Properties

Label 2432.2.c.d.1217.2
Level $2432$
Weight $2$
Character 2432.1217
Analytic conductor $19.420$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2432,2,Mod(1217,2432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2432.1217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2432 = 2^{7} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2432.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.4196177716\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2432.1217
Dual form 2432.2.c.d.1217.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.00000i q^{5} +4.46410 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -2.00000i q^{5} +4.46410 q^{7} +2.00000 q^{9} -4.00000i q^{11} -4.46410i q^{13} -2.00000 q^{15} -7.92820 q^{17} -1.00000i q^{19} -4.46410i q^{21} -6.46410 q^{23} +1.00000 q^{25} -5.00000i q^{27} +2.46410i q^{29} -4.92820 q^{31} -4.00000 q^{33} -8.92820i q^{35} -2.00000i q^{37} -4.46410 q^{39} +6.92820 q^{41} +4.92820i q^{43} -4.00000i q^{45} +6.92820 q^{47} +12.9282 q^{49} +7.92820i q^{51} +12.4641i q^{53} -8.00000 q^{55} -1.00000 q^{57} -9.92820i q^{59} +8.92820i q^{61} +8.92820 q^{63} -8.92820 q^{65} +5.92820i q^{67} +6.46410i q^{69} -14.0000 q^{71} +7.00000 q^{73} -1.00000i q^{75} -17.8564i q^{77} -2.00000 q^{79} +1.00000 q^{81} -2.92820i q^{83} +15.8564i q^{85} +2.46410 q^{87} +0.928203 q^{89} -19.9282i q^{91} +4.92820i q^{93} -2.00000 q^{95} +12.9282 q^{97} -8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 8 q^{9} - 8 q^{15} - 4 q^{17} - 12 q^{23} + 4 q^{25} + 8 q^{31} - 16 q^{33} - 4 q^{39} + 24 q^{49} - 32 q^{55} - 4 q^{57} + 8 q^{63} - 8 q^{65} - 56 q^{71} + 28 q^{73} - 8 q^{79} + 4 q^{81} - 4 q^{87} - 24 q^{89} - 8 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times\).

\(n\) \(1407\) \(1921\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 4.46410 1.68727 0.843636 0.536916i \(-0.180411\pi\)
0.843636 + 0.536916i \(0.180411\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 4.46410i − 1.23812i −0.785344 0.619060i \(-0.787514\pi\)
0.785344 0.619060i \(-0.212486\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −7.92820 −1.92287 −0.961436 0.275029i \(-0.911312\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) − 4.46410i − 0.974147i
\(22\) 0 0
\(23\) −6.46410 −1.34786 −0.673929 0.738796i \(-0.735395\pi\)
−0.673929 + 0.738796i \(0.735395\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) 2.46410i 0.457572i 0.973477 + 0.228786i \(0.0734756\pi\)
−0.973477 + 0.228786i \(0.926524\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) − 8.92820i − 1.50914i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −4.46410 −0.714828
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 4.92820i 0.751544i 0.926712 + 0.375772i \(0.122622\pi\)
−0.926712 + 0.375772i \(0.877378\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 12.9282 1.84689
\(50\) 0 0
\(51\) 7.92820i 1.11017i
\(52\) 0 0
\(53\) 12.4641i 1.71208i 0.516913 + 0.856038i \(0.327081\pi\)
−0.516913 + 0.856038i \(0.672919\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) − 9.92820i − 1.29254i −0.763108 0.646271i \(-0.776328\pi\)
0.763108 0.646271i \(-0.223672\pi\)
\(60\) 0 0
\(61\) 8.92820i 1.14314i 0.820554 + 0.571570i \(0.193665\pi\)
−0.820554 + 0.571570i \(0.806335\pi\)
\(62\) 0 0
\(63\) 8.92820 1.12485
\(64\) 0 0
\(65\) −8.92820 −1.10741
\(66\) 0 0
\(67\) 5.92820i 0.724245i 0.932130 + 0.362123i \(0.117948\pi\)
−0.932130 + 0.362123i \(0.882052\pi\)
\(68\) 0 0
\(69\) 6.46410i 0.778186i
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) − 17.8564i − 2.03493i
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 2.92820i − 0.321412i −0.987002 0.160706i \(-0.948623\pi\)
0.987002 0.160706i \(-0.0513771\pi\)
\(84\) 0 0
\(85\) 15.8564i 1.71987i
\(86\) 0 0
\(87\) 2.46410 0.264179
\(88\) 0 0
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) − 19.9282i − 2.08904i
\(92\) 0 0
\(93\) 4.92820i 0.511031i
