) 0 0
\(901\) 37.3305 1.24366
\(902\) 0 0
\(903\) 0.697224 0.0232022
\(904\) 0 0
\(905\) 81.7527 2.71755
\(906\) 0 0
\(907\) −17.2750 −0.573608 −0.286804 0.957989i \(-0.592593\pi\)
−0.286804 + 0.957989i \(0.592593\pi\)
\(908\) 0 0
\(909\) 5.51388 0.182884
\(910\) 0 0
\(911\) 17.3944 0.576304 0.288152 0.957585i \(-0.406959\pi\)
0.288152 + 0.957585i \(0.406959\pi\)
\(912\) 0 0
\(913\) −2.87217 −0.0950548
\(914\) 0 0
\(915\) −25.4222 −0.840432
\(916\) 0 0
\(917\) −14.8167 −0.489289
\(918\) 0 0
\(919\) −28.0278 −0.924550 −0.462275 0.886737i \(-0.652967\pi\)
−0.462275 + 0.886737i \(0.652967\pi\)
\(920\) 0 0
\(921\) 18.3944 0.606118
\(922\) 0 0
\(923\) −2.72498 −0.0896938
\(924\) 0 0
\(925\) −21.6972 −0.713400
\(926\) 0 0
\(927\) 17.2111 0.565287
\(928\) 0 0
\(929\) −38.3028 −1.25667 −0.628337 0.777942i \(-0.716264\pi\)
−0.628337 + 0.777942i \(0.716264\pi\)
\(930\) 0 0
\(931\) −7.60555 −0.249262
\(932\) 0 0
\(933\) 15.1194 0.494988
\(934\) 0 0
\(935\) −52.5416 −1.71830
\(936\) 0 0
\(937\) −42.0278 −1.37299 −0.686493 0.727136i \(-0.740851\pi\)
−0.686493 + 0.727136i \(0.740851\pi\)
\(938\) 0 0
\(939\) 1.57779 0.0514894
\(940\) 0 0
\(941\) 9.21110 0.300273 0.150137 0.988665i \(-0.452029\pi\)
0.150137 + 0.988665i \(0.452029\pi\)
\(942\) 0 0
\(943\) 3.60555 0.117413
\(944\) 0 0
\(945\) −4.30278 −0.139969
\(946\) 0 0
\(947\) 38.8444 1.26227 0.631137 0.775671i \(-0.282589\pi\)
0.631137 + 0.775671i \(0.282589\pi\)
\(948\) 0 0
\(949\) −14.3305 −0.465189
\(950\) 0 0
\(951\) 16.3028 0.528654
\(952\) 0 0
\(953\) −5.72498 −0.185450 −0.0927252 0.995692i \(-0.529558\pi\)
−0.0927252 + 0.995692i \(0.529558\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) 0 0
\(957\) 2.57779 0.0833283
\(958\) 0 0
\(959\) −1.60555 −0.0518460
\(960\) 0 0
\(961\) −18.0000 −0.580645
\(962\) 0 0
\(963\) −3.30278 −0.106430
\(964\) 0 0
\(965\) 64.6611 2.08151
\(966\) 0 0
\(967\) 56.0555 1.80262 0.901312 0.433171i \(-0.142605\pi\)
0.901312 + 0.433171i \(0.142605\pi\)
\(968\) 0 0
\(969\) −57.8444 −1.85823
\(970\) 0 0
\(971\) 52.9361 1.69880 0.849400 0.527750i \(-0.176964\pi\)
0.849400 + 0.527750i \(0.176964\pi\)
\(972\) 0 0
\(973\) −10.1194 −0.324414
\(974\) 0 0
\(975\) 17.6056 0.563829
\(976\) 0 0
\(977\) −17.3028 −0.553565 −0.276782 0.960933i \(-0.589268\pi\)
−0.276782 + 0.960933i \(0.589268\pi\)
\(978\) 0 0
\(979\) −4.03616 −0.128996
\(980\) 0 0
\(981\) 13.5139 0.431465
\(982\) 0 0
\(983\) 31.4222 1.00221 0.501106 0.865386i \(-0.332927\pi\)
0.501106 + 0.865386i \(0.332927\pi\)
\(984\) 0 0
\(985\) −90.0833 −2.87029
\(986\) 0 0
\(987\) 2.60555 0.0829356
\(988\) 0 0
\(989\) −0.697224 −0.0221704
\(990\) 0 0
\(991\) 51.9638 1.65069 0.825343 0.564632i \(-0.190982\pi\)
0.825343 + 0.564632i \(0.190982\pi\)
\(992\) 0 0
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) −38.3305 −1.21516
\(996\) 0 0
\(997\) −24.8167 −0.785951 −0.392976 0.919549i \(-0.628554\pi\)
−0.392976 + 0.919549i \(0.628554\pi\)
\(998\) 0 0
\(999\) 1.60555 0.0507974
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 1932.2.a.c.1.1 2
3.2 odd 2 5796.2.a.m.1.2 2
4.3 odd 2 7728.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.c.1.1 2 1.1 even 1 trivial
5796.2.a.m.1.2 2 3.2 odd 2
7728.2.a.bf.1.1 2 4.3 odd 2