Properties

Label 3234.2.e.a.2155.10
Level $3234$
Weight $2$
Character 3234.2155
Analytic conductor $25.824$
Analytic rank $0$
Dimension $16$
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] = [3234,2,Mod(2155,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.2155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (of order \(2\), degree \(1\), 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: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + 130215 x^{8} - 239606 x^{7} + 378750 x^{6} - 477124 x^{5} + \cdots + 13417 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 462)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2155.10
Root \(0.500000 + 3.32851i\) of defining polynomial
Character \(\chi\) \(=\) 3234.2155
Dual form 3234.2.e.a.2155.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -2.46248i q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -2.46248i q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +2.46248 q^{10} +(3.20979 - 0.834997i) q^{11} -1.00000i q^{12} -1.32035 q^{13} +2.46248 q^{15} +1.00000 q^{16} +4.47479 q^{17} -1.00000i q^{18} +4.43177 q^{19} +2.46248i q^{20} +(0.834997 + 3.20979i) q^{22} -8.28995 q^{23} +1.00000 q^{24} -1.06381 q^{25} -1.32035i q^{26} -1.00000i q^{27} -1.44409i q^{29} +2.46248i q^{30} -2.71125i q^{31} +1.00000i q^{32} +(0.834997 + 3.20979i) q^{33} +4.47479i q^{34} +1.00000 q^{36} -4.93126 q^{37} +4.43177i q^{38} -1.32035i q^{39} -2.46248 q^{40} -8.18233 q^{41} -9.63797i q^{43} +(-3.20979 + 0.834997i) q^{44} +2.46248i q^{45} -8.28995i q^{46} -0.767488i q^{47} +1.00000i q^{48} -1.06381i q^{50} +4.47479i q^{51} +1.32035 q^{52} -1.89194 q^{53} +1.00000 q^{54} +(-2.05616 - 7.90406i) q^{55} +4.43177i q^{57} +1.44409 q^{58} -7.78781i q^{59} -2.46248 q^{60} +4.47886 q^{61} +2.71125 q^{62} -1.00000 q^{64} +3.25135i q^{65} +(-3.20979 + 0.834997i) q^{66} -1.72369 q^{67} -4.47479 q^{68} -8.28995i q^{69} -10.3771 q^{71} +1.00000i q^{72} +11.1724 q^{73} -4.93126i q^{74} -1.06381i q^{75} -4.43177 q^{76} +1.32035 q^{78} +2.92867i q^{79} -2.46248i q^{80} +1.00000 q^{81} -8.18233i q^{82} +10.1118 q^{83} -11.0191i q^{85} +9.63797 q^{86} +1.44409 q^{87} +(-0.834997 - 3.20979i) q^{88} -12.6544i q^{89} -2.46248 q^{90} +8.28995 q^{92} +2.71125 q^{93} +0.767488 q^{94} -10.9132i q^{95} -1.00000 q^{96} +1.37808i q^{97} +(-3.20979 + 0.834997i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9} - 4 q^{10} + 8 q^{11} - 4 q^{15} + 16 q^{16} + 20 q^{19} + 2 q^{22} + 8 q^{23} + 16 q^{24} - 20 q^{25} + 2 q^{33} + 16 q^{36} - 28 q^{37} + 4 q^{40} + 32 q^{41} - 8 q^{44} + 16 q^{54} - 14 q^{55} + 4 q^{60} - 56 q^{61} - 8 q^{62} - 16 q^{64} - 8 q^{66} + 32 q^{67} - 56 q^{71} + 88 q^{73} - 20 q^{76} + 16 q^{81} + 8 q^{83} + 24 q^{86} - 2 q^{88} + 4 q^{90} - 8 q^{92} - 8 q^{93} - 28 q^{94} - 16 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(1079\) \(2059\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) 2.46248i 1.10126i −0.834751 0.550628i \(-0.814388\pi\)
0.834751 0.550628i \(-0.185612\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 2.46248 0.778705
\(11\) 3.20979 0.834997i 0.967789 0.251761i
\(12\) 1.00000i 0.288675i
\(13\) −1.32035 −0.366201 −0.183100 0.983094i \(-0.558613\pi\)
−0.183100 + 0.983094i \(0.558613\pi\)
\(14\) 0 0
\(15\) 2.46248 0.635810
\(16\) 1.00000 0.250000
\(17\) 4.47479 1.08529 0.542647 0.839961i \(-0.317422\pi\)
0.542647 + 0.839961i \(0.317422\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.43177 1.01672 0.508359 0.861145i \(-0.330252\pi\)
0.508359 + 0.861145i \(0.330252\pi\)
\(20\) 2.46248i 0.550628i
\(21\) 0 0
\(22\) 0.834997 + 3.20979i 0.178022 + 0.684330i
\(23\) −8.28995 −1.72857 −0.864287 0.502999i \(-0.832230\pi\)
−0.864287 + 0.502999i \(0.832230\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.06381 −0.212763
\(26\) 1.32035i 0.258943i
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 1.44409i 0.268162i −0.990970 0.134081i \(-0.957192\pi\)
0.990970 0.134081i \(-0.0428082\pi\)
\(30\) 2.46248i 0.449585i
\(31\) 2.71125i 0.486955i −0.969907 0.243477i \(-0.921712\pi\)
0.969907 0.243477i \(-0.0782881\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0.834997 + 3.20979i 0.145354 + 0.558753i
\(34\) 4.47479i 0.767419i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.93126 −0.810695 −0.405347 0.914163i \(-0.632849\pi\)
−0.405347 + 0.914163i \(0.632849\pi\)
\(38\) 4.43177i 0.718929i
\(39\) 1.32035i 0.211426i
\(40\) −2.46248 −0.389352
\(41\) −8.18233 −1.27787 −0.638933 0.769263i \(-0.720624\pi\)
−0.638933 + 0.769263i \(0.720624\pi\)
\(42\) 0 0
\(43\) 9.63797i 1.46978i −0.678188 0.734888i \(-0.737235\pi\)
0.678188 0.734888i \(-0.262765\pi\)
\(44\) −3.20979 + 0.834997i −0.483895 + 0.125881i
\(45\) 2.46248i 0.367085i
\(46\) 8.28995i 1.22229i
\(47\) 0.767488i 0.111950i −0.998432 0.0559748i \(-0.982173\pi\)
0.998432 0.0559748i \(-0.0178266\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0 0
\(50\) 1.06381i 0.150446i
\(51\) 4.47479i 0.626595i
\(52\) 1.32035 0.183100
\(53\) −1.89194 −0.259878 −0.129939 0.991522i \(-0.541478\pi\)
−0.129939 + 0.991522i \(0.541478\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.05616 7.90406i −0.277253 1.06578i
\(56\) 0 0
\(57\) 4.43177i 0.587003i
\(58\) 1.44409 0.189619
\(59\) 7.78781i 1.01389i −0.861980 0.506943i \(-0.830775\pi\)