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 12.9282 1.31266 0.656330 0.754474i \(-0.272108\pi\)
0.656330 + 0.754474i \(0.272108\pi\)
\(98\) 0 0
\(99\) − 8.00000i − 0.804030i
\(100\) 0 0
\(101\) − 9.85641i − 0.980749i −0.871512 0.490375i \(-0.836860\pi\)
0.871512 0.490375i \(-0.163140\pi\)
\(102\) 0 0
\(103\) 2.92820 0.288524 0.144262 0.989539i \(-0.453919\pi\)
0.144262 + 0.989539i \(0.453919\pi\)
\(104\) 0 0
\(105\) −8.92820 −0.871303
\(106\) 0 0
\(107\) 13.9282i 1.34649i 0.739419 + 0.673245i \(0.235100\pi\)
−0.739419 + 0.673245i \(0.764900\pi\)
\(108\) 0 0
\(109\) 7.39230i 0.708054i 0.935235 + 0.354027i \(0.115188\pi\)
−0.935235 + 0.354027i \(0.884812\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) 12.9282i 1.20556i
\(116\) 0 0
\(117\) − 8.92820i − 0.825413i
\(118\) 0 0
\(119\) −35.3923 −3.24441
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 6.92820i − 0.624695i
\(124\) 0 0
\(125\) − 12.0000i − 1.07331i
\(126\) 0 0
\(127\) −0.928203 −0.0823647 −0.0411824 0.999152i \(-0.513112\pi\)
−0.0411824 + 0.999152i \(0.513112\pi\)
\(128\) 0 0
\(129\) 4.92820 0.433904
\(130\) 0 0
\(131\) − 7.07180i − 0.617866i −0.951084 0.308933i \(-0.900028\pi\)
0.951084 0.308933i \(-0.0999718\pi\)
\(132\) 0 0
\(133\) − 4.46410i − 0.387087i
\(134\) 0 0
\(135\) −10.0000 −0.860663
\(136\) 0 0
\(137\) −8.85641 −0.756654 −0.378327 0.925672i \(-0.623500\pi\)
−0.378327 + 0.925672i \(0.623500\pi\)
\(138\) 0 0
\(139\) − 15.8564i − 1.34492i −0.740132 0.672461i \(-0.765237\pi\)
0.740132 0.672461i \(-0.234763\pi\)
\(140\) 0 0
\(141\) − 6.92820i − 0.583460i
\(142\) 0 0
\(143\) −17.8564 −1.49323
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) 0 0
\(147\) − 12.9282i − 1.06630i
\(148\) 0 0
\(149\) 10.9282i 0.895273i 0.894216 + 0.447637i \(0.147734\pi\)
−0.894216 + 0.447637i \(0.852266\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −15.8564 −1.28191
\(154\) 0 0
\(155\) 9.85641i 0.791686i
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) 12.4641 0.988468
\(160\) 0 0
\(161\) −28.8564 −2.27420
\(162\) 0 0
\(163\) − 23.8564i − 1.86858i −0.356517 0.934289i \(-0.616036\pi\)
0.356517 0.934289i \(-0.383964\pi\)
\(164\) 0 0
\(165\) 8.00000i 0.622799i
\(166\) 0 0
\(167\) −5.07180 −0.392467 −0.196234 0.980557i \(-0.562871\pi\)
−0.196234 + 0.980557i \(0.562871\pi\)
\(168\) 0 0
\(169\) −6.92820 −0.532939
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) 10.0000i 0.760286i 0.924928 + 0.380143i \(0.124125\pi\)
−0.924928 + 0.380143i \(0.875875\pi\)
\(174\) 0 0
\(175\) 4.46410 0.337454
\(176\) 0 0
\(177\) −9.92820 −0.746249
\(178\) 0 0
\(179\) − 25.8564i − 1.93260i −0.257421 0.966299i \(-0.582873\pi\)
0.257421 0.966299i \(-0.417127\pi\)
\(180\) 0 0
\(181\) − 10.0000i − 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) 8.92820 0.659992
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 31.7128i 2.31907i
\(188\) 0 0
\(189\) − 22.3205i − 1.62358i
\(190\) 0 0
\(191\) 3.53590 0.255849 0.127924 0.991784i \(-0.459169\pi\)
0.127924 + 0.991784i \(0.459169\pi\)
\(192\) 0 0
\(193\) 9.85641 0.709480 0.354740 0.934965i \(-0.384569\pi\)
0.354740 + 0.934965i \(0.384569\pi\)
\(194\) 0 0
\(195\) 8.92820i 0.639362i
\(196\) 0 0
\(197\) − 6.92820i − 0.493614i −0.969065 0.246807i \(-0.920619\pi\)
0.969065 0.246807i \(-0.0793814\pi\)
\(198\) 0 0
\(199\) 13.5359 0.959534 0.479767 0.877396i \(-0.340721\pi\)
0.479767 + 0.877396i \(0.340721\pi\)
\(200\) 0 0
\(201\) 5.92820 0.418143
\(202\) 0 0
\(203\) 11.0000i 0.772049i
\(204\) 0 0
\(205\) − 13.8564i − 0.967773i
\(206\) 0 0
\(207\) −12.9282 −0.898572
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 24.8564i 1.71119i 0.517649 + 0.855593i \(0.326807\pi\)
−0.517649 + 0.855593i \(0.673193\pi\)
\(212\) 0 0
\(213\) 14.0000i 0.959264i
\(214\) 0 0
\(215\) 9.85641 0.672201
\(216\) 0 0
\(217\) −22.0000 −1.49346
\(218\) 0 0
\(219\) − 7.00000i − 0.473016i
\(220\) 0 0
\(221\) 35.3923i 2.38074i
\(222\) 0 0
\(223\) 26.9282 1.80325 0.901623 0.432523i \(-0.142377\pi\)
