Properties

Label 1134.2.a.c.1.1
Level $1134$
Weight $2$
Character 1134.1
Self dual yes
Analytic conductor $9.055$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1134,2,Mod(1,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1134.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.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: \(9.05503558921\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1134.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} -7.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} -4.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} +5.00000 q^{34} -2.00000 q^{35} +2.00000 q^{37} +7.00000 q^{38} -2.00000 q^{40} +3.00000 q^{41} -1.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} +12.0000 q^{53} +2.00000 q^{55} +1.00000 q^{56} +4.00000 q^{58} -7.00000 q^{59} -12.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +13.0000 q^{67} -5.00000 q^{68} +2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{73} -2.00000 q^{74} -7.00000 q^{76} -1.00000 q^{77} -6.00000 q^{79} +2.00000 q^{80} -3.00000 q^{82} +16.0000 q^{83} -10.0000 q^{85} +1.00000 q^{86} -1.00000 q^{88} -6.00000 q^{89} +6.00000 q^{91} +4.00000 q^{92} -14.0000 q^{95} -5.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.00000 0.857493
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 7.00000 1.13555
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) −5.00000 −0.606339
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) −14.0000 −1.43637
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 7.00000 0.644402
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 12.0000 1.08643
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) −13.0000 −1.12303
\(135\) 0 0
\(136\) 5.00000 0.428746
\(137\) −19.0000 −1.62328 −0.811640 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −6.00000 −0.501745
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) −1.00000 −0.0827606
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −24.0000 −1.96616 −0.983078 0.183186i \(-0.941359\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 7.00000 0.567775
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −16.0000 −1.24184
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 10.0000 0.766965
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) 0 0
\(190\) 14.0000 1.01567
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 17.0000 1.22369 0.611843 0.790979i \(-0.290428\pi\)
0.611843 + 0.790979i \(0.290428\pi\)
\(194\) 5.00000 0.358979
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 4.00000 0.281439
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −7.00000 −0.484200
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) −29.0000 −1.89985 −0.949927 0.312473i \(-0.898843\pi\)
−0.949927 + 0.312473i \(0.898843\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.00000 −0.455661
\(237\) 0 0
\(238\) −5.00000 −0.324102
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 42.0000 2.67240
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 4.00000 0.247121
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) −7.00000 −0.429198
\(267\) 0 0
\(268\) 13.0000 0.794101
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 19.0000 1.14783
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 5.00000 0.299880
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) 1.00000 0.0585206
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) 24.0000 1.39028
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 12.0000 0.681554
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 35.0000 1.94745
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) 26.0000 1.42053
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −10.0000 −0.542326
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 2.00000 0.105556 0.0527780 0.998606i \(-0.483192\pi\)
0.0527780 + 0.998606i \(0.483192\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −4.00000 −0.207950
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) −14.0000 −0.718185
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −17.0000 −0.865277
\(387\) 0 0
\(388\) −5.00000 −0.253837
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 9.00000 0.449439 0.224719 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) 7.00000 0.344447
\(414\) 0 0
\(415\) 32.0000 1.57082
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 7.00000 0.342381
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −28.0000 −1.33942
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) 7.00000 0.332580 0.166290 0.986077i \(-0.446821\pi\)
0.166290 + 0.986077i \(0.446821\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 17.0000 0.802280 0.401140 0.916017i \(-0.368614\pi\)
0.401140 + 0.916017i \(0.368614\pi\)
\(450\) 0 0
\(451\) 3.00000 0.141264
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) −3.00000 −0.140797
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 29.0000 1.34340
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 0 0
\(472\) 7.00000 0.322201
\(473\) −1.00000 −0.0459800
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 5.00000 0.229175
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −23.0000 −1.04762
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 12.0000 0.543214
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) −42.0000 −1.88967
