Properties

Label 1425.2.c.d.799.1
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
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 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.d.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} -4.00000i q^{23} -3.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} -5.00000i q^{32} +2.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} -1.00000i q^{38} -4.00000 q^{39} -2.00000i q^{42} -10.0000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -4.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} -1.00000i q^{57} +4.00000i q^{58} -4.00000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} +16.0000i q^{67} -2.00000i q^{68} -4.00000 q^{69} +3.00000i q^{72} -2.00000i q^{73} +1.00000 q^{76} -4.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +2.00000 q^{84} -10.0000 q^{86} +4.00000i q^{87} +6.00000i q^{88} +8.00000 q^{91} -4.00000i q^{92} -12.0000 q^{94} -5.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000i 0.426401i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 2.00000i 0.348155i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 4.00000i − 0.554700i
\(53\) − 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
\(56\) 6.00000 0.801784
\(57\) − 1.00000i − 0.132453i
\(58\) 4.00000i 0.525226i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 16.0000i 1.95471i 0.211604 + 0.977356i \(0.432131\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) − 4.00000i − 0.455842i
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 4.00000i 0.428845i
\(88\) 6.00000i 0.639602i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 2.00000i 0.198030i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −12.0000 −1.17670
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 2.00000i − 0.188982i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 4.00000i 0.369800i
\(118\) 4.00000i 0.368230i
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 2.00000i − 0.181071i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 2.00000i 0.173422i
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 4.00000i 0.340503i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 8.00000i 0.668994i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) 2.00000i 0.161690i
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 6.00000i − 0.462910i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 10.0000i − 0.762493i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) − 2.00000i − 0.147844i
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) − 12.0000i − 0.875190i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 7.00000i 0.505181i
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) − 14.0000i − 0.985037i
\(203\) − 8.00000i − 0.561490i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000i 0.278019i
\(208\) 4.00000i 0.277350i
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 10.0000i 0.677285i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 24.0000i 1.60716i 0.595198 + 0.803579i \(0.297074\pi\)
−0.595198 + 0.803579i \(0.702926\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 12.0000i 0.787839i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) − 8.00000i − 0.519656i
\(238\) − 4.00000i − 0.259281i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 7.00000i 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.00000i − 0.254514i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 22.0000 1.38863 0.694314 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 8.00000i 0.502956i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 10.0000i 0.622573i
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 14.0000i 0.864923i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 16.0000i 0.977356i
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 8.00000i − 0.484182i
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 8.00000i 0.479808i
\(279\) 0 0
\(280\) 0 0
\(281\) 28.0000 1.67034 0.835170 0.549992i \(-0.185369\pi\)
0.835170 + 0.549992i \(0.185369\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 26.0000i 1.54554i 0.634686 + 0.772770i \(0.281129\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 5.00000i 0.294628i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) − 2.00000i − 0.117041i
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) − 18.0000i − 1.04271i
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 20.0000 1.15278
\(302\) − 24.0000i − 1.38104i
\(303\) − 14.0000i − 0.804279i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 12.0000i 0.679366i
\(313\) − 14.0000i − 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) − 8.00000i − 0.445823i
\(323\) − 2.00000i − 0.111283i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 16.0000i − 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000i 0.0540738i
\(343\) 20.0000i 1.07990i
\(344\) −30.0000 −1.61749
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 4.00000i 0.214423i
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 10.0000i 0.533002i
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) − 12.0000i − 0.634220i
\(359\) 2.00000 0.105556 0.0527780 0.998606i \(-0.483192\pi\)
0.0527780 + 0.998606i \(0.483192\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 10.0000i − 0.525588i
\(363\) 7.00000i 0.367405i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) 16.0000i 0.824042i
\(378\) 2.00000i 0.102869i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 18.0000i 0.920960i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 10.0000i 0.508329i
\(388\) 16.0000i 0.812277i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) − 9.00000i − 0.454569i
\(393\) 14.0000i 0.706207i
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 12.0000i 0.601506i