0.861980 0.506943i \(-0.169225\pi\)
\(60\) −2.46248 −0.317905
\(61\) 4.47886 0.573459 0.286729 0.958012i \(-0.407432\pi\)
0.286729 + 0.958012i \(0.407432\pi\)
\(62\) 2.71125 0.344329
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.25135i 0.403280i
\(66\) −3.20979 + 0.834997i −0.395098 + 0.102781i
\(67\) −1.72369 −0.210582 −0.105291 0.994441i \(-0.533577\pi\)
−0.105291 + 0.994441i \(0.533577\pi\)
\(68\) −4.47479 −0.542647
\(69\) 8.28995i 0.997992i
\(70\) 0 0
\(71\) −10.3771 −1.23153 −0.615766 0.787929i \(-0.711153\pi\)
−0.615766 + 0.787929i \(0.711153\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 11.1724 1.30764 0.653818 0.756652i \(-0.273166\pi\)
0.653818 + 0.756652i \(0.273166\pi\)
\(74\) 4.93126i 0.573248i
\(75\) 1.06381i 0.122839i
\(76\) −4.43177 −0.508359
\(77\) 0 0
\(78\) 1.32035 0.149501
\(79\) 2.92867i 0.329501i 0.986335 + 0.164750i \(0.0526819\pi\)
−0.986335 + 0.164750i \(0.947318\pi\)
\(80\) 2.46248i 0.275314i
\(81\) 1.00000 0.111111
\(82\) 8.18233i 0.903587i
\(83\) 10.1118 1.10991 0.554957 0.831879i \(-0.312735\pi\)
0.554957 + 0.831879i \(0.312735\pi\)
\(84\) 0 0
\(85\) 11.0191i 1.19519i
\(86\) 9.63797 1.03929
\(87\) 1.44409 0.154823
\(88\) −0.834997 3.20979i −0.0890110 0.342165i
\(89\) 12.6544i 1.34136i −0.741745 0.670682i \(-0.766002\pi\)
0.741745 0.670682i \(-0.233998\pi\)
\(90\) −2.46248 −0.259568
\(91\) 0 0
\(92\) 8.28995 0.864287
\(93\) 2.71125 0.281143
\(94\) 0.767488 0.0791603
\(95\) 10.9132i 1.11967i
\(96\) −1.00000 −0.102062
\(97\) 1.37808i 0.139923i 0.997550 + 0.0699613i \(0.0222876\pi\)
−0.997550 + 0.0699613i \(0.977712\pi\)
\(98\) 0 0
\(99\) −3.20979 + 0.834997i −0.322596 + 0.0839204i
\(100\) 1.06381 0.106381
\(101\) 7.56722 0.752967 0.376483 0.926423i \(-0.377133\pi\)
0.376483 + 0.926423i \(0.377133\pi\)
\(102\) −4.47479 −0.443070
\(103\) 16.4958i 1.62538i 0.582697 + 0.812690i \(0.301998\pi\)
−0.582697 + 0.812690i \(0.698002\pi\)
\(104\) 1.32035i 0.129471i
\(105\) 0 0
\(106\) 1.89194i 0.183762i
\(107\) 5.76864i 0.557675i −0.960338 0.278838i \(-0.910051\pi\)
0.960338 0.278838i \(-0.0899492\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 8.63196i 0.826792i −0.910551 0.413396i \(-0.864342\pi\)
0.910551 0.413396i \(-0.135658\pi\)
\(110\) 7.90406 2.05616i 0.753622 0.196048i
\(111\) 4.93126i 0.468055i
\(112\) 0 0
\(113\) 17.4933 1.64563 0.822817 0.568306i \(-0.192401\pi\)
0.822817 + 0.568306i \(0.192401\pi\)
\(114\) −4.43177 −0.415074
\(115\) 20.4138i 1.90360i
\(116\) 1.44409i 0.134081i
\(117\) 1.32035 0.122067
\(118\) 7.78781 0.716926
\(119\) 0 0
\(120\) 2.46248i 0.224793i
\(121\) 9.60556 5.36034i 0.873233 0.487303i
\(122\) 4.47886i 0.405497i
\(123\) 8.18233i 0.737776i
\(124\) 2.71125i 0.243477i
\(125\) 9.69279i 0.866949i
\(126\) 0 0
\(127\) 12.0959i 1.07333i −0.843794 0.536667i \(-0.819683\pi\)
0.843794 0.536667i \(-0.180317\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 9.63797 0.848576
\(130\) −3.25135 −0.285162
\(131\) −17.2415 −1.50640 −0.753200 0.657792i \(-0.771491\pi\)
−0.753200 + 0.657792i \(0.771491\pi\)
\(132\) −0.834997 3.20979i −0.0726772 0.279377i
\(133\) 0 0
\(134\) 1.72369i 0.148904i
\(135\) −2.46248 −0.211937
\(136\) 4.47479i 0.383710i
\(137\) 2.50957 0.214407 0.107204 0.994237i \(-0.465810\pi\)
0.107204 + 0.994237i \(0.465810\pi\)
\(138\) 8.28995 0.705687
\(139\) 19.9532 1.69241 0.846206 0.532856i \(-0.178881\pi\)
0.846206 + 0.532856i \(0.178881\pi\)
\(140\) 0 0
\(141\) 0.767488 0.0646341
\(142\) 10.3771i 0.870824i
\(143\) −4.23807 + 1.10249i −0.354405 + 0.0921951i
\(144\) −1.00000 −0.0833333
\(145\) −3.55606 −0.295314
\(146\) 11.1724i 0.924638i
\(147\) 0 0
\(148\) 4.93126 0.405347
\(149\) 9.78029i 0.801232i −0.916246 0.400616i \(-0.868796\pi\)
0.916246 0.400616i \(-0.131204\pi\)
\(150\) 1.06381 0.0868600
\(151\) 2.67337i 0.217556i −0.994066 0.108778i \(-0.965306\pi\)
0.994066 0.108778i \(-0.0346937\pi\)
\(152\) 4.43177i 0.359464i
\(153\) −4.47479 −0.361765
\(154\) 0 0
\(155\) −6.67640 −0.536261
\(156\) 1.32035i 0.105713i
\(157\) 15.8345i 1.26373i −0.775079 0.631865i \(-0.782290\pi\)
0.775079 0.631865i \(-0.217710\pi\)
\(158\) −2.92867 −0.232992
\(159\) 1.89194i 0.150041i
\(160\) 2.46248 0.194676
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) −20.3615 −1.59484 −0.797418 0.603428i \(-0.793801\pi\)
−0.797418 + 0.603428i \(0.793801\pi\)
\(164\) 8.18233 0.638933
\(165\) 7.90406 2.05616i 0.615330 0.160072i
\(166\) 10.1118i 0.784827i
\(167\) 16.1119 1.24678 0.623389 0.781912i \(-0.285755\pi\)
0.623389 + 0.781912i \(0.285755\pi\)
\(168\) 0 0
\(169\) −11.2567 −0.865897
\(170\) 11.0191 0.845124
\(171\) −4.43177 −0.338906
\(172\) 9.63797i 0.734888i
\(173\) 7.24694 0.550975 0.275487 0.961305i \(-0.411161\pi\)
0.275487 + 0.961305i \(0.411161\pi\)
\(174\) 1.44409i 0.109477i
\(175\) 0 0
\(176\) 3.20979 0.834997i 0.241947 0.0629403i
\(177\) 7.78781 0.585367
\(178\) 12.6544 0.948487
\(179\) 11.6016 0.867141 0.433571 0.901120i \(-0.357253\pi\)
0.433571 + 0.901120i \(0.357253\pi\)
\(180\) 2.46248i 0.183543i
\(181\) 7.74423i 0.575624i 0.957687 + 0.287812i \(0.0929279\pi\)
−0.957687 + 0.287812i \(0.907072\pi\)
\(182\) 0 0
\(183\) 4.47886i 0.331087i
\(184\) 8.28995i 0.611143i
\(185\) 12.1431i 0.892782i
\(186\) 2.71125i 0.198798i
\(187\) 14.3631 3.73643i 1.05034 0.273235i
\(188\) 0.767488i 0.0559748i
\(189\) 0 0
