Properties

Label 4730.2.a.w.1.6
Level $4730$
Weight $2$
Character 4730.1
Self dual yes
Analytic conductor $37.769$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4730,2,Mod(1,4730)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4730.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4730.a (trivial)

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: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 7x^{5} + 41x^{4} - 6x^{3} - 28x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.771947\) of defining polynomial
Character \(\chi\) \(=\) 4730.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.228053 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.228053 q^{6} -2.81890 q^{7} -1.00000 q^{8} -2.94799 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.228053 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.228053 q^{6} -2.81890 q^{7} -1.00000 q^{8} -2.94799 q^{9} -1.00000 q^{10} +1.00000 q^{11} -0.228053 q^{12} +1.12433 q^{13} +2.81890 q^{14} -0.228053 q^{15} +1.00000 q^{16} -1.66554 q^{17} +2.94799 q^{18} +7.42051 q^{19} +1.00000 q^{20} +0.642859 q^{21} -1.00000 q^{22} -0.875665 q^{23} +0.228053 q^{24} +1.00000 q^{25} -1.12433 q^{26} +1.35646 q^{27} -2.81890 q^{28} -4.72931 q^{29} +0.228053 q^{30} -2.66066 q^{31} -1.00000 q^{32} -0.228053 q^{33} +1.66554 q^{34} -2.81890 q^{35} -2.94799 q^{36} -5.63729 q^{37} -7.42051 q^{38} -0.256408 q^{39} -1.00000 q^{40} +7.80977 q^{41} -0.642859 q^{42} -1.00000 q^{43} +1.00000 q^{44} -2.94799 q^{45} +0.875665 q^{46} +13.2060 q^{47} -0.228053 q^{48} +0.946220 q^{49} -1.00000 q^{50} +0.379832 q^{51} +1.12433 q^{52} -3.40667 q^{53} -1.35646 q^{54} +1.00000 q^{55} +2.81890 q^{56} -1.69227 q^{57} +4.72931 q^{58} +0.759347 q^{59} -0.228053 q^{60} +4.85201 q^{61} +2.66066 q^{62} +8.31011 q^{63} +1.00000 q^{64} +1.12433 q^{65} +0.228053 q^{66} -2.24757 q^{67} -1.66554 q^{68} +0.199698 q^{69} +2.81890 q^{70} +7.41158 q^{71} +2.94799 q^{72} +2.51101 q^{73} +5.63729 q^{74} -0.228053 q^{75} +7.42051 q^{76} -2.81890 q^{77} +0.256408 q^{78} -10.7456 q^{79} +1.00000 q^{80} +8.53463 q^{81} -7.80977 q^{82} -12.0367 q^{83} +0.642859 q^{84} -1.66554 q^{85} +1.00000 q^{86} +1.07853 q^{87} -1.00000 q^{88} -5.21429 q^{89} +2.94799 q^{90} -3.16939 q^{91} -0.875665 q^{92} +0.606772 q^{93} -13.2060 q^{94} +7.42051 q^{95} +0.228053 q^{96} -6.50507 q^{97} -0.946220 q^{98} -2.94799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 7 q^{3} + 8 q^{4} + 8 q^{5} + 7 q^{6} - 6 q^{7} - 8 q^{8} + 7 q^{9} - 8 q^{10} + 8 q^{11} - 7 q^{12} - 2 q^{13} + 6 q^{14} - 7 q^{15} + 8 q^{16} - 8 q^{17} - 