0.901623 + 0.432523i \(0.142377\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) 1.92820i 0.127979i 0.997951 + 0.0639897i \(0.0203825\pi\)
−0.997951 + 0.0639897i \(0.979618\pi\)
\(228\) 0 0
\(229\) − 3.07180i − 0.202990i −0.994836 0.101495i \(-0.967637\pi\)
0.994836 0.101495i \(-0.0323626\pi\)
\(230\) 0 0
\(231\) −17.8564 −1.17487
\(232\) 0 0
\(233\) 15.8564 1.03879 0.519394 0.854535i \(-0.326158\pi\)
0.519394 + 0.854535i \(0.326158\pi\)
\(234\) 0 0
\(235\) − 13.8564i − 0.903892i
\(236\) 0 0
\(237\) 2.00000i 0.129914i
\(238\) 0 0
\(239\) 16.3205 1.05569 0.527843 0.849342i \(-0.323001\pi\)
0.527843 + 0.849342i \(0.323001\pi\)
\(240\) 0 0
\(241\) −24.7846 −1.59652 −0.798259 0.602315i \(-0.794245\pi\)
−0.798259 + 0.602315i \(0.794245\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) − 25.8564i − 1.65191i
\(246\) 0 0
\(247\) −4.46410 −0.284044
\(248\) 0 0
\(249\) −2.92820 −0.185567
\(250\) 0 0
\(251\) 12.9282i 0.816021i 0.912977 + 0.408010i \(0.133777\pi\)
−0.912977 + 0.408010i \(0.866223\pi\)
\(252\) 0 0
\(253\) 25.8564i 1.62558i
\(254\) 0 0
\(255\) 15.8564 0.992967
\(256\) 0 0
\(257\) −0.928203 −0.0578997 −0.0289499 0.999581i \(-0.509216\pi\)
−0.0289499 + 0.999581i \(0.509216\pi\)
\(258\) 0 0
\(259\) − 8.92820i − 0.554772i
\(260\) 0 0
\(261\) 4.92820i 0.305048i
\(262\) 0 0
\(263\) −2.92820 −0.180561 −0.0902804 0.995916i \(-0.528776\pi\)
−0.0902804 + 0.995916i \(0.528776\pi\)
\(264\) 0 0
\(265\) 24.9282 1.53133
\(266\) 0 0
\(267\) − 0.928203i − 0.0568051i
\(268\) 0 0
\(269\) 7.85641i 0.479014i 0.970895 + 0.239507i \(0.0769857\pi\)
−0.970895 + 0.239507i \(0.923014\pi\)
\(270\) 0 0
\(271\) 4.60770 0.279898 0.139949 0.990159i \(-0.455306\pi\)
0.139949 + 0.990159i \(0.455306\pi\)
\(272\) 0 0
\(273\) −19.9282 −1.20611
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 24.9282i − 1.49779i −0.662688 0.748895i \(-0.730585\pi\)
0.662688 0.748895i \(-0.269415\pi\)
\(278\) 0 0
\(279\) −9.85641 −0.590088
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) − 18.9282i − 1.12516i −0.826741 0.562582i \(-0.809808\pi\)
0.826741 0.562582i \(-0.190192\pi\)
\(284\) 0 0
\(285\) 2.00000i 0.118470i
\(286\) 0 0
\(287\) 30.9282 1.82563
\(288\) 0 0
\(289\) 45.8564 2.69744
\(290\) 0 0
\(291\) − 12.9282i − 0.757865i
\(292\) 0 0
\(293\) 8.32051i 0.486089i 0.970015 + 0.243045i \(0.0781462\pi\)
−0.970015 + 0.243045i \(0.921854\pi\)
\(294\) 0 0
\(295\) −19.8564 −1.15608
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) 28.8564i 1.66881i
\(300\) 0 0
\(301\) 22.0000i 1.26806i
\(302\) 0 0
\(303\) −9.85641 −0.566236
\(304\) 0 0
\(305\) 17.8564 1.02245
\(306\) 0 0
\(307\) − 1.85641i − 0.105951i −0.998596 0.0529754i \(-0.983130\pi\)
0.998596 0.0529754i \(-0.0168705\pi\)
\(308\) 0 0
\(309\) − 2.92820i − 0.166580i
\(310\) 0 0
\(311\) 7.39230 0.419179 0.209590 0.977789i \(-0.432787\pi\)
0.209590 + 0.977789i \(0.432787\pi\)
\(312\) 0 0
\(313\) 16.8564 0.952780 0.476390 0.879234i \(-0.341945\pi\)
0.476390 + 0.879234i \(0.341945\pi\)
\(314\) 0 0
\(315\) − 17.8564i − 1.00609i
\(316\) 0 0
\(317\) − 14.4641i − 0.812385i −0.913788 0.406192i \(-0.866856\pi\)
0.913788 0.406192i \(-0.133144\pi\)
\(318\) 0 0
\(319\) 9.85641 0.551853
\(320\) 0 0
\(321\) 13.9282 0.777396
\(322\) 0 0
\(323\) 7.92820i 0.441137i
\(324\) 0 0
\(325\) − 4.46410i − 0.247624i
\(326\) 0 0
\(327\) 7.39230 0.408795
\(328\) 0 0
\(329\) 30.9282 1.70513
\(330\) 0 0
\(331\) − 9.78461i − 0.537811i −0.963167 0.268905i \(-0.913338\pi\)
0.963167 0.268905i \(-0.0866619\pi\)
\(332\) 0 0
\(333\) − 4.00000i − 0.219199i
\(334\) 0 0
\(335\) 11.8564 0.647785
\(336\) 0 0
\(337\) −7.07180 −0.385225 −0.192613 0.981275i \(-0.561696\pi\)
−0.192613 + 0.981275i \(0.561696\pi\)
\(338\) 0 0
\(339\) − 12.9282i − 0.702164i
\(340\) 0 0
\(341\) 19.7128i 1.06751i
\(342\) 0 0
\(343\) 26.4641 1.42893
\(344\) 0 0
\(345\) 12.9282 0.696031
\(346\) 0 0
\(347\) − 27.8564i − 1.49541i −0.664031 0.747705i \(-0.731156\pi\)
0.664031 0.747705i \(-0.268844\pi\)