\(190\) 10.9132 0.791724
\(191\) −21.2314 −1.53625 −0.768125 0.640300i \(-0.778810\pi\)
−0.768125 + 0.640300i \(0.778810\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 3.51327i 0.252891i 0.991974 + 0.126445i \(0.0403569\pi\)
−0.991974 + 0.126445i \(0.959643\pi\)
\(194\) −1.37808 −0.0989402
\(195\) −3.25135 −0.232834
\(196\) 0 0
\(197\) 21.9715i 1.56540i −0.622397 0.782702i \(-0.713841\pi\)
0.622397 0.782702i \(-0.286159\pi\)
\(198\) −0.834997 3.20979i −0.0593407 0.228110i
\(199\) 1.44180i 0.102206i −0.998693 0.0511032i \(-0.983726\pi\)
0.998693 0.0511032i \(-0.0162737\pi\)
\(200\) 1.06381i 0.0752229i
\(201\) 1.72369i 0.121580i
\(202\) 7.56722i 0.532428i
\(203\) 0 0
\(204\) 4.47479i 0.313298i
\(205\) 20.1488i 1.40726i
\(206\) −16.4958 −1.14932
\(207\) 8.28995 0.576191
\(208\) −1.32035 −0.0915501
\(209\) 14.2251 3.70052i 0.983970 0.255970i
\(210\) 0 0
\(211\) 0.163763i 0.0112739i 0.999984 + 0.00563697i \(0.00179431\pi\)
−0.999984 + 0.00563697i \(0.998206\pi\)
\(212\) 1.89194 0.129939
\(213\) 10.3771i 0.711025i
\(214\) 5.76864 0.394336
\(215\) −23.7333 −1.61860
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 8.63196 0.584630
\(219\) 11.1724i 0.754964i
\(220\) 2.05616 + 7.90406i 0.138627 + 0.532891i
\(221\) −5.90830 −0.397436
\(222\) 4.93126 0.330965
\(223\) 27.3847i 1.83382i 0.399099 + 0.916908i \(0.369323\pi\)
−0.399099 + 0.916908i \(0.630677\pi\)
\(224\) 0 0
\(225\) 1.06381 0.0709209
\(226\) 17.4933i 1.16364i
\(227\) −3.90255 −0.259021 −0.129511 0.991578i \(-0.541341\pi\)
−0.129511 + 0.991578i \(0.541341\pi\)
\(228\) 4.43177i 0.293501i
\(229\) 5.04055i 0.333089i −0.986034 0.166544i \(-0.946739\pi\)
0.986034 0.166544i \(-0.0532609\pi\)
\(230\) −20.4138 −1.34605
\(231\) 0 0
\(232\) −1.44409 −0.0948095
\(233\) 21.2457i 1.39185i −0.718113 0.695927i \(-0.754994\pi\)
0.718113 0.695927i \(-0.245006\pi\)
\(234\) 1.32035i 0.0863143i
\(235\) −1.88992 −0.123285
\(236\) 7.78781i 0.506943i
\(237\) −2.92867 −0.190237
\(238\) 0 0
\(239\) 5.64110i 0.364893i −0.983216 0.182446i \(-0.941598\pi\)
0.983216 0.182446i \(-0.0584016\pi\)
\(240\) 2.46248 0.158952
\(241\) −7.77466 −0.500810 −0.250405 0.968141i \(-0.580564\pi\)
−0.250405 + 0.968141i \(0.580564\pi\)
\(242\) 5.36034 + 9.60556i 0.344576 + 0.617469i
\(243\) 1.00000i 0.0641500i
\(244\) −4.47886 −0.286729
\(245\) 0 0
\(246\) 8.18233 0.521686
\(247\) −5.85152 −0.372323
\(248\) −2.71125 −0.172164
\(249\) 10.1118i 0.640809i
\(250\) 9.69279 0.613026
\(251\) 22.5045i 1.42047i −0.703963 0.710236i \(-0.748588\pi\)
0.703963 0.710236i \(-0.251412\pi\)
\(252\) 0 0
\(253\) −26.6090 + 6.92208i −1.67290 + 0.435188i
\(254\) 12.0959 0.758962
\(255\) 11.0191 0.690041
\(256\) 1.00000 0.0625000
\(257\) 3.01578i 0.188119i 0.995567 + 0.0940597i \(0.0299845\pi\)
−0.995567 + 0.0940597i \(0.970016\pi\)
\(258\) 9.63797i 0.600034i
\(259\) 0 0
\(260\) 3.25135i 0.201640i
\(261\) 1.44409i 0.0893872i
\(262\) 17.2415i 1.06519i
\(263\) 28.1819i 1.73777i −0.495011 0.868887i \(-0.664836\pi\)
0.495011 0.868887i \(-0.335164\pi\)
\(264\) 3.20979 0.834997i 0.197549 0.0513905i
\(265\) 4.65887i 0.286192i
\(266\) 0 0
\(267\) 12.6544 0.774437
\(268\) 1.72369 0.105291
\(269\) 0.604546i 0.0368598i 0.999830 + 0.0184299i \(0.00586675\pi\)
−0.999830 + 0.0184299i \(0.994133\pi\)
\(270\) 2.46248i 0.149862i
\(271\) 19.7931 1.20234 0.601172 0.799120i \(-0.294701\pi\)
0.601172 + 0.799120i \(0.294701\pi\)
\(272\) 4.47479 0.271324
\(273\) 0 0
\(274\) 2.50957i 0.151609i
\(275\) −3.41462 + 0.888281i −0.205909 + 0.0535654i
\(276\) 8.28995i 0.498996i
\(277\) 2.16446i 0.130050i −0.997884 0.0650249i \(-0.979287\pi\)
0.997884 0.0650249i \(-0.0207127\pi\)
\(278\) 19.9532i 1.19672i
\(279\) 2.71125i 0.162318i
\(280\) 0 0
\(281\) 29.4916i 1.75932i 0.475601 + 0.879661i \(0.342231\pi\)
−0.475601 + 0.879661i \(0.657769\pi\)
\(282\) 0.767488i 0.0457032i
\(283\) 28.3880 1.68749 0.843745 0.536744i \(-0.180346\pi\)
0.843745 + 0.536744i \(0.180346\pi\)
\(284\) 10.3771 0.615766
\(285\) 10.9132 0.646440
\(286\) −1.10249 4.23807i −0.0651918 0.250602i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 3.02370 0.177865
\(290\) 3.55606i 0.208819i
\(291\) −1.37808 −0.0807843
\(292\) −11.1724 −0.653818
\(293\) 9.39403 0.548805 0.274403 0.961615i \(-0.411520\pi\)
0.274403 + 0.961615i \(0.411520\pi\)
\(294\) 0 0
\(295\) −19.1773 −1.11655
\(296\) 4.93126i 0.286624i
\(297\) −0.834997 3.20979i −0.0484515 0.186251i
\(298\) 9.78029 0.566557
\(299\) 10.9457 0.633005
\(300\) 1.06381i 0.0614193i
\(301\) 0 0
\(302\) 2.67337 0.153835
\(303\) 7.56722i 0.434726i
\(304\) 4.43177 0.254180
\(305\) 11.0291i 0.631524i
\(306\) 4.47479i 0.255806i
\(307\) −26.9732 −1.53944 −0.769722 0.638380i \(-0.779605\pi\)
−0.769722 + 0.638380i \(0.779605\pi\)
\(308\) 0 0
\(309\) −16.4958 −0.938413
\(310\) 6.67640i 0.379194i
\(311\) 0.125330i 0.00710683i −0.999994 0.00355342i \(-0.998869\pi\)
0.999994 0.00355342i \(-0.00113109\pi\)
\(312\) −1.32035 −0.0747504
\(313\) 7.72987i 0.436918i −0.975846 0.218459i \(-0.929897\pi\)
0.975846 0.218459i \(-0.0701030\pi\)
\(314\) 15.8345 0.893591
\(315\) 0 0
\(316\) 2.92867i 0.164750i
\(317\) 24.3802 1.36933 0.684665 0.728858i \(-0.259949\pi\)
0.684665 + 0.728858i \(0.259949\pi\)
\(318\) 1.89194 0.106095
\(319\) −1.20582 4.63525i −0.0675127 0.259524i
\(320\) 2.46248i 0.137657i