7 q^{18} + 8 q^{20} + 14 q^{21} - 8 q^{22} - 18 q^{23} + 7 q^{24} + 8 q^{25} + 2 q^{26} - 22 q^{27} - 6 q^{28} + 8 q^{29} + 7 q^{30} - 11 q^{31} - 8 q^{32} - 7 q^{33} + 8 q^{34} - 6 q^{35} + 7 q^{36} - 17 q^{37} - 6 q^{39} - 8 q^{40} + 12 q^{41} - 14 q^{42} - 8 q^{43} + 8 q^{44} + 7 q^{45} + 18 q^{46} - 19 q^{47} - 7 q^{48} - 2 q^{49} - 8 q^{50} - q^{51} - 2 q^{52} - 7 q^{53} + 22 q^{54} + 8 q^{55} + 6 q^{56} - 3 q^{57} - 8 q^{58} + q^{59} - 7 q^{60} + 6 q^{61} + 11 q^{62} - 15 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} - 22 q^{67} - 8 q^{68} + 8 q^{69} + 6 q^{70} - 14 q^{71} - 7 q^{72} - 13 q^{73} + 17 q^{74} - 7 q^{75} - 6 q^{77} + 6 q^{78} - 8 q^{79} + 8 q^{80} + 28 q^{81} - 12 q^{82} - 4 q^{83} + 14 q^{84} - 8 q^{85} + 8 q^{86} - 30 q^{87} - 8 q^{88} + 5 q^{89} - 7 q^{90} - 8 q^{91} - 18 q^{92} + q^{93} + 19 q^{94} + 7 q^{96} - 23 q^{97} + 2 q^{98} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.228053 −0.131666 −0.0658332 0.997831i \(-0.520971\pi\)
−0.0658332 + 0.997831i \(0.520971\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.228053 0.0931022
\(7\) −2.81890 −1.06545 −0.532723 0.846290i \(-0.678831\pi\)
−0.532723 + 0.846290i \(0.678831\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.94799 −0.982664
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −0.228053 −0.0658332
\(13\) 1.12433 0.311834 0.155917 0.987770i \(-0.450167\pi\)
0.155917 + 0.987770i \(0.450167\pi\)
\(14\) 2.81890 0.753384
\(15\) −0.228053 −0.0588830
\(16\) 1.00000 0.250000
\(17\) −1.66554 −0.403953 −0.201977 0.979390i \(-0.564736\pi\)
−0.201977 + 0.979390i \(0.564736\pi\)
\(18\) 2.94799 0.694848
\(19\) 7.42051 1.70238 0.851190 0.524857i \(-0.175881\pi\)
0.851190 + 0.524857i \(0.175881\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.642859 0.140283
\(22\) −1.00000 −0.213201
\(23\) −0.875665 −0.182589 −0.0912944 0.995824i \(-0.529100\pi\)
−0.0912944 + 0.995824i \(0.529100\pi\)
\(24\) 0.228053 0.0465511
\(25\) 1.00000 0.200000
\(26\) −1.12433 −0.220500
\(27\) 1.35646 0.261050
\(28\) −2.81890 −0.532723
\(29\) −4.72931 −0.878210 −0.439105 0.898436i \(-0.644704\pi\)
−0.439105 + 0.898436i \(0.644704\pi\)
\(30\) 0.228053 0.0416366
\(31\) −2.66066 −0.477869 −0.238935 0.971036i \(-0.576798\pi\)
−0.238935 + 0.971036i \(0.576798\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.228053 −0.0396989
\(34\) 1.66554 0.285638
\(35\) −2.81890 −0.476482
\(36\) −2.94799 −0.491332
\(37\) −5.63729 −0.926765 −0.463383 0.886158i \(-0.653364\pi\)
−0.463383 + 0.886158i \(0.653364\pi\)
\(38\) −7.42051 −1.20376
\(39\) −0.256408 −0.0410581
