Properties

Label 9200.2.a.ci.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $3$
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] = [9200,2,Mod(1,9200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, 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("9200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.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: \(73.4623698596\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.93543 q^{3} +4.93543 q^{7} +0.745898 q^{9} +O(q^{10})\) \(q-1.93543 q^{3} +4.93543 q^{7} +0.745898 q^{9} -0.745898 q^{11} -1.74590 q^{13} +6.10856 q^{17} -5.44364 q^{19} -9.55220 q^{21} -1.00000 q^{23} +4.36266 q^{27} -1.66492 q^{29} +1.61676 q^{31} +1.44364 q^{33} -4.34625 q^{37} +3.37907 q^{39} -6.95184 q^{41} +5.01641 q^{43} +2.68133 q^{47} +17.3585 q^{49} -11.8227 q^{51} -13.7417 q^{53} +10.5358 q^{57} -12.2171 q^{59} -13.9794 q^{61} +3.68133 q^{63} +13.1044 q^{67} +1.93543 q^{69} +9.67716 q^{71} -5.69774 q^{73} -3.68133 q^{77} -10.3791 q^{79} -10.6813 q^{81} +0.637339 q^{83} +3.22235 q^{87} +2.72532 q^{89} -8.61676 q^{91} -3.12914 q^{93} -7.12497 q^{97} -0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 7 q^{7} + 3 q^{9} - 3 q^{11} - 6 q^{13} + 5 q^{17} - 7 q^{19} - 6 q^{21} - 3 q^{23} - q^{27} - q^{29} - 10 q^{31} - 5 q^{33} - 2 q^{37} - 7 q^{39} - 10 q^{41} + 12 q^{43} + q^{47} + 6 q^{49} - 9 q^{51} - 10 q^{53} + 12 q^{57} - 10 q^{59} - 13 q^{61} + 4 q^{63} - 6 q^{67} - 2 q^{69} - 10 q^{71} - 7 q^{73} - 4 q^{77} - 14 q^{79} - 25 q^{81} + 16 q^{83} - 5 q^{87} - 20 q^{89} - 11 q^{91} - 25 q^{93} - 5 q^{97} - 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.93543 −1.11742 −0.558711 0.829362i \(-0.688704\pi\)
−0.558711 + 0.829362i \(0.688704\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.93543 1.86542 0.932709 0.360630i \(-0.117438\pi\)
0.932709 + 0.360630i \(0.117438\pi\)
\(8\) 0 0
\(9\) 0.745898 0.248633
\(10\) 0 0
\(11\) −0.745898 −0.224897 −0.112448 0.993658i \(-0.535869\pi\)
−0.112448 + 0.993658i \(0.535869\pi\)
\(12\) 0 0
\(13\) −1.74590 −0.484225 −0.242113 0.970248i \(-0.577840\pi\)
−0.242113 + 0.970248i \(0.577840\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.10856 1.48154 0.740772 0.671757i \(-0.234460\pi\)
0.740772 + 0.671757i \(0.234460\pi\)
\(18\) 0 0
\(19\) −5.44364 −1.24886 −0.624428 0.781083i \(-0.714668\pi\)
−0.624428 + 0.781083i \(0.714668\pi\)
\(20\) 0 0
\(21\) −9.55220 −2.08446
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.36266 0.839595
\(28\) 0 0
\(29\) −1.66492 −0.309169 −0.154584 0.987980i \(-0.549404\pi\)
−0.154584 + 0.987980i \(0.549404\pi\)
\(30\) 0 0
\(31\) 1.61676 0.290379 0.145190 0.989404i \(-0.453621\pi\)
0.145190 + 0.989404i \(0.453621\pi\)
\(32\) 0 0
\(33\) 1.44364 0.251305
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.34625 −0.714520 −0.357260 0.934005i \(-0.616289\pi\)
−0.357260 + 0.934005i \(0.616289\pi\)
\(38\) 0 0
\(39\) 3.37907 0.541084
\(40\) 0 0
\(41\) −6.95184 −1.08569 −0.542847 0.839831i \(-0.682654\pi\)
−0.542847 + 0.839831i \(0.682654\pi\)
\(42\) 0 0
\(43\) 5.01641 0.764995 0.382497 0.923957i \(-0.375064\pi\)
0.382497 + 0.923957i \(0.375064\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.68133 0.391112 0.195556 0.980693i \(-0.437349\pi\)
0.195556 + 0.980693i \(0.437349\pi\)
\(48\) 0 0
\(49\) 17.3585 2.47978
\(50\) 0 0
\(51\) −11.8227 −1.65551
\(52\) 0 0
\(53\) −13.7417 −1.88757 −0.943786 0.330558i \(-0.892763\pi\)
−0.943786 + 0.330558i \(0.892763\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.5358 1.39550
\(58\) 0 0
\(59\) −12.2171 −1.59053 −0.795267 0.606260i \(-0.792669\pi\)
−0.795267 + 0.606260i \(0.792669\pi\)
\(60\) 0 0
\(61\) −13.9794 −1.78988 −0.894941 0.446185i \(-0.852782\pi\)
−0.894941 + 0.446185i \(0.852782\pi\)
\(62\) 0 0
\(63\) 3.68133 0.463804
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.1044 1.60096 0.800478 0.599362i \(-0.204579\pi\)
0.800478 + 0.599362i \(0.204579\pi\)
\(68\) 0 0
\(69\) 1.93543 0.232999
\(70\) 0 0
\(71\) 9.67716 1.14847 0.574234 0.818691i \(-0.305300\pi\)
0.574234 + 0.818691i \(0.305300\pi\)