\(321\) 5.76864 0.321974
\(322\) 0 0
\(323\) 19.8312 1.10344
\(324\) −1.00000 −0.0555556
\(325\) 1.40461 0.0779138
\(326\) 20.3615i 1.12772i
\(327\) 8.63196 0.477349
\(328\) 8.18233i 0.451794i
\(329\) 0 0
\(330\) 2.05616 + 7.90406i 0.113188 + 0.435104i
\(331\) −21.4532 −1.17918 −0.589588 0.807704i \(-0.700710\pi\)
−0.589588 + 0.807704i \(0.700710\pi\)
\(332\) −10.1118 −0.554957
\(333\) 4.93126 0.270232
\(334\) 16.1119i 0.881606i
\(335\) 4.24455i 0.231905i
\(336\) 0 0
\(337\) 30.0086i 1.63467i 0.576163 + 0.817335i \(0.304549\pi\)
−0.576163 + 0.817335i \(0.695451\pi\)
\(338\) 11.2567i 0.612282i
\(339\) 17.4933i 0.950108i
\(340\) 11.0191i 0.597593i
\(341\) −2.26388 8.70255i −0.122596 0.471269i
\(342\) 4.43177i 0.239643i
\(343\) 0 0
\(344\) −9.63797 −0.519644
\(345\) −20.4138 −1.09904
\(346\) 7.24694i 0.389598i
\(347\) 34.5340i 1.85388i 0.375205 + 0.926942i \(0.377572\pi\)
−0.375205 + 0.926942i \(0.622428\pi\)
\(348\) −1.44409 −0.0774116
\(349\) −4.77961 −0.255847 −0.127923 0.991784i \(-0.540831\pi\)
−0.127923 + 0.991784i \(0.540831\pi\)
\(350\) 0 0
\(351\) 1.32035i 0.0704753i
\(352\) 0.834997 + 3.20979i 0.0445055 + 0.171083i
\(353\) 13.5771i 0.722634i −0.932443 0.361317i \(-0.882327\pi\)
0.932443 0.361317i \(-0.117673\pi\)
\(354\) 7.78781i 0.413917i
\(355\) 25.5533i 1.35623i
\(356\) 12.6544i 0.670682i
\(357\) 0 0
\(358\) 11.6016i 0.613161i
\(359\) 1.72779i 0.0911890i −0.998960 0.0455945i \(-0.985482\pi\)
0.998960 0.0455945i \(-0.0145182\pi\)
\(360\) 2.46248 0.129784
\(361\) 0.640626 0.0337172
\(362\) −7.74423 −0.407028
\(363\) 5.36034 + 9.60556i 0.281345 + 0.504161i
\(364\) 0 0
\(365\) 27.5119i 1.44004i
\(366\) −4.47886 −0.234114
\(367\) 25.4861i 1.33036i 0.746681 + 0.665182i \(0.231646\pi\)
−0.746681 + 0.665182i \(0.768354\pi\)
\(368\) −8.28995 −0.432143
\(369\) 8.18233 0.425955
\(370\) −12.1431 −0.631292
\(371\) 0 0
\(372\) −2.71125 −0.140572
\(373\) 33.3334i 1.72594i 0.505258 + 0.862968i \(0.331397\pi\)
−0.505258 + 0.862968i \(0.668603\pi\)
\(374\) 3.73643 + 14.3631i 0.193206 + 0.742700i
\(375\) 9.69279 0.500533
\(376\) −0.767488 −0.0395801
\(377\) 1.90672i 0.0982010i
\(378\) 0 0
\(379\) −19.2656 −0.989610 −0.494805 0.869004i \(-0.664760\pi\)
−0.494805 + 0.869004i \(0.664760\pi\)
\(380\) 10.9132i 0.559833i
\(381\) 12.0959 0.619690
\(382\) 21.2314i 1.08629i
\(383\) 5.15509i 0.263413i 0.991289 + 0.131706i \(0.0420456\pi\)
−0.991289 + 0.131706i \(0.957954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −3.51327 −0.178821
\(387\) 9.63797i 0.489925i
\(388\) 1.37808i 0.0699613i
\(389\) −5.16455 −0.261853 −0.130927 0.991392i \(-0.541795\pi\)
−0.130927 + 0.991392i \(0.541795\pi\)
\(390\) 3.25135i 0.164638i
\(391\) −37.0957 −1.87601
\(392\) 0 0
\(393\) 17.2415i 0.869720i
\(394\) 21.9715 1.10691
\(395\) 7.21179 0.362864
\(396\) 3.20979 0.834997i 0.161298 0.0419602i
\(397\) 6.34291i 0.318342i 0.987251 + 0.159171i \(0.0508821\pi\)
−0.987251 + 0.159171i \(0.949118\pi\)
\(398\) 1.44180 0.0722708
\(399\) 0 0
\(400\) −1.06381 −0.0531907
\(401\) 22.0015 1.09870 0.549351 0.835591i \(-0.314875\pi\)
0.549351 + 0.835591i \(0.314875\pi\)
\(402\) 1.72369 0.0859698
\(403\) 3.57981i 0.178323i
\(404\) −7.56722 −0.376483
\(405\) 2.46248i 0.122362i
\(406\) 0 0
\(407\) −15.8283 + 4.11759i −0.784582 + 0.204101i
\(408\) 4.47479 0.221535
\(409\) 14.1189 0.698136 0.349068 0.937097i \(-0.386498\pi\)
0.349068 + 0.937097i \(0.386498\pi\)
\(410\) −20.1488 −0.995080
\(411\) 2.50957i 0.123788i
\(412\) 16.4958i 0.812690i
\(413\) 0 0
\(414\) 8.28995i 0.407429i
\(415\) 24.9001i 1.22230i
\(416\) 1.32035i 0.0647357i
\(417\) 19.9532i 0.977114i
\(418\) 3.70052 + 14.2251i 0.180998 + 0.695772i
\(419\) 22.7760i 1.11268i −0.830955 0.556340i \(-0.812205\pi\)
0.830955 0.556340i \(-0.187795\pi\)
\(420\) 0 0
\(421\) −6.80960 −0.331880 −0.165940 0.986136i \(-0.553066\pi\)
−0.165940 + 0.986136i \(0.553066\pi\)
\(422\) −0.163763 −0.00797188
\(423\) 0.767488i 0.0373165i
\(424\) 1.89194i 0.0918808i
\(425\) −4.76033 −0.230910
\(426\) 10.3771 0.502771
\(427\) 0 0
\(428\) 5.76864i 0.278838i
\(429\) −1.10249 4.23807i −0.0532288 0.204616i
\(430\) 23.7333i 1.14452i
\(431\) 30.8640i 1.48667i 0.668921 + 0.743334i \(0.266757\pi\)
−0.668921 + 0.743334i \(0.733243\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 32.2967i 1.55208i −0.630683 0.776040i \(-0.717225\pi\)
0.630683 0.776040i \(-0.282775\pi\)
\(434\) 0 0
\(435\) 3.55606i 0.170500i
\(436\) 8.63196i 0.413396i
\(437\) −36.7392 −1.75747
\(438\) −11.1724 −0.533840
\(439\) −28.6715 −1.36842 −0.684208 0.729287i \(-0.739852\pi\)
−0.684208 + 0.729287i \(0.739852\pi\)
\(440\) −7.90406 + 2.05616i −0.376811 + 0.0980238i
\(441\) 0 0
\(442\) 5.90830i 0.281029i
\(443\) 34.9816 1.66203 0.831013 0.556253i \(-0.187762\pi\)
0.831013 + 0.556253i \(0.187762\pi\)
\(444\) 4.93126i 0.234027i
\(445\) −31.1612 −1.47718
\(446\) −27.3847 −1.29670
\(447\) 9.78029 0.462592
\(448\) 0 0
\(449\) −3.57926 −0.168916 −0.0844579 0.996427i \(-0.526916\pi\)
−0.0844579 + 0.996427i \(0.526916\pi\)
\(450\) 1.06381i 0.0501486i
\(451\) −26.2636 + 6.83222i −1.23670 + 0.321717i
\(452\) −17.4933 −0.822817
\(453\) 2.67337 0.125606
\(454\) 3.90255i 0.183156i
\(455\) 0 0
\(456\) 4.43177 0.207537
\(457\) 20.3994i 0.954242i 0.878838 + 0.477121i \(0.158320\pi\)
−0.878838 + 0.477121i \(0.841680\pi\)
\(458\) 5.04055 0.235529