\(348\) 0 0
\(349\) − 14.0000i − 0.749403i −0.927146 0.374701i \(-0.877745\pi\)
0.927146 0.374701i \(-0.122255\pi\)
\(350\) 0 0
\(351\) −22.3205 −1.19138
\(352\) 0 0
\(353\) 10.0718 0.536068 0.268034 0.963410i \(-0.413626\pi\)
0.268034 + 0.963410i \(0.413626\pi\)
\(354\) 0 0
\(355\) 28.0000i 1.48609i
\(356\) 0 0
\(357\) 35.3923i 1.87316i
\(358\) 0 0
\(359\) −24.4641 −1.29117 −0.645583 0.763690i \(-0.723386\pi\)
−0.645583 + 0.763690i \(0.723386\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) − 14.0000i − 0.732793i
\(366\) 0 0
\(367\) 13.0718 0.682342 0.341171 0.940001i \(-0.389176\pi\)
0.341171 + 0.940001i \(0.389176\pi\)
\(368\) 0 0
\(369\) 13.8564 0.721336
\(370\) 0 0
\(371\) 55.6410i 2.88874i
\(372\) 0 0
\(373\) − 11.3923i − 0.589871i −0.955517 0.294936i \(-0.904702\pi\)
0.955517 0.294936i \(-0.0952982\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 11.0000 0.566529
\(378\) 0 0
\(379\) 6.85641i 0.352190i 0.984373 + 0.176095i \(0.0563466\pi\)
−0.984373 + 0.176095i \(0.943653\pi\)
\(380\) 0 0
\(381\) 0.928203i 0.0475533i
\(382\) 0 0
\(383\) −24.7846 −1.26643 −0.633217 0.773974i \(-0.718266\pi\)
−0.633217 + 0.773974i \(0.718266\pi\)
\(384\) 0 0
\(385\) −35.7128 −1.82009
\(386\) 0 0
\(387\) 9.85641i 0.501029i
\(388\) 0 0
\(389\) − 25.8564i − 1.31097i −0.755207 0.655486i \(-0.772464\pi\)
0.755207 0.655486i \(-0.227536\pi\)
\(390\) 0 0
\(391\) 51.2487 2.59176
\(392\) 0 0
\(393\) −7.07180 −0.356725
\(394\) 0 0
\(395\) 4.00000i 0.201262i
\(396\) 0 0
\(397\) 8.92820i 0.448094i 0.974578 + 0.224047i \(0.0719269\pi\)
−0.974578 + 0.224047i \(0.928073\pi\)
\(398\) 0 0
\(399\) −4.46410 −0.223485
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 22.0000i 1.09590i
\(404\) 0 0
\(405\) − 2.00000i − 0.0993808i
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 8.85641i 0.436854i
\(412\) 0 0
\(413\) − 44.3205i − 2.18087i
\(414\) 0 0
\(415\) −5.85641 −0.287480
\(416\) 0 0
\(417\) −15.8564 −0.776492
\(418\) 0 0
\(419\) 30.7846i 1.50393i 0.659205 + 0.751963i \(0.270893\pi\)
−0.659205 + 0.751963i \(0.729107\pi\)
\(420\) 0 0
\(421\) − 11.5359i − 0.562225i −0.959675 0.281113i \(-0.909297\pi\)
0.959675 0.281113i \(-0.0907035\pi\)
\(422\) 0 0
\(423\) 13.8564 0.673722
\(424\) 0 0
\(425\) −7.92820 −0.384574
\(426\) 0 0
\(427\) 39.8564i 1.92879i
\(428\) 0 0
\(429\) 17.8564i 0.862115i
\(430\) 0 0
\(431\) 12.9282 0.622730 0.311365 0.950290i \(-0.399214\pi\)
0.311365 + 0.950290i \(0.399214\pi\)
\(432\) 0 0
\(433\) −24.7846 −1.19107 −0.595536 0.803328i \(-0.703060\pi\)
−0.595536 + 0.803328i \(0.703060\pi\)
\(434\) 0 0
\(435\) − 4.92820i − 0.236289i
\(436\) 0 0
\(437\) 6.46410i 0.309220i
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 25.8564 1.23126
\(442\) 0 0
\(443\) − 17.8564i − 0.848383i −0.905572 0.424192i \(-0.860558\pi\)
0.905572 0.424192i \(-0.139442\pi\)
\(444\) 0 0
\(445\) − 1.85641i − 0.0880021i
\(446\) 0 0
\(447\) 10.9282 0.516886
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) − 27.7128i − 1.30495i
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) −39.8564 −1.86850
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 39.6410i 1.85028i
\(460\) 0 0
\(461\) − 13.8564i − 0.645357i −0.946509 0.322679i \(-0.895417\pi\)
0.946509 0.322679i \(-0.104583\pi\)
\(462\) 0 0
\(463\) 16.7846 0.780047 0.390023 0.920805i \(-0.372467\pi\)
0.390023 + 0.920805i \(0.372467\pi\)
\(464\) 0 0
\(465\) 9.85641 0.457080
\(466\) 0 0
\(467\) − 2.78461i − 0.128856i −0.997922 0.0644282i \(-0.979478\pi\)
0.997922 0.0644282i \(-0.0205223\pi\)
\(468\) 0 0
\(469\) 26.4641i 1.22200i
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 19.7128 0.906396
\(474\) 0 0
\(475\) − 1.00000i − 0.0458831i
\(476\) 0 0
\(477\) 24.9282i 1.14138i
\(478\) 0 0
\(479\) −16.7846 −0.766908 −0.383454 0.923560i \(-0.625266\pi\)
−0.383454 + 0.923560i \(0.625266\pi\)
\(480\) 0 0
\(481\) −8.92820 −0.407091
\(482\) 0 0
\(483\) 28.8564i 1.31301i
\(484\) 0 0
\(485\) − 25.8564i − 1.17408i