\(72\) 0 0
\(73\) −5.69774 −0.666870 −0.333435 0.942773i \(-0.608208\pi\)
−0.333435 + 0.942773i \(0.608208\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.68133 −0.419527
\(78\) 0 0
\(79\) −10.3791 −1.16774 −0.583868 0.811848i \(-0.698462\pi\)
−0.583868 + 0.811848i \(0.698462\pi\)
\(80\) 0 0
\(81\) −10.6813 −1.18681
\(82\) 0 0
\(83\) 0.637339 0.0699570 0.0349785 0.999388i \(-0.488864\pi\)
0.0349785 + 0.999388i \(0.488864\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.22235 0.345472
\(88\) 0 0
\(89\) 2.72532 0.288884 0.144442 0.989513i \(-0.453861\pi\)
0.144442 + 0.989513i \(0.453861\pi\)
\(90\) 0 0
\(91\) −8.61676 −0.903282
\(92\) 0 0
\(93\) −3.12914 −0.324476
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.12497 −0.723431 −0.361715 0.932289i \(-0.617809\pi\)
−0.361715 + 0.932289i \(0.617809\pi\)
\(98\) 0 0
\(99\) −0.556364 −0.0559167
\(100\) 0 0
\(101\) 14.9753 1.49009 0.745047 0.667012i \(-0.232427\pi\)
0.745047 + 0.667012i \(0.232427\pi\)
\(102\) 0 0
\(103\) −10.8667 −1.07073 −0.535364 0.844622i \(-0.679825\pi\)
−0.535364 + 0.844622i \(0.679825\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.49180 −0.724259 −0.362130 0.932128i \(-0.617950\pi\)
−0.362130 + 0.932128i \(0.617950\pi\)
\(108\) 0 0
\(109\) −18.3309 −1.75578 −0.877891 0.478860i \(-0.841050\pi\)
−0.877891 + 0.478860i \(0.841050\pi\)
\(110\) 0 0
\(111\) 8.41188 0.798420
\(112\) 0 0
\(113\) −0.379068 −0.0356597 −0.0178299 0.999841i \(-0.505676\pi\)
−0.0178299 + 0.999841i \(0.505676\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.30226 −0.120394
\(118\) 0 0
\(119\) 30.1484 2.76370
\(120\) 0 0
\(121\) −10.4436 −0.949421
\(122\) 0 0
\(123\) 13.4548 1.21318
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.64852 0.235018 0.117509 0.993072i \(-0.462509\pi\)
0.117509 + 0.993072i \(0.462509\pi\)
\(128\) 0 0
\(129\) −9.70892 −0.854822
\(130\) 0 0
\(131\) 5.53579 0.483664 0.241832 0.970318i \(-0.422252\pi\)
0.241832 + 0.970318i \(0.422252\pi\)
\(132\) 0 0
\(133\) −26.8667 −2.32964
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.16896 −0.527050 −0.263525 0.964653i \(-0.584885\pi\)
−0.263525 + 0.964653i \(0.584885\pi\)
\(138\) 0 0
\(139\) 20.7693 1.76163 0.880815 0.473460i \(-0.156995\pi\)
0.880815 + 0.473460i \(0.156995\pi\)
\(140\) 0 0
\(141\) −5.18953 −0.437038
\(142\) 0 0
\(143\) 1.30226 0.108901
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −33.5962 −2.77097
\(148\) 0 0
\(149\) −1.06457 −0.0872128 −0.0436064 0.999049i \(-0.513885\pi\)
−0.0436064 + 0.999049i \(0.513885\pi\)
\(150\) 0 0
\(151\) 12.6608 1.03032 0.515159 0.857095i \(-0.327733\pi\)
0.515159 + 0.857095i \(0.327733\pi\)
\(152\) 0 0
\(153\) 4.55636 0.368360
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.8873 −1.18813 −0.594067 0.804416i \(-0.702479\pi\)
−0.594067 + 0.804416i \(0.702479\pi\)
\(158\) 0 0
\(159\) 26.5962 2.10921
\(160\) 0 0
\(161\) −4.93543 −0.388967
\(162\) 0 0
\(163\) 6.72949 0.527094 0.263547 0.964646i \(-0.415108\pi\)
0.263547 + 0.964646i \(0.415108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.98359 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(168\) 0 0
\(169\) −9.95184 −0.765526
\(170\) 0 0
\(171\) −4.06040 −0.310506
\(172\) 0 0
\(173\) 15.8503 1.20508 0.602538 0.798091i \(-0.294156\pi\)
0.602538 + 0.798091i \(0.294156\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 23.6454 1.77730
\(178\) 0 0
\(179\) −3.02759 −0.226292 −0.113146 0.993578i \(-0.536093\pi\)
−0.113146 + 0.993578i \(0.536093\pi\)
\(180\) 0 0
\(181\) 1.82270 0.135481 0.0677403 0.997703i \(-0.478421\pi\)
0.0677403 + 0.997703i \(0.478421\pi\)
\(182\) 0 0
\(183\) 27.0562 2.00005
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.55636 −0.333194
\(188\) 0 0
\(189\) 21.5316 1.56619
\(190\) 0 0
\(191\) −19.6126 −1.41912 −0.709559 0.704646i \(-0.751106\pi\)
−0.709559 + 0.704646i \(0.751106\pi\)
\(192\) 0 0
\(193\) 0.335076 0.0241193 0.0120597 0.999927i \(-0.496161\pi\)
0.0120597 + 0.999927i \(0.496161\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.5316 1.46282 0.731409 0.681939i \(-0.238863\pi\)