\(459\) 4.47479i 0.208865i
\(460\) 20.4138i 0.951800i
\(461\) 13.9967 0.651892 0.325946 0.945388i \(-0.394317\pi\)
0.325946 + 0.945388i \(0.394317\pi\)
\(462\) 0 0
\(463\) 33.6499 1.56384 0.781922 0.623376i \(-0.214239\pi\)
0.781922 + 0.623376i \(0.214239\pi\)
\(464\) 1.44409i 0.0670404i
\(465\) 6.67640i 0.309611i
\(466\) 21.2457 0.984189
\(467\) 9.08969i 0.420620i 0.977635 + 0.210310i \(0.0674474\pi\)
−0.977635 + 0.210310i \(0.932553\pi\)
\(468\) −1.32035 −0.0610334
\(469\) 0 0
\(470\) 1.88992i 0.0871757i
\(471\) 15.8345 0.729614
\(472\) −7.78781 −0.358463
\(473\) −8.04768 30.9359i −0.370033 1.42243i
\(474\) 2.92867i 0.134518i
\(475\) −4.71458 −0.216320
\(476\) 0 0
\(477\) 1.89194 0.0866260
\(478\) 5.64110 0.258018
\(479\) −39.5655 −1.80779 −0.903897 0.427751i \(-0.859306\pi\)
−0.903897 + 0.427751i \(0.859306\pi\)
\(480\) 2.46248i 0.112396i
\(481\) 6.51102 0.296877
\(482\) 7.77466i 0.354126i
\(483\) 0 0
\(484\) −9.60556 + 5.36034i −0.436616 + 0.243652i
\(485\) 3.39349 0.154090
\(486\) −1.00000 −0.0453609
\(487\) −3.20923 −0.145424 −0.0727120 0.997353i \(-0.523165\pi\)
−0.0727120 + 0.997353i \(0.523165\pi\)
\(488\) 4.47886i 0.202748i
\(489\) 20.3615i 0.920779i
\(490\) 0 0
\(491\) 19.0209i 0.858401i −0.903209 0.429200i \(-0.858795\pi\)
0.903209 0.429200i \(-0.141205\pi\)
\(492\) 8.18233i 0.368888i
\(493\) 6.46201i 0.291034i
\(494\) 5.85152i 0.263272i
\(495\) 2.05616 + 7.90406i 0.0924177 + 0.355261i
\(496\) 2.71125i 0.121739i
\(497\) 0 0
\(498\) −10.1118 −0.453120
\(499\) 25.6082 1.14638 0.573190 0.819422i \(-0.305706\pi\)
0.573190 + 0.819422i \(0.305706\pi\)
\(500\) 9.69279i 0.433475i
\(501\) 16.1119i 0.719828i
\(502\) 22.5045 1.00443
\(503\) −2.73335 −0.121874 −0.0609370 0.998142i \(-0.519409\pi\)
−0.0609370 + 0.998142i \(0.519409\pi\)
\(504\) 0 0
\(505\) 18.6341i 0.829209i
\(506\) −6.92208 26.6090i −0.307724 1.18292i
\(507\) 11.2567i 0.499926i
\(508\) 12.0959i 0.536667i
\(509\) 16.8094i 0.745064i −0.928019 0.372532i \(-0.878490\pi\)
0.928019 0.372532i \(-0.121510\pi\)
\(510\) 11.0191i 0.487933i
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 4.43177i 0.195668i
\(514\) −3.01578 −0.133021
\(515\) 40.6206 1.78996
\(516\) −9.63797 −0.424288
\(517\) −0.640850 2.46348i −0.0281845 0.108344i
\(518\) 0 0
\(519\) 7.24694i 0.318105i
\(520\) 3.25135 0.142581
\(521\) 35.5653i 1.55814i −0.626936 0.779071i \(-0.715691\pi\)
0.626936 0.779071i \(-0.284309\pi\)
\(522\) −1.44409 −0.0632063
\(523\) −3.81853 −0.166973 −0.0834863 0.996509i \(-0.526605\pi\)
−0.0834863 + 0.996509i \(0.526605\pi\)
\(524\) 17.2415 0.753200
\(525\) 0 0
\(526\) 28.1819 1.22879
\(527\) 12.1323i 0.528489i
\(528\) 0.834997 + 3.20979i 0.0363386 + 0.139688i
\(529\) 45.7232 1.98797
\(530\) −4.65887 −0.202368
\(531\) 7.78781i 0.337962i
\(532\) 0 0
\(533\) 10.8036 0.467955
\(534\) 12.6544i 0.547609i
\(535\) −14.2052 −0.614143
\(536\) 1.72369i 0.0744520i
\(537\) 11.6016i 0.500644i
\(538\) −0.604546 −0.0260638
\(539\) 0 0
\(540\) 2.46248 0.105968
\(541\) 24.8436i 1.06811i −0.845450 0.534055i \(-0.820667\pi\)
0.845450 0.534055i \(-0.179333\pi\)
\(542\) 19.7931i 0.850186i
\(543\) −7.74423 −0.332337
\(544\) 4.47479i 0.191855i
\(545\) −21.2560 −0.910509
\(546\) 0 0
\(547\) 44.4066i 1.89869i 0.314237 + 0.949345i \(0.398251\pi\)
−0.314237 + 0.949345i \(0.601749\pi\)
\(548\) −2.50957 −0.107204
\(549\) −4.47886 −0.191153
\(550\) −0.888281 3.41462i −0.0378764 0.145600i
\(551\) 6.39990i 0.272645i
\(552\) −8.28995 −0.352844
\(553\) 0 0
\(554\) 2.16446 0.0919590
\(555\) −12.1431 −0.515448
\(556\) −19.9532 −0.846206
\(557\) 27.6823i 1.17294i 0.809972 + 0.586469i \(0.199482\pi\)
−0.809972 + 0.586469i \(0.800518\pi\)
\(558\) −2.71125 −0.114776
\(559\) 12.7255i 0.538233i
\(560\) 0 0
\(561\) 3.73643 + 14.3631i 0.157752 + 0.606412i
\(562\) −29.4916 −1.24403
\(563\) −11.0353 −0.465083 −0.232541 0.972587i \(-0.574704\pi\)
−0.232541 + 0.972587i \(0.574704\pi\)
\(564\) −0.767488 −0.0323171
\(565\) 43.0770i 1.81226i
\(566\) 28.3880i 1.19324i
\(567\) 0 0
\(568\) 10.3771i 0.435412i
\(569\) 40.6462i 1.70398i −0.523559 0.851989i \(-0.675396\pi\)
0.523559 0.851989i \(-0.324604\pi\)
\(570\) 10.9132i 0.457102i
\(571\) 17.7550i 0.743022i 0.928429 + 0.371511i \(0.121160\pi\)
−0.928429 + 0.371511i \(0.878840\pi\)
\(572\) 4.23807 1.10249i 0.177203 0.0460975i
\(573\) 21.2314i 0.886954i
\(574\) 0 0
\(575\) 8.81895 0.367776
\(576\) 1.00000 0.0416667
\(577\) 1.73555i 0.0722517i 0.999347 + 0.0361259i \(0.0115017\pi\)
−0.999347 + 0.0361259i \(0.988498\pi\)
\(578\) 3.02370i 0.125769i
\(579\) −3.51327 −0.146007
\(580\) 3.55606 0.147657
\(581\) 0 0
\(582\) 1.37808i 0.0571231i
\(583\) −6.07274 + 1.57977i −0.251507 + 0.0654272i
\(584\) 11.1724i 0.462319i
\(585\) 3.25135i 0.134427i
\(586\) 9.39403i 0.388064i
\(587\) 0.837774i 0.0345786i 0.999851 + 0.0172893i \(0.00550363\pi\)
−0.999851 + 0.0172893i \(0.994496\pi\)
\(588\) 0 0
\(589\) 12.0156i 0.495096i
\(590\) 19.1773i 0.789518i
\(591\) 21.9715 0.903786
\(592\) −4.93126 −0.202674
\(593\) −26.0199 −1.06851 −0.534255 0.845323i \(-0.679408\pi\)
−0.534255 + 0.845323i \(0.679408\pi\)
\(594\) 3.20979 0.834997i 0.131699 0.0342603i
\(595\) 0 0
\(596\) 9.78029i 0.400616i
\(597\) 1.44180 0.0590089
\(598\) 10.9457i 0.447602i
\(599\) 11.2416 0.459320 0.229660 0.973271i \(-0.426239\pi\)
0.229660 + 0.973271i \(0.426239\pi\)
