Properties

Label 845.2.c.e.506.3
Level $845$
Weight $2$
Character 845.506
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4$
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] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 506.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 845.506
Dual form 845.2.c.e.506.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.26795i q^{6} -2.00000i q^{7} +1.73205i q^{8} -2.46410 q^{9} +O(q^{10})\) \(q+1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.26795i q^{6} -2.00000i q^{7} +1.73205i q^{8} -2.46410 q^{9} +1.73205 q^{10} +1.26795i q^{11} +0.732051 q^{12} +3.46410 q^{14} +0.732051i q^{15} -5.00000 q^{16} -3.46410 q^{17} -4.26795i q^{18} +4.19615i q^{19} +1.00000i q^{20} +1.46410i q^{21} -2.19615 q^{22} -4.73205 q^{23} -1.26795i q^{24} -1.00000 q^{25} +4.00000 q^{27} +2.00000i q^{28} -9.46410 q^{29} -1.26795 q^{30} -0.196152i q^{31} -5.19615i q^{32} -0.928203i q^{33} -6.00000i q^{34} -2.00000 q^{35} +2.46410 q^{36} +4.00000i q^{37} -7.26795 q^{38} +1.73205 q^{40} -3.46410i q^{41} -2.53590 q^{42} -10.1962 q^{43} -1.26795i q^{44} +2.46410i q^{45} -8.19615i q^{46} -6.00000i q^{47} +3.66025 q^{48} +3.00000 q^{49} -1.73205i q^{50} +2.53590 q^{51} -10.3923 q^{53} +6.92820i q^{54} +1.26795 q^{55} +3.46410 q^{56} -3.07180i q^{57} -16.3923i q^{58} +15.1244i q^{59} -0.732051i q^{60} +12.3923 q^{61} +0.339746 q^{62} +4.92820i q^{63} -1.00000 q^{64} +1.60770 q^{66} -14.3923i q^{67} +3.46410 q^{68} +3.46410 q^{69} -3.46410i q^{70} +1.26795i q^{71} -4.26795i q^{72} +4.00000i q^{73} -6.92820 q^{74} +0.732051 q^{75} -4.19615i q^{76} +2.53590 q^{77} +12.3923 q^{79} +5.00000i q^{80} +4.46410 q^{81} +6.00000 q^{82} -6.00000i q^{83} -1.46410i q^{84} +3.46410i q^{85} -17.6603i q^{86} +6.92820 q^{87} -2.19615 q^{88} -0.928203i q^{89} -4.26795 q^{90} +4.73205 q^{92} +0.143594i q^{93} +10.3923 q^{94} +4.19615 q^{95} +3.80385i q^{96} +2.00000i q^{97} +5.19615i q^{98} -3.12436i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49} + 24 q^{51} + 12 q^{55} + 8 q^{61} + 36 q^{62} - 4 q^{64} + 48 q^{66} - 4 q^{75} + 24 q^{77} + 8 q^{79} + 4 q^{81} + 24 q^{82} + 12 q^{88} - 24 q^{90} + 12 q^{92} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.26795i − 0.517638i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.73205i 0.612372i
\(9\) −2.46410 −0.821367
\(10\) 1.73205 0.547723
\(11\) 1.26795i 0.382301i 0.981561 + 0.191151i \(0.0612219\pi\)
−0.981561 + 0.191151i \(0.938778\pi\)
\(12\) 0.732051 0.211325
\(13\) 0 0
\(14\) 3.46410 0.925820
\(15\) 0.732051i 0.189015i
\(16\) −5.00000 −1.25000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) − 4.26795i − 1.00597i
\(19\) 4.19615i 0.962663i 0.876539 + 0.481332i \(0.159847\pi\)
−0.876539 + 0.481332i \(0.840153\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 1.46410i 0.319493i
\(22\) −2.19615 −0.468221
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) − 1.26795i − 0.258819i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 2.00000i 0.377964i
\(29\) −9.46410 −1.75744 −0.878720 0.477338i \(-0.841602\pi\)
−0.878720 + 0.477338i \(0.841602\pi\)
\(30\) −1.26795 −0.231495
\(31\) − 0.196152i − 0.0352300i −0.999845 0.0176150i \(-0.994393\pi\)
0.999845 0.0176150i \(-0.00560732\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) − 0.928203i − 0.161579i
\(34\) − 6.00000i − 1.02899i
\(35\) −2.00000 −0.338062
\(36\) 2.46410 0.410684
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) −7.26795 −1.17902
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) − 3.46410i − 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) −2.53590 −0.391298
\(43\) −10.1962 −1.55490 −0.777449 0.628946i \(-0.783487\pi\)
−0.777449 + 0.628946i \(0.783487\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) 2.46410i 0.367327i
\(46\) − 8.19615i − 1.20846i
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 3.66025 0.528312
\(49\) 3.00000 0.428571
\(50\) − 1.73205i − 0.244949i
\(51\) 2.53590 0.355097
\(52\) 0 0
\(53\) −10.3923 −1.42749 −0.713746 0.700404i \(-0.753003\pi\)
−0.713746 + 0.700404i \(0.753003\pi\)
\(54\) 6.92820i 0.942809i
\(55\) 1.26795 0.170970
\(56\) 3.46410 0.462910
\(57\) − 3.07180i − 0.406869i
\(58\) − 16.3923i − 2.15242i
\(59\) 15.1244i 1.96902i 0.175319 + 0.984512i \(0.443904\pi\)
−0.175319 + 0.984512i \(0.556096\pi\)
\(60\) − 0.732051i − 0.0945074i
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0.339746 0.0431478
\(63\) 4.92820i 0.620895i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.60770 0.197894
\(67\) − 14.3923i − 1.75830i −0.476545 0.879150i \(-0.658111\pi\)
0.476545 0.879150i \(-0.341889\pi\)
\(68\) 3.46410 0.420084
\(69\) 3.46410 0.417029
\(70\) − 3.46410i − 0.414039i
\(71\) 1.26795i 0.150478i 0.997166 + 0.0752389i \(0.0239720\pi\)
−0.997166 + 0.0752389i \(0.976028\pi\)
\(72\) − 4.26795i − 0.502983i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0.732051 0.0845299
\(76\) − 4.19615i − 0.481332i
\(77\) 2.53590 0.288992
\(78\) 0 0
\(79\) 12.3923 1.39424 0.697122 0.716953i \(-0.254464\pi\)
0.697122 + 0.716953i \(0.254464\pi\)