0.731409 + 0.681939i \(0.238863\pi\)
\(198\) 0 0
\(199\) −16.7253 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(200\) 0 0
\(201\) −25.3627 −1.78894
\(202\) 0 0
\(203\) −8.21712 −0.576729
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.745898 −0.0518435
\(208\) 0 0
\(209\) 4.06040 0.280864
\(210\) 0 0
\(211\) 13.3955 0.922183 0.461091 0.887353i \(-0.347458\pi\)
0.461091 + 0.887353i \(0.347458\pi\)
\(212\) 0 0
\(213\) −18.7295 −1.28332
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.97942 0.541679
\(218\) 0 0
\(219\) 11.0276 0.745175
\(220\) 0 0
\(221\) −10.6649 −0.717400
\(222\) 0 0
\(223\) 1.27468 0.0853587 0.0426794 0.999089i \(-0.486411\pi\)
0.0426794 + 0.999089i \(0.486411\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.57978 0.237598 0.118799 0.992918i \(-0.462096\pi\)
0.118799 + 0.992918i \(0.462096\pi\)
\(228\) 0 0
\(229\) 10.4119 0.688036 0.344018 0.938963i \(-0.388212\pi\)
0.344018 + 0.938963i \(0.388212\pi\)
\(230\) 0 0
\(231\) 7.12497 0.468788
\(232\) 0 0
\(233\) −3.06040 −0.200493 −0.100247 0.994963i \(-0.531963\pi\)
−0.100247 + 0.994963i \(0.531963\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 20.0880 1.30485
\(238\) 0 0
\(239\) 19.1895 1.24127 0.620634 0.784100i \(-0.286875\pi\)
0.620634 + 0.784100i \(0.286875\pi\)
\(240\) 0 0
\(241\) 0.346255 0.0223042 0.0111521 0.999938i \(-0.496450\pi\)
0.0111521 + 0.999938i \(0.496450\pi\)
\(242\) 0 0
\(243\) 7.58501 0.486579
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.50403 0.604727
\(248\) 0 0
\(249\) −1.23353 −0.0781715
\(250\) 0 0
\(251\) −14.3861 −0.908041 −0.454021 0.890991i \(-0.650011\pi\)
−0.454021 + 0.890991i \(0.650011\pi\)
\(252\) 0 0
\(253\) 0.745898 0.0468942
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.28586 −0.0802095 −0.0401047 0.999195i \(-0.512769\pi\)
−0.0401047 + 0.999195i \(0.512769\pi\)
\(258\) 0 0
\(259\) −21.4506 −1.33288
\(260\) 0 0
\(261\) −1.24186 −0.0768694
\(262\) 0 0
\(263\) 28.2294 1.74070 0.870348 0.492437i \(-0.163894\pi\)
0.870348 + 0.492437i \(0.163894\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.27468 −0.322805
\(268\) 0 0
\(269\) −19.4395 −1.18525 −0.592623 0.805480i \(-0.701907\pi\)
−0.592623 + 0.805480i \(0.701907\pi\)
\(270\) 0 0
\(271\) 6.42723 0.390426 0.195213 0.980761i \(-0.437460\pi\)
0.195213 + 0.980761i \(0.437460\pi\)
\(272\) 0 0
\(273\) 16.6772 1.00935
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.08514 −0.245452 −0.122726 0.992441i \(-0.539164\pi\)
−0.122726 + 0.992441i \(0.539164\pi\)
\(278\) 0 0
\(279\) 1.20594 0.0721978
\(280\) 0 0
\(281\) −6.38741 −0.381041 −0.190520 0.981683i \(-0.561018\pi\)
−0.190520 + 0.981683i \(0.561018\pi\)
\(282\) 0 0
\(283\) −21.4835 −1.27706 −0.638530 0.769597i \(-0.720457\pi\)
−0.638530 + 0.769597i \(0.720457\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −34.3103 −2.02527
\(288\) 0 0
\(289\) 20.3145 1.19497
\(290\) 0 0
\(291\) 13.7899 0.808378
\(292\) 0 0
\(293\) 16.1208 0.941787 0.470894 0.882190i \(-0.343932\pi\)
0.470894 + 0.882190i \(0.343932\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.25410 −0.188822
\(298\) 0 0
\(299\) 1.74590 0.100968
\(300\) 0 0
\(301\) 24.7581 1.42704
\(302\) 0 0
\(303\) −28.9836 −1.66506
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.96302 −0.397400 −0.198700 0.980060i \(-0.563672\pi\)
−0.198700 + 0.980060i \(0.563672\pi\)
\(308\) 0 0
\(309\) 21.0318 1.19645
\(310\) 0 0
\(311\) 4.14031 0.234776 0.117388 0.993086i \(-0.462548\pi\)
0.117388 + 0.993086i \(0.462548\pi\)
\(312\) 0 0
\(313\) −6.26528 −0.354135 −0.177067 0.984199i \(-0.556661\pi\)
−0.177067 + 0.984199i \(0.556661\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.53996 −0.535817 −0.267909 0.963444i \(-0.586333\pi\)
−0.267909 + 0.963444i \(0.586333\pi\)
\(318\) 0 0
\(319\) 1.24186 0.0695310
\(320\) 0 0
\(321\) 14.4999 0.809304
\(322\) 0 0
\(323\) −33.2528 −1.85023
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 35.4782 1.96195
\(328\) 0 0
\(329\) 13.2335 0.729588
\(330\) 0 0
\(331\) 14.3023 0.786123 0.393062 0.919512i \(-0.371416\pi\)
0.393062 + 0.919512i \(0.371416\pi\)