\(600\) −1.06381 −0.0434300
\(601\) 11.4274 0.466134 0.233067 0.972461i \(-0.425124\pi\)
0.233067 + 0.972461i \(0.425124\pi\)
\(602\) 0 0
\(603\) 1.72369 0.0701940
\(604\) 2.67337i 0.108778i
\(605\) −13.1997 23.6535i −0.536645 0.961652i
\(606\) −7.56722 −0.307397
\(607\) −10.1805 −0.413215 −0.206608 0.978424i \(-0.566242\pi\)
−0.206608 + 0.978424i \(0.566242\pi\)
\(608\) 4.43177i 0.179732i
\(609\) 0 0
\(610\) 11.0291 0.446555
\(611\) 1.01336i 0.0409960i
\(612\) 4.47479 0.180882
\(613\) 24.9802i 1.00894i −0.863429 0.504471i \(-0.831687\pi\)
0.863429 0.504471i \(-0.168313\pi\)
\(614\) 26.9732i 1.08855i
\(615\) −20.1488 −0.812479
\(616\) 0 0
\(617\) −28.5114 −1.14783 −0.573914 0.818916i \(-0.694576\pi\)
−0.573914 + 0.818916i \(0.694576\pi\)
\(618\) 16.4958i 0.663558i
\(619\) 24.3658i 0.979344i 0.871907 + 0.489672i \(0.162883\pi\)
−0.871907 + 0.489672i \(0.837117\pi\)
\(620\) 6.67640 0.268131
\(621\) 8.28995i 0.332664i
\(622\) 0.125330 0.00502529
\(623\) 0 0
\(624\) 1.32035i 0.0528565i
\(625\) −29.1874 −1.16749
\(626\) 7.72987 0.308948
\(627\) 3.70052 + 14.2251i 0.147785 + 0.568095i
\(628\) 15.8345i 0.631865i
\(629\) −22.0663 −0.879843
\(630\) 0 0
\(631\) −18.2463 −0.726374 −0.363187 0.931716i \(-0.618311\pi\)
−0.363187 + 0.931716i \(0.618311\pi\)
\(632\) 2.92867 0.116496
\(633\) −0.163763 −0.00650901
\(634\) 24.3802i 0.968262i
\(635\) −29.7858 −1.18202
\(636\) 1.89194i 0.0750204i
\(637\) 0 0
\(638\) 4.63525 1.20582i 0.183511 0.0477387i
\(639\) 10.3771 0.410511
\(640\) −2.46248 −0.0973381
\(641\) −19.3987 −0.766202 −0.383101 0.923706i \(-0.625144\pi\)
−0.383101 + 0.923706i \(0.625144\pi\)
\(642\) 5.76864i 0.227670i
\(643\) 11.3901i 0.449182i 0.974453 + 0.224591i \(0.0721047\pi\)
−0.974453 + 0.224591i \(0.927895\pi\)
\(644\) 0 0
\(645\) 23.7333i 0.934498i
\(646\) 19.8312i 0.780250i
\(647\) 17.1756i 0.675243i −0.941282 0.337621i \(-0.890378\pi\)
0.941282 0.337621i \(-0.109622\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −6.50280 24.9973i −0.255257 0.981228i
\(650\) 1.40461i 0.0550934i
\(651\) 0 0
\(652\) 20.3615 0.797418
\(653\) −32.3964 −1.26777 −0.633884 0.773428i \(-0.718540\pi\)
−0.633884 + 0.773428i \(0.718540\pi\)
\(654\) 8.63196i 0.337537i
\(655\) 42.4569i 1.65893i
\(656\) −8.18233 −0.319466
\(657\) −11.1724 −0.435878
\(658\) 0 0
\(659\) 8.52103i 0.331932i −0.986131 0.165966i \(-0.946926\pi\)
0.986131 0.165966i \(-0.0530742\pi\)
\(660\) −7.90406 + 2.05616i −0.307665 + 0.0800361i
\(661\) 39.3852i 1.53191i −0.642897 0.765953i \(-0.722268\pi\)
0.642897 0.765953i \(-0.277732\pi\)
\(662\) 21.4532i 0.833803i
\(663\) 5.90830i 0.229460i
\(664\) 10.1118i 0.392414i
\(665\) 0 0
\(666\) 4.93126i 0.191083i
\(667\) 11.9715i 0.463537i
\(668\) −16.1119 −0.623389
\(669\) −27.3847 −1.05875
\(670\) −4.24455 −0.163981
\(671\) 14.3762 3.73983i 0.554987 0.144375i
\(672\) 0 0
\(673\) 23.4185i 0.902716i 0.892343 + 0.451358i \(0.149060\pi\)
−0.892343 + 0.451358i \(0.850940\pi\)
\(674\) −30.0086 −1.15589
\(675\) 1.06381i 0.0409462i
\(676\) 11.2567 0.432949
\(677\) −29.6821 −1.14078 −0.570388 0.821376i \(-0.693207\pi\)
−0.570388 + 0.821376i \(0.693207\pi\)
\(678\) −17.4933 −0.671828
\(679\) 0 0
\(680\) −11.0191 −0.422562
\(681\) 3.90255i 0.149546i
\(682\) 8.70255 2.26388i 0.333238 0.0866886i
\(683\) 28.9457 1.10757 0.553787 0.832658i \(-0.313182\pi\)
0.553787 + 0.832658i \(0.313182\pi\)
\(684\) 4.43177 0.169453
\(685\) 6.17977i 0.236117i
\(686\) 0 0
\(687\) 5.04055 0.192309
\(688\) 9.63797i 0.367444i
\(689\) 2.49803 0.0951675
\(690\) 20.4138i 0.777142i
\(691\) 27.9501i 1.06327i 0.846973 + 0.531636i \(0.178422\pi\)
−0.846973 + 0.531636i \(0.821578\pi\)
\(692\) −7.24694 −0.275487
\(693\) 0 0
\(694\) −34.5340 −1.31089
\(695\) 49.1345i 1.86378i
\(696\) 1.44409i 0.0547383i
\(697\) −36.6142 −1.38686
\(698\) 4.77961i 0.180911i
\(699\) 21.2457 0.803587
\(700\) 0 0
\(701\) 13.2062i 0.498793i 0.968401 + 0.249397i \(0.0802323\pi\)
−0.968401 + 0.249397i \(0.919768\pi\)
\(702\) −1.32035 −0.0498336
\(703\) −21.8542 −0.824249
\(704\) −3.20979 + 0.834997i −0.120974 + 0.0314701i
\(705\) 1.88992i 0.0711786i
\(706\) 13.5771 0.510979
\(707\) 0 0
\(708\) −7.78781 −0.292684
\(709\) 14.7808 0.555104 0.277552 0.960711i \(-0.410477\pi\)
0.277552 + 0.960711i \(0.410477\pi\)
\(710\) −25.5533 −0.959000
\(711\) 2.92867i 0.109834i
\(712\) −12.6544 −0.474244
\(713\) 22.4761i 0.841737i
\(714\) 0 0
\(715\) 2.71487 + 10.4362i 0.101530 + 0.390290i
\(716\) −11.6016 −0.433571
\(717\) 5.64110 0.210671
\(718\) 1.72779 0.0644804
\(719\) 3.71055i 0.138380i 0.997603 + 0.0691900i \(0.0220415\pi\)
−0.997603 + 0.0691900i \(0.977959\pi\)
\(720\) 2.46248i 0.0917713i
\(721\) 0 0
\(722\) 0.640626i 0.0238416i
\(723\) 7.77466i 0.289143i
\(724\) 7.74423i 0.287812i
\(725\) 1.53625i 0.0570548i
\(726\) −9.60556 + 5.36034i −0.356496 + 0.198941i
\(727\) 16.9392i 0.628241i 0.949383 + 0.314120i \(0.101710\pi\)
−0.949383 + 0.314120i \(0.898290\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 27.5119 1.01826
\(731\) 43.1278i 1.59514i
\(732\) 4.47886i 0.165543i
\(733\) 11.3990 0.421032 0.210516 0.977590i \(-0.432486\pi\)
0.210516 + 0.977590i \(0.432486\pi\)
\(734\) −25.4861 −0.940709
\(735\) 0 0
\(736\) 8.28995i 0.305572i
\(737\) −5.53269 + 1.43928i −0.203799 + 0.0530164i
\(738\) 8.18233i 0.301196i