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −29.0000 −1.29822 −0.649109 0.760695i \(-0.724858\pi\)
−0.649109 + 0.760695i \(0.724858\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −3.00000 −0.133897
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −1.00000 −0.0442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 15.0000 0.661622
\(515\) 28.0000 1.23383
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −24.0000 −1.04249
\(531\) 0 0
\(532\) 7.00000 0.303488
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) −13.0000 −0.561514
\(537\) 0 0
\(538\) −20.0000 −0.862261
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) 6.00000 0.257722
\(543\) 0 0
\(544\) 5.00000 0.214373
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −21.0000 −0.897895 −0.448948 0.893558i \(-0.648201\pi\)
−0.448948 + 0.893558i \(0.648201\pi\)
\(548\) −19.0000 −0.811640
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 28.0000 1.19284
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −5.00000 −0.212047
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 31.0000 1.30649 0.653247 0.757145i \(-0.273406\pi\)
0.653247 + 0.757145i \(0.273406\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −15.0000 −0.628833 −0.314416 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) −33.0000 −1.38101 −0.690504 0.723329i \(-0.742611\pi\)
−0.690504 + 0.723329i \(0.742611\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −35.0000 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −8.00000 −0.332182
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −1.00000 −0.0413803
\(585\) 0 0
\(586\) 0 0
\(587\) −47.0000 −1.93990 −0.969949 0.243309i \(-0.921767\pi\)
−0.969949 + 0.243309i \(0.921767\pi\)
\(588\) 0 0
\(589\) 42.0000 1.73058
\(590\) 14.0000 0.576371
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 10.0000 0.409960
\(596\) −24.0000 −0.983078
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −20.0000 −0.813116
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) 0 0
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −17.0000 −0.684394 −0.342197 0.939628i \(-0.611171\pi\)
−0.342197 + 0.939628i \(0.611171\pi\)
\(618\) 0 0
\(619\) 37.0000 1.48716 0.743578 0.668649i \(-0.233127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(620\) −12.0000 −0.481932
\(621\) 0 0
\(622\) −2.00000 −0.0801927
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 17.0000 0.679457
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 6.00000 0.238667
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) −24.0000 −0.952411
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 1.00000 0.0394976 0.0197488 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) 0 0
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −35.0000 −1.37706
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) −7.00000 −0.274774
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 14.0000 0.542897
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) −26.0000 −1.00447
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 9.00000 0.346667
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 5.00000 0.191882
\(680\) 10.0000 0.383482
\(681\) 0 0
\(682\) 6.00000 0.229752
\(683\) 39.0000 1.49229 0.746147 0.665782i \(-0.231902\pi\)
0.746147 + 0.665782i \(0.231902\pi\)
\(684\) 0 0
\(685\) −38.0000 −1.45191
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) −72.0000 −2.74298
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) −2.00000 −0.0746393
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) −30.0000 −1.11648
\(723\) 0 0
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) 5.00000 0.184932
\(732\) 0 0
\(733\) −18.0000 −0.664845 −0.332423 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 13.0000 0.478861
\(738\) 0 0
\(739\) −33.0000 −1.21392 −0.606962 0.794731i \(-0.707612\pi\)
−0.606962 + 0.794731i \(0.707612\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) −48.0000 −1.75858
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) −5.00000 −0.182818
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −48.0000 −1.74459 −0.872295 0.488980i \(-0.837369\pi\)
−0.872295 + 0.488980i \(0.837369\pi\)
\(758\) 17.0000 0.617468
\(759\) 0 0
\(760\) 14.0000 0.507833
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 42.0000 1.51653
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) 17.0000 0.611843
\(773\) −52.0000 −1.87031 −0.935155 0.354239i \(-0.884740\pi\)
−0.935155 + 0.354239i \(0.884740\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 5.00000 0.179490
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) −21.0000 −0.752403
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 20.0000 0.715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 72.0000 2.55679
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −9.00000 −0.317801
\(803\) 1.00000 0.0352892
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) −36.0000 −1.26805
\(807\) 0 0
\(808\) 4.00000 0.140720
\(809\) −43.0000 −1.51180 −0.755900 0.654687i \(-0.772800\pi\)
−0.755900 + 0.654687i \(0.772800\pi\)
\(810\) 0 0
\(811\) 31.0000 1.08856 0.544279 0.838905i \(-0.316803\pi\)
0.544279 + 0.838905i \(0.316803\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 7.00000 0.244899
\(818\) −11.0000 −0.384606
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −7.00000 −0.243561