\(40\) −1.00000 −0.158114
\(41\) 7.80977 1.21968 0.609841 0.792524i \(-0.291233\pi\)
0.609841 + 0.792524i \(0.291233\pi\)
\(42\) −0.642859 −0.0991953
\(43\) −1.00000 −0.152499
\(44\) 1.00000 0.150756
\(45\) −2.94799 −0.439461
\(46\) 0.875665 0.129110
\(47\) 13.2060 1.92629 0.963146 0.268978i \(-0.0866858\pi\)
0.963146 + 0.268978i \(0.0866858\pi\)
\(48\) −0.228053 −0.0329166
\(49\) 0.946220 0.135174
\(50\) −1.00000 −0.141421
\(51\) 0.379832 0.0531871
\(52\) 1.12433 0.155917
\(53\) −3.40667 −0.467943 −0.233971 0.972244i \(-0.575172\pi\)
−0.233971 + 0.972244i \(0.575172\pi\)
\(54\) −1.35646 −0.184590
\(55\) 1.00000 0.134840
\(56\) 2.81890 0.376692
\(57\) −1.69227 −0.224146
\(58\) 4.72931 0.620988
\(59\) 0.759347 0.0988586 0.0494293 0.998778i \(-0.484260\pi\)
0.0494293 + 0.998778i \(0.484260\pi\)
\(60\) −0.228053 −0.0294415
\(61\) 4.85201 0.621236 0.310618 0.950535i \(-0.399464\pi\)
0.310618 + 0.950535i \(0.399464\pi\)
\(62\) 2.66066 0.337905
\(63\) 8.31011 1.04697
\(64\) 1.00000 0.125000
\(65\) 1.12433 0.139457
\(66\) 0.228053 0.0280714
\(67\) −2.24757 −0.274585 −0.137292 0.990531i \(-0.543840\pi\)
−0.137292 + 0.990531i \(0.543840\pi\)
\(68\) −1.66554 −0.201977
\(69\) 0.199698 0.0240408
\(70\) 2.81890 0.336923
\(71\) 7.41158 0.879593 0.439797 0.898097i \(-0.355051\pi\)
0.439797 + 0.898097i \(0.355051\pi\)
\(72\) 2.94799 0.347424
\(73\) 2.51101 0.293892 0.146946 0.989145i \(-0.453056\pi\)
0.146946 + 0.989145i \(0.453056\pi\)
\(74\) 5.63729 0.655322
\(75\) −0.228053 −0.0263333
\(76\) 7.42051 0.851190
\(77\) −2.81890 −0.321244
\(78\) 0.256408 0.0290325
\(79\) −10.7456 −1.20898 −0.604490 0.796613i \(-0.706623\pi\)
−0.604490 + 0.796613i \(0.706623\pi\)
\(80\) 1.00000 0.111803
\(81\) 8.53463 0.948292
\(82\) −7.80977 −0.862445
\(83\) −12.0367 −1.32120 −0.660602 0.750737i \(-0.729699\pi\)
−0.660602 + 0.750737i \(0.729699\pi\)
\(84\) 0.642859 0.0701417
\(85\) −1.66554 −0.180653
\(86\) 1.00000 0.107833
\(87\) 1.07853 0.115631
\(88\) −1.00000 −0.106600
\(89\) −5.21429 −0.552714 −0.276357 0.961055i \(-0.589127\pi\)
−0.276357 + 0.961055i \(0.589127\pi\)
\(90\) 2.94799 0.310746
\(91\) −3.16939 −0.332242
\(92\) −0.875665 −0.0912944
\(93\) 0.606772 0.0629194
\(94\) −13.2060 −1.36209
\(95\) 7.42051 0.761328
\(96\) 0.228053 0.0232756
\(97\) −6.50507 −0.660490 −0.330245 0.943895i \(-0.607131\pi\)
−0.330245 + 0.943895i \(0.607131\pi\)
\(98\) −0.946220 −0.0955827
\(99\) −2.94799 −0.296284
\(100\) 1.00000 0.100000
\(101\) 4.52576 0.450330 0.225165 0.974321i \(-0.427708\pi\)
0.225165 + 0.974321i \(0.427708\pi\)
\(102\) −0.379832 −0.0376089