\(486\) 0 0
\(487\) −34.6410 −1.56973 −0.784867 0.619664i \(-0.787269\pi\)
−0.784867 + 0.619664i \(0.787269\pi\)
\(488\) 0 0
\(489\) −23.8564 −1.07882
\(490\) 0 0
\(491\) 14.1436i 0.638291i 0.947706 + 0.319146i \(0.103396\pi\)
−0.947706 + 0.319146i \(0.896604\pi\)
\(492\) 0 0
\(493\) − 19.5359i − 0.879853i
\(494\) 0 0
\(495\) −16.0000 −0.719147
\(496\) 0 0
\(497\) −62.4974 −2.80339
\(498\) 0 0
\(499\) 22.7846i 1.01998i 0.860181 + 0.509990i \(0.170351\pi\)
−0.860181 + 0.509990i \(0.829649\pi\)
\(500\) 0 0
\(501\) 5.07180i 0.226591i
\(502\) 0 0
\(503\) 22.3205 0.995222 0.497611 0.867400i \(-0.334211\pi\)
0.497611 + 0.867400i \(0.334211\pi\)
\(504\) 0 0
\(505\) −19.7128 −0.877209
\(506\) 0 0
\(507\) 6.92820i 0.307692i
\(508\) 0 0
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 0 0
\(511\) 31.2487 1.38236
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) − 5.85641i − 0.258064i
\(516\) 0 0
\(517\) − 27.7128i − 1.21881i
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) −13.0718 −0.572686 −0.286343 0.958127i \(-0.592440\pi\)
−0.286343 + 0.958127i \(0.592440\pi\)
\(522\) 0 0
\(523\) 31.0000i 1.35554i 0.735276 + 0.677768i \(0.237052\pi\)
−0.735276 + 0.677768i \(0.762948\pi\)
\(524\) 0 0
\(525\) − 4.46410i − 0.194829i
\(526\) 0 0
\(527\) 39.0718 1.70199
\(528\) 0 0
\(529\) 18.7846 0.816722
\(530\) 0 0
\(531\) − 19.8564i − 0.861695i
\(532\) 0 0
\(533\) − 30.9282i − 1.33965i
\(534\) 0 0
\(535\) 27.8564 1.20434
\(536\) 0 0
\(537\) −25.8564 −1.11579
\(538\) 0 0
\(539\) − 51.7128i − 2.22743i
\(540\) 0 0
\(541\) − 12.7846i − 0.549653i −0.961494 0.274827i \(-0.911380\pi\)
0.961494 0.274827i \(-0.0886205\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 14.7846 0.633303
\(546\) 0 0
\(547\) 6.14359i 0.262681i 0.991337 + 0.131341i \(0.0419281\pi\)
−0.991337 + 0.131341i \(0.958072\pi\)
\(548\) 0 0
\(549\) 17.8564i 0.762093i
\(550\) 0 0
\(551\) 2.46410 0.104974
\(552\) 0 0
\(553\) −8.92820 −0.379666
\(554\) 0 0
\(555\) 4.00000i 0.169791i
\(556\) 0 0
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 0 0
\(559\) 22.0000 0.930501
\(560\) 0 0
\(561\) 31.7128 1.33892
\(562\) 0 0
\(563\) − 31.7128i − 1.33654i −0.743921 0.668268i \(-0.767036\pi\)
0.743921 0.668268i \(-0.232964\pi\)
\(564\) 0 0
\(565\) − 25.8564i − 1.08779i
\(566\) 0 0
\(567\) 4.46410 0.187475
\(568\) 0 0
\(569\) −8.78461 −0.368270 −0.184135 0.982901i \(-0.558948\pi\)
−0.184135 + 0.982901i \(0.558948\pi\)
\(570\) 0 0
\(571\) − 28.7846i − 1.20460i −0.798270 0.602299i \(-0.794251\pi\)
0.798270 0.602299i \(-0.205749\pi\)
\(572\) 0 0
\(573\) − 3.53590i − 0.147714i
\(574\) 0 0
\(575\) −6.46410 −0.269572
\(576\) 0 0
\(577\) 11.7846 0.490600 0.245300 0.969447i \(-0.421114\pi\)
0.245300 + 0.969447i \(0.421114\pi\)
\(578\) 0 0
\(579\) − 9.85641i − 0.409618i
\(580\) 0 0
\(581\) − 13.0718i − 0.542310i
\(582\) 0 0
\(583\) 49.8564 2.06484
\(584\) 0 0
\(585\) −17.8564 −0.738272
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 4.92820i 0.203063i
\(590\) 0 0
\(591\) −6.92820 −0.284988
\(592\) 0 0
\(593\) 16.1436 0.662938 0.331469 0.943466i \(-0.392456\pi\)
0.331469 + 0.943466i \(0.392456\pi\)
\(594\) 0 0
\(595\) 70.7846i 2.90189i
\(596\) 0 0
\(597\) − 13.5359i − 0.553987i
\(598\) 0 0
\(599\) −9.85641 −0.402722 −0.201361 0.979517i \(-0.564536\pi\)
−0.201361 + 0.979517i \(0.564536\pi\)
\(600\) 0 0
\(601\) −27.8564 −1.13629 −0.568143 0.822930i \(-0.692338\pi\)
−0.568143 + 0.822930i \(0.692338\pi\)
\(602\) 0 0
\(603\) 11.8564i 0.482830i
\(604\) 0 0
\(605\) 10.0000i 0.406558i
\(606\) 0 0
\(607\) 39.8564 1.61772 0.808861 0.588000i \(-0.200085\pi\)
0.808861 + 0.588000i \(0.200085\pi\)
\(608\) 0 0
\(609\) 11.0000 0.445742
\(610\) 0 0
\(611\) − 30.9282i − 1.25122i
\(612\) 0 0
\(613\) 34.6410i 1.39914i 0.714565 + 0.699569i \(0.246625\pi\)
−0.714565 + 0.699569i \(0.753375\pi\)
\(614\) 0 0
\(615\) −13.8564 −0.558744
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) 27.8564i 1.11964i 0.828613 + 0.559822i \(0.189130\pi\)