\(80\) 5.00000i 0.559017i
\(81\) 4.46410 0.496011
\(82\) 6.00000 0.662589
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) − 1.46410i − 0.159747i
\(85\) 3.46410i 0.375735i
\(86\) − 17.6603i − 1.90435i
\(87\) 6.92820 0.742781
\(88\) −2.19615 −0.234111
\(89\) − 0.928203i − 0.0983893i −0.998789 0.0491947i \(-0.984335\pi\)
0.998789 0.0491947i \(-0.0156655\pi\)
\(90\) −4.26795 −0.449881
\(91\) 0 0
\(92\) 4.73205 0.493350
\(93\) 0.143594i 0.0148900i
\(94\) 10.3923 1.07188
\(95\) 4.19615 0.430516
\(96\) 3.80385i 0.388229i
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 5.19615i 0.524891i
\(99\) − 3.12436i − 0.314010i
\(100\) 1.00000 0.100000
\(101\) −12.9282 −1.28640 −0.643202 0.765696i \(-0.722395\pi\)
−0.643202 + 0.765696i \(0.722395\pi\)
\(102\) 4.39230i 0.434903i
\(103\) −10.1962 −1.00466 −0.502328 0.864677i \(-0.667523\pi\)
−0.502328 + 0.864677i \(0.667523\pi\)
\(104\) 0 0
\(105\) 1.46410 0.142882
\(106\) − 18.0000i − 1.74831i
\(107\) 0.339746 0.0328445 0.0164222 0.999865i \(-0.494772\pi\)
0.0164222 + 0.999865i \(0.494772\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 2.19615i 0.209395i
\(111\) − 2.92820i − 0.277933i
\(112\) 10.0000i 0.944911i
\(113\) 15.4641 1.45474 0.727370 0.686245i \(-0.240742\pi\)
0.727370 + 0.686245i \(0.240742\pi\)
\(114\) 5.32051 0.498311
\(115\) 4.73205i 0.441266i
\(116\) 9.46410 0.878720
\(117\) 0 0
\(118\) −26.1962 −2.41155
\(119\) 6.92820i 0.635107i
\(120\) −1.26795 −0.115747
\(121\) 9.39230 0.853846
\(122\) 21.4641i 1.94327i
\(123\) 2.53590i 0.228654i
\(124\) 0.196152i 0.0176150i
\(125\) 1.00000i 0.0894427i
\(126\) −8.53590 −0.760438
\(127\) −5.80385 −0.515008 −0.257504 0.966277i \(-0.582900\pi\)
−0.257504 + 0.966277i \(0.582900\pi\)
\(128\) − 12.1244i − 1.07165i
\(129\) 7.46410 0.657178
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0.928203i 0.0807897i
\(133\) 8.39230 0.727705
\(134\) 24.9282 2.15347
\(135\) − 4.00000i − 0.344265i
\(136\) − 6.00000i − 0.514496i
\(137\) 12.9282i 1.10453i 0.833668 + 0.552265i \(0.186237\pi\)
−0.833668 + 0.552265i \(0.813763\pi\)
\(138\) 6.00000i 0.510754i
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 2.00000 0.169031
\(141\) 4.39230i 0.369899i
\(142\) −2.19615 −0.184297
\(143\) 0 0
\(144\) 12.3205 1.02671
\(145\) 9.46410i 0.785951i
\(146\) −6.92820 −0.573382
\(147\) −2.19615 −0.181136
\(148\) − 4.00000i − 0.328798i
\(149\) 19.8564i 1.62670i 0.581775 + 0.813350i \(0.302359\pi\)
−0.581775 + 0.813350i \(0.697641\pi\)
\(150\) 1.26795i 0.103528i
\(151\) 12.1962i 0.992509i 0.868177 + 0.496254i \(0.165292\pi\)
−0.868177 + 0.496254i \(0.834708\pi\)
\(152\) −7.26795 −0.589509
\(153\) 8.53590 0.690086
\(154\) 4.39230i 0.353942i
\(155\) −0.196152 −0.0157553
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 21.4641i 1.70759i
\(159\) 7.60770 0.603329
\(160\) −5.19615 −0.410792
\(161\) 9.46410i 0.745876i
\(162\) 7.73205i 0.607487i
\(163\) − 6.39230i − 0.500684i −0.968157 0.250342i \(-0.919457\pi\)
0.968157 0.250342i \(-0.0805431\pi\)
\(164\) 3.46410i 0.270501i
\(165\) −0.928203 −0.0722605
\(166\) 10.3923 0.806599
\(167\) − 12.9282i − 1.00041i −0.865906 0.500207i \(-0.833257\pi\)
0.865906 0.500207i \(-0.166743\pi\)
\(168\) −2.53590 −0.195649
\(169\) 0 0
\(170\) −6.00000 −0.460179
\(171\) − 10.3397i − 0.790700i
\(172\) 10.1962 0.777449
\(173\) −15.4641 −1.17571 −0.587857 0.808965i \(-0.700028\pi\)
−0.587857 + 0.808965i \(0.700028\pi\)
\(174\) 12.0000i 0.909718i
\(175\) 2.00000i 0.151186i
\(176\) − 6.33975i − 0.477876i
\(177\) − 11.0718i − 0.832207i
\(178\) 1.60770 0.120502
\(179\) 5.07180 0.379084 0.189542 0.981873i \(-0.439300\pi\)
0.189542 + 0.981873i \(0.439300\pi\)
\(180\) − 2.46410i − 0.183663i
\(181\) 20.3923 1.51575 0.757874 0.652401i \(-0.226238\pi\)
0.757874 + 0.652401i \(0.226238\pi\)
\(182\) 0 0
\(183\) −9.07180 −0.670607
\(184\) − 8.19615i − 0.604228i
\(185\) 4.00000 0.294086
\(186\) −0.248711 −0.0182364
\(187\) − 4.39230i − 0.321197i
\(188\) 6.00000i 0.437595i
\(189\) − 8.00000i − 0.581914i
\(190\) 7.26795i 0.527272i
\(191\) −18.9282 −1.36960 −0.684798 0.728733i \(-0.740110\pi\)
−0.684798 + 0.728733i \(0.740110\pi\)
\(192\) 0.732051 0.0528312
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) −3.46410 −0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 0.928203i 0.0661317i 0.999453 + 0.0330659i \(0.0105271\pi\)
−0.999453 + 0.0330659i \(0.989473\pi\)
\(198\) 5.41154 0.384582
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) − 1.73205i − 0.122474i
\(201\) 10.5359i 0.743145i
\(202\) − 22.3923i − 1.57552i
\(203\) 18.9282i 1.32850i
\(204\) −2.53590 −0.177548
\(205\) −3.46410 −0.241943
\(206\) − 17.6603i − 1.23045i
\(207\) 11.6603 0.810444
\(208\) 0 0
\(209\) −5.32051 −0.368027
\(210\) 2.53590i 0.174994i
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.3923 0.713746
\(213\) − 0.928203i − 0.0635994i
\(214\) 0.588457i 0.0402261i
\(215\) 10.1962i 0.695372i
\(216\) 6.92820i 0.471405i
\(217\) −0.392305 −0.0266314
\(218\) −3.46410 −0.234619
\(219\) − 2.92820i − 0.197870i
\(220\) −1.26795 −0.0854851
\(221\) 0 0
\(222\) 5.07180 0.340397