\(103\) −2.63398 −0.259534 −0.129767 0.991545i \(-0.541423\pi\)
−0.129767 + 0.991545i \(0.541423\pi\)
\(104\) −1.12433 −0.110250
\(105\) 0.642859 0.0627366
\(106\) 3.40667 0.330885
\(107\) −9.33164 −0.902124 −0.451062 0.892493i \(-0.648955\pi\)
−0.451062 + 0.892493i \(0.648955\pi\)
\(108\) 1.35646 0.130525
\(109\) −4.97087 −0.476123 −0.238061 0.971250i \(-0.576512\pi\)
−0.238061 + 0.971250i \(0.576512\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 1.28560 0.122024
\(112\) −2.81890 −0.266361
\(113\) −8.23164 −0.774367 −0.387184 0.922003i \(-0.626552\pi\)
−0.387184 + 0.922003i \(0.626552\pi\)
\(114\) 1.69227 0.158495
\(115\) −0.875665 −0.0816562
\(116\) −4.72931 −0.439105
\(117\) −3.31453 −0.306428
\(118\) −0.759347 −0.0699036
\(119\) 4.69500 0.430390
\(120\) 0.228053 0.0208183
\(121\) 1.00000 0.0909091
\(122\) −4.85201 −0.439280
\(123\) −1.78104 −0.160591
\(124\) −2.66066 −0.238935
\(125\) 1.00000 0.0894427
\(126\) −8.31011 −0.740323
\(127\) −20.1915 −1.79171 −0.895854 0.444348i \(-0.853435\pi\)
−0.895854 + 0.444348i \(0.853435\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.228053 0.0200789
\(130\) −1.12433 −0.0986107
\(131\) −17.5653 −1.53469 −0.767345 0.641235i \(-0.778422\pi\)
−0.767345 + 0.641235i \(0.778422\pi\)
\(132\) −0.228053 −0.0198495
\(133\) −20.9177 −1.81379
\(134\) 2.24757 0.194161
\(135\) 1.35646 0.116745
\(136\) 1.66554 0.142819
\(137\) −10.6093 −0.906413 −0.453206 0.891406i \(-0.649720\pi\)
−0.453206 + 0.891406i \(0.649720\pi\)
\(138\) −0.199698 −0.0169994
\(139\) −1.26492 −0.107289 −0.0536445 0.998560i \(-0.517084\pi\)
−0.0536445 + 0.998560i \(0.517084\pi\)
\(140\) −2.81890 −0.238241
\(141\) −3.01167 −0.253628
\(142\) −7.41158 −0.621966
\(143\) 1.12433 0.0940216
\(144\) −2.94799 −0.245666
\(145\) −4.72931 −0.392748
\(146\) −2.51101 −0.207813
\(147\) −0.215788 −0.0177979
\(148\) −5.63729 −0.463383
\(149\) 9.56881 0.783907 0.391954 0.919985i \(-0.371799\pi\)
0.391954 + 0.919985i \(0.371799\pi\)
\(150\) 0.228053 0.0186204
\(151\) 15.7568 1.28227 0.641136 0.767428i \(-0.278464\pi\)
0.641136 + 0.767428i \(0.278464\pi\)
\(152\) −7.42051 −0.601882
\(153\) 4.91000 0.396950
\(154\) 2.81890 0.227154
\(155\) −2.66066 −0.213710
\(156\) −0.256408 −0.0205291
\(157\) −4.47588 −0.357214 −0.178607 0.983920i \(-0.557159\pi\)
−0.178607 + 0.983920i \(0.557159\pi\)
\(158\) 10.7456 0.854877
\(159\) 0.776902 0.0616123
\(160\) −1.00000 −0.0790569
\(161\) 2.46842 0.194539
\(162\) −8.53463 −0.670544
\(163\) 15.5677 1.21936 0.609679 0.792648i \(-0.291298\pi\)
0.609679 + 0.792648i \(0.291298\pi\)
\(164\) 7.80977 0.609841
\(165\) −0.228053 −0.0177539