\(739\) 5.65173i 0.207902i −0.994582 0.103951i \(-0.966851\pi\)
0.994582 0.103951i \(-0.0331485\pi\)
\(740\) 12.1431i 0.446391i
\(741\) 5.85152i 0.214961i
\(742\) 0 0
\(743\) 2.21080i 0.0811065i −0.999177 0.0405533i \(-0.987088\pi\)
0.999177 0.0405533i \(-0.0129120\pi\)
\(744\) 2.71125i 0.0993992i
\(745\) −24.0838 −0.882361
\(746\) −33.3334 −1.22042
\(747\) −10.1118 −0.369971
\(748\) −14.3631 + 3.73643i −0.525168 + 0.136618i
\(749\) 0 0
\(750\) 9.69279i 0.353930i
\(751\) 5.87413 0.214350 0.107175 0.994240i \(-0.465819\pi\)
0.107175 + 0.994240i \(0.465819\pi\)
\(752\) 0.767488i 0.0279874i
\(753\) 22.5045 0.820110
\(754\) −1.90672 −0.0694386
\(755\) −6.58313 −0.239585
\(756\) 0 0
\(757\) 15.3568 0.558152 0.279076 0.960269i \(-0.409972\pi\)
0.279076 + 0.960269i \(0.409972\pi\)
\(758\) 19.2656i 0.699760i
\(759\) −6.92208 26.6090i −0.251256 0.965847i
\(760\) −10.9132 −0.395862
\(761\) 22.4927 0.815359 0.407680 0.913125i \(-0.366338\pi\)
0.407680 + 0.913125i \(0.366338\pi\)
\(762\) 12.0959i 0.438187i
\(763\) 0 0
\(764\) 21.2314 0.768125
\(765\) 11.0191i 0.398395i
\(766\) −5.15509 −0.186261
\(767\) 10.2827i 0.371286i
\(768\) 1.00000i 0.0360844i
\(769\) 39.6302 1.42910 0.714551 0.699583i \(-0.246631\pi\)
0.714551 + 0.699583i \(0.246631\pi\)
\(770\) 0 0
\(771\) −3.01578 −0.108611
\(772\) 3.51327i 0.126445i
\(773\) 15.7052i 0.564877i 0.959285 + 0.282438i \(0.0911433\pi\)
−0.959285 + 0.282438i \(0.908857\pi\)
\(774\) −9.63797 −0.346430
\(775\) 2.88426i 0.103606i
\(776\) 1.37808 0.0494701
\(777\) 0 0
\(778\) 5.16455i 0.185158i
\(779\) −36.2622 −1.29923
\(780\) 3.25135 0.116417
\(781\) −33.3083 + 8.66483i −1.19186 + 0.310052i
\(782\) 37.0957i 1.32654i
\(783\) −1.44409 −0.0516077
\(784\) 0 0
\(785\) −38.9921 −1.39169
\(786\) 17.2415 0.614985
\(787\) 41.3222 1.47298 0.736488 0.676450i \(-0.236483\pi\)
0.736488 + 0.676450i \(0.236483\pi\)
\(788\) 21.9715i 0.782702i
\(789\) 28.1819 1.00330
\(790\) 7.21179i 0.256584i
\(791\) 0 0
\(792\) 0.834997 + 3.20979i 0.0296703 + 0.114055i
\(793\) −5.91368 −0.210001
\(794\) −6.34291 −0.225102
\(795\) −4.65887 −0.165233
\(796\) 1.44180i 0.0511032i
\(797\) 20.2076i 0.715789i 0.933762 + 0.357894i \(0.116505\pi\)
−0.933762 + 0.357894i \(0.883495\pi\)
\(798\) 0 0
\(799\) 3.43434i 0.121498i
\(800\) 1.06381i 0.0376115i
\(801\) 12.6544i 0.447121i
\(802\) 22.0015i 0.776900i
\(803\) 35.8612 9.32896i 1.26552 0.329212i
\(804\) 1.72369i 0.0607898i
\(805\) 0 0
\(806\) −3.57981 −0.126093
\(807\) −0.604546 −0.0212810
\(808\) 7.56722i 0.266214i
\(809\) 33.7222i 1.18561i 0.805346 + 0.592805i \(0.201980\pi\)
−0.805346 + 0.592805i \(0.798020\pi\)
\(810\) 2.46248 0.0865228
\(811\) −9.84154 −0.345583 −0.172792 0.984958i \(-0.555279\pi\)
−0.172792 + 0.984958i \(0.555279\pi\)
\(812\) 0 0
\(813\) 19.7931i 0.694174i
\(814\) −4.11759 15.8283i −0.144321 0.554783i
\(815\) 50.1398i 1.75632i
\(816\) 4.47479i 0.156649i
\(817\) 42.7133i 1.49435i
\(818\) 14.1189i 0.493657i
\(819\) 0 0
\(820\) 20.1488i 0.703628i
\(821\) 8.79863i 0.307074i −0.988143 0.153537i \(-0.950933\pi\)
0.988143 0.153537i \(-0.0490665\pi\)
\(822\) −2.50957 −0.0875314
\(823\) −32.8876 −1.14639 −0.573195 0.819419i \(-0.694296\pi\)
−0.573195 + 0.819419i \(0.694296\pi\)
\(824\) 16.4958 0.574658
\(825\) −0.888281 3.41462i −0.0309260 0.118882i
\(826\) 0 0
\(827\) 43.5923i 1.51585i 0.652339 + 0.757927i \(0.273788\pi\)
−0.652339 + 0.757927i \(0.726212\pi\)
\(828\) −8.28995 −0.288096
\(829\) 31.6573i 1.09950i 0.835328 + 0.549752i \(0.185278\pi\)
−0.835328 + 0.549752i \(0.814722\pi\)
\(830\) 24.9001 0.864295
\(831\) 2.16446 0.0750842
\(832\) 1.32035 0.0457751
\(833\) 0 0
\(834\) −19.9532 −0.690924
\(835\) 39.6753i 1.37302i
\(836\) −14.2251 + 3.70052i −0.491985 + 0.127985i
\(837\) −2.71125 −0.0937145
\(838\) 22.7760 0.786784
\(839\) 33.1104i 1.14310i 0.820568 + 0.571549i \(0.193657\pi\)
−0.820568 + 0.571549i \(0.806343\pi\)
\(840\) 0 0
\(841\) 26.9146 0.928089
\(842\) 6.80960i 0.234674i
\(843\) −29.4916 −1.01575
\(844\) 0.163763i 0.00563697i
\(845\) 27.7193i 0.953574i
\(846\) −0.767488 −0.0263868
\(847\) 0 0
\(848\) −1.89194 −0.0649695
\(849\) 28.3880i 0.974273i
\(850\) 4.76033i 0.163278i
\(851\) 40.8799 1.40135
\(852\) 10.3771i 0.355513i
\(853\) 32.8121 1.12347 0.561733 0.827318i \(-0.310135\pi\)
0.561733 + 0.827318i \(0.310135\pi\)
\(854\) 0 0
\(855\) 10.9132i 0.373222i
\(856\) −5.76864 −0.197168
\(857\) −11.6763 −0.398853 −0.199427 0.979913i \(-0.563908\pi\)
−0.199427 + 0.979913i \(0.563908\pi\)
\(858\) 4.23807 1.10249i 0.144685 0.0376385i
\(859\) 40.7175i 1.38926i −0.719366 0.694631i \(-0.755568\pi\)
0.719366 0.694631i \(-0.244432\pi\)
\(860\) 23.7333 0.809299
\(861\) 0 0
\(862\) −30.8640 −1.05123
\(863\) −18.0888 −0.615749 −0.307874 0.951427i \(-0.599618\pi\)
−0.307874 + 0.951427i \(0.599618\pi\)
\(864\) 1.00000 0.0340207
\(865\) 17.8455i 0.606764i
\(866\) 32.2967 1.09749
\(867\) 3.02370i 0.102690i
\(868\) 0 0
\(869\) 2.44543 + 9.40042i 0.0829555 + 0.318887i
\(870\) 3.55606 0.120562
\(871\) 2.27588 0.0771153
\(872\) −8.63196 −0.292315
\(873\) 1.37808i 0.0466408i
\(874\) 36.7392i 1.24272i
\(875\) 0 0
\(876\) 11.1724i 0.377482i
\(877\) 6.62096i 0.223574i 0.993732 + 0.111787i \(0.0356574\pi\)
−0.993732 + 0.111787i \(0.964343\pi\)
\(878\) 28.6715i 0.967616i
\(879\) 9.39403i 0.316853i
\(880\) −2.05616 7.90406i −0.0693133 0.266446i