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) −10.3923 −0.694365
\(225\) 2.46410 0.164273
\(226\) 26.7846i 1.78169i
\(227\) 3.46410i 0.229920i 0.993370 + 0.114960i \(0.0366741\pi\)
−0.993370 + 0.114960i \(0.963326\pi\)
\(228\) 3.07180i 0.203435i
\(229\) 14.3923i 0.951070i 0.879697 + 0.475535i \(0.157746\pi\)
−0.879697 + 0.475535i \(0.842254\pi\)
\(230\) −8.19615 −0.540438
\(231\) −1.85641 −0.122143
\(232\) − 16.3923i − 1.07621i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) − 15.1244i − 0.984512i
\(237\) −9.07180 −0.589277
\(238\) −12.0000 −0.777844
\(239\) − 3.80385i − 0.246050i −0.992404 0.123025i \(-0.960740\pi\)
0.992404 0.123025i \(-0.0392596\pi\)
\(240\) − 3.66025i − 0.236268i
\(241\) − 18.3923i − 1.18475i −0.805661 0.592376i \(-0.798190\pi\)
0.805661 0.592376i \(-0.201810\pi\)
\(242\) 16.2679i 1.04574i
\(243\) −15.2679 −0.979439
\(244\) −12.3923 −0.793336
\(245\) − 3.00000i − 0.191663i
\(246\) −4.39230 −0.280043
\(247\) 0 0
\(248\) 0.339746 0.0215739
\(249\) 4.39230i 0.278351i
\(250\) −1.73205 −0.109545
\(251\) 14.5359 0.917498 0.458749 0.888566i \(-0.348298\pi\)
0.458749 + 0.888566i \(0.348298\pi\)
\(252\) − 4.92820i − 0.310448i
\(253\) − 6.00000i − 0.377217i
\(254\) − 10.0526i − 0.630754i
\(255\) − 2.53590i − 0.158804i
\(256\) 19.0000 1.18750
\(257\) 7.85641 0.490069 0.245035 0.969514i \(-0.421201\pi\)
0.245035 + 0.969514i \(0.421201\pi\)
\(258\) 12.9282i 0.804875i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 23.3205 1.44350
\(262\) 0 0
\(263\) 4.73205 0.291791 0.145895 0.989300i \(-0.453394\pi\)
0.145895 + 0.989300i \(0.453394\pi\)
\(264\) 1.60770 0.0989468
\(265\) 10.3923i 0.638394i
\(266\) 14.5359i 0.891253i
\(267\) 0.679492i 0.0415842i
\(268\) 14.3923i 0.879150i
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 6.92820 0.421637
\(271\) 20.9808i 1.27449i 0.770661 + 0.637245i \(0.219926\pi\)
−0.770661 + 0.637245i \(0.780074\pi\)
\(272\) 17.3205 1.05021
\(273\) 0 0
\(274\) −22.3923 −1.35277
\(275\) − 1.26795i − 0.0764602i
\(276\) −3.46410 −0.208514
\(277\) 5.60770 0.336934 0.168467 0.985707i \(-0.446118\pi\)
0.168467 + 0.985707i \(0.446118\pi\)
\(278\) − 14.5359i − 0.871805i
\(279\) 0.483340i 0.0289368i
\(280\) − 3.46410i − 0.207020i
\(281\) − 1.60770i − 0.0959071i −0.998850 0.0479535i \(-0.984730\pi\)
0.998850 0.0479535i \(-0.0152699\pi\)
\(282\) −7.60770 −0.453032
\(283\) −1.41154 −0.0839075 −0.0419538 0.999120i \(-0.513358\pi\)
−0.0419538 + 0.999120i \(0.513358\pi\)
\(284\) − 1.26795i − 0.0752389i
\(285\) −3.07180 −0.181958
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 12.8038i 0.754474i
\(289\) −5.00000 −0.294118
\(290\) −16.3923 −0.962589
\(291\) − 1.46410i − 0.0858272i
\(292\) − 4.00000i − 0.234082i
\(293\) 18.9282i 1.10580i 0.833248 + 0.552899i \(0.186478\pi\)
−0.833248 + 0.552899i \(0.813522\pi\)
\(294\) − 3.80385i − 0.221845i
\(295\) 15.1244 0.880574
\(296\) −6.92820 −0.402694
\(297\) 5.07180i 0.294295i
\(298\) −34.3923 −1.99229
\(299\) 0 0
\(300\) −0.732051 −0.0422650
\(301\) 20.3923i 1.17539i
\(302\) −21.1244 −1.21557
\(303\) 9.46410 0.543698
\(304\) − 20.9808i − 1.20333i
\(305\) − 12.3923i − 0.709581i
\(306\) 14.7846i 0.845180i
\(307\) − 22.7846i − 1.30039i −0.759769 0.650193i \(-0.774688\pi\)
0.759769 0.650193i \(-0.225312\pi\)
\(308\) −2.53590 −0.144496
\(309\) 7.46410 0.424618
\(310\) − 0.339746i − 0.0192963i
\(311\) −4.39230 −0.249065 −0.124532 0.992216i \(-0.539743\pi\)
−0.124532 + 0.992216i \(0.539743\pi\)
\(312\) 0 0
\(313\) 6.39230 0.361314 0.180657 0.983546i \(-0.442178\pi\)
0.180657 + 0.983546i \(0.442178\pi\)
\(314\) − 17.3205i − 0.977453i
\(315\) 4.92820 0.277673
\(316\) −12.3923 −0.697122
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 13.1769i 0.738925i
\(319\) − 12.0000i − 0.671871i
\(320\) 1.00000i 0.0559017i
\(321\) −0.248711 −0.0138817
\(322\) −16.3923 −0.913507
\(323\) − 14.5359i − 0.808799i
\(324\) −4.46410 −0.248006
\(325\) 0 0
\(326\) 11.0718 0.613210
\(327\) − 1.46410i − 0.0809650i
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) − 1.60770i − 0.0885007i
\(331\) − 28.5885i − 1.57136i −0.618631 0.785682i \(-0.712312\pi\)
0.618631 0.785682i \(-0.287688\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 9.85641i − 0.540128i
\(334\) 22.3923 1.22525
\(335\) −14.3923 −0.786336
\(336\) − 7.32051i − 0.399366i
\(337\) 5.60770 0.305471 0.152735 0.988267i \(-0.451192\pi\)
0.152735 + 0.988267i \(0.451192\pi\)
\(338\) 0 0
\(339\) −11.3205 −0.614846
\(340\) − 3.46410i − 0.187867i
\(341\) 0.248711 0.0134685
\(342\) 17.9090 0.968406
\(343\) − 20.0000i − 1.07990i
\(344\) − 17.6603i − 0.952177i
\(345\) − 3.46410i − 0.186501i
\(346\) − 26.7846i − 1.43995i
\(347\) 11.6603 0.625955 0.312978 0.949761i \(-0.398674\pi\)
0.312978 + 0.949761i \(0.398674\pi\)
\(348\) −6.92820 −0.371391
\(349\) − 6.39230i − 0.342172i −0.985256 0.171086i \(-0.945272\pi\)
0.985256 0.171086i \(-0.0547276\pi\)
\(350\) −3.46410 −0.185164
\(351\) 0 0
\(352\) 6.58846 0.351166
\(353\) 27.7128i 1.47500i 0.675345 + 0.737502i \(0.263995\pi\)
−0.675345 + 0.737502i \(0.736005\pi\)
\(354\) 19.1769 1.01924