\(166\) 12.0367 0.934232
\(167\) −6.01893 −0.465759 −0.232879 0.972506i \(-0.574815\pi\)
−0.232879 + 0.972506i \(0.574815\pi\)
\(168\) −0.642859 −0.0495977
\(169\) −11.7359 −0.902759
\(170\) 1.66554 0.127741
\(171\) −21.8756 −1.67287
\(172\) −1.00000 −0.0762493
\(173\) −4.00241 −0.304298 −0.152149 0.988358i \(-0.548619\pi\)
−0.152149 + 0.988358i \(0.548619\pi\)
\(174\) −1.07853 −0.0817633
\(175\) −2.81890 −0.213089
\(176\) 1.00000 0.0753778
\(177\) −0.173171 −0.0130164
\(178\) 5.21429 0.390828
\(179\) −0.139202 −0.0104044 −0.00520221 0.999986i \(-0.501656\pi\)
−0.00520221 + 0.999986i \(0.501656\pi\)
\(180\) −2.94799 −0.219730
\(181\) 6.05904 0.450365 0.225182 0.974317i \(-0.427702\pi\)
0.225182 + 0.974317i \(0.427702\pi\)
\(182\) 3.16939 0.234931
\(183\) −1.10651 −0.0817959
\(184\) 0.875665 0.0645549
\(185\) −5.63729 −0.414462
\(186\) −0.606772 −0.0444907
\(187\) −1.66554 −0.121796
\(188\) 13.2060 0.963146
\(189\) −3.82372 −0.278135
\(190\) −7.42051 −0.538340
\(191\) −16.1843 −1.17106 −0.585528 0.810653i \(-0.699113\pi\)
−0.585528 + 0.810653i \(0.699113\pi\)
\(192\) −0.228053 −0.0164583
\(193\) 25.5558 1.83954 0.919772 0.392452i \(-0.128373\pi\)
0.919772 + 0.392452i \(0.128373\pi\)
\(194\) 6.50507 0.467037
\(195\) −0.256408 −0.0183617
\(196\) 0.946220 0.0675872
\(197\) 19.6077 1.39699 0.698495 0.715615i \(-0.253853\pi\)
0.698495 + 0.715615i \(0.253853\pi\)
\(198\) 2.94799 0.209505
\(199\) −21.6580 −1.53529 −0.767647 0.640873i \(-0.778573\pi\)
−0.767647 + 0.640873i \(0.778573\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0.512565 0.0361536
\(202\) −4.52576 −0.318431
\(203\) 13.3315 0.935685
\(204\) 0.379832 0.0265935
\(205\) 7.80977 0.545458
\(206\) 2.63398 0.183518
\(207\) 2.58145 0.179423
\(208\) 1.12433 0.0779586
\(209\) 7.42051 0.513287
\(210\) −0.642859 −0.0443615
\(211\) 12.3193 0.848098 0.424049 0.905639i \(-0.360608\pi\)
0.424049 + 0.905639i \(0.360608\pi\)
\(212\) −3.40667 −0.233971
\(213\) −1.69023 −0.115813
\(214\) 9.33164 0.637898
\(215\) −1.00000 −0.0681994
\(216\) −1.35646 −0.0922952
\(217\) 7.50016 0.509144
\(218\) 4.97087 0.336670
\(219\) −0.572644 −0.0386957
\(220\) 1.00000 0.0674200
\(221\) −1.87263 −0.125966
\(222\) −1.28560 −0.0862839
\(223\) −20.5666 −1.37724 −0.688620 0.725123i \(-0.741783\pi\)
−0.688620 + 0.725123i \(0.741783\pi\)
\(224\) 2.81890 0.188346
\(225\) −2.94799 −0.196533
\(226\) 8.23164 0.547560
\(227\) −24.9903 −1.65866 −0.829331 0.558757i \(-0.811278\pi\)
−0.829331 + 0.558757i \(0.811278\pi\)
\(228\) −1.69227 −0.112073
\(229\) −23.3892 −1.54560 −0.772801 0.634648i \(-0.781145\pi\)
−0.772801 + 0.634648i \(0.781145\pi\)
