Properties

Label 2496.2.c.f.961.1
Level $2496$
Weight $2$
Character 2496.961
Analytic conductor $19.931$
Analytic rank $0$
Dimension $2$
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] = [2496,2,Mod(961,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.c (of order \(2\), degree \(1\), not 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: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2496.961
Dual form 2496.2.c.f.961.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} -2.00000 q^{17} +6.00000i q^{19} +2.00000i q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +10.0000i q^{31} -4.00000 q^{35} -8.00000i q^{37} +(-3.00000 - 2.00000i) q^{39} +10.0000i q^{41} -4.00000 q^{43} -2.00000i q^{45} -12.0000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +6.00000 q^{53} -6.00000i q^{57} -4.00000i q^{59} -2.00000 q^{61} -2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +2.00000i q^{67} -4.00000 q^{69} -4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} +4.00000i q^{83} +4.00000i q^{85} -10.0000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} -10.0000i q^{93} +12.0000 q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 20 q^{29} - 8 q^{35} - 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 8 q^{65} - 8 q^{69} - 2 q^{75} + 2 q^{81} - 20 q^{87} + 8 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000i 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 3.00000 + 2.00000i 0.832050 + 0.554700i
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) −3.00000 2.00000i −0.480384 0.320256i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 12.0000i 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 4.00000i 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 4.00000 6.00000i 0.496139 0.744208i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 4.00000 6.00000i 0.419314 0.628971i
\(92\) 0 0
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 12.0000i 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i −0.981480 0.191565i \(-0.938644\pi\)
0.981480 0.191565i \(-0.0613564\pi\)
\(110\) 0 0
\(111\) 8.00000i 0.759326i
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) 3.00000 + 2.00000i 0.277350 + 0.184900i
\(118\) 0 0
\(119\) 4.00000i 0.366679i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 2.00000i 0.172133i
\(136\) 0 0
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 12.0000i 1.01058i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 20.0000i 1.66091i
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i 0.913475 + 0.406894i \(0.133388\pi\)
−0.913475 + 0.406894i \(0.866612\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 8.00000i 0.630488i
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) 0 0
\(171\) 6.00000i 0.458831i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −16.0000 −1.17634
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) −4.00000 + 6.00000i −0.286446 + 0.429669i
\(196\) 0 0
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 2.00000i 0.141069i
\(202\) 0 0
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) 20.0000 1.39686
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) 0 0
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) −6.00000 4.00000i −0.403604 0.269069i
\(222\) 0 0
\(223\) 14.0000i 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.00000i 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000i 1.03495i −0.855697 0.517477i \(-0.826871\pi\)
0.855697 0.517477i \(-0.173129\pi\)
\(240\) 0 0
\(241\) 20.0000i 1.28831i 0.764894 + 0.644157i \(0.222792\pi\)
−0.764894 + 0.644157i \(0.777208\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.00000i 0.383326i
\(246\) 0 0
\(247\) −12.0000 + 18.0000i −0.763542 + 1.14531i
\(248\) 0 0
\(249\) 4.00000i 0.253490i
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.00000i 0.250490i
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 10.0000i 0.607457i −0.952759 0.303728i \(-0.901768\pi\)
0.952759 0.303728i \(-0.0982315\pi\)
\(272\) 0 0
\(273\) −4.00000 + 6.00000i −0.242091 + 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 10.0000i 0.598684i
\(280\) 0 0
\(281\) 10.0000i 0.596550i −0.954480 0.298275i \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 12.0000i 0.703452i
\(292\) 0 0
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 + 8.00000i 0.693978 + 0.462652i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 4.00000i 0.229039i
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 3.00000 + 2.00000i 0.166410 + 0.110940i