\(332\) 0 0
\(333\) −3.24186 −0.177653
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.34731 0.182340 0.0911699 0.995835i \(-0.470939\pi\)
0.0911699 + 0.995835i \(0.470939\pi\)
\(338\) 0 0
\(339\) 0.733661 0.0398470
\(340\) 0 0
\(341\) −1.20594 −0.0653054
\(342\) 0 0
\(343\) 51.1236 2.76042
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −23.9383 −1.28507 −0.642537 0.766255i \(-0.722118\pi\)
−0.642537 + 0.766255i \(0.722118\pi\)
\(348\) 0 0
\(349\) −11.9149 −0.637788 −0.318894 0.947790i \(-0.603311\pi\)
−0.318894 + 0.947790i \(0.603311\pi\)
\(350\) 0 0
\(351\) −7.61676 −0.406553
\(352\) 0 0
\(353\) 25.2447 1.34364 0.671820 0.740714i \(-0.265513\pi\)
0.671820 + 0.740714i \(0.265513\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −58.3502 −3.08822
\(358\) 0 0
\(359\) −20.4671 −1.08021 −0.540105 0.841598i \(-0.681615\pi\)
−0.540105 + 0.841598i \(0.681615\pi\)
\(360\) 0 0
\(361\) 10.6332 0.559641
\(362\) 0 0
\(363\) 20.2130 1.06090
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −36.4999 −1.90528 −0.952639 0.304104i \(-0.901643\pi\)
−0.952639 + 0.304104i \(0.901643\pi\)
\(368\) 0 0
\(369\) −5.18537 −0.269939
\(370\) 0 0
\(371\) −67.8214 −3.52111
\(372\) 0 0
\(373\) 5.57978 0.288910 0.144455 0.989511i \(-0.453857\pi\)
0.144455 + 0.989511i \(0.453857\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.90679 0.149707
\(378\) 0 0
\(379\) −36.6168 −1.88088 −0.940438 0.339964i \(-0.889585\pi\)
−0.940438 + 0.339964i \(0.889585\pi\)
\(380\) 0 0
\(381\) −5.12603 −0.262614
\(382\) 0 0
\(383\) −28.7581 −1.46947 −0.734736 0.678353i \(-0.762694\pi\)
−0.734736 + 0.678353i \(0.762694\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.74173 0.190203
\(388\) 0 0
\(389\) 26.3913 1.33809 0.669046 0.743221i \(-0.266703\pi\)
0.669046 + 0.743221i \(0.266703\pi\)
\(390\) 0 0
\(391\) −6.10856 −0.308923
\(392\) 0 0
\(393\) −10.7141 −0.540457
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.08931 −0.0546710 −0.0273355 0.999626i \(-0.508702\pi\)
−0.0273355 + 0.999626i \(0.508702\pi\)
\(398\) 0 0
\(399\) 51.9987 2.60319
\(400\) 0 0
\(401\) −22.5962 −1.12840 −0.564200 0.825638i \(-0.690815\pi\)
−0.564200 + 0.825638i \(0.690815\pi\)
\(402\) 0 0
\(403\) −2.82270 −0.140609
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.24186 0.160693
\(408\) 0 0
\(409\) −16.0510 −0.793671 −0.396835 0.917890i \(-0.629892\pi\)
−0.396835 + 0.917890i \(0.629892\pi\)
\(410\) 0 0
\(411\) 11.9396 0.588937
\(412\) 0 0
\(413\) −60.2968 −2.96701
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −40.1976 −1.96849
\(418\) 0 0
\(419\) 24.4342 1.19369 0.596845 0.802356i \(-0.296421\pi\)
0.596845 + 0.802356i \(0.296421\pi\)
\(420\) 0 0
\(421\) −11.3473 −0.553034 −0.276517 0.961009i \(-0.589180\pi\)
−0.276517 + 0.961009i \(0.589180\pi\)
\(422\) 0 0
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −68.9945 −3.33888
\(428\) 0 0
\(429\) −2.52044 −0.121688
\(430\) 0 0
\(431\) −7.83805 −0.377546 −0.188773 0.982021i \(-0.560451\pi\)
−0.188773 + 0.982021i \(0.560451\pi\)
\(432\) 0 0
\(433\) 37.9023 1.82147 0.910735 0.412990i \(-0.135516\pi\)
0.910735 + 0.412990i \(0.135516\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.44364 0.260404
\(438\) 0 0
\(439\) 17.7704 0.848134 0.424067 0.905631i \(-0.360602\pi\)
0.424067 + 0.905631i \(0.360602\pi\)
\(440\) 0 0
\(441\) 12.9477 0.616556
\(442\) 0 0
\(443\) 16.9466 0.805158 0.402579 0.915385i \(-0.368114\pi\)
0.402579 + 0.915385i \(0.368114\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.06040 0.0974535
\(448\) 0 0
\(449\) −21.0890 −0.995253 −0.497627 0.867391i \(-0.665795\pi\)
−0.497627 + 0.867391i \(0.665795\pi\)
\(450\) 0 0
\(451\) 5.18537 0.244169
\(452\) 0 0
\(453\) −24.5040 −1.15130
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13.1372 −0.614532 −0.307266 0.951624i \(-0.599414\pi\)
−0.307266 + 0.951624i \(0.599414\pi\)
\(458\) 0 0
\(459\) 26.6496 1.24390
\(460\) 0 0
\(461\) −14.5439 −0.677375 −0.338687 0.940899i \(-0.609983\pi\)
−0.338687 + 0.940899i \(0.609983\pi\)
\(462\) 0 0
\(463\) 19.4283 0.902909 0.451455 0.892294i \(-0.350905\pi\)