\(355\) 1.26795 0.0672958
\(356\) 0.928203i 0.0491947i
\(357\) − 5.07180i − 0.268428i
\(358\) 8.78461i 0.464281i
\(359\) − 8.19615i − 0.432576i −0.976330 0.216288i \(-0.930605\pi\)
0.976330 0.216288i \(-0.0693951\pi\)
\(360\) −4.26795 −0.224941
\(361\) 1.39230 0.0732792
\(362\) 35.3205i 1.85640i
\(363\) −6.87564 −0.360878
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) − 15.7128i − 0.821322i
\(367\) 22.1962 1.15863 0.579315 0.815104i \(-0.303320\pi\)
0.579315 + 0.815104i \(0.303320\pi\)
\(368\) 23.6603 1.23338
\(369\) 8.53590i 0.444361i
\(370\) 6.92820i 0.360180i
\(371\) 20.7846i 1.07908i
\(372\) − 0.143594i − 0.00744498i
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 7.60770 0.393385
\(375\) − 0.732051i − 0.0378029i
\(376\) 10.3923 0.535942
\(377\) 0 0
\(378\) 13.8564 0.712697
\(379\) − 32.9808i − 1.69411i −0.531507 0.847054i \(-0.678374\pi\)
0.531507 0.847054i \(-0.321626\pi\)
\(380\) −4.19615 −0.215258
\(381\) 4.24871 0.217668
\(382\) − 32.7846i − 1.67741i
\(383\) − 0.928203i − 0.0474290i −0.999719 0.0237145i \(-0.992451\pi\)
0.999719 0.0237145i \(-0.00754926\pi\)
\(384\) 8.87564i 0.452933i
\(385\) − 2.53590i − 0.129241i
\(386\) −17.3205 −0.881591
\(387\) 25.1244 1.27714
\(388\) − 2.00000i − 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 16.3923 0.828994
\(392\) 5.19615i 0.262445i
\(393\) 0 0
\(394\) −1.60770 −0.0809945
\(395\) − 12.3923i − 0.623525i
\(396\) 3.12436i 0.157005i
\(397\) 12.7846i 0.641641i 0.947140 + 0.320821i \(0.103959\pi\)
−0.947140 + 0.320821i \(0.896041\pi\)
\(398\) − 34.6410i − 1.73640i
\(399\) −6.14359 −0.307564
\(400\) 5.00000 0.250000
\(401\) 23.0718i 1.15215i 0.817397 + 0.576075i \(0.195416\pi\)
−0.817397 + 0.576075i \(0.804584\pi\)
\(402\) −18.2487 −0.910163
\(403\) 0 0
\(404\) 12.9282 0.643202
\(405\) − 4.46410i − 0.221823i
\(406\) −32.7846 −1.62707
\(407\) −5.07180 −0.251400
\(408\) 4.39230i 0.217451i
\(409\) − 38.3923i − 1.89838i −0.314708 0.949189i \(-0.601906\pi\)
0.314708 0.949189i \(-0.398094\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) − 9.46410i − 0.466830i
\(412\) 10.1962 0.502328
\(413\) 30.2487 1.48844
\(414\) 20.1962i 0.992587i
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 6.14359 0.300853
\(418\) − 9.21539i − 0.450739i
\(419\) −9.46410 −0.462352 −0.231176 0.972912i \(-0.574257\pi\)
−0.231176 + 0.972912i \(0.574257\pi\)
\(420\) −1.46410 −0.0714408
\(421\) 10.7846i 0.525610i 0.964849 + 0.262805i \(0.0846475\pi\)
−0.964849 + 0.262805i \(0.915352\pi\)
\(422\) 13.8564i 0.674519i
\(423\) 14.7846i 0.718852i
\(424\) − 18.0000i − 0.874157i
\(425\) 3.46410 0.168034
\(426\) 1.60770 0.0778931
\(427\) − 24.7846i − 1.19941i
\(428\) −0.339746 −0.0164222
\(429\) 0 0
\(430\) −17.6603 −0.851653
\(431\) 19.5167i 0.940084i 0.882644 + 0.470042i \(0.155761\pi\)
−0.882644 + 0.470042i \(0.844239\pi\)
\(432\) −20.0000 −0.962250
\(433\) 6.78461 0.326048 0.163024 0.986622i \(-0.447875\pi\)
0.163024 + 0.986622i \(0.447875\pi\)
\(434\) − 0.679492i − 0.0326167i
\(435\) − 6.92820i − 0.332182i
\(436\) − 2.00000i − 0.0957826i
\(437\) − 19.8564i − 0.949861i
\(438\) 5.07180 0.242340
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 2.19615i 0.104697i
\(441\) −7.39230 −0.352015
\(442\) 0 0
\(443\) 34.9808 1.66199 0.830993 0.556283i \(-0.187773\pi\)
0.830993 + 0.556283i \(0.187773\pi\)
\(444\) 2.92820i 0.138966i
\(445\) −0.928203 −0.0440011
\(446\) −3.46410 −0.164030
\(447\) − 14.5359i − 0.687524i
\(448\) 2.00000i 0.0944911i
\(449\) − 27.4641i − 1.29611i −0.761593 0.648056i \(-0.775582\pi\)
0.761593 0.648056i \(-0.224418\pi\)
\(450\) 4.26795i 0.201193i
\(451\) 4.39230 0.206826
\(452\) −15.4641 −0.727370
\(453\) − 8.92820i − 0.419484i
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 5.32051 0.249156
\(457\) − 30.7846i − 1.44004i −0.693952 0.720022i \(-0.744132\pi\)
0.693952 0.720022i \(-0.255868\pi\)
\(458\) −24.9282 −1.16482
\(459\) −13.8564 −0.646762
\(460\) − 4.73205i − 0.220633i
\(461\) 3.46410i 0.161339i 0.996741 + 0.0806696i \(0.0257059\pi\)
−0.996741 + 0.0806696i \(0.974294\pi\)
\(462\) − 3.21539i − 0.149593i
\(463\) − 18.3923i − 0.854763i −0.904071 0.427381i \(-0.859436\pi\)
0.904071 0.427381i \(-0.140564\pi\)
\(464\) 47.3205 2.19680
\(465\) 0.143594 0.00665899
\(466\) 10.3923i 0.481414i
\(467\) −38.1962 −1.76751 −0.883754 0.467953i \(-0.844992\pi\)
−0.883754 + 0.467953i \(0.844992\pi\)
\(468\) 0 0
\(469\) −28.7846 −1.32915
\(470\) − 10.3923i − 0.479361i
\(471\) 7.32051 0.337311
\(472\) −26.1962 −1.20578
\(473\) − 12.9282i − 0.594439i
\(474\) − 15.7128i − 0.721713i
\(475\) − 4.19615i − 0.192533i
\(476\) − 6.92820i − 0.317554i
\(477\) 25.6077 1.17250
\(478\) 6.58846 0.301349
\(479\) − 18.3397i − 0.837964i −0.907994 0.418982i \(-0.862387\pi\)
0.907994 0.418982i \(-0.137613\pi\)
\(480\) 3.80385 0.173621
\(481\) 0 0
\(482\) 31.8564 1.45102
\(483\) − 6.92820i − 0.315244i
\(484\) −9.39230 −0.426923
\(485\) 2.00000 0.0908153
\(486\) − 26.4449i − 1.19956i
\(487\) − 5.60770i − 0.254109i −0.991896 0.127054i \(-0.959448\pi\)
0.991896 0.127054i \(-0.0405523\pi\)
\(488\) 21.4641i 0.971634i