\(230\) 0.875665 0.0577397
\(231\) 0.642859 0.0422970
\(232\) 4.72931 0.310494
\(233\) −7.48056 −0.490068 −0.245034 0.969514i \(-0.578799\pi\)
−0.245034 + 0.969514i \(0.578799\pi\)
\(234\) 3.31453 0.216678
\(235\) 13.2060 0.861464
\(236\) 0.759347 0.0494293
\(237\) 2.45057 0.159182
\(238\) −4.69500 −0.304332
\(239\) 10.7522 0.695499 0.347750 0.937587i \(-0.386946\pi\)
0.347750 + 0.937587i \(0.386946\pi\)
\(240\) −0.228053 −0.0147208
\(241\) 10.8505 0.698944 0.349472 0.936947i \(-0.386361\pi\)
0.349472 + 0.936947i \(0.386361\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −6.01572 −0.385909
\(244\) 4.85201 0.310618
\(245\) 0.946220 0.0604518
\(246\) 1.78104 0.113555
\(247\) 8.34313 0.530861
\(248\) 2.66066 0.168952
\(249\) 2.74501 0.173958
\(250\) −1.00000 −0.0632456
\(251\) −1.64877 −0.104069 −0.0520346 0.998645i \(-0.516571\pi\)
−0.0520346 + 0.998645i \(0.516571\pi\)
\(252\) 8.31011 0.523487
\(253\) −0.875665 −0.0550526
\(254\) 20.1915 1.26693
\(255\) 0.379832 0.0237860
\(256\) 1.00000 0.0625000
\(257\) −21.6789 −1.35229 −0.676146 0.736767i \(-0.736351\pi\)
−0.676146 + 0.736767i \(0.736351\pi\)
\(258\) −0.228053 −0.0141980
\(259\) 15.8910 0.987418
\(260\) 1.12433 0.0697283
\(261\) 13.9420 0.862986
\(262\) 17.5653 1.08519
\(263\) −15.0436 −0.927631 −0.463815 0.885932i \(-0.653520\pi\)
−0.463815 + 0.885932i \(0.653520\pi\)
\(264\) 0.228053 0.0140357
\(265\) −3.40667 −0.209270
\(266\) 20.9177 1.28255
\(267\) 1.18914 0.0727739
\(268\) −2.24757 −0.137292
\(269\) 17.5715 1.07135 0.535677 0.844423i \(-0.320056\pi\)
0.535677 + 0.844423i \(0.320056\pi\)
\(270\) −1.35646 −0.0825513
\(271\) −5.85055 −0.355396 −0.177698 0.984085i \(-0.556865\pi\)
−0.177698 + 0.984085i \(0.556865\pi\)
\(272\) −1.66554 −0.100988
\(273\) 0.722789 0.0437452
\(274\) 10.6093 0.640930
\(275\) 1.00000 0.0603023
\(276\) 0.199698 0.0120204
\(277\) 5.20593 0.312794 0.156397 0.987694i \(-0.450012\pi\)
0.156397 + 0.987694i \(0.450012\pi\)
\(278\) 1.26492 0.0758648
\(279\) 7.84362 0.469585
\(280\) 2.81890 0.168462
\(281\) 0.709992 0.0423546 0.0211773 0.999776i \(-0.493259\pi\)
0.0211773 + 0.999776i \(0.493259\pi\)
\(282\) 3.01167 0.179342
\(283\) −12.9171 −0.767843 −0.383921 0.923366i \(-0.625427\pi\)
−0.383921 + 0.923366i \(0.625427\pi\)
\(284\) 7.41158 0.439797
\(285\) −1.69227 −0.100241
\(286\) −1.12433 −0.0664833
\(287\) −22.0150 −1.29950
\(288\) 2.94799 0.173712
\(289\) −14.2260 −0.836822
\(290\) 4.72931 0.277714
\(291\) 1.48350 0.0869644
\(292\) 2.51101 0.146946
\(293\) 31.7028 1.85210 0.926049 0.377402i \(-0.123183\pi\)
0.926049 + 0.377402i \(0.123183\pi\)