−0.828613 + 0.559822i \(0.810870\pi\)
\(620\) 0 0
\(621\) 32.3205i 1.29698i
\(622\) 0 0
\(623\) 4.14359 0.166010
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 0 0
\(629\) 15.8564i 0.632236i
\(630\) 0 0
\(631\) −38.6410 −1.53827 −0.769137 0.639084i \(-0.779314\pi\)
−0.769137 + 0.639084i \(0.779314\pi\)
\(632\) 0 0
\(633\) 24.8564 0.987953
\(634\) 0 0
\(635\) 1.85641i 0.0736692i
\(636\) 0 0
\(637\) − 57.7128i − 2.28666i
\(638\) 0 0
\(639\) −28.0000 −1.10766
\(640\) 0 0
\(641\) 8.78461 0.346971 0.173486 0.984836i \(-0.444497\pi\)
0.173486 + 0.984836i \(0.444497\pi\)
\(642\) 0 0
\(643\) 32.9282i 1.29856i 0.760549 + 0.649281i \(0.224930\pi\)
−0.760549 + 0.649281i \(0.775070\pi\)
\(644\) 0 0
\(645\) − 9.85641i − 0.388096i
\(646\) 0 0
\(647\) 13.5359 0.532151 0.266076 0.963952i \(-0.414273\pi\)
0.266076 + 0.963952i \(0.414273\pi\)
\(648\) 0 0
\(649\) −39.7128 −1.55886
\(650\) 0 0
\(651\) 22.0000i 0.862248i
\(652\) 0 0
\(653\) 6.78461i 0.265502i 0.991149 + 0.132751i \(0.0423811\pi\)
−0.991149 + 0.132751i \(0.957619\pi\)
\(654\) 0 0
\(655\) −14.1436 −0.552636
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 2.85641i 0.111270i 0.998451 + 0.0556349i \(0.0177183\pi\)
−0.998451 + 0.0556349i \(0.982282\pi\)
\(660\) 0 0
\(661\) 30.4641i 1.18492i 0.805601 + 0.592458i \(0.201842\pi\)
−0.805601 + 0.592458i \(0.798158\pi\)
\(662\) 0 0
\(663\) 35.3923 1.37452
\(664\) 0 0
\(665\) −8.92820 −0.346221
\(666\) 0 0
\(667\) − 15.9282i − 0.616742i
\(668\) 0 0
\(669\) − 26.9282i − 1.04110i
\(670\) 0 0
\(671\) 35.7128 1.37868
\(672\) 0 0
\(673\) 9.07180 0.349692 0.174846 0.984596i \(-0.444057\pi\)
0.174846 + 0.984596i \(0.444057\pi\)
\(674\) 0 0
\(675\) − 5.00000i − 0.192450i
\(676\) 0 0
\(677\) − 7.39230i − 0.284109i −0.989859 0.142055i \(-0.954629\pi\)
0.989859 0.142055i \(-0.0453709\pi\)
\(678\) 0 0
\(679\) 57.7128 2.21481
\(680\) 0 0
\(681\) 1.92820 0.0738889
\(682\) 0 0
\(683\) 17.8564i 0.683256i 0.939835 + 0.341628i \(0.110978\pi\)
−0.939835 + 0.341628i \(0.889022\pi\)
\(684\) 0 0
\(685\) 17.7128i 0.676772i
\(686\) 0 0
\(687\) −3.07180 −0.117196
\(688\) 0 0
\(689\) 55.6410 2.11975
\(690\) 0 0
\(691\) 7.85641i 0.298872i 0.988771 + 0.149436i \(0.0477458\pi\)
−0.988771 + 0.149436i \(0.952254\pi\)
\(692\) 0 0
\(693\) − 35.7128i − 1.35662i
\(694\) 0 0
\(695\) −31.7128 −1.20294
\(696\) 0 0
\(697\) −54.9282 −2.08055
\(698\) 0 0
\(699\) − 15.8564i − 0.599744i
\(700\) 0 0
\(701\) − 7.71281i − 0.291309i −0.989336 0.145654i \(-0.953471\pi\)
0.989336 0.145654i \(-0.0465288\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −13.8564 −0.521862
\(706\) 0 0
\(707\) − 44.0000i − 1.65479i
\(708\) 0 0
\(709\) 10.7846i 0.405025i 0.979280 + 0.202512i \(0.0649106\pi\)
−0.979280 + 0.202512i \(0.935089\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 31.8564 1.19303
\(714\) 0 0
\(715\) 35.7128i 1.33558i
\(716\) 0 0
\(717\) − 16.3205i − 0.609501i
\(718\) 0 0
\(719\) 13.3923 0.499449 0.249724 0.968317i \(-0.419660\pi\)
0.249724 + 0.968317i \(0.419660\pi\)
\(720\) 0 0
\(721\) 13.0718 0.486819
\(722\) 0 0
\(723\) 24.7846i 0.921750i
\(724\) 0 0
\(725\) 2.46410i 0.0915144i
\(726\) 0 0
\(727\) −0.320508 −0.0118870 −0.00594349 0.999982i \(-0.501892\pi\)
−0.00594349 + 0.999982i \(0.501892\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 39.0718i − 1.44512i
\(732\) 0 0
\(733\) − 43.8564i − 1.61987i −0.586517 0.809937i \(-0.699501\pi\)
0.586517 0.809937i \(-0.300499\pi\)
\(734\) 0 0
\(735\) −25.8564 −0.953728
\(736\) 0 0
\(737\) 23.7128 0.873473
\(738\) 0 0
\(739\) 9.85641i 0.362574i 0.983430 + 0.181287i \(0.0580263\pi\)
−0.983430 + 0.181287i \(0.941974\pi\)
\(740\) 0 0
\(741\) 4.46410i 0.163993i
\(742\) 0 0
\(743\) 16.9282 0.621036 0.310518 0.950568i \(-0.399498\pi\)
0.310518 + 0.950568i \(0.399498\pi\)
\(744\) 0 0
\(745\) 21.8564 0.800757
\(746\) 0 0
\(747\) − 5.85641i − 0.214275i
\(748\) 0 0
\(749\) 62.1769i 2.27190i
\(750\) 0 0
\(751\) 33.0718 1.20681 0.603404 0.797436i \(-0.293811\pi\)
