Properties

Label 4650.2.d.v.3349.2
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4650,2,Mod(3349,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4650.3349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (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: \(37.1304369399\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.v.3349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +4.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +8.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{36} +2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} +4.00000i q^{42} +4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -2.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} +8.00000i q^{58} +12.0000 q^{59} -8.00000 q^{61} +1.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} +8.00000i q^{77} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -16.0000i q^{83} -4.00000 q^{84} -4.00000 q^{86} -8.00000i q^{87} -2.00000i q^{88} +2.00000 q^{89} -8.00000 q^{91} +6.00000i q^{92} -1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} +2.00000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} - 8 q^{14} + 2 q^{16} + 8 q^{21} - 2 q^{24} - 4 q^{26} + 16 q^{29} + 2 q^{31} + 2 q^{36} + 4 q^{39} + 12 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} - 2 q^{54} + 8 q^{56} + 24 q^{59} - 16 q^{61} - 2 q^{64} + 4 q^{66} - 12 q^{69} - 16 q^{71} - 4 q^{74} - 8 q^{79} + 2 q^{81} - 8 q^{84} - 8 q^{86} + 4 q^{89} - 16 q^{91} - 24 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 2.00000i 0.426401i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 8.00000i 1.05045i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 2.00000i 0.226455i
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 8.00000i − 0.857690i
\(88\) − 2.00000i − 0.213201i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 6.00000i 0.625543i
\(93\) − 1.00000i − 0.103695i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000i 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) − 2.00000i − 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 8.00000i − 0.724286i
\(123\) − 6.00000i − 0.541002i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) − 8.00000i − 0.671345i
\(143\) 4.00000i 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 9.00000i 0.742307i
\(148\) − 2.00000i − 0.164399i
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) − 4.00000i − 0.318223i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 1.00000i 0.0785674i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) − 12.0000i − 0.901975i
\(178\) 2.00000i 0.149906i
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) 8.00000i 0.591377i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) − 12.0000i − 0.875190i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) − 10.0000i − 0.703598i
\(203\) 32.0000i 2.24596i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 6.00000i 0.417029i
\(208\) 2.00000i 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 8.00000i 0.548151i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 4.00000i 0.271538i
\(218\) 18.0000i 1.21911i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 2.00000i 0.134231i
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) − 8.00000i − 0.525226i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) − 1.00000i − 0.0635001i
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 4.00000i 0.251976i
\(253\) − 12.0000i − 0.754434i
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) − 4.00000i − 0.247121i
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) − 4.00000i − 0.244339i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 8.00000i 0.484182i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 18.0000i − 1.07957i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 24.0000i 1.41668i
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) − 4.00000i − 0.234082i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 2.00000i 0.116052i
\(298\) − 14.0000i − 0.810998i
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 20.0000i 1.15087i
\(303\) 10.0000i 0.574485i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) − 8.00000i − 0.455842i
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) − 4.00000i − 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 22.0000i 1.23564i 0.786318 + 0.617822i \(0.211985\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) − 2.00000i − 0.112154i
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 24.0000i 1.33747i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 18.0000i − 0.995402i
\(328\) − 6.00000i − 0.331295i
\(329\) −48.0000 −2.64633
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 16.0000i 0.878114i
\(333\) − 2.00000i − 0.109599i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 16.0000i 0.871576i 0.900049 + 0.435788i \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) − 16.0000i − 0.858925i −0.903085 0.429463i \(-0.858703\pi\)
0.903085 0.429463i \(-0.141297\pi\)
\(348\) 8.00000i 0.428845i
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 2.00000i 0.106600i
\(353\) 4.00000i 0.212899i 0.994318 + 0.106449i \(0.0339482\pi\)
−0.994318 + 0.106449i \(0.966052\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) − 14.0000i − 0.739923i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 12.0000i − 0.630706i
\(363\) 7.00000i 0.367405i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) − 6.00000i − 0.312772i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 1.00000i 0.0518476i
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 16.0000i 0.824042i
\(378\) − 4.00000i − 0.205738i
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 8.00000i 0.409316i
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) − 4.00000i − 0.203331i
