Properties

Label 1050.2.g.h.799.2
Level $1050$
Weight $2$
Character 1050.799
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(799,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, 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("1050.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.g (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: \(8.38429221223\)
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 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.799
Dual form 1050.2.g.h.799.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} +1.00000i q^{42} -8.00000i q^{43} -12.0000i q^{47} -1.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +1.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} +2.00000 q^{61} -4.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} +6.00000i q^{68} +1.00000i q^{72} -14.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -12.0000i q^{83} -1.00000 q^{84} +8.00000 q^{86} -6.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} +4.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +14.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{14} + 2 q^{16} + 8 q^{19} + 2 q^{21} - 2 q^{24} + 4 q^{26} + 12 q^{29} - 8 q^{31} + 12 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} - 2 q^{49} - 12 q^{51} - 2 q^{54} + 2 q^{56} + 24 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} - 2 q^{84} + 16 q^{86} - 12 q^{89} + 4 q^{91} + 24 q^{94} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 1.00000i − 0.188982i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000i 0.154303i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000i 0.277350i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 6.00000i 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 1.00000i 0.0944911i
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000i 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000i 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 1.00000i 0.0824786i
\(148\) − 2.00000i − 0.164399i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 16.0000i 1.27289i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 8.00000i 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) − 6.00000i − 0.449719i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000i 0.148250i
\(183\) − 2.00000i − 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 12.0000i 0.875190i
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 6.00000i − 0.422159i
\(203\) 6.00000i 0.421117i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) − 2.00000i − 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 4.00000i − 0.271538i
\(218\) − 14.0000i − 0.948200i
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 2.00000i 0.134231i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 16.0000i − 1.03931i
\(238\) 6.00000i 0.388922i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 8.00000i − 0.509028i
\(248\) 4.00000i 0.254000i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000i 0.741362i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 6.00000i 0.367194i
\(268\) − 8.00000i − 0.488678i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) − 2.00000i − 0.121046i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) − 1.00000i − 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 14.0000i 0.819288i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) − 18.0000i − 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) − 16.0000i − 0.920697i
\(303\) 6.00000i 0.344691i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 14.0000i 0.774202i
\(328\) − 6.00000i − 0.331295i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) − 4.00000i − 0.216295i
\(343\) − 1.00000i − 0.0539949i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 6.00000i 0.321634i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) − 6.00000i − 0.317554i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 2.00000i 0.105118i
\(363\) 11.0000i 0.577350i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) − 4.00000i − 0.207390i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 12.0000i − 0.618031i
\(378\) − 1.00000i − 0.0514344i
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 8.00000i 0.406663i
\(388\) − 14.0000i − 0.710742i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 0.0505076i
\(393\) − 12.0000i − 0.605320i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 8.00000i 0.398508i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 6.00000i 0.297044i
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 16.0000i − 0.788263i
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 28.0000i − 1.36302i
\(423\) 12.0000i 0.583460i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) − 14.0000i − 0.668946i
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) − 12.0000i − 0.570782i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 18.0000i 0.851371i
\(448\) − 1.00000i − 0.0472456i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 18.0000i 0.846649i
\(453\) 16.0000i 0.751746i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) − 12.0000i − 0.552345i
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 26.0000i 1.18427i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 6.00000i 0.270501i
\(493\) − 36.0000i − 1.62136i
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) − 12.0000i − 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 12.0000i 0.535586i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) − 8.00000i − 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) − 2.00000i − 0.0878750i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) − 4.00000i − 0.173422i
\(533\) − 12.0000i − 0.519778i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 20.0000i 0.859074i
\(543\) − 2.00000i − 0.0858282i
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 4.00000i 0.169334i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) − 30.0000i − 1.26547i
\(563\) 36.0000i 1.51722i 0.651546 + 0.758610i \(0.274121\pi\)
−0.651546 + 0.758610i \(0.725879\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 14.0000i 0.580319i
\(583\) 0 0
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 1.00000i − 0.0412393i
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 2.00000i 0.0821995i
\(593\) − 42.0000i − 1.72473i −0.506284 0.862367i \(-0.668981\pi\)
0.506284 0.862367i \(-0.331019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) − 4.00000i − 0.163709i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 8.00000i 0.326056i