0.451455 + 0.892294i \(0.350905\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.61259 −0.167171 −0.0835855 0.996501i \(-0.526637\pi\)
−0.0835855 + 0.996501i \(0.526637\pi\)
\(468\) 0 0
\(469\) 64.6758 2.98645
\(470\) 0 0
\(471\) 28.8133 1.32765
\(472\) 0 0
\(473\) −3.74173 −0.172045
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.2499 −0.469312
\(478\) 0 0
\(479\) −5.13720 −0.234725 −0.117362 0.993089i \(-0.537444\pi\)
−0.117362 + 0.993089i \(0.537444\pi\)
\(480\) 0 0
\(481\) 7.58812 0.345988
\(482\) 0 0
\(483\) 9.55220 0.434640
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −24.7693 −1.12240 −0.561202 0.827679i \(-0.689661\pi\)
−0.561202 + 0.827679i \(0.689661\pi\)
\(488\) 0 0
\(489\) −13.0245 −0.588987
\(490\) 0 0
\(491\) −3.54413 −0.159944 −0.0799721 0.996797i \(-0.525483\pi\)
−0.0799721 + 0.996797i \(0.525483\pi\)
\(492\) 0 0
\(493\) −10.1703 −0.458047
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 47.7610 2.14237
\(498\) 0 0
\(499\) −31.5358 −1.41174 −0.705868 0.708344i \(-0.749443\pi\)
−0.705868 + 0.708344i \(0.749443\pi\)
\(500\) 0 0
\(501\) 5.77454 0.257988
\(502\) 0 0
\(503\) −32.2346 −1.43727 −0.718635 0.695388i \(-0.755233\pi\)
−0.718635 + 0.695388i \(0.755233\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.2611 0.855416
\(508\) 0 0
\(509\) 0.302263 0.0133976 0.00669878 0.999978i \(-0.497868\pi\)
0.00669878 + 0.999978i \(0.497868\pi\)
\(510\) 0 0
\(511\) −28.1208 −1.24399
\(512\) 0 0
\(513\) −23.7487 −1.04853
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) −30.6772 −1.34658
\(520\) 0 0
\(521\) −30.3463 −1.32949 −0.664747 0.747069i \(-0.731461\pi\)
−0.664747 + 0.747069i \(0.731461\pi\)
\(522\) 0 0
\(523\) −42.5878 −1.86224 −0.931118 0.364717i \(-0.881166\pi\)
−0.931118 + 0.364717i \(0.881166\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.87609 0.430209
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.11273 −0.395459
\(532\) 0 0
\(533\) 12.1372 0.525721
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.85969 0.252864
\(538\) 0 0
\(539\) −12.9477 −0.557696
\(540\) 0 0
\(541\) 38.9313 1.67379 0.836893 0.547367i \(-0.184370\pi\)
0.836893 + 0.547367i \(0.184370\pi\)
\(542\) 0 0
\(543\) −3.52772 −0.151389
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 16.6004 0.709780 0.354890 0.934908i \(-0.384518\pi\)
0.354890 + 0.934908i \(0.384518\pi\)
\(548\) 0 0
\(549\) −10.4272 −0.445023
\(550\) 0 0
\(551\) 9.06324 0.386107
\(552\) 0 0
\(553\) −51.2252 −2.17832
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.4506 −0.739408 −0.369704 0.929150i \(-0.620541\pi\)
−0.369704 + 0.929150i \(0.620541\pi\)
\(558\) 0 0
\(559\) −8.75814 −0.370430
\(560\) 0 0
\(561\) 8.81853 0.372319
\(562\) 0 0
\(563\) 8.25827 0.348045 0.174022 0.984742i \(-0.444324\pi\)
0.174022 + 0.984742i \(0.444324\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −52.7170 −2.21391
\(568\) 0 0
\(569\) −41.4178 −1.73633 −0.868163 0.496279i \(-0.834699\pi\)
−0.868163 + 0.496279i \(0.834699\pi\)
\(570\) 0 0
\(571\) −7.19059 −0.300917 −0.150458 0.988616i \(-0.548075\pi\)
−0.150458 + 0.988616i \(0.548075\pi\)
\(572\) 0 0
\(573\) 37.9588 1.58575
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −39.3103 −1.63651 −0.818255 0.574855i \(-0.805058\pi\)
−0.818255 + 0.574855i \(0.805058\pi\)
\(578\) 0 0
\(579\) −0.648517 −0.0269515
\(580\) 0 0
\(581\) 3.14554 0.130499
\(582\) 0 0
\(583\) 10.2499 0.424509
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.7899 1.18829 0.594143 0.804359i \(-0.297491\pi\)
0.594143 + 0.804359i \(0.297491\pi\)
\(588\) 0 0
\(589\) −8.80107 −0.362642
\(590\) 0 0
\(591\) −39.7376 −1.63458
\(592\) 0 0
\(593\) 40.8873 1.67904 0.839519 0.543330i \(-0.182837\pi\)
0.839519 + 0.543330i \(0.182837\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 32.3707 1.32485
\(598\) 0 0
\(599\) −19.1801 −0.783679 −0.391840 0.920034i \(-0.628161\pi\)
−0.391840 + 0.920034i \(0.628161\pi\)
\(600\) 0 0
\(601\) 3.36683 0.137336 0.0686679 0.997640i \(-0.478125\pi\)
0.0686679 + 0.997640i \(0.478125\pi\)
\(602\) 0 0
\(603\) 9.77454 0.398050