\(388\) − 2.00000i − 0.101535i
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000i 0.454569i
\(393\) 4.00000i 0.201773i
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 12.0000i 0.601506i
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 2.00000i 0.0996271i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 4.00000i 0.197066i
\(413\) 48.0000i 2.36193i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 18.0000i 0.881464i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) − 12.0000i − 0.583460i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) − 32.0000i − 1.54859i
\(428\) − 4.00000i − 0.193347i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 4.00000i − 0.192228i −0.995370 0.0961139i \(-0.969359\pi\)
0.995370 0.0961139i \(-0.0306413\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 4.00000i 0.191127i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) 14.0000i 0.662177i
\(448\) − 4.00000i − 0.188982i
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) − 18.0000i − 0.846649i
\(453\) − 20.0000i − 0.939682i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) − 8.00000i − 0.374224i −0.982339 0.187112i \(-0.940087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) − 4.00000i − 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 8.00000i 0.372194i
\(463\) − 42.0000i − 1.95191i −0.217982 0.975953i \(-0.569947\pi\)
0.217982 0.975953i \(-0.430053\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 12.0000i − 0.552345i
\(473\) 8.00000i 0.367840i
\(474\) −4.00000 −0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) − 8.00000i − 0.365911i
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 10.0000i 0.455488i
\(483\) − 24.0000i − 1.09204i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 32.0000i − 1.43540i
\(498\) − 16.0000i − 0.716977i
\(499\) 34.0000 1.52205 0.761025 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 2.00000i 0.0892644i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) − 9.00000i − 0.399704i
\(508\) − 6.00000i − 0.266207i
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 24.0000i 1.05552i
\(518\) − 8.00000i − 0.351500i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 8.00000i − 0.350150i
\(523\) − 12.0000i − 0.524723i −0.964970 0.262362i \(-0.915499\pi\)
0.964970 0.262362i \(-0.0845013\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 14.0000i 0.604145i
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 12.0000i 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 6.00000i 0.255377i
\(553\) − 16.0000i − 0.680389i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 18.0000 0.763370
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 4.00000i 0.167984i
\(568\) 8.00000i 0.335673i
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) − 8.00000i − 0.334205i
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 42.0000i − 1.74848i −0.485491 0.874241i \(-0.661359\pi\)
0.485491 0.874241i \(-0.338641\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 64.0000 2.65517
\(582\) 2.00000i 0.0829027i
\(583\) − 4.00000i − 0.165663i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 2.00000i 0.0821995i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) − 12.0000i − 0.491127i
\(598\) 12.0000i 0.490716i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) − 4.00000i − 0.162893i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 0 0
\(609\) 32.0000 1.29671
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 24.0000i 0.962312i
\(623\) 8.00000i 0.320513i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) 18.0000i 0.718278i
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 4.00000i 0.159111i
\(633\) 20.0000i 0.794929i
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) − 18.0000i − 0.713186i
\(638\) 16.0000i 0.633446i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 4.00000i 0.157867i
\(643\) − 44.0000i − 1.73519i −0.497271 0.867595i \(-0.665665\pi\)
0.497271 0.867595i \(-0.334335\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) − 14.0000i − 0.550397i −0.961387 0.275198i \(-0.911256\pi\)
0.961387 0.275198i \(-0.0887435\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) − 12.0000i − 0.469956i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) − 4.00000i − 0.156055i
\(658\) − 48.0000i − 1.87123i
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 26.0000i 1.01052i
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) − 48.0000i − 1.85857i
\(668\) − 18.0000i − 0.696441i
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −16.0000 −0.617673
\(672\) 4.00000i 0.154303i
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 50.0000i − 1.92166i −0.277145 0.960828i \(-0.589388\pi\)
0.277145 0.960828i \(-0.410612\pi\)
\(678\) 18.0000i 0.691286i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 2.00000i 0.0765840i
\(683\) − 20.0000i − 0.765279i −0.923898 0.382639i \(-0.875015\pi\)
0.923898 0.382639i \(-0.124985\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 4.00000i 0.152610i
\(688\) 4.00000i 0.152499i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) − 18.0000i − 0.684257i
\(693\) − 8.00000i − 0.303895i
\(694\) 16.0000 0.607352
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) − 2.00000i − 0.0757011i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −4.00000 −0.150542
\(707\) − 40.0000i − 1.50435i
\(708\) 12.0000i 0.450988i
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) − 2.00000i − 0.0749532i
\(713\) − 6.00000i − 0.224702i
\(714\) 0 0
\(715\) 0 0
\(716\) 14.0000 0.523205
\(717\) 8.00000i 0.298765i
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 19.0000i − 0.707107i
\(723\) − 10.0000i − 0.371904i