\(326\) 0 0
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 8.00000i 0.430706i
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 16.0000i 0.856460i 0.903670 + 0.428230i \(0.140863\pi\)
−0.903670 + 0.428230i \(0.859137\pi\)
\(350\) 0 0
\(351\) −3.00000 2.00000i −0.160128 0.106752i
\(352\) 0 0
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 4.00000i 0.211112i 0.994413 + 0.105556i \(0.0336622\pi\)
−0.994413 + 0.105556i \(0.966338\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 10.0000i 0.520579i
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 12.0000i 0.619677i
\(376\) 0 0
\(377\) 30.0000 + 20.0000i 1.54508 + 1.03005i
\(378\) 0 0
\(379\) 34.0000i 1.74646i −0.487306 0.873231i \(-0.662020\pi\)
0.487306 0.873231i \(-0.337980\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 4.00000i 0.204390i −0.994764 0.102195i \(-0.967413\pi\)
0.994764 0.102195i \(-0.0325866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000i 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 30.0000i 1.49813i −0.662497 0.749064i \(-0.730503\pi\)
0.662497 0.749064i \(-0.269497\pi\)
\(402\) 0 0
\(403\) −20.0000 + 30.0000i −0.996271 + 1.49441i
\(404\) 0 0
\(405\) 2.00000i 0.0993808i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000i 0.197787i 0.995098 + 0.0988936i \(0.0315304\pi\)
−0.995098 + 0.0988936i \(0.968470\pi\)
\(410\) 0 0
\(411\) 2.00000i 0.0986527i
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i −0.873195 0.487370i \(-0.837956\pi\)
0.873195 0.487370i \(-0.162044\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000i 0.963366i −0.876346 0.481683i \(-0.840026\pi\)
0.876346 0.481683i \(-0.159974\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 20.0000i 0.958927i
\(436\) 0 0
\(437\) 24.0000i 1.14808i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) 6.00000i 0.283158i −0.989927 0.141579i \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 10.0000i 0.469841i
\(454\) 0 0
\(455\) −12.0000 8.00000i −0.562569 0.375046i
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i 0.990233 + 0.139422i \(0.0445244\pi\)
−0.990233 + 0.139422i \(0.955476\pi\)
\(464\) 0 0
\(465\) −20.0000 −0.927478
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000i 0.275299i
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 24.0000i 1.09659i 0.836286 + 0.548294i \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) 0 0
\(481\) 16.0000 24.0000i 0.729537 1.09431i
\(482\) 0 0
\(483\) 8.00000i 0.364013i
\(484\) 0 0
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 0 0
\(489\) 14.0000i 0.633102i
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000i 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 4.00000i 0.177998i
\(506\) 0 0
\(507\) −5.00000 12.0000i −0.222058 0.532939i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) 6.00000i 0.264906i
\(514\) 0 0
\(515\) 32.0000i 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 2.00000i 0.0872872i
\(526\) 0 0
\(527\) 20.0000i 0.871214i
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) −20.0000 + 30.0000i −0.866296 + 1.29944i
\(534\) 0 0
\(535\) 16.0000i 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000i 0.859867i 0.902861 + 0.429934i \(0.141463\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 60.0000i 2.55609i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) −12.0000 8.00000i −0.507546 0.338364i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000 0.674320 0.337160 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) 28.0000i 1.17797i
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 8.00000i 0.333044i 0.986038 + 0.166522i \(0.0532537\pi\)
−0.986038 + 0.166522i \(0.946746\pi\)
\(578\) 0 0
\(579\) 16.0000i 0.664937i
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.00000 6.00000i 0.165380 0.248069i
\(586\) 0 0
\(587\) 28.0000i 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) 0 0
\(591\) 22.0000i 0.904959i
\(592\) 0 0
\(593\) 26.0000i 1.06769i 0.845582 + 0.533846i \(0.179254\pi\)
−0.845582 + 0.533846i \(0.820746\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) 22.0000i 0.894427i
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 20.0000i 0.810441i
\(610\) 0 0
\(611\) 24.0000 36.0000i 0.970936 1.45640i
\(612\) 0 0
\(613\) 16.0000i 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) −20.0000 −0.806478
\(616\) 0 0
\(617\) 22.0000i 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) 0 0