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) −32.0000 −1.11074
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) 40.0000 1.38426
\(836\) −7.00000 −0.242100
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 12.0000 0.413547
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 46.0000 1.58245
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −5.00000 −0.171499
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) −3.00000 −0.102538
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 29.0000 0.989467 0.494734 0.869045i \(-0.335266\pi\)
0.494734 + 0.869045i \(0.335266\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 0 0
\(862\) 0 0
\(863\) 38.0000 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 25.0000 0.849535
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −78.0000 −2.64293
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 28.0000 0.947114
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 16.0000 0.540282 0.270141 0.962821i \(-0.412930\pi\)
0.270141 + 0.962821i \(0.412930\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 53.0000 1.78359 0.891796 0.452438i \(-0.149446\pi\)
0.891796 + 0.452438i \(0.149446\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) −7.00000 −0.235170
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) 48.0000 1.60446
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −17.0000 −0.567297
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) −3.00000 −0.0998891
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) 0 0
\(907\) −27.0000 −0.896520 −0.448260 0.893903i \(-0.647956\pi\)
−0.448260 + 0.893903i \(0.647956\pi\)
\(908\) 3.00000 0.0995585
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) 4.00000 0.131306
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) −29.0000 −0.949927
\(933\) 0 0
\(934\) 13.0000 0.425373
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 13.0000 0.424465
\(939\) 0 0
\(940\) 0 0
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) 1.00000 0.0325128
\(947\) −37.0000 −1.20234 −0.601169 0.799122i \(-0.705298\pi\)
−0.601169 + 0.799122i \(0.705298\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) −7.00000 −0.227110
\(951\) 0 0
\(952\) −5.00000 −0.162051
\(953\) −35.0000 −1.13376 −0.566881 0.823800i \(-0.691850\pi\)
−0.566881 + 0.823800i \(0.691850\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 20.0000 0.646171
\(959\) 19.0000 0.613542
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 12.0000 0.386896
\(963\) 0 0
\(964\) 23.0000 0.740780
\(965\) 34.0000 1.09450
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 10.0000 0.320421
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) −15.0000 −0.479893 −0.239946 0.970786i \(-0.577130\pi\)
−0.239946 + 0.970786i \(0.577130\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −33.0000 −1.05307
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) 20.0000 0.637253
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) 42.0000 1.33620
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 28.0000 0.887660
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 29.0000 0.917979
\(999\) 0 0
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 1134.2.a.c.1.1 1
3.2 odd 2 1134.2.a.f.1.1 1
4.3 odd 2 9072.2.a.t.1.1 1
7.6 odd 2 7938.2.a.e.1.1 1
9.2 odd 6 378.2.f.b.253.1 2
9.4 even 3 126.2.f.b.43.1 2
9.5 odd 6 378.2.f.b.127.1 2
9.7 even 3 126.2.f.b.85.1 yes 2
12.11 even 2 9072.2.a.f.1.1 1
21.20 even 2 7938.2.a.bb.1.1 1
36.7 odd 6 1008.2.r.a.337.1 2
36.11 even 6 3024.2.r.c.1009.1 2
36.23 even 6 3024.2.r.c.2017.1 2
36.31 odd 6 1008.2.r.a.673.1 2
63.2 odd 6 2646.2.h.b.361.1 2
63.4 even 3 882.2.h.h.79.1 2
63.5 even 6 2646.2.e.h.2125.1 2
63.11 odd 6 2646.2.e.i.1549.1 2
63.13 odd 6 882.2.f.f.295.1 2
63.16 even 3 882.2.h.h.67.1 2
63.20 even 6 2646.2.f.b.1765.1 2
63.23 odd 6 2646.2.e.i.2125.1 2
63.25 even 3 882.2.e.a.373.1 2
63.31 odd 6 882.2.h.g.79.1 2
63.32 odd 6 2646.2.h.b.667.1 2
63.34 odd 6 882.2.f.f.589.1 2
63.38 even 6 2646.2.e.h.1549.1 2
63.40 odd 6 882.2.e.e.655.1 2
63.41 even 6 2646.2.f.b.883.1 2
63.47 even 6 2646.2.h.c.361.1 2
63.52 odd 6 882.2.e.e.373.1 2
63.58 even 3 882.2.e.a.655.1 2
63.59 even 6 2646.2.h.c.667.1 2
63.61 odd 6 882.2.h.g.67.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.b.43.1 2 9.4 even 3
126.2.f.b.85.1 yes 2 9.7 even 3
378.2.f.b.127.1 2 9.5 odd 6
378.2.f.b.253.1 2 9.2 odd 6
882.2.e.a.373.1 2 63.25 even 3
882.2.e.a.655.1 2 63.58 even 3
882.2.e.e.373.1 2 63.52 odd 6
882.2.e.e.655.1 2 63.40 odd 6
882.2.f.f.295.1 2 63.13 odd 6
882.2.f.f.589.1 2 63.34 odd 6
882.2.h.g.67.1 2 63.61 odd 6
882.2.h.g.79.1 2 63.31 odd 6
882.2.h.h.67.1 2 63.16 even 3
882.2.h.h.79.1 2 63.4 even 3
1008.2.r.a.337.1 2 36.7 odd 6
1008.2.r.a.673.1 2 36.31 odd 6
1134.2.a.c.1.1 1 1.1 even 1 trivial
1134.2.a.f.1.1 1 3.2 odd 2
2646.2.e.h.1549.1 2 63.38 even 6
2646.2.e.h.2125.1 2 63.5 even 6
2646.2.e.i.1549.1 2 63.11 odd 6
2646.2.e.i.2125.1 2 63.23 odd 6
2646.2.f.b.883.1 2 63.41 even 6
2646.2.f.b.1765.1 2 63.20 even 6
2646.2.h.b.361.1 2 63.2 odd 6
2646.2.h.b.667.1 2 63.32 odd 6
2646.2.h.c.361.1 2 63.47 even 6
2646.2.h.c.667.1 2 63.59 even 6
3024.2.r.c.1009.1 2 36.11 even 6
3024.2.r.c.2017.1 2 36.23 even 6
7938.2.a.e.1.1 1 7.6 odd 2
7938.2.a.bb.1.1 1 21.20 even 2
9072.2.a.f.1.1 1 12.11 even 2
9072.2.a.t.1.1 1 4.3 odd 2