\(603\) − 8.00000i − 0.325785i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) − 6.00000i − 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 6.00000i − 0.240385i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 28.0000i 1.11290i
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) − 16.0000i − 0.626608i
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 14.0000i 0.546192i
\(658\) 12.0000i 0.467809i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 12.0000i 0.466041i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 1.00000i 0.0385758i
\(673\) − 50.0000i − 1.92736i −0.267063 0.963679i \(-0.586053\pi\)
0.267063 0.963679i \(-0.413947\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) − 18.0000i − 0.691286i
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 2.00000i 0.0763048i
\(688\) − 8.00000i − 0.304997i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 36.0000i − 1.36360i
\(698\) − 26.0000i − 0.984115i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) − 6.00000i − 0.225653i
\(708\) 12.0000i 0.450988i
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 6.00000i 0.224860i
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) − 3.00000i − 0.111648i
\(723\) − 26.0000i − 0.966950i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 2.00000i 0.0739221i
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) − 6.00000i − 0.220863i
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 6.00000i 0.220267i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) − 12.0000i − 0.437304i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 8.00000i 0.289809i
\(763\) − 14.0000i − 0.506834i
\(764\) 0 0
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) − 24.0000i − 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) − 22.0000i − 0.791797i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 2.00000i 0.0717496i
\(778\) 30.0000i 1.07555i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) − 4.00000i − 0.142044i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 4.00000i 0.141598i
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 18.0000i 0.633630i
\(808\) 6.00000i 0.211079i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) − 20.0000i − 0.701431i
\(814\) 0 0
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) − 32.0000i − 1.11954i
\(818\) 22.0000i 0.769212i
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 8.00000i − 0.278862i −0.990232 0.139431i \(-0.955473\pi\)
0.990232 0.139431i \(-0.0445274\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 2.00000i 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 12.0000i 0.414533i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 10.0000i − 0.344623i
\(843\) 30.0000i 1.03325i
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) − 11.0000i − 0.377964i
\(848\) − 6.00000i − 0.206041i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 24.0000i 0.817443i
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 19.0000i 0.645274i
\(868\) 4.00000i 0.135769i
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 14.0000i 0.474100i
\(873\) − 14.0000i − 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) 28.0000i 0.944954i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) 40.0000i 1.34611i 0.739594 + 0.673054i \(0.235018\pi\)
−0.739594 + 0.673054i \(0.764982\pi\)
\(884\) 12.0000 0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) − 48.0000i − 1.60626i
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 6.00000i 0.200223i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 8.00000i − 0.266223i
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) − 16.0000i − 0.531271i −0.964073 0.265636i \(-0.914418\pi\)
0.964073 0.265636i \(-0.0855818\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 12.0000i 0.396275i
\(918\) 6.00000i 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) − 30.0000i − 0.987997i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) − 16.0000i − 0.525509i
\(928\) 6.00000i 0.196960i
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) − 34.0000i − 1.11073i −0.831606 0.555366i \(-0.812578\pi\)
0.831606 0.555366i \(-0.187422\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 16.0000i 0.519656i
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) − 6.00000i − 0.194461i
\(953\) 6.00000i 0.194359i 0.995267 + 0.0971795i \(0.0309821\pi\)
−0.995267 + 0.0971795i \(0.969018\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 24.0000i 0.775405i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 4.00000i 0.128965i
\(963\) − 12.0000i − 0.386695i
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 56.0000i 1.80084i 0.435023 + 0.900419i \(0.356740\pi\)
−0.435023 + 0.900419i \(0.643260\pi\)
\(968\) 11.0000i 0.353553i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 4.00000i 0.128234i
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 16.0000i 0.511624i
\(979\) 0 0
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 24.0000i 0.765871i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) − 12.0000i − 0.381964i
\(988\) 8.00000i 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 4.00000i 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 4.00000i 0.126618i
\(999\) −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 1050.2.g.h.799.2 2
3.2 odd 2 3150.2.g.o.2899.1 2
5.2 odd 4 1050.2.a.a.1.1 1
5.3 odd 4 210.2.a.d.1.1 1
5.4 even 2 inner 1050.2.g.h.799.1 2
15.2 even 4 3150.2.a.ba.1.1 1
15.8 even 4 630.2.a.f.1.1 1
15.14 odd 2 3150.2.g.o.2899.2 2
20.3 even 4 1680.2.a.b.1.1 1
20.7 even 4 8400.2.a.cn.1.1 1
35.3 even 12 1470.2.i.h.961.1 2
35.13 even 4 1470.2.a.m.1.1 1
35.18 odd 12 1470.2.i.d.961.1 2
35.23 odd 12 1470.2.i.d.361.1 2
35.27 even 4 7350.2.a.bd.1.1 1
35.33 even 12 1470.2.i.h.361.1 2
40.3 even 4 6720.2.a.cc.1.1 1
40.13 odd 4 6720.2.a.bb.1.1 1
60.23 odd 4 5040.2.a.ba.1.1 1
105.83 odd 4 4410.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.d.1.1 1 5.3 odd 4
630.2.a.f.1.1 1 15.8 even 4
1050.2.a.a.1.1 1 5.2 odd 4
1050.2.g.h.799.1 2 5.4 even 2 inner
1050.2.g.h.799.2 2 1.1 even 1 trivial
1470.2.a.m.1.1 1 35.13 even 4
1470.2.i.d.361.1 2 35.23 odd 12
1470.2.i.d.961.1 2 35.18 odd 12
1470.2.i.h.361.1 2 35.33 even 12
1470.2.i.h.961.1 2 35.3 even 12
1680.2.a.b.1.1 1 20.3 even 4
3150.2.a.ba.1.1 1 15.2 even 4
3150.2.g.o.2899.1 2 3.2 odd 2
3150.2.g.o.2899.2 2 15.14 odd 2
4410.2.a.f.1.1 1 105.83 odd 4
5040.2.a.ba.1.1 1 60.23 odd 4
6720.2.a.bb.1.1 1 40.13 odd 4
6720.2.a.cc.1.1 1 40.3 even 4
7350.2.a.bd.1.1 1 35.27 even 4
8400.2.a.cn.1.1 1 20.7 even 4