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) 6.00000i 0.297044i
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 8.00000i − 0.394132i
\(413\) − 8.00000i − 0.393654i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) 8.00000i 0.391762i
\(418\) 2.00000i 0.0978232i
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 12.0000i 0.583460i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 12.0000i 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 36.0000i − 1.73005i −0.501729 0.865025i \(-0.667303\pi\)
0.501729 0.865025i \(-0.332697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 4.00000i − 0.191346i
\(438\) 2.00000i 0.0955637i
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000i 0.380521i
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) − 18.0000i − 0.851371i
\(448\) − 14.0000i − 0.661438i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.0000i 0.658505i
\(453\) − 24.0000i − 1.12762i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) 14.0000i 0.654892i 0.944870 + 0.327446i \(0.106188\pi\)
−0.944870 + 0.327446i \(0.893812\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 4.00000i 0.186097i
\(463\) − 22.0000i − 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 16.0000i − 0.740392i −0.928954 0.370196i \(-0.879291\pi\)
0.928954 0.370196i \(-0.120709\pi\)
\(468\) 4.00000i 0.184900i
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 12.0000i 0.552345i
\(473\) 20.0000i 0.919601i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 2.00000i 0.0915737i
\(478\) − 6.00000i − 0.274434i
\(479\) −38.0000 −1.73626 −0.868132 0.496333i \(-0.834679\pi\)
−0.868132 + 0.496333i \(0.834679\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 26.0000i − 1.18427i
\(483\) − 8.00000i − 0.364013i
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) − 22.0000i − 0.981908i
\(503\) − 8.00000i − 0.356702i −0.983967 0.178351i \(-0.942924\pi\)
0.983967 0.178351i \(-0.0570763\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 3.00000i 0.133235i
\(508\) − 4.00000i − 0.177471i
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 11.0000i 0.486136i
\(513\) 1.00000i 0.0441511i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 24.0000i 1.05552i
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 2.00000i 0.0867110i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 48.0000 2.07328
\(537\) − 12.0000i − 0.517838i
\(538\) 4.00000i 0.172452i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) − 12.0000i − 0.515444i
\(543\) − 10.0000i − 0.429141i
\(544\) −10.0000 −0.428746
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 16.0000i − 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 12.0000i 0.510754i
\(553\) 16.0000i 0.680389i
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 38.0000i 1.61011i 0.593199 + 0.805056i \(0.297865\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) − 28.0000i − 1.18111i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 26.0000 1.09286
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 42.0000i − 1.74848i −0.485491 0.874241i \(-0.661359\pi\)
0.485491 0.874241i \(-0.338641\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) − 16.0000i − 0.663221i
\(583\) 4.00000i 0.165663i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 32.0000i 1.32078i 0.750922 + 0.660391i \(0.229609\pi\)
−0.750922 + 0.660391i \(0.770391\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 12.0000i 0.491127i
\(598\) 16.0000i 0.654289i
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) − 20.0000i − 0.815139i
\(603\) − 16.0000i − 0.651570i
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) − 5.00000i − 0.202777i
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 2.00000i 0.0808452i
\(613\) − 18.0000i − 0.727013i −0.931592 0.363507i \(-0.881579\pi\)
0.931592 0.363507i \(-0.118421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 8.00000i 0.321807i
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 18.0000i 0.721734i
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 2.00000i 0.0798723i
\(628\) 18.0000i 0.718278i
\(629\) 0 0
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) − 24.0000i − 0.954669i
\(633\) 12.0000i 0.476957i
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) − 12.0000i − 0.475457i
\(638\) − 8.00000i − 0.316723i
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 26.0000i 1.02534i 0.858586 + 0.512670i \(0.171344\pi\)
−0.858586 + 0.512670i \(0.828656\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 3.00000i − 0.117851i
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) − 6.00000i − 0.234978i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) − 24.0000i − 0.935617i
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 8.00000i 0.310694i
\(664\) −36.0000 −1.39707
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 8.00000i 0.309529i
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) − 10.0000i − 0.385758i
\(673\) 24.0000i 0.925132i 0.886585 + 0.462566i \(0.153071\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 50.0000i 1.92166i 0.277145 + 0.960828i \(0.410612\pi\)
−0.277145 + 0.960828i \(0.589388\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) − 14.0000i − 0.534133i
\(688\) 10.0000i 0.381246i
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 4.00000i 0.151947i
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) − 18.0000i − 0.681310i
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 0 0
\(704\) 14.0000 0.527645
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 28.0000i 1.05305i
\(708\) 4.00000i 0.150329i
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) − 6.00000i − 0.224074i
\(718\) − 2.00000i − 0.0746393i
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 1.00000i − 0.0372161i