0.603404 + 0.797436i \(0.293811\pi\)
\(752\) 0 0
\(753\) 12.9282 0.471130
\(754\) 0 0
\(755\) 32.0000i 1.16460i
\(756\) 0 0
\(757\) 1.21539i 0.0441741i 0.999756 + 0.0220871i \(0.00703110\pi\)
−0.999756 + 0.0220871i \(0.992969\pi\)
\(758\) 0 0
\(759\) 25.8564 0.938528
\(760\) 0 0
\(761\) 48.7128 1.76584 0.882919 0.469525i \(-0.155575\pi\)
0.882919 + 0.469525i \(0.155575\pi\)
\(762\) 0 0
\(763\) 33.0000i 1.19468i
\(764\) 0 0
\(765\) 31.7128i 1.14658i
\(766\) 0 0
\(767\) −44.3205 −1.60032
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 0.928203i 0.0334284i
\(772\) 0 0
\(773\) 54.4641i 1.95894i 0.201596 + 0.979469i \(0.435387\pi\)
−0.201596 + 0.979469i \(0.564613\pi\)
\(774\) 0 0
\(775\) −4.92820 −0.177026
\(776\) 0 0
\(777\) −8.92820 −0.320298
\(778\) 0 0
\(779\) − 6.92820i − 0.248229i
\(780\) 0 0
\(781\) 56.0000i 2.00384i
\(782\) 0 0
\(783\) 12.3205 0.440299
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 19.7846i 0.705245i 0.935766 + 0.352623i \(0.114710\pi\)
−0.935766 + 0.352623i \(0.885290\pi\)
\(788\) 0 0
\(789\) 2.92820i 0.104247i
\(790\) 0 0
\(791\) 57.7128 2.05203
\(792\) 0 0
\(793\) 39.8564 1.41534
\(794\) 0 0
\(795\) − 24.9282i − 0.884112i
\(796\) 0 0
\(797\) 30.1769i 1.06892i 0.845193 + 0.534461i \(0.179485\pi\)
−0.845193 + 0.534461i \(0.820515\pi\)
\(798\) 0 0
\(799\) −54.9282 −1.94322
\(800\) 0 0
\(801\) 1.85641 0.0655929
\(802\) 0 0
\(803\) − 28.0000i − 0.988099i
\(804\) 0 0
\(805\) 57.7128i 2.03411i
\(806\) 0 0
\(807\) 7.85641 0.276559
\(808\) 0 0
\(809\) −18.0718 −0.635371 −0.317685 0.948196i \(-0.602906\pi\)
−0.317685 + 0.948196i \(0.602906\pi\)
\(810\) 0 0
\(811\) 1.14359i 0.0401570i 0.999798 + 0.0200785i \(0.00639161\pi\)
−0.999798 + 0.0200785i \(0.993608\pi\)
\(812\) 0 0
\(813\) − 4.60770i − 0.161599i
\(814\) 0 0
\(815\) −47.7128 −1.67131
\(816\) 0 0
\(817\) 4.92820 0.172416
\(818\) 0 0
\(819\) − 39.8564i − 1.39270i
\(820\) 0 0
\(821\) 44.6410i 1.55798i 0.627035 + 0.778991i \(0.284268\pi\)
−0.627035 + 0.778991i \(0.715732\pi\)
\(822\) 0 0
\(823\) 23.5359 0.820410 0.410205 0.911993i \(-0.365457\pi\)
0.410205 + 0.911993i \(0.365457\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) − 14.7128i − 0.511615i −0.966728 0.255807i \(-0.917659\pi\)
0.966728 0.255807i \(-0.0823413\pi\)
\(828\) 0 0
\(829\) − 12.3205i − 0.427909i −0.976844 0.213954i \(-0.931366\pi\)
0.976844 0.213954i \(-0.0686344\pi\)
\(830\) 0 0
\(831\) −24.9282 −0.864750
\(832\) 0 0
\(833\) −102.497 −3.55133
\(834\) 0 0
\(835\) 10.1436i 0.351034i
\(836\) 0 0
\(837\) 24.6410i 0.851718i
\(838\) 0 0
\(839\) −22.0000 −0.759524 −0.379762 0.925084i \(-0.623994\pi\)
−0.379762 + 0.925084i \(0.623994\pi\)
\(840\) 0 0
\(841\) 22.9282 0.790628
\(842\) 0 0
\(843\) − 4.00000i − 0.137767i
\(844\) 0 0
\(845\) 13.8564i 0.476675i
\(846\) 0 0
\(847\) −22.3205 −0.766942
\(848\) 0 0
\(849\) −18.9282 −0.649614
\(850\) 0 0
\(851\) 12.9282i 0.443173i
\(852\) 0 0
\(853\) 13.0718i 0.447570i 0.974639 + 0.223785i \(0.0718413\pi\)
−0.974639 + 0.223785i \(0.928159\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −49.8564 −1.70306 −0.851531 0.524304i \(-0.824326\pi\)
−0.851531 + 0.524304i \(0.824326\pi\)
\(858\) 0 0
\(859\) − 21.0718i − 0.718960i −0.933153 0.359480i \(-0.882954\pi\)
0.933153 0.359480i \(-0.117046\pi\)
\(860\) 0 0
\(861\) − 30.9282i − 1.05403i
\(862\) 0 0
\(863\) −14.7846 −0.503274 −0.251637 0.967822i \(-0.580969\pi\)
−0.251637 + 0.967822i \(0.580969\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) − 45.8564i − 1.55737i
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) 26.4641 0.896702
\(872\) 0 0
\(873\) 25.8564 0.875107
\(874\) 0 0
\(875\) − 53.5692i − 1.81097i
\(876\) 0 0
\(877\) − 31.2487i − 1.05519i −0.849495 0.527597i \(-0.823093\pi\)
0.849495 0.527597i \(-0.176907\pi\)
\(878\) 0 0
\(879\) 8.32051 0.280644
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) − 49.7128i − 1.67297i −0.547990 0.836485i \(-0.684607\pi\)
0.547990 0.836485i \(-0.315393\pi\)