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13.1924 0.535462 0.267731 0.963494i \(-0.413726\pi\)
0.267731 + 0.963494i \(0.413726\pi\)
\(608\) 0 0
\(609\) 15.9037 0.644450
\(610\) 0 0
\(611\) −4.68133 −0.189386
\(612\) 0 0
\(613\) −32.6207 −1.31754 −0.658768 0.752346i \(-0.728922\pi\)
−0.658768 + 0.752346i \(0.728922\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −41.7212 −1.67963 −0.839815 0.542872i \(-0.817337\pi\)
−0.839815 + 0.542872i \(0.817337\pi\)
\(618\) 0 0
\(619\) 0.339245 0.0136354 0.00681771 0.999977i \(-0.497830\pi\)
0.00681771 + 0.999977i \(0.497830\pi\)
\(620\) 0 0
\(621\) −4.36266 −0.175068
\(622\) 0 0
\(623\) 13.4506 0.538889
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −7.85863 −0.313843
\(628\) 0 0
\(629\) −26.5494 −1.05859
\(630\) 0 0
\(631\) −3.71725 −0.147982 −0.0739908 0.997259i \(-0.523574\pi\)
−0.0739908 + 0.997259i \(0.523574\pi\)
\(632\) 0 0
\(633\) −25.9260 −1.03047
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −30.3062 −1.20077
\(638\) 0 0
\(639\) 7.21818 0.285547
\(640\) 0 0
\(641\) −22.0328 −0.870244 −0.435122 0.900372i \(-0.643295\pi\)
−0.435122 + 0.900372i \(0.643295\pi\)
\(642\) 0 0
\(643\) −24.6842 −0.973449 −0.486724 0.873556i \(-0.661808\pi\)
−0.486724 + 0.873556i \(0.661808\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.1731 1.26486 0.632428 0.774619i \(-0.282058\pi\)
0.632428 + 0.774619i \(0.282058\pi\)
\(648\) 0 0
\(649\) 9.11273 0.357706
\(650\) 0 0
\(651\) −15.4436 −0.605284
\(652\) 0 0
\(653\) 26.9466 1.05450 0.527251 0.849709i \(-0.323223\pi\)
0.527251 + 0.849709i \(0.323223\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.24993 −0.165806
\(658\) 0 0
\(659\) 19.7417 0.769029 0.384514 0.923119i \(-0.374369\pi\)
0.384514 + 0.923119i \(0.374369\pi\)
\(660\) 0 0
\(661\) −1.28692 −0.0500552 −0.0250276 0.999687i \(-0.507967\pi\)
−0.0250276 + 0.999687i \(0.507967\pi\)
\(662\) 0 0
\(663\) 20.6412 0.801639
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.66492 0.0644661
\(668\) 0 0
\(669\) −2.46705 −0.0953817
\(670\) 0 0
\(671\) 10.4272 0.402539
\(672\) 0 0
\(673\) −12.4970 −0.481725 −0.240862 0.970559i \(-0.577430\pi\)
−0.240862 + 0.970559i \(0.577430\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.69251 0.103482 0.0517408 0.998661i \(-0.483523\pi\)
0.0517408 + 0.998661i \(0.483523\pi\)
\(678\) 0 0
\(679\) −35.1648 −1.34950
\(680\) 0 0
\(681\) −6.92842 −0.265498
\(682\) 0 0
\(683\) −2.72115 −0.104122 −0.0520610 0.998644i \(-0.516579\pi\)
−0.0520610 + 0.998644i \(0.516579\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −20.1515 −0.768827
\(688\) 0 0
\(689\) 23.9917 0.914010
\(690\) 0 0
\(691\) −38.3051 −1.45719 −0.728597 0.684942i \(-0.759827\pi\)
−0.728597 + 0.684942i \(0.759827\pi\)
\(692\) 0 0
\(693\) −2.74590 −0.104308
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −42.4657 −1.60850
\(698\) 0 0
\(699\) 5.92319 0.224036
\(700\) 0 0
\(701\) 39.8472 1.50501 0.752504 0.658588i \(-0.228846\pi\)
0.752504 + 0.658588i \(0.228846\pi\)
\(702\) 0 0
\(703\) 23.6594 0.892332
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 73.9094 2.77965
\(708\) 0 0
\(709\) −44.8995 −1.68624 −0.843118 0.537728i \(-0.819283\pi\)
−0.843118 + 0.537728i \(0.819283\pi\)
\(710\) 0 0
\(711\) −7.74173 −0.290338
\(712\) 0 0
\(713\) −1.61676 −0.0605482
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −37.1400 −1.38702
\(718\) 0 0
\(719\) 31.5400 1.17624 0.588121 0.808773i \(-0.299868\pi\)
0.588121 + 0.808773i \(0.299868\pi\)
\(720\) 0 0
\(721\) −53.6318 −1.99735
\(722\) 0 0
\(723\) −0.670152 −0.0249232
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −48.9219 −1.81441 −0.907206 0.420687i \(-0.861789\pi\)
−0.907206 + 0.420687i \(0.861789\pi\)
\(728\) 0 0
\(729\) 17.3637 0.643101
\(730\) 0 0
\(731\) 30.6430 1.13337
\(732\) 0 0
\(733\) 14.1536 0.522776 0.261388 0.965234i \(-0.415820\pi\)
0.261388 + 0.965234i \(0.415820\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.77454 −0.360050
\(738\) 0 0
\(739\) −4.46421 −0.164219 −0.0821093 0.996623i \(-0.526166\pi\)
−0.0821093 + 0.996623i \(0.526166\pi\)