\(881\) 16.5800i 0.558594i −0.960205 0.279297i \(-0.909899\pi\)
0.960205 0.279297i \(-0.0901014\pi\)
\(882\) 0 0
\(883\) 44.3526 1.49258 0.746292 0.665619i \(-0.231832\pi\)
0.746292 + 0.665619i \(0.231832\pi\)
\(884\) 5.90830 0.198718
\(885\) 19.1773i 0.644639i
\(886\) 34.9816i 1.17523i
\(887\) −32.5838 −1.09406 −0.547028 0.837114i \(-0.684241\pi\)
−0.547028 + 0.837114i \(0.684241\pi\)
\(888\) −4.93126 −0.165482
\(889\) 0 0
\(890\) 31.1612i 1.04453i
\(891\) 3.20979 0.834997i 0.107532 0.0279735i
\(892\) 27.3847i 0.916908i
\(893\) 3.40133i 0.113821i
\(894\) 9.78029i 0.327102i
\(895\) 28.5686i 0.954944i
\(896\) 0 0
\(897\) 10.9457i 0.365465i
\(898\) 3.57926i 0.119441i
\(899\) −3.91530 −0.130583
\(900\) −1.06381 −0.0354604
\(901\) −8.46603 −0.282044
\(902\) −6.83222 26.2636i −0.227488 0.874482i
\(903\) 0 0
\(904\) 17.4933i 0.581820i
\(905\) 19.0700 0.633909
\(906\) 2.67337i 0.0888168i
\(907\) −41.0943 −1.36451 −0.682257 0.731112i \(-0.739002\pi\)
−0.682257 + 0.731112i \(0.739002\pi\)
\(908\) 3.90255 0.129511
\(909\) −7.56722 −0.250989
\(910\) 0 0
\(911\) 28.2471 0.935868 0.467934 0.883764i \(-0.344999\pi\)
0.467934 + 0.883764i \(0.344999\pi\)
\(912\) 4.43177i 0.146751i
\(913\) 32.4568 8.44332i 1.07416 0.279433i
\(914\) −20.3994 −0.674751
\(915\) 11.0291 0.364611
\(916\) 5.04055i 0.166544i
\(917\) 0 0
\(918\) 4.47479 0.147690
\(919\) 39.1121i 1.29019i 0.764102 + 0.645095i \(0.223182\pi\)
−0.764102 + 0.645095i \(0.776818\pi\)
\(920\) 20.4138 0.673024
\(921\) 26.9732i 0.888798i
\(922\) 13.9967i 0.460957i
\(923\) 13.7014 0.450988
\(924\) 0 0
\(925\) 5.24594 0.172486
\(926\) 33.6499i 1.10581i
\(927\) 16.4958i 0.541793i
\(928\) 1.44409 0.0474047
\(929\) 24.6597i 0.809058i 0.914525 + 0.404529i \(0.132565\pi\)
−0.914525 + 0.404529i \(0.867435\pi\)
\(930\) 6.67640 0.218928
\(931\) 0 0
\(932\) 21.2457i 0.695927i
\(933\) 0.125330 0.00410313
\(934\) −9.08969 −0.297424
\(935\) −9.20090 35.3690i −0.300901 1.15669i
\(936\) 1.32035i 0.0431572i
\(937\) 10.7483 0.351133 0.175566 0.984468i \(-0.443824\pi\)
0.175566 + 0.984468i \(0.443824\pi\)
\(938\) 0 0
\(939\) 7.72987 0.252255
\(940\) 1.88992 0.0616425
\(941\) −33.8736 −1.10425 −0.552125 0.833762i \(-0.686183\pi\)
−0.552125 + 0.833762i \(0.686183\pi\)
\(942\) 15.8345i 0.515915i
\(943\) 67.8311 2.20888
\(944\) 7.78781i 0.253471i
\(945\) 0 0
\(946\) 30.9359 8.04768i 1.00581 0.261652i
\(947\) 20.7718 0.674994 0.337497 0.941327i \(-0.390420\pi\)
0.337497 + 0.941327i \(0.390420\pi\)
\(948\) 2.92867 0.0951187
\(949\) −14.7516 −0.478857
\(950\) 4.71458i 0.152961i
\(951\) 24.3802i 0.790583i
\(952\) 0 0
\(953\) 1.28486i 0.0416205i 0.999783 + 0.0208103i \(0.00662459\pi\)
−0.999783 + 0.0208103i \(0.993375\pi\)
\(954\) 1.89194i 0.0612539i
\(955\) 52.2819i 1.69180i
\(956\) 5.64110i 0.182446i
\(957\) 4.63525 1.20582i 0.149836 0.0389785i
\(958\) 39.5655i 1.27830i
\(959\) 0 0
\(960\) −2.46248 −0.0794762
\(961\) 23.6491 0.762875
\(962\) 6.51102i 0.209924i
\(963\) 5.76864i 0.185892i
\(964\) 7.77466 0.250405
\(965\) 8.65137 0.278497
\(966\) 0 0
\(967\) 44.2584i 1.42325i 0.702558 + 0.711627i \(0.252041\pi\)
−0.702558 + 0.711627i \(0.747959\pi\)
\(968\) −5.36034 9.60556i −0.172288 0.308734i
\(969\) 19.8312i 0.637071i
\(970\) 3.39349i 0.108958i
\(971\) 39.8945i 1.28028i −0.768260 0.640138i \(-0.778877\pi\)
0.768260 0.640138i \(-0.221123\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 3.20923i 0.102830i
\(975\) 1.40461i 0.0449835i
\(976\) 4.47886 0.143365
\(977\) −1.60685 −0.0514076 −0.0257038 0.999670i \(-0.508183\pi\)
−0.0257038 + 0.999670i \(0.508183\pi\)
\(978\) 20.3615 0.651089
\(979\) −10.5664 40.6180i −0.337703 1.29816i
\(980\) 0 0
\(981\) 8.63196i 0.275597i
\(982\) 19.0209 0.606981
\(983\) 1.13364i 0.0361574i −0.999837 0.0180787i \(-0.994245\pi\)
0.999837 0.0180787i \(-0.00575494\pi\)
\(984\) −8.18233 −0.260843
\(985\) −54.1044 −1.72391
\(986\) 6.46201 0.205792
\(987\) 0 0
\(988\) 5.85152 0.186162
\(989\) 79.8982i 2.54062i
\(990\) −7.90406 + 2.05616i −0.251207 + 0.0653492i
\(991\) −5.28668 −0.167937 −0.0839684 0.996468i \(-0.526759\pi\)
−0.0839684 + 0.996468i \(0.526759\pi\)
\(992\) 2.71125 0.0860822
\(993\) 21.4532i 0.680798i
\(994\) 0 0
\(995\) −3.55040 −0.112555
\(996\) 10.1118i 0.320404i
\(997\) 1.09038 0.0345327 0.0172663 0.999851i \(-0.494504\pi\)
0.0172663 + 0.999851i \(0.494504\pi\)
\(998\) 25.6082i 0.810613i
\(999\) 4.93126i 0.156018i
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 3234.2.e.a.2155.10 16
7.2 even 3 462.2.p.a.241.5 16
7.3 odd 6 462.2.p.b.439.1 yes 16
7.6 odd 2 3234.2.e.b.2155.15 16
11.10 odd 2 3234.2.e.b.2155.2 16
21.2 odd 6 1386.2.bk.a.703.4 16
21.17 even 6 1386.2.bk.b.901.8 16
77.10 even 6 462.2.p.a.439.5 yes 16
77.65 odd 6 462.2.p.b.241.1 yes 16
77.76 even 2 inner 3234.2.e.a.2155.7 16
231.65 even 6 1386.2.bk.b.703.8 16
231.164 odd 6 1386.2.bk.a.901.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.p.a.241.5 16 7.2 even 3
462.2.p.a.439.5 yes 16 77.10 even 6
462.2.p.b.241.1 yes 16 77.65 odd 6
462.2.p.b.439.1 yes 16 7.3 odd 6
1386.2.bk.a.703.4 16 21.2 odd 6
1386.2.bk.a.901.4 16 231.164 odd 6
1386.2.bk.b.703.8 16 231.65 even 6
1386.2.bk.b.901.8 16 21.17 even 6
3234.2.e.a.2155.7 16 77.76 even 2 inner
3234.2.e.a.2155.10 16 1.1 even 1 trivial
3234.2.e.b.2155.2 16 11.10 odd 2
3234.2.e.b.2155.15 16 7.6 odd 2