\(884\) 0 0
\(885\) 19.8564i 0.667466i
\(886\) 0 0
\(887\) −12.9282 −0.434087 −0.217043 0.976162i \(-0.569641\pi\)
−0.217043 + 0.976162i \(0.569641\pi\)
\(888\) 0 0
\(889\) −4.14359 −0.138972
\(890\) 0 0
\(891\) − 4.00000i − 0.134005i
\(892\) 0 0
\(893\) − 6.92820i − 0.231843i
\(894\) 0 0
\(895\) −51.7128 −1.72857
\(896\) 0 0
\(897\) 28.8564 0.963487
\(898\) 0 0
\(899\) − 12.1436i − 0.405012i
\(900\) 0 0
\(901\) − 98.8179i − 3.29210i
\(902\) 0 0
\(903\) 22.0000 0.732114
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 13.1436i 0.436426i 0.975901 + 0.218213i \(0.0700227\pi\)
−0.975901 + 0.218213i \(0.929977\pi\)
\(908\) 0 0
\(909\) − 19.7128i − 0.653833i
\(910\) 0 0
\(911\) 7.85641 0.260294 0.130147 0.991495i \(-0.458455\pi\)
0.130147 + 0.991495i \(0.458455\pi\)
\(912\) 0 0
\(913\) −11.7128 −0.387638
\(914\) 0 0
\(915\) − 17.8564i − 0.590315i
\(916\) 0 0
\(917\) − 31.5692i − 1.04251i
\(918\) 0 0
\(919\) 5.39230 0.177876 0.0889379 0.996037i \(-0.471653\pi\)
0.0889379 + 0.996037i \(0.471653\pi\)
\(920\) 0 0
\(921\) −1.85641 −0.0611707
\(922\) 0 0
\(923\) 62.4974i 2.05713i
\(924\) 0 0
\(925\) − 2.00000i − 0.0657596i
\(926\) 0 0
\(927\) 5.85641 0.192350
\(928\) 0 0
\(929\) 8.71281 0.285858 0.142929 0.989733i \(-0.454348\pi\)
0.142929 + 0.989733i \(0.454348\pi\)
\(930\) 0 0
\(931\) − 12.9282i − 0.423705i
\(932\) 0 0
\(933\) − 7.39230i − 0.242013i
\(934\) 0 0
\(935\) 63.4256 2.07424
\(936\) 0 0
\(937\) 29.9282 0.977712 0.488856 0.872365i \(-0.337414\pi\)
0.488856 + 0.872365i \(0.337414\pi\)
\(938\) 0 0
\(939\) − 16.8564i − 0.550088i
\(940\) 0 0
\(941\) 41.1051i 1.33999i 0.742366 + 0.669994i \(0.233703\pi\)
−0.742366 + 0.669994i \(0.766297\pi\)
\(942\) 0 0
\(943\) −44.7846 −1.45839
\(944\) 0 0
\(945\) −44.6410 −1.45217
\(946\) 0 0
\(947\) − 16.9282i − 0.550093i −0.961431 0.275046i \(-0.911307\pi\)
0.961431 0.275046i \(-0.0886932\pi\)
\(948\) 0 0
\(949\) − 31.2487i − 1.01438i
\(950\) 0 0
\(951\) −14.4641 −0.469031
\(952\) 0 0
\(953\) 7.21539 0.233729 0.116865 0.993148i \(-0.462716\pi\)
0.116865 + 0.993148i \(0.462716\pi\)
\(954\) 0 0
\(955\) − 7.07180i − 0.228838i
\(956\) 0 0
\(957\) − 9.85641i − 0.318612i
\(958\) 0 0
\(959\) −39.5359 −1.27668
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 0 0
\(963\) 27.8564i 0.897660i
\(964\) 0 0
\(965\) − 19.7128i − 0.634578i
\(966\) 0 0
\(967\) −24.7846 −0.797019 −0.398510 0.917164i \(-0.630472\pi\)
−0.398510 + 0.917164i \(0.630472\pi\)
\(968\) 0 0
\(969\) 7.92820 0.254691
\(970\) 0 0
\(971\) 25.8564i 0.829772i 0.909873 + 0.414886i \(0.136178\pi\)
−0.909873 + 0.414886i \(0.863822\pi\)
\(972\) 0 0
\(973\) − 70.7846i − 2.26925i
\(974\) 0 0
\(975\) −4.46410 −0.142966
\(976\) 0 0
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) − 3.71281i − 0.118662i
\(980\) 0 0
\(981\) 14.7846i 0.472036i
\(982\) 0 0
\(983\) −20.6410 −0.658346 −0.329173 0.944270i \(-0.606770\pi\)
−0.329173 + 0.944270i \(0.606770\pi\)
\(984\) 0 0
\(985\) −13.8564 −0.441502
\(986\) 0 0
\(987\) − 30.9282i − 0.984456i
\(988\) 0 0
\(989\) − 31.8564i − 1.01297i
\(990\) 0 0
\(991\) 43.5692 1.38402 0.692011 0.721887i \(-0.256725\pi\)
0.692011 + 0.721887i \(0.256725\pi\)
\(992\) 0 0
\(993\) −9.78461 −0.310505
\(994\) 0 0
\(995\) − 27.0718i − 0.858234i
\(996\) 0 0
\(997\) 42.7846i 1.35500i 0.735522 + 0.677501i \(0.236937\pi\)
−0.735522 + 0.677501i \(0.763063\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 2432.2.c.d.1217.2 yes 4
4.3 odd 2 2432.2.c.c.1217.3 yes 4
8.3 odd 2 2432.2.c.c.1217.1 4
8.5 even 2 inner 2432.2.c.d.1217.4 yes 4
16.3 odd 4 4864.2.a.s.1.2 2
16.5 even 4 4864.2.a.u.1.1 2
16.11 odd 4 4864.2.a.x.1.2 2
16.13 even 4 4864.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.c.1217.1 4 8.3 odd 2
2432.2.c.c.1217.3 yes 4 4.3 odd 2
2432.2.c.d.1217.2 yes 4 1.1 even 1 trivial
2432.2.c.d.1217.4 yes 4 8.5 even 2 inner
4864.2.a.s.1.2 2 16.3 odd 4
4864.2.a.u.1.1 2 16.5 even 4
4864.2.a.v.1.1 2 16.13 even 4
4864.2.a.x.1.2 2 16.11 odd 4