\(723\) − 26.0000i − 0.966950i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 14.0000i 0.519231i 0.965712 + 0.259616i \(0.0835959\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(728\) − 24.0000i − 0.889499i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) − 2.00000i − 0.0739221i
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) − 32.0000i − 1.17874i
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) − 4.00000i − 0.146845i
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 12.0000i 0.439057i
\(748\) 4.00000i 0.146254i
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 22.0000i − 0.801725i
\(754\) 16.0000 0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000i 0.144905i
\(763\) − 20.0000i − 0.724049i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 16.0000i 0.577727i
\(768\) 17.0000i 0.613435i
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 4.00000i 0.143963i
\(773\) 14.0000i 0.503545i 0.967786 + 0.251773i \(0.0810135\pi\)
−0.967786 + 0.251773i \(0.918987\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) 48.0000 1.72310
\(777\) 0 0
\(778\) − 30.0000i − 1.07555i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000i 0.286079i
\(783\) − 4.00000i − 0.142948i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −28.0000 −0.995565
\(792\) − 6.00000i − 0.213201i
\(793\) − 8.00000i − 0.284088i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) 14.0000i 0.495905i 0.968772 + 0.247953i \(0.0797578\pi\)
−0.968772 + 0.247953i \(0.920242\pi\)
\(798\) − 2.00000i − 0.0707992i
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) − 12.0000i − 0.423735i
\(803\) 4.00000i 0.141157i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) 4.00000i 0.140807i
\(808\) − 42.0000i − 1.47755i
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) − 8.00000i − 0.280745i
\(813\) − 12.0000i − 0.420858i
\(814\) 0 0
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) − 10.0000i − 0.349856i
\(818\) − 26.0000i − 0.909069i
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 6.00000i 0.209274i
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) 4.00000i 0.139010i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 28.0000i 0.970725i
\(833\) − 6.00000i − 0.207888i
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) − 26.0000i − 0.898155i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 34.0000i 1.17172i
\(843\) − 28.0000i − 0.964371i
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) − 14.0000i − 0.481046i
\(848\) 2.00000i 0.0686803i
\(849\) 26.0000 0.892318
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) − 8.00000i − 0.273115i
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 4.00000i 0.136241i
\(863\) 40.0000i 1.36162i 0.732462 + 0.680808i \(0.238371\pi\)
−0.732462 + 0.680808i \(0.761629\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −36.0000 −1.22333
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 64.0000 2.16856
\(872\) 30.0000i 1.01593i
\(873\) − 16.0000i − 0.541518i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) − 12.0000i − 0.405211i −0.979260 0.202606i \(-0.935059\pi\)
0.979260 0.202606i \(-0.0649409\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −10.0000 −0.336909 −0.168454 0.985709i \(-0.553878\pi\)
−0.168454 + 0.985709i \(0.553878\pi\)
\(882\) 3.00000i 0.101015i
\(883\) − 46.0000i − 1.54802i −0.633171 0.774012i \(-0.718247\pi\)
0.633171 0.774012i \(-0.281753\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 24.0000i 0.803579i
\(893\) − 12.0000i − 0.401565i
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 6.00000 0.200446
\(897\) 16.0000i 0.534224i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) − 20.0000i − 0.665558i
\(904\) 42.0000 1.39690
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 24.0000i 0.794284i
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) − 28.0000i − 0.924641i
\(918\) − 2.00000i − 0.0660098i
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 6.00000i − 0.197599i
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) 8.00000i 0.262754i
\(928\) 20.0000i 0.656532i
\(929\) −10.0000 −0.328089 −0.164045 0.986453i \(-0.552454\pi\)
−0.164045 + 0.986453i \(0.552454\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) − 18.0000i − 0.589610i
\(933\) 18.0000i 0.589294i
\(934\) −16.0000 −0.523536
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) 32.0000i 1.04484i
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) − 12.0000i − 0.388922i
\(953\) 46.0000i 1.49009i 0.667016 + 0.745043i \(0.267571\pi\)
−0.667016 + 0.745043i \(0.732429\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) − 8.00000i − 0.258603i
\(958\) 38.0000i 1.22772i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) − 34.0000i − 1.09337i −0.837340 0.546683i \(-0.815890\pi\)
0.837340 0.546683i \(-0.184110\pi\)
\(968\) 21.0000i 0.674966i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 6.00000i 0.191859i
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 6.00000i 0.191468i
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.00000 0.254772
\(987\) − 24.0000i − 0.763928i
\(988\) − 4.00000i − 0.127257i
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 32.0000i 1.01294i
\(999\) 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 1425.2.c.d.799.1 2
5.2 odd 4 285.2.a.b.1.1 1
5.3 odd 4 1425.2.a.d.1.1 1
5.4 even 2 inner 1425.2.c.d.799.2 2
15.2 even 4 855.2.a.b.1.1 1
15.8 even 4 4275.2.a.o.1.1 1
20.7 even 4 4560.2.a.v.1.1 1
95.37 even 4 5415.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 1 5.2 odd 4
855.2.a.b.1.1 1 15.2 even 4
1425.2.a.d.1.1 1 5.3 odd 4
1425.2.c.d.799.1 2 1.1 even 1 trivial
1425.2.c.d.799.2 2 5.4 even 2 inner
4275.2.a.o.1.1 1 15.8 even 4
4560.2.a.v.1.1 1 20.7 even 4
5415.2.a.c.1.1 1 95.37 even 4