\(740\) 0 0
\(741\) −18.3944 −0.675736
\(742\) 0 0
\(743\) −34.2898 −1.25797 −0.628985 0.777418i \(-0.716529\pi\)
−0.628985 + 0.777418i \(0.716529\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.475390 0.0173936
\(748\) 0 0
\(749\) −36.9753 −1.35105
\(750\) 0 0
\(751\) 8.28275 0.302242 0.151121 0.988515i \(-0.451712\pi\)
0.151121 + 0.988515i \(0.451712\pi\)
\(752\) 0 0
\(753\) 27.8433 1.01467
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.32985 −0.121025 −0.0605127 0.998167i \(-0.519274\pi\)
−0.0605127 + 0.998167i \(0.519274\pi\)
\(758\) 0 0
\(759\) −1.44364 −0.0524007
\(760\) 0 0
\(761\) −18.4077 −0.667279 −0.333640 0.942701i \(-0.608277\pi\)
−0.333640 + 0.942701i \(0.608277\pi\)
\(762\) 0 0
\(763\) −90.4710 −3.27527
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.3298 0.770176
\(768\) 0 0
\(769\) 51.4506 1.85536 0.927679 0.373379i \(-0.121801\pi\)
0.927679 + 0.373379i \(0.121801\pi\)
\(770\) 0 0
\(771\) 2.48869 0.0896279
\(772\) 0 0
\(773\) −17.4506 −0.627656 −0.313828 0.949480i \(-0.601612\pi\)
−0.313828 + 0.949480i \(0.601612\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 41.5163 1.48939
\(778\) 0 0
\(779\) 37.8433 1.35588
\(780\) 0 0
\(781\) −7.21818 −0.258287
\(782\) 0 0
\(783\) −7.26350 −0.259576
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −21.2335 −0.756893 −0.378447 0.925623i \(-0.623542\pi\)
−0.378447 + 0.925623i \(0.623542\pi\)
\(788\) 0 0
\(789\) −54.6360 −1.94509
\(790\) 0 0
\(791\) −1.87086 −0.0665203
\(792\) 0 0
\(793\) 24.4067 0.866706
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.3215 0.896934 0.448467 0.893799i \(-0.351970\pi\)
0.448467 + 0.893799i \(0.351970\pi\)
\(798\) 0 0
\(799\) 16.3791 0.579450
\(800\) 0 0
\(801\) 2.03281 0.0718259
\(802\) 0 0
\(803\) 4.24993 0.149977
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 37.6238 1.32442
\(808\) 0 0
\(809\) 6.14137 0.215919 0.107960 0.994155i \(-0.465568\pi\)
0.107960 + 0.994155i \(0.465568\pi\)
\(810\) 0 0
\(811\) −15.5051 −0.544457 −0.272229 0.962233i \(-0.587761\pi\)
−0.272229 + 0.962233i \(0.587761\pi\)
\(812\) 0 0
\(813\) −12.4395 −0.436271
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −27.3075 −0.955368
\(818\) 0 0
\(819\) −6.42723 −0.224586
\(820\) 0 0
\(821\) −21.4835 −0.749778 −0.374889 0.927070i \(-0.622319\pi\)
−0.374889 + 0.927070i \(0.622319\pi\)
\(822\) 0 0
\(823\) 40.1565 1.39977 0.699883 0.714258i \(-0.253235\pi\)
0.699883 + 0.714258i \(0.253235\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.0573 1.32338 0.661691 0.749777i \(-0.269839\pi\)
0.661691 + 0.749777i \(0.269839\pi\)
\(828\) 0 0
\(829\) −51.2580 −1.78026 −0.890132 0.455703i \(-0.849388\pi\)
−0.890132 + 0.455703i \(0.849388\pi\)
\(830\) 0 0
\(831\) 7.90652 0.274274
\(832\) 0 0
\(833\) 106.035 3.67391
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.05339 0.243801
\(838\) 0 0
\(839\) −45.4200 −1.56807 −0.784035 0.620716i \(-0.786842\pi\)
−0.784035 + 0.620716i \(0.786842\pi\)
\(840\) 0 0
\(841\) −26.2280 −0.904415
\(842\) 0 0
\(843\) 12.3624 0.425783
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −51.5439 −1.77107
\(848\) 0 0
\(849\) 41.5798 1.42701
\(850\) 0 0
\(851\) 4.34625 0.148988
\(852\) 0 0
\(853\) −40.0867 −1.37254 −0.686270 0.727346i \(-0.740753\pi\)
−0.686270 + 0.727346i \(0.740753\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.63423 0.124143 0.0620715 0.998072i \(-0.480229\pi\)
0.0620715 + 0.998072i \(0.480229\pi\)
\(858\) 0 0
\(859\) 37.7529 1.28811 0.644056 0.764978i \(-0.277250\pi\)
0.644056 + 0.764978i \(0.277250\pi\)
\(860\) 0 0
\(861\) 66.4053 2.26309
\(862\) 0 0
\(863\) −3.94555 −0.134308 −0.0671541 0.997743i \(-0.521392\pi\)
−0.0671541 + 0.997743i \(0.521392\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −39.3173 −1.33529
\(868\) 0 0
\(869\) 7.74173 0.262620
\(870\) 0 0
\(871\) −22.8789 −0.775223
\(872\) 0 0
\(873\) −5.31450 −0.179869
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.64958 0.292075 0.146038 0.989279i \(-0.453348\pi\)
0.146038 + 0.989279i \(0.453348\pi\)
\(878\) 0 0
\(879\) −31.2007 −1.05237
\(880\) 0 0