\(619\) 26.0000i 1.04503i 0.852631 + 0.522514i \(0.175006\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.0000i 0.637962i
\(630\) 0 0
\(631\) 10.0000i 0.398094i 0.979990 + 0.199047i \(0.0637846\pi\)
−0.979990 + 0.199047i \(0.936215\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 16.0000i 0.634941i
\(636\) 0 0
\(637\) 9.00000 + 6.00000i 0.356593 + 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 6.00000i 0.236617i −0.992977 0.118308i \(-0.962253\pi\)
0.992977 0.118308i \(-0.0377472\pi\)
\(644\) 0 0
\(645\) 8.00000i 0.315000i
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 16.0000i 0.625172i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 40.0000i 1.55582i 0.628376 + 0.777910i \(0.283720\pi\)
−0.628376 + 0.777910i \(0.716280\pi\)
\(662\) 0 0
\(663\) 6.00000 + 4.00000i 0.233021 + 0.155347i
\(664\) 0 0
\(665\) 24.0000i 0.930680i
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 0 0
\(669\) 14.0000i 0.541271i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 8.00000i 0.306561i
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 4.00000i 0.152610i
\(688\) 0 0
\(689\) 18.0000 + 12.0000i 0.685745 + 0.457164i
\(690\) 0 0
\(691\) 10.0000i 0.380418i 0.981744 + 0.190209i \(0.0609166\pi\)
−0.981744 + 0.190209i \(0.939083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 40.0000i 1.51729i
\(696\) 0 0
\(697\) 20.0000i 0.757554i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) 24.0000 0.903892
\(706\) 0 0
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) 36.0000i 1.35201i 0.736898 + 0.676004i \(0.236290\pi\)
−0.736898 + 0.676004i \(0.763710\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 40.0000i 1.49801i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 32.0000i 1.19174i
\(722\) 0 0
\(723\) 20.0000i 0.743808i
\(724\) 0 0
\(725\) 10.0000 0.371391
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 44.0000i 1.62518i 0.582838 + 0.812589i \(0.301942\pi\)
−0.582838 + 0.812589i \(0.698058\pi\)
\(734\) 0 0
\(735\) 6.00000i 0.221313i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 26.0000i 0.956425i 0.878244 + 0.478213i \(0.158715\pi\)
−0.878244 + 0.478213i \(0.841285\pi\)
\(740\) 0 0
\(741\) 12.0000 18.0000i 0.440831 0.661247i
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 0 0
\(745\) −28.0000 −1.02584
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 16.0000i 0.584627i
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000i 1.08750i 0.839248 + 0.543750i \(0.182996\pi\)
−0.839248 + 0.543750i \(0.817004\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 4.00000i 0.144620i
\(766\) 0 0
\(767\) 8.00000 12.0000i 0.288863 0.433295i
\(768\) 0 0
\(769\) 24.0000i 0.865462i 0.901523 + 0.432731i \(0.142450\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 6.00000i 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 10.0000i 0.359211i
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) −60.0000 −2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) 4.00000i 0.142766i
\(786\) 0 0
\(787\) 38.0000i 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 28.0000i 0.995565i
\(792\) 0 0
\(793\) −6.00000 4.00000i −0.213066 0.142044i
\(794\) 0 0
\(795\) 12.0000i 0.425596i
\(796\) 0 0
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 24.0000i 0.849059i
\(800\) 0 0
\(801\) 6.00000i 0.212000i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) 10.0000i 0.351147i −0.984466 0.175574i \(-0.943822\pi\)
0.984466 0.175574i \(-0.0561780\pi\)
\(812\) 0 0
\(813\) 10.0000i 0.350715i
\(814\) 0 0
\(815\) 28.0000 0.980797
\(816\) 0 0
\(817\) 24.0000i 0.839654i
\(818\) 0 0
\(819\) 4.00000 6.00000i 0.139771 0.209657i
\(820\) 0 0
\(821\) 30.0000i 1.04701i −0.852023 0.523504i \(-0.824625\pi\)
0.852023 0.523504i \(-0.175375\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000i 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 10.0000i 0.345651i
\(838\) 0 0
\(839\) 16.0000i 0.552381i −0.961103 0.276191i \(-0.910928\pi\)
0.961103 0.276191i \(-0.0890721\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 10.0000i 0.344418i
\(844\) 0 0
\(845\) 24.0000 10.0000i 0.825625 0.344010i
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 32.0000i 1.09695i
\(852\) 0 0
\(853\) 56.0000i 1.91740i −0.284413 0.958702i \(-0.591799\pi\)
0.284413 0.958702i \(-0.408201\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) 0 0
\(863\) 44.0000i 1.49778i −0.662696 0.748889i \(-0.730588\pi\)
0.662696 0.748889i \(-0.269412\pi\)
\(864\) 0 0