\(489\) 4.67949i 0.211614i
\(490\) 5.19615 0.234738
\(491\) 9.46410 0.427109 0.213554 0.976931i \(-0.431496\pi\)
0.213554 + 0.976931i \(0.431496\pi\)
\(492\) − 2.53590i − 0.114327i
\(493\) 32.7846 1.47654
\(494\) 0 0
\(495\) −3.12436 −0.140429
\(496\) 0.980762i 0.0440375i
\(497\) 2.53590 0.113751
\(498\) −7.60770 −0.340909
\(499\) 12.9808i 0.581099i 0.956860 + 0.290549i \(0.0938380\pi\)
−0.956860 + 0.290549i \(0.906162\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 9.46410i 0.422825i
\(502\) 25.1769i 1.12370i
\(503\) −25.5167 −1.13773 −0.568866 0.822430i \(-0.692618\pi\)
−0.568866 + 0.822430i \(0.692618\pi\)
\(504\) −8.53590 −0.380219
\(505\) 12.9282i 0.575297i
\(506\) 10.3923 0.461994
\(507\) 0 0
\(508\) 5.80385 0.257504
\(509\) 32.5359i 1.44213i 0.692868 + 0.721064i \(0.256347\pi\)
−0.692868 + 0.721064i \(0.743653\pi\)
\(510\) 4.39230 0.194495
\(511\) 8.00000 0.353899
\(512\) 8.66025i 0.382733i
\(513\) 16.7846i 0.741059i
\(514\) 13.6077i 0.600210i
\(515\) 10.1962i 0.449296i
\(516\) −7.46410 −0.328589
\(517\) 7.60770 0.334586
\(518\) 13.8564i 0.608816i
\(519\) 11.3205 0.496915
\(520\) 0 0
\(521\) −7.60770 −0.333299 −0.166650 0.986016i \(-0.553295\pi\)
−0.166650 + 0.986016i \(0.553295\pi\)
\(522\) 40.3923i 1.76792i
\(523\) −13.8038 −0.603600 −0.301800 0.953371i \(-0.597588\pi\)
−0.301800 + 0.953371i \(0.597588\pi\)
\(524\) 0 0
\(525\) − 1.46410i − 0.0638986i
\(526\) 8.19615i 0.357369i
\(527\) 0.679492i 0.0295991i
\(528\) 4.64102i 0.201974i
\(529\) −0.607695 −0.0264215
\(530\) −18.0000 −0.781870
\(531\) − 37.2679i − 1.61729i
\(532\) −8.39230 −0.363853
\(533\) 0 0
\(534\) −1.17691 −0.0509301
\(535\) − 0.339746i − 0.0146885i
\(536\) 24.9282 1.07673
\(537\) −3.71281 −0.160220
\(538\) 13.6077i 0.586669i
\(539\) 3.80385i 0.163843i
\(540\) 4.00000i 0.172133i
\(541\) 5.60770i 0.241094i 0.992708 + 0.120547i \(0.0384648\pi\)
−0.992708 + 0.120547i \(0.961535\pi\)
\(542\) −36.3397 −1.56093
\(543\) −14.9282 −0.640631
\(544\) 18.0000i 0.771744i
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −1.80385 −0.0771270 −0.0385635 0.999256i \(-0.512278\pi\)
−0.0385635 + 0.999256i \(0.512278\pi\)
\(548\) − 12.9282i − 0.552265i
\(549\) −30.5359 −1.30324
\(550\) 2.19615 0.0936443
\(551\) − 39.7128i − 1.69182i
\(552\) 6.00000i 0.255377i
\(553\) − 24.7846i − 1.05395i
\(554\) 9.71281i 0.412658i
\(555\) −2.92820 −0.124295
\(556\) 8.39230 0.355913
\(557\) 25.8564i 1.09557i 0.836619 + 0.547786i \(0.184529\pi\)
−0.836619 + 0.547786i \(0.815471\pi\)
\(558\) −0.837169 −0.0354402
\(559\) 0 0
\(560\) 10.0000 0.422577
\(561\) 3.21539i 0.135754i
\(562\) 2.78461 0.117462
\(563\) −16.0526 −0.676535 −0.338267 0.941050i \(-0.609841\pi\)
−0.338267 + 0.941050i \(0.609841\pi\)
\(564\) − 4.39230i − 0.184949i
\(565\) − 15.4641i − 0.650580i
\(566\) − 2.44486i − 0.102765i
\(567\) − 8.92820i − 0.374949i
\(568\) −2.19615 −0.0921485
\(569\) 9.46410 0.396756 0.198378 0.980126i \(-0.436433\pi\)
0.198378 + 0.980126i \(0.436433\pi\)
\(570\) − 5.32051i − 0.222852i
\(571\) −15.6077 −0.653162 −0.326581 0.945169i \(-0.605897\pi\)
−0.326581 + 0.945169i \(0.605897\pi\)
\(572\) 0 0
\(573\) 13.8564 0.578860
\(574\) − 12.0000i − 0.500870i
\(575\) 4.73205 0.197340
\(576\) 2.46410 0.102671
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) − 8.66025i − 0.360219i
\(579\) − 7.32051i − 0.304230i
\(580\) − 9.46410i − 0.392975i
\(581\) −12.0000 −0.497844
\(582\) 2.53590 0.105116
\(583\) − 13.1769i − 0.545732i
\(584\) −6.92820 −0.286691
\(585\) 0 0
\(586\) −32.7846 −1.35432
\(587\) − 15.4641i − 0.638272i −0.947709 0.319136i \(-0.896607\pi\)
0.947709 0.319136i \(-0.103393\pi\)
\(588\) 2.19615 0.0905678
\(589\) 0.823085 0.0339146
\(590\) 26.1962i 1.07848i
\(591\) − 0.679492i − 0.0279506i
\(592\) − 20.0000i − 0.821995i
\(593\) 14.7846i 0.607131i 0.952811 + 0.303566i \(0.0981771\pi\)
−0.952811 + 0.303566i \(0.901823\pi\)
\(594\) −8.78461 −0.360437
\(595\) 6.92820 0.284029
\(596\) − 19.8564i − 0.813350i
\(597\) 14.6410 0.599217
\(598\) 0 0
\(599\) −28.3923 −1.16008 −0.580039 0.814589i \(-0.696963\pi\)
−0.580039 + 0.814589i \(0.696963\pi\)
\(600\) 1.26795i 0.0517638i
\(601\) −39.5692 −1.61406 −0.807031 0.590509i \(-0.798927\pi\)
−0.807031 + 0.590509i \(0.798927\pi\)
\(602\) −35.3205 −1.43956
\(603\) 35.4641i 1.44421i
\(604\) − 12.1962i − 0.496254i
\(605\) − 9.39230i − 0.381851i
\(606\) 16.3923i 0.665892i
\(607\) −26.9808 −1.09512 −0.547558 0.836768i \(-0.684442\pi\)
−0.547558 + 0.836768i \(0.684442\pi\)
\(608\) 21.8038 0.884263
\(609\) − 13.8564i − 0.561490i
\(610\) 21.4641 0.869056
\(611\) 0 0
\(612\) −8.53590 −0.345043
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 39.4641 1.59264
\(615\) 2.53590 0.102257
\(616\) 4.39230i 0.176971i
\(617\) − 21.7128i − 0.874125i −0.899431 0.437062i \(-0.856019\pi\)
0.899431 0.437062i \(-0.143981\pi\)
\(618\) 12.9282i 0.520049i
\(619\) 44.9808i 1.80793i 0.427607 + 0.903965i \(0.359357\pi\)
−0.427607 + 0.903965i \(0.640643\pi\)
\(620\) 0.196152 0.00787767
\(621\) −18.9282 −0.759563
\(622\) − 7.60770i − 0.305041i
