Properties

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -3.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} +5.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} -1.00000 q^{36} +1.00000i q^{38} -4.00000 q^{39} +2.00000i q^{42} +10.0000i q^{43} -2.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} +4.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} +6.00000 q^{56} +1.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} +2.00000i q^{73} +1.00000 q^{76} +4.00000i q^{77} -4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{84} -10.0000 q^{86} -4.00000i q^{87} -6.00000i q^{88} +8.00000 q^{91} +4.00000i q^{92} -12.0000 q^{94} -5.00000 q^{96} -16.0000i q^{97} +3.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 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 \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 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 \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 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 \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 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 \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) −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 \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 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 \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 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 \(-0.567869\pi\)
0.211604 0.977356i \(-0.432131\pi\)
\(68\) 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 \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 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 \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 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 \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 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 \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) −12.0000 −1.17670
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 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 \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) −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 \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 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 \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 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 \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 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 \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 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 \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 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 \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) − 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 \(-0.639842\pi\)
−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 \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 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 \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 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 \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) − 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 \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 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 \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) − 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 \(-0.718871\pi\)
0.634686 0.772770i \(-0.281129\pi\)
\(284\) 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 \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) −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 \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) − 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 \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) − 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 \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) − 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 \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 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 \(0.0840516\pi\)
−0.965339 + 0.260998i \(0.915948\pi\)
\(368\) − 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 \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 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 \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) −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 \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) − 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 \(0.332697\pi\)
−0.501729 + 0.865025i \(0.667303\pi\)
\(434\) 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 \(-0.875893\pi\)
0.924949 0.380091i \(-0.124107\pi\)
\(444\) 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 \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 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 \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 16.0000i 0.740392i 0.928954 + 0.370196i \(0.120709\pi\)
−0.928954 + 0.370196i \(0.879291\pi\)
\(468\) − 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 \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 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 \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) −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 \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) −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 \(0.111123\pi\)
−0.939680 + 0.342055i \(0.888877\pi\)
\(548\) 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 \(-0.702135\pi\)
0.593199 0.805056i \(-0.297865\pi\)
\(558\) 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 \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) −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 \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) 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 \(-0.770391\pi\)
0.750922 0.660391i \(-0.229609\pi\)
\(588\) 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 \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) −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 \(-0.894733\pi\)
0.945814 0.324710i \(-0.105267\pi\)
\(608\) 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 \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) − 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 \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 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 \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 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 \(-0.846929\pi\)
0.886585 0.462566i \(-0.153071\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) − 50.0000i − 1.92166i −0.277145 0.960828i \(-0.589388\pi\)
0.277145 0.960828i \(-0.410612\pi\)
\(678\) 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 \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −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 \(-0.916404\pi\)
0.965712 0.259616i \(-0.0835959\pi\)
\(728\) 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 \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) −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 \(0.0468799\pi\)
−0.989174 + 0.146746i \(0.953120\pi\)
\(744\) 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 \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 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 \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 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 \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 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 \(-0.920242\pi\)
0.968772 0.247953i \(-0.0797578\pi\)
\(798\) 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 \(0.0784612\pi\)
−0.969774 + 0.244005i \(0.921539\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) − 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 \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 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 \(-0.761629\pi\)
0.732462 0.680808i \(-0.238371\pi\)
\(864\) 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 \(0.0649409\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 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 \(0.281753\pi\)
−0.633171 + 0.774012i \(0.718247\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 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 \(0.331616\pi\)
−0.504664 + 0.863316i \(0.668384\pi\)
\(908\) 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 \(-0.759347\pi\)
0.727564 0.686040i \(-0.240653\pi\)
\(938\) − 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 \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 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 \(-0.732429\pi\)
0.667016 0.745043i \(-0.267571\pi\)
\(954\) 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 \(0.184110\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(968\) − 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 \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) − 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 \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) 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 \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) − 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.799.2 2
5.2 odd 4 1425.2.a.d.1.1 1
5.3 odd 4 285.2.a.b.1.1 1
5.4 even 2 inner 1425.2.c.d.799.1 2
15.2 even 4 4275.2.a.o.1.1 1
15.8 even 4 855.2.a.b.1.1 1
20.3 even 4 4560.2.a.v.1.1 1
95.18 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 5.3 odd 4
855.2.a.b.1.1 1 15.8 even 4
1425.2.a.d.1.1 1 5.2 odd 4
1425.2.c.d.799.1 2 5.4 even 2 inner
1425.2.c.d.799.2 2 1.1 even 1 trivial
4275.2.a.o.1.1 1 15.2 even 4
4560.2.a.v.1.1 1 20.3 even 4
5415.2.a.c.1.1 1 95.18 even 4