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) − 24.0000i − 0.890111i −0.895503 0.445055i \(-0.853184\pi\)
0.895503 0.445055i \(-0.146816\pi\)
\(728\) 8.00000i 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 8.00000i − 0.295689i
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 8.00000i 0.294684i
\(738\) − 6.00000i − 0.220863i
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.00000i 0.293689i
\(743\) 26.0000i 0.953847i 0.878945 + 0.476924i \(0.158248\pi\)
−0.878945 + 0.476924i \(0.841752\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 16.0000i 0.585409i
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 2.00000i − 0.0728841i
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 18.0000i 0.654221i 0.944986 + 0.327111i \(0.106075\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 6.00000i 0.217357i
\(763\) 72.0000i 2.60658i
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) − 22.0000i − 0.791797i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 8.00000i 0.286998i
\(778\) 36.0000i 1.29066i
\(779\) 0 0
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 16.0000i 0.570338i 0.958477 + 0.285169i \(0.0920498\pi\)
−0.958477 + 0.285169i \(0.907950\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −72.0000 −2.56003
\(792\) 2.00000i 0.0710669i
\(793\) − 16.0000i − 0.568177i
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) − 10.0000i − 0.353112i
\(803\) 8.00000i 0.282314i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) 10.0000i 0.351799i
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) − 32.0000i − 1.12298i
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 30.0000i − 1.04893i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 12.0000i 0.418548i
\(823\) − 22.0000i − 0.766872i −0.923567 0.383436i \(-0.874741\pi\)
0.923567 0.383436i \(-0.125259\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) 16.0000i 0.556375i 0.960527 + 0.278187i \(0.0897336\pi\)
−0.960527 + 0.278187i \(0.910266\pi\)
\(828\) − 6.00000i − 0.208514i
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) − 2.00000i − 0.0693375i
\(833\) 0 0
\(834\) −18.0000 −0.623289
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00000i 0.0345651i
\(838\) − 12.0000i − 0.414533i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 18.0000i 0.620321i
\(843\) − 6.00000i − 0.206651i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) − 28.0000i − 0.962091i
\(848\) − 2.00000i − 0.0686803i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) − 8.00000i − 0.274075i
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) − 16.0000i − 0.544962i
\(863\) − 18.0000i − 0.612727i −0.951915 0.306364i \(-0.900888\pi\)
0.951915 0.306364i \(-0.0991123\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) − 17.0000i − 0.577350i
\(868\) − 4.00000i − 0.135769i
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 18.0000i − 0.609557i
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 18.0000i − 0.607817i −0.952701 0.303908i \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 9.00000i 0.303046i
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 2.00000i 0.0669650i
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) − 12.0000i − 0.400668i
\(898\) 34.0000i 1.13459i
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 0 0
\(902\) 12.0000i 0.399556i
\(903\) 16.0000i 0.532447i
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) − 4.00000i − 0.132745i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) − 32.0000i − 1.05905i
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 24.0000i 0.790398i
\(923\) − 16.0000i − 0.526646i
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 42.0000 1.38021
\(927\) 4.00000i 0.131377i
\(928\) 8.00000i 0.262613i
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000i 0.196537i
\(933\) − 24.0000i − 0.785725i
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 36.0000 1.17357 0.586783 0.809744i \(-0.300394\pi\)
0.586783 + 0.809744i \(0.300394\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) − 36.0000i − 1.17232i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) − 4.00000i − 0.129914i
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 4.00000i 0.129573i 0.997899 + 0.0647864i \(0.0206366\pi\)
−0.997899 + 0.0647864i \(0.979363\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) − 16.0000i − 0.517207i
\(958\) 8.00000i 0.258468i
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) − 4.00000i − 0.128965i
\(963\) − 4.00000i − 0.128898i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 24.0000 0.772187
\(967\) − 2.00000i − 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 0 0
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 72.0000i − 2.30821i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 12.0000i 0.383718i
\(979\) 4.00000 0.127841
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) − 42.0000i − 1.34027i
\(983\) − 54.0000i − 1.72233i −0.508323 0.861166i \(-0.669735\pi\)
0.508323 0.861166i \(-0.330265\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) 48.0000i 1.52786i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) − 26.0000i − 0.825085i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) − 6.00000i − 0.190022i −0.995476 0.0950110i \(-0.969711\pi\)
0.995476 0.0950110i \(-0.0302886\pi\)
\(998\) 34.0000i 1.07625i
\(999\) −2.00000 −0.0632772
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 4650.2.d.v.3349.2 2
5.2 odd 4 930.2.a.c.1.1 1
5.3 odd 4 4650.2.a.bw.1.1 1
5.4 even 2 inner 4650.2.d.v.3349.1 2
15.2 even 4 2790.2.a.o.1.1 1
20.7 even 4 7440.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.c.1.1 1 5.2 odd 4
2790.2.a.o.1.1 1 15.2 even 4
4650.2.a.bw.1.1 1 5.3 odd 4
4650.2.d.v.3349.1 2 5.4 even 2 inner
4650.2.d.v.3349.2 2 1.1 even 1 trivial
7440.2.a.bb.1.1 1 20.7 even 4