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 11.0718i 0.442518i
\(627\) 3.89488 0.155547
\(628\) 10.0000 0.399043
\(629\) − 13.8564i − 0.552491i
\(630\) 8.53590i 0.340078i
\(631\) − 16.1962i − 0.644759i −0.946611 0.322379i \(-0.895517\pi\)
0.946611 0.322379i \(-0.104483\pi\)
\(632\) 21.4641i 0.853796i
\(633\) −5.85641 −0.232771
\(634\) −41.5692 −1.65092
\(635\) 5.80385i 0.230319i
\(636\) −7.60770 −0.301665
\(637\) 0 0
\(638\) 20.7846 0.822871
\(639\) − 3.12436i − 0.123598i
\(640\) −12.1244 −0.479257
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) − 0.430781i − 0.0170016i
\(643\) 34.7846i 1.37177i 0.727709 + 0.685886i \(0.240585\pi\)
−0.727709 + 0.685886i \(0.759415\pi\)
\(644\) − 9.46410i − 0.372938i
\(645\) − 7.46410i − 0.293899i
\(646\) 25.1769 0.990572
\(647\) 16.0526 0.631091 0.315546 0.948910i \(-0.397812\pi\)
0.315546 + 0.948910i \(0.397812\pi\)
\(648\) 7.73205i 0.303744i
\(649\) −19.1769 −0.752760
\(650\) 0 0
\(651\) 0.287187 0.0112557
\(652\) 6.39230i 0.250342i
\(653\) −19.8564 −0.777041 −0.388521 0.921440i \(-0.627014\pi\)
−0.388521 + 0.921440i \(0.627014\pi\)
\(654\) 2.53590 0.0991615
\(655\) 0 0
\(656\) 17.3205i 0.676252i
\(657\) − 9.85641i − 0.384535i
\(658\) − 20.7846i − 0.810268i
\(659\) −14.5359 −0.566238 −0.283119 0.959085i \(-0.591369\pi\)
−0.283119 + 0.959085i \(0.591369\pi\)
\(660\) 0.928203 0.0361303
\(661\) 30.7846i 1.19738i 0.800980 + 0.598691i \(0.204312\pi\)
−0.800980 + 0.598691i \(0.795688\pi\)
\(662\) 49.5167 1.92452
\(663\) 0 0
\(664\) 10.3923 0.403300
\(665\) − 8.39230i − 0.325440i
\(666\) 17.0718 0.661519
\(667\) 44.7846 1.73407
\(668\) 12.9282i 0.500207i
\(669\) − 1.46410i − 0.0566054i
\(670\) − 24.9282i − 0.963061i
\(671\) 15.7128i 0.606586i
\(672\) 7.60770 0.293473
\(673\) −6.39230 −0.246405 −0.123203 0.992382i \(-0.539317\pi\)
−0.123203 + 0.992382i \(0.539317\pi\)
\(674\) 9.71281i 0.374124i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −10.3923 −0.399409 −0.199704 0.979856i \(-0.563998\pi\)
−0.199704 + 0.979856i \(0.563998\pi\)
\(678\) − 19.6077i − 0.753029i
\(679\) 4.00000 0.153506
\(680\) −6.00000 −0.230089
\(681\) − 2.53590i − 0.0971758i
\(682\) 0.430781i 0.0164954i
\(683\) − 39.4641i − 1.51005i −0.655695 0.755026i \(-0.727624\pi\)
0.655695 0.755026i \(-0.272376\pi\)
\(684\) 10.3397i 0.395350i
\(685\) 12.9282 0.493961
\(686\) 34.6410 1.32260
\(687\) − 10.5359i − 0.401970i
\(688\) 50.9808 1.94362
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) 45.7654i 1.74100i 0.492171 + 0.870498i \(0.336203\pi\)
−0.492171 + 0.870498i \(0.663797\pi\)
\(692\) 15.4641 0.587857
\(693\) −6.24871 −0.237369
\(694\) 20.1962i 0.766635i
\(695\) 8.39230i 0.318338i
\(696\) 12.0000i 0.454859i
\(697\) 12.0000i 0.454532i
\(698\) 11.0718 0.419074
\(699\) −4.39230 −0.166132
\(700\) − 2.00000i − 0.0755929i
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −16.7846 −0.633044
\(704\) − 1.26795i − 0.0477876i
\(705\) 4.39230 0.165424
\(706\) −48.0000 −1.80650
\(707\) 25.8564i 0.972430i
\(708\) 11.0718i 0.416104i
\(709\) − 9.60770i − 0.360825i −0.983591 0.180412i \(-0.942257\pi\)
0.983591 0.180412i \(-0.0577432\pi\)
\(710\) 2.19615i 0.0824201i
\(711\) −30.5359 −1.14519
\(712\) 1.60770 0.0602509
\(713\) 0.928203i 0.0347615i
\(714\) 8.78461 0.328756
\(715\) 0 0
\(716\) −5.07180 −0.189542
\(717\) 2.78461i 0.103993i
\(718\) 14.1962 0.529796
\(719\) −1.85641 −0.0692323 −0.0346161 0.999401i \(-0.511021\pi\)
−0.0346161 + 0.999401i \(0.511021\pi\)
\(720\) − 12.3205i − 0.459158i
\(721\) 20.3923i 0.759449i
\(722\) 2.41154i 0.0897483i
\(723\) 13.4641i 0.500735i
\(724\) −20.3923 −0.757874
\(725\) 9.46410 0.351488
\(726\) − 11.9090i − 0.441983i
\(727\) −13.4115 −0.497407 −0.248703 0.968580i \(-0.580004\pi\)
−0.248703 + 0.968580i \(0.580004\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 6.92820i 0.256424i
\(731\) 35.3205 1.30638
\(732\) 9.07180 0.335303
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) 38.4449i 1.41903i
\(735\) 2.19615i 0.0810063i
\(736\) 24.5885i 0.906343i
\(737\) 18.2487 0.672200
\(738\) −14.7846 −0.544229
\(739\) 7.80385i 0.287069i 0.989645 + 0.143535i \(0.0458468\pi\)
−0.989645 + 0.143535i \(0.954153\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) − 43.8564i − 1.60894i −0.593996 0.804468i \(-0.702451\pi\)
0.593996 0.804468i \(-0.297549\pi\)
\(744\) −0.248711 −0.00911820
\(745\) 19.8564 0.727482
\(746\) − 17.3205i − 0.634149i
\(747\) 14.7846i 0.540941i
\(748\) 4.39230i 0.160599i
\(749\) − 0.679492i − 0.0248281i
\(750\) 1.26795 0.0462990
\(751\) −15.6077 −0.569533 −0.284766 0.958597i \(-0.591916\pi\)
−0.284766 + 0.958597i \(0.591916\pi\)
\(752\) 30.0000i 1.09399i
\(753\) −10.6410 −0.387780
\(754\) 0 0
\(755\) 12.1962 0.443863
\(756\) 8.00000i 0.290957i
\(757\) 18.3923 0.668480 0.334240 0.942488i \(-0.391520\pi\)
0.334240 + 0.942488i \(0.391520\pi\)
\(758\) 57.1244 2.07485
\(759\) 4.39230i 0.159431i
\(760\) 7.26795i 0.263636i
\(761\) − 7.85641i − 0.284795i −0.989810 0.142397i \(-0.954519\pi\)
0.989810 0.142397i \(-0.0454810\pi\)
\(762\) 7.35898i 0.266588i
\(763\) 4.00000 0.144810
\(764\) 18.9282 0.684798