\(865\) 12.0000i 0.408012i
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 + 6.00000i −0.135535 + 0.203302i
\(872\) 0 0
\(873\) 12.0000i 0.406138i
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) 8.00000i 0.270141i −0.990836 0.135070i \(-0.956874\pi\)
0.990836 0.135070i \(-0.0431261\pi\)
\(878\) 0 0
\(879\) 14.0000i 0.472208i
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 16.0000i 0.536623i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 72.0000 2.40939
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12.0000 8.00000i −0.400668 0.267112i
\(898\) 0 0
\(899\) 100.000i 3.33519i
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 44.0000i 1.46261i
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 4.00000i 0.132236i
\(916\) 0 0
\(917\) 16.0000i 0.528367i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 2.00000i 0.0659022i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000i 0.263038i
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 6.00000i 0.196854i −0.995144 0.0984268i \(-0.968619\pi\)
0.995144 0.0984268i \(-0.0313810\pi\)
\(930\) 0 0
\(931\) 18.0000i 0.589926i
\(932\) 0 0
\(933\) 28.0000 0.916679
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 10.0000i 0.325991i 0.986627 + 0.162995i \(0.0521156\pi\)
−0.986627 + 0.162995i \(0.947884\pi\)
\(942\) 0 0
\(943\) 40.0000i 1.30258i
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 0 0
\(949\) 8.00000 12.0000i 0.259691 0.389536i
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 24.0000i 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 0 0
\(965\) 32.0000 1.03012
\(966\) 0 0
\(967\) 22.0000i 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 0 0
\(969\) 12.0000i 0.385496i
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 40.0000i 1.28234i
\(974\) 0 0
\(975\) −3.00000 2.00000i −0.0960769 0.0640513i
\(976\) 0 0
\(977\) 42.0000i 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.00000i 0.127710i
\(982\) 0 0
\(983\) 24.0000i 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 44.0000 1.40196
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 10.0000i 0.317340i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 0 0
\(999\) 8.00000i 0.253109i
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 2496.2.c.f.961.1 2
4.3 odd 2 2496.2.c.m.961.1 2
8.3 odd 2 624.2.c.a.337.2 2
8.5 even 2 78.2.b.a.25.2 yes 2
13.12 even 2 inner 2496.2.c.f.961.2 2
24.5 odd 2 234.2.b.a.181.1 2
24.11 even 2 1872.2.c.b.1585.1 2
40.13 odd 4 1950.2.f.g.649.2 2
40.29 even 2 1950.2.b.c.1351.1 2
40.37 odd 4 1950.2.f.d.649.1 2
52.51 odd 2 2496.2.c.m.961.2 2
56.13 odd 2 3822.2.c.d.883.2 2
104.5 odd 4 1014.2.a.g.1.1 1
104.21 odd 4 1014.2.a.b.1.1 1
104.29 even 6 1014.2.i.c.823.1 4
104.37 odd 12 1014.2.e.e.529.1 2
104.45 odd 12 1014.2.e.b.991.1 2
104.51 odd 2 624.2.c.a.337.1 2
104.61 even 6 1014.2.i.c.361.2 4
104.69 even 6 1014.2.i.c.361.1 4
104.77 even 2 78.2.b.a.25.1 2
104.83 even 4 8112.2.a.j.1.1 1
104.85 odd 12 1014.2.e.e.991.1 2
104.93 odd 12 1014.2.e.b.529.1 2
104.99 even 4 8112.2.a.g.1.1 1
104.101 even 6 1014.2.i.c.823.2 4
312.5 even 4 3042.2.a.c.1.1 1
312.77 odd 2 234.2.b.a.181.2 2
312.125 even 4 3042.2.a.n.1.1 1
312.155 even 2 1872.2.c.b.1585.2 2
520.77 odd 4 1950.2.f.g.649.1 2
520.389 even 2 1950.2.b.c.1351.2 2
520.493 odd 4 1950.2.f.d.649.2 2
728.181 odd 2 3822.2.c.d.883.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 2 104.77 even 2
78.2.b.a.25.2 yes 2 8.5 even 2
234.2.b.a.181.1 2 24.5 odd 2
234.2.b.a.181.2 2 312.77 odd 2
624.2.c.a.337.1 2 104.51 odd 2
624.2.c.a.337.2 2 8.3 odd 2
1014.2.a.b.1.1 1 104.21 odd 4
1014.2.a.g.1.1 1 104.5 odd 4
1014.2.e.b.529.1 2 104.93 odd 12
1014.2.e.b.991.1 2 104.45 odd 12
1014.2.e.e.529.1 2 104.37 odd 12
1014.2.e.e.991.1 2 104.85 odd 12
1014.2.i.c.361.1 4 104.69 even 6
1014.2.i.c.361.2 4 104.61 even 6
1014.2.i.c.823.1 4 104.29 even 6
1014.2.i.c.823.2 4 104.101 even 6
1872.2.c.b.1585.1 2 24.11 even 2
1872.2.c.b.1585.2 2 312.155 even 2
1950.2.b.c.1351.1 2 40.29 even 2
1950.2.b.c.1351.2 2 520.389 even 2
1950.2.f.d.649.1 2 40.37 odd 4
1950.2.f.d.649.2 2 520.493 odd 4
1950.2.f.g.649.1 2 520.77 odd 4
1950.2.f.g.649.2 2 40.13 odd 4
2496.2.c.f.961.1 2 1.1 even 1 trivial
2496.2.c.f.961.2 2 13.12 even 2 inner
2496.2.c.m.961.1 2 4.3 odd 2
2496.2.c.m.961.2 2 52.51 odd 2
3042.2.a.c.1.1 1 312.5 even 4
3042.2.a.n.1.1 1 312.125 even 4
3822.2.c.d.883.1 2 728.181 odd 2
3822.2.c.d.883.2 2 56.13 odd 2
8112.2.a.g.1.1 1 104.99 even 4
8112.2.a.j.1.1 1 104.83 even 4