\(881\) −56.1760 −1.89262 −0.946308 0.323266i \(-0.895219\pi\)
−0.946308 + 0.323266i \(0.895219\pi\)
\(882\) 0 0
\(883\) −12.1054 −0.407381 −0.203690 0.979035i \(-0.565294\pi\)
−0.203690 + 0.979035i \(0.565294\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.7058 −0.829540 −0.414770 0.909926i \(-0.636138\pi\)
−0.414770 + 0.909926i \(0.636138\pi\)
\(888\) 0 0
\(889\) 13.0716 0.438407
\(890\) 0 0
\(891\) 7.96719 0.266911
\(892\) 0 0
\(893\) −14.5962 −0.488443
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −3.37907 −0.112824
\(898\) 0 0
\(899\) −2.69179 −0.0897761
\(900\) 0 0
\(901\) −83.9422 −2.79652
\(902\) 0 0
\(903\) −47.9177 −1.59460
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.7170 0.754305 0.377153 0.926151i \(-0.376903\pi\)
0.377153 + 0.926151i \(0.376903\pi\)
\(908\) 0 0
\(909\) 11.1700 0.370486
\(910\) 0 0
\(911\) 12.3156 0.408033 0.204016 0.978967i \(-0.434600\pi\)
0.204016 + 0.978967i \(0.434600\pi\)
\(912\) 0 0
\(913\) −0.475390 −0.0157331
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.3215 0.902236
\(918\) 0 0
\(919\) −46.2088 −1.52429 −0.762144 0.647408i \(-0.775853\pi\)
−0.762144 + 0.647408i \(0.775853\pi\)
\(920\) 0 0
\(921\) 13.4764 0.444064
\(922\) 0 0
\(923\) −16.8953 −0.556117
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.10545 −0.266218
\(928\) 0 0
\(929\) 9.12391 0.299346 0.149673 0.988736i \(-0.452178\pi\)
0.149673 + 0.988736i \(0.452178\pi\)
\(930\) 0 0
\(931\) −94.4933 −3.09689
\(932\) 0 0
\(933\) −8.01330 −0.262344
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.9190 1.10809 0.554043 0.832488i \(-0.313084\pi\)
0.554043 + 0.832488i \(0.313084\pi\)
\(938\) 0 0
\(939\) 12.1260 0.395718
\(940\) 0 0
\(941\) −37.2926 −1.21570 −0.607852 0.794050i \(-0.707969\pi\)
−0.607852 + 0.794050i \(0.707969\pi\)
\(942\) 0 0
\(943\) 6.95184 0.226383
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.7047 1.09526 0.547629 0.836722i \(-0.315531\pi\)
0.547629 + 0.836722i \(0.315531\pi\)
\(948\) 0 0
\(949\) 9.94767 0.322915
\(950\) 0 0
\(951\) 18.4639 0.598734
\(952\) 0 0
\(953\) 48.4465 1.56934 0.784668 0.619917i \(-0.212834\pi\)
0.784668 + 0.619917i \(0.212834\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2.40354 −0.0776955
\(958\) 0 0
\(959\) −30.4465 −0.983168
\(960\) 0 0
\(961\) −28.3861 −0.915680
\(962\) 0 0
\(963\) −5.58812 −0.180075
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −28.2528 −0.908548 −0.454274 0.890862i \(-0.650101\pi\)
−0.454274 + 0.890862i \(0.650101\pi\)
\(968\) 0 0
\(969\) 64.3585 2.06749
\(970\) 0 0
\(971\) −1.65553 −0.0531284 −0.0265642 0.999647i \(-0.508457\pi\)
−0.0265642 + 0.999647i \(0.508457\pi\)
\(972\) 0 0
\(973\) 102.506 3.28618
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.1630 0.357136 0.178568 0.983928i \(-0.442854\pi\)
0.178568 + 0.983928i \(0.442854\pi\)
\(978\) 0 0
\(979\) −2.03281 −0.0649690
\(980\) 0 0
\(981\) −13.6730 −0.436545
\(982\) 0 0
\(983\) 15.0615 0.480386 0.240193 0.970725i \(-0.422789\pi\)
0.240193 + 0.970725i \(0.422789\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −25.6126 −0.815258
\(988\) 0 0
\(989\) −5.01641 −0.159512
\(990\) 0 0
\(991\) 43.8971 1.39444 0.697219 0.716858i \(-0.254421\pi\)
0.697219 + 0.716858i \(0.254421\pi\)
\(992\) 0 0
\(993\) −27.6811 −0.878432
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −38.5655 −1.22138 −0.610691 0.791869i \(-0.709108\pi\)
−0.610691 + 0.791869i \(0.709108\pi\)
\(998\) 0 0
\(999\) −18.9612 −0.599907
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 9200.2.a.ci.1.1 3
4.3 odd 2 4600.2.a.v.1.3 3
5.4 even 2 1840.2.a.q.1.3 3
20.3 even 4 4600.2.e.q.4049.5 6
20.7 even 4 4600.2.e.q.4049.2 6
20.19 odd 2 920.2.a.i.1.1 3
40.19 odd 2 7360.2.a.bw.1.3 3
40.29 even 2 7360.2.a.cf.1.1 3
60.59 even 2 8280.2.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.1 3 20.19 odd 2
1840.2.a.q.1.3 3 5.4 even 2
4600.2.a.v.1.3 3 4.3 odd 2
4600.2.e.q.4049.2 6 20.7 even 4
4600.2.e.q.4049.5 6 20.3 even 4
7360.2.a.bw.1.3 3 40.19 odd 2
7360.2.a.cf.1.1 3 40.29 even 2
8280.2.a.bl.1.3 3 60.59 even 2
9200.2.a.ci.1.1 3 1.1 even 1 trivial