\(765\) − 8.53590i − 0.308616i
\(766\) 1.60770 0.0580884
\(767\) 0 0
\(768\) −13.9090 −0.501897
\(769\) − 6.78461i − 0.244659i −0.992490 0.122330i \(-0.960963\pi\)
0.992490 0.122330i \(-0.0390365\pi\)
\(770\) 4.39230 0.158288
\(771\) −5.75129 −0.207128
\(772\) − 10.0000i − 0.359908i
\(773\) − 6.92820i − 0.249190i −0.992208 0.124595i \(-0.960237\pi\)
0.992208 0.124595i \(-0.0397632\pi\)
\(774\) 43.5167i 1.56417i
\(775\) 0.196152i 0.00704600i
\(776\) −3.46410 −0.124354
\(777\) −5.85641 −0.210097
\(778\) − 10.3923i − 0.372582i
\(779\) 14.5359 0.520803
\(780\) 0 0
\(781\) −1.60770 −0.0575279
\(782\) 28.3923i 1.01531i
\(783\) −37.8564 −1.35288
\(784\) −15.0000 −0.535714
\(785\) 10.0000i 0.356915i
\(786\) 0 0
\(787\) 51.5692i 1.83824i 0.393973 + 0.919122i \(0.371100\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(788\) − 0.928203i − 0.0330659i
\(789\) −3.46410 −0.123325
\(790\) 21.4641 0.763658
\(791\) − 30.9282i − 1.09968i
\(792\) 5.41154 0.192291
\(793\) 0 0
\(794\) −22.1436 −0.785847
\(795\) − 7.60770i − 0.269817i
\(796\) 20.0000 0.708881
\(797\) −28.6410 −1.01452 −0.507258 0.861794i \(-0.669341\pi\)
−0.507258 + 0.861794i \(0.669341\pi\)
\(798\) − 10.6410i − 0.376688i
\(799\) 20.7846i 0.735307i
\(800\) 5.19615i 0.183712i
\(801\) 2.28719i 0.0808138i
\(802\) −39.9615 −1.41109
\(803\) −5.07180 −0.178980
\(804\) − 10.5359i − 0.371572i
\(805\) 9.46410 0.333566
\(806\) 0 0
\(807\) −5.75129 −0.202455
\(808\) − 22.3923i − 0.787759i
\(809\) −9.46410 −0.332740 −0.166370 0.986063i \(-0.553205\pi\)
−0.166370 + 0.986063i \(0.553205\pi\)
\(810\) 7.73205 0.271677
\(811\) 28.1962i 0.990101i 0.868864 + 0.495050i \(0.164850\pi\)
−0.868864 + 0.495050i \(0.835150\pi\)
\(812\) − 18.9282i − 0.664250i
\(813\) − 15.3590i − 0.538663i
\(814\) − 8.78461i − 0.307900i
\(815\) −6.39230 −0.223913
\(816\) −12.6795 −0.443871
\(817\) − 42.7846i − 1.49684i
\(818\) 66.4974 2.32503
\(819\) 0 0
\(820\) 3.46410 0.120972
\(821\) − 40.6410i − 1.41838i −0.705017 0.709191i \(-0.749061\pi\)
0.705017 0.709191i \(-0.250939\pi\)
\(822\) 16.3923 0.571747
\(823\) 46.5885 1.62397 0.811986 0.583677i \(-0.198387\pi\)
0.811986 + 0.583677i \(0.198387\pi\)
\(824\) − 17.6603i − 0.615224i
\(825\) 0.928203i 0.0323159i
\(826\) 52.3923i 1.82296i
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) −11.6603 −0.405222
\(829\) 20.3923 0.708254 0.354127 0.935197i \(-0.384778\pi\)
0.354127 + 0.935197i \(0.384778\pi\)
\(830\) − 10.3923i − 0.360722i
\(831\) −4.10512 −0.142405
\(832\) 0 0
\(833\) −10.3923 −0.360072
\(834\) 10.6410i 0.368468i
\(835\) −12.9282 −0.447399
\(836\) 5.32051 0.184014
\(837\) − 0.784610i − 0.0271201i
\(838\) − 16.3923i − 0.566263i
\(839\) − 17.6603i − 0.609700i −0.952400 0.304850i \(-0.901394\pi\)
0.952400 0.304850i \(-0.0986063\pi\)
\(840\) 2.53590i 0.0874968i
\(841\) 60.5692 2.08859
\(842\) −18.6795 −0.643738
\(843\) 1.17691i 0.0405351i
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −25.6077 −0.880411
\(847\) − 18.7846i − 0.645447i
\(848\) 51.9615 1.78437
\(849\) 1.03332 0.0354635
\(850\) 6.00000i 0.205798i
\(851\) − 18.9282i − 0.648850i
\(852\) 0.928203i 0.0317997i
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 42.9282 1.46897
\(855\) −10.3397 −0.353612
\(856\) 0.588457i 0.0201131i
\(857\) −47.5692 −1.62493 −0.812467 0.583007i \(-0.801876\pi\)
−0.812467 + 0.583007i \(0.801876\pi\)
\(858\) 0 0
\(859\) 45.1769 1.54142 0.770708 0.637188i \(-0.219903\pi\)
0.770708 + 0.637188i \(0.219903\pi\)
\(860\) − 10.1962i − 0.347686i
\(861\) 5.07180 0.172846
\(862\) −33.8038 −1.15136
\(863\) 2.78461i 0.0947892i 0.998876 + 0.0473946i \(0.0150918\pi\)
−0.998876 + 0.0473946i \(0.984908\pi\)
\(864\) − 20.7846i − 0.707107i
\(865\) 15.4641i 0.525795i
\(866\) 11.7513i 0.399325i
\(867\) 3.66025 0.124309
\(868\) 0.392305 0.0133157
\(869\) 15.7128i 0.533021i
\(870\) 12.0000 0.406838
\(871\) 0 0
\(872\) −3.46410 −0.117309
\(873\) − 4.92820i − 0.166794i
\(874\) 34.3923 1.16334
\(875\) 2.00000 0.0676123
\(876\) 2.92820i 0.0989348i
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) − 55.4256i − 1.87052i
\(879\) − 13.8564i − 0.467365i
\(880\) −6.33975 −0.213713
\(881\) 12.6795 0.427183 0.213591 0.976923i \(-0.431484\pi\)
0.213591 + 0.976923i \(0.431484\pi\)
\(882\) − 12.8038i − 0.431128i
\(883\) −34.1962 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(884\) 0 0
\(885\) −11.0718 −0.372174
\(886\) 60.5885i 2.03551i
\(887\) 17.9090 0.601324 0.300662 0.953731i \(-0.402792\pi\)
0.300662 + 0.953731i \(0.402792\pi\)
\(888\) 5.07180 0.170198
\(889\) 11.6077i 0.389310i
\(890\) − 1.60770i − 0.0538901i
\(891\) 5.66025i 0.189626i
\(892\) − 2.00000i − 0.0669650i
\(893\) 25.1769 0.842513
\(894\) 25.1769 0.842042
\(895\) − 5.07180i − 0.169531i
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 47.5692 1.58741
\(899\) 1.85641i 0.0619146i
\(900\) −2.46410 −0.0821367
\(901\) 36.0000 1.19933
\(902\) 7.60770i 0.253309i
\(903\) − 14.9282i − 0.496779i
\(904\) 26.7846i 0.890843i
\(905\) − 20.3923i − 0.677863i
\(906\) 15.4641 0.513760
\(907\) −39.7654 −1.32039 −0.660194 0.751095i \(-0.729526\pi\)
−0.660194 + 0.751095i \(0.729526\pi\)
\(908\) − 3.46410i − 0.114960i
\(909\) 31.8564 1.05661
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 15.3590i 0.508587i
\(913\) 7.60770 0.251778
\(914\) 53.3205 1.76369
\(915\) 9.07180i 0.299904i
\(916\) − 14.3923i − 0.475535i
\(917\) 0 0
\(918\) − 24.0000i − 0.792118i
\(919\) −53.1769 −1.75414 −0.877072 0.480358i \(-0.840507\pi\)
−0.877072 + 0.480358i \(0.840507\pi\)
\(920\) −8.19615 −0.270219
\(921\) 16.6795i 0.549608i
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) 1.85641 0.0610713
\(925\) − 4.00000i − 0.131519i
\(926\) 31.8564 1.04687
\(927\) 25.1244 0.825192
\(928\) 49.1769i 1.61431i
\(929\) 51.4641i 1.68848i 0.535963 + 0.844241i \(0.319948\pi\)
−0.535963 + 0.844241i \(0.680052\pi\)
\(930\) 0.248711i 0.00815557i
\(931\) 12.5885i 0.412570i
\(932\) −6.00000 −0.196537
\(933\) 3.21539 0.105267
\(934\) − 66.1577i − 2.16475i
\(935\) −4.39230 −0.143644
\(936\) 0 0
\(937\) −6.78461 −0.221644 −0.110822 0.993840i \(-0.535348\pi\)
−0.110822 + 0.993840i \(0.535348\pi\)
\(938\) − 49.8564i − 1.62787i
\(939\) −4.67949 −0.152709
\(940\) 6.00000 0.195698
\(941\) − 31.1769i − 1.01634i −0.861257 0.508169i \(-0.830322\pi\)
0.861257 0.508169i \(-0.169678\pi\)
\(942\) 12.6795i 0.413120i
\(943\) 16.3923i 0.533807i
\(944\) − 75.6218i − 2.46128i
\(945\) −8.00000 −0.260240
\(946\) 22.3923 0.728037
\(947\) 28.6410i 0.930708i 0.885125 + 0.465354i \(0.154073\pi\)
−0.885125 + 0.465354i \(0.845927\pi\)
\(948\) 9.07180 0.294638
\(949\) 0 0
\(950\) 7.26795 0.235803
\(951\) − 17.5692i − 0.569721i
\(952\) −12.0000 −0.388922
\(953\) 12.9282 0.418786 0.209393 0.977832i \(-0.432851\pi\)
0.209393 + 0.977832i \(0.432851\pi\)
\(954\) 44.3538i 1.43601i
\(955\) 18.9282i 0.612502i
\(956\) 3.80385i 0.123025i
\(957\) 8.78461i 0.283966i
\(958\) 31.7654 1.02629
\(959\) 25.8564 0.834947
\(960\) − 0.732051i − 0.0236268i
\(961\) 30.9615 0.998759
\(962\) 0 0
\(963\) −0.837169 −0.0269774
\(964\) 18.3923i 0.592376i
\(965\) 10.0000 0.321911
\(966\) 12.0000 0.386094
\(967\) − 29.6077i − 0.952119i −0.879413 0.476060i \(-0.842065\pi\)
0.879413 0.476060i \(-0.157935\pi\)
\(968\) 16.2679i 0.522872i
\(969\) 10.6410i 0.341839i
\(970\) 3.46410i 0.111226i
\(971\) 5.07180 0.162762 0.0813809 0.996683i \(-0.474067\pi\)
0.0813809 + 0.996683i \(0.474067\pi\)
\(972\) 15.2679 0.489720
\(973\) 16.7846i 0.538090i
\(974\) 9.71281 0.311219
\(975\) 0 0
\(976\) −61.9615 −1.98334
\(977\) 39.7128i 1.27053i 0.772296 + 0.635263i \(0.219108\pi\)
−0.772296 + 0.635263i \(0.780892\pi\)
\(978\) −8.10512 −0.259173
\(979\) 1.17691 0.0376144
\(980\) 3.00000i 0.0958315i
\(981\) − 4.92820i − 0.157345i
\(982\) 16.3923i 0.523099i
\(983\) 13.6077i 0.434018i 0.976170 + 0.217009i \(0.0696301\pi\)
−0.976170 + 0.217009i \(0.930370\pi\)
\(984\) −4.39230 −0.140022
\(985\) 0.928203 0.0295750
\(986\) 56.7846i 1.80839i
\(987\) 8.78461 0.279617
\(988\) 0 0
\(989\) 48.2487 1.53422
\(990\) − 5.41154i − 0.171990i
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −1.01924 −0.0323608
\(993\) 20.9282i 0.664136i
\(994\) 4.39230i 0.139315i
\(995\) 20.0000i 0.634043i
\(996\) − 4.39230i − 0.139176i
\(997\) 54.3923 1.72262 0.861311 0.508078i \(-0.169644\pi\)
0.861311 + 0.508078i \(0.169644\pi\)
\(998\) −22.4833 −0.711698
\(999\) 16.0000i 0.506218i
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 845.2.c.e.506.3 4
13.2 odd 12 845.2.e.e.191.1 4
13.3 even 3 845.2.m.a.316.2 4
13.4 even 6 845.2.m.a.361.2 4
13.5 odd 4 65.2.a.c.1.2 2
13.6 odd 12 845.2.e.e.146.1 4
13.7 odd 12 845.2.e.f.146.2 4
13.8 odd 4 845.2.a.d.1.1 2
13.9 even 3 845.2.m.c.361.2 4
13.10 even 6 845.2.m.c.316.2 4
13.11 odd 12 845.2.e.f.191.2 4
13.12 even 2 inner 845.2.c.e.506.1 4
39.5 even 4 585.2.a.k.1.1 2
39.8 even 4 7605.2.a.be.1.2 2
52.31 even 4 1040.2.a.h.1.2 2
65.18 even 4 325.2.b.e.274.2 4
65.34 odd 4 4225.2.a.w.1.2 2
65.44 odd 4 325.2.a.g.1.1 2
65.57 even 4 325.2.b.e.274.3 4
91.83 even 4 3185.2.a.k.1.2 2
104.5 odd 4 4160.2.a.y.1.2 2
104.83 even 4 4160.2.a.bj.1.1 2
143.109 even 4 7865.2.a.h.1.1 2
156.83 odd 4 9360.2.a.cm.1.2 2
195.44 even 4 2925.2.a.z.1.2 2
195.83 odd 4 2925.2.c.v.2224.3 4
195.122 odd 4 2925.2.c.v.2224.2 4
260.239 even 4 5200.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.2 2 13.5 odd 4
325.2.a.g.1.1 2 65.44 odd 4
325.2.b.e.274.2 4 65.18 even 4
325.2.b.e.274.3 4 65.57 even 4
585.2.a.k.1.1 2 39.5 even 4
845.2.a.d.1.1 2 13.8 odd 4
845.2.c.e.506.1 4 13.12 even 2 inner
845.2.c.e.506.3 4 1.1 even 1 trivial
845.2.e.e.146.1 4 13.6 odd 12
845.2.e.e.191.1 4 13.2 odd 12
845.2.e.f.146.2 4 13.7 odd 12
845.2.e.f.191.2 4 13.11 odd 12
845.2.m.a.316.2 4 13.3 even 3
845.2.m.a.361.2 4 13.4 even 6
845.2.m.c.316.2 4 13.10 even 6
845.2.m.c.361.2 4 13.9 even 3
1040.2.a.h.1.2 2 52.31 even 4
2925.2.a.z.1.2 2 195.44 even 4
2925.2.c.v.2224.2 4 195.122 odd 4
2925.2.c.v.2224.3 4 195.83 odd 4
3185.2.a.k.1.2 2 91.83 even 4
4160.2.a.y.1.2 2 104.5 odd 4
4160.2.a.bj.1.1 2 104.83 even 4
4225.2.a.w.1.2 2 65.34 odd 4
5200.2.a.ca.1.1 2 260.239 even 4
7605.2.a.be.1.2 2 39.8 even 4
7865.2.a.h.1.1 2 143.109 even 4
9360.2.a.cm.1.2 2 156.83 odd 4