\(294\) 0.215788 0.0125850
\(295\) 0.759347 0.0442109
\(296\) 5.63729 0.327661
\(297\) 1.35646 0.0787096
\(298\) −9.56881 −0.554306
\(299\) −0.984541 −0.0569375
\(300\) −0.228053 −0.0131666
\(301\) 2.81890 0.162479
\(302\) −15.7568 −0.906703
\(303\) −1.03211 −0.0592933
\(304\) 7.42051 0.425595
\(305\) 4.85201 0.277825
\(306\) −4.91000 −0.280686
\(307\) −12.5516 −0.716360 −0.358180 0.933653i \(-0.616603\pi\)
−0.358180 + 0.933653i \(0.616603\pi\)
\(308\) −2.81890 −0.160622
\(309\) 0.600688 0.0341719
\(310\) 2.66066 0.151116
\(311\) −31.6930 −1.79715 −0.898573 0.438823i \(-0.855395\pi\)
−0.898573 + 0.438823i \(0.855395\pi\)
\(312\) 0.256408 0.0145162
\(313\) −29.8594 −1.68775 −0.843877 0.536536i \(-0.819732\pi\)
−0.843877 + 0.536536i \(0.819732\pi\)
\(314\) 4.47588 0.252589
\(315\) 8.31011 0.468221
\(316\) −10.7456 −0.604490
\(317\) 21.6465 1.21579 0.607894 0.794018i \(-0.292014\pi\)
0.607894 + 0.794018i \(0.292014\pi\)
\(318\) −0.776902 −0.0435665
\(319\) −4.72931 −0.264790
\(320\) 1.00000 0.0559017
\(321\) 2.12811 0.118779
\(322\) −2.46842 −0.137559
\(323\) −12.3592 −0.687682
\(324\) 8.53463 0.474146
\(325\) 1.12433 0.0623669
\(326\) −15.5677 −0.862216
\(327\) 1.13362 0.0626894
\(328\) −7.80977 −0.431223
\(329\) −37.2264 −2.05236
\(330\) 0.228053 0.0125539
\(331\) −30.4783 −1.67524 −0.837621 0.546252i \(-0.816054\pi\)
−0.837621 + 0.546252i \(0.816054\pi\)
\(332\) −12.0367 −0.660602
\(333\) 16.6187 0.910699
\(334\) 6.01893 0.329341
\(335\) −2.24757 −0.122798
\(336\) 0.642859 0.0350708
\(337\) −23.7603 −1.29431 −0.647154 0.762359i \(-0.724041\pi\)
−0.647154 + 0.762359i \(0.724041\pi\)
\(338\) 11.7359 0.638347
\(339\) 1.87725 0.101958
\(340\) −1.66554 −0.0903267
\(341\) −2.66066 −0.144083
\(342\) 21.8756 1.18290
\(343\) 17.0650 0.921425
\(344\) 1.00000 0.0539164
\(345\) 0.199698 0.0107514
\(346\) 4.00241 0.215171
\(347\) −14.2525 −0.765112 −0.382556 0.923932i \(-0.624956\pi\)
−0.382556 + 0.923932i \(0.624956\pi\)
\(348\) 1.07853 0.0578154
\(349\) −5.27164 −0.282184 −0.141092 0.989996i \(-0.545061\pi\)
−0.141092 + 0.989996i \(0.545061\pi\)
\(350\) 2.81890 0.150677
\(351\) 1.52511 0.0814044
\(352\) −1.00000 −0.0533002
\(353\) −21.3297 −1.13526 −0.567632 0.823282i \(-0.692140\pi\)
−0.567632 + 0.823282i \(0.692140\pi\)
\(354\) 0.173171 0.00920395
\(355\) 7.41158 0.393366
\(356\) −5.21429 −0.276357
\(357\) −1.07071 −0.0566679
\(358\) 0.139202 0.00735704
\(359\) 4.53953 0.239587 0.119794 0.992799i \(-0.461777\pi\)
0.119794 + 0.992799i \(0.461777\pi\)
\(360\) 2.94799 0.155373
\(361\) 36.0639 1.89810
\(362\) −6.05904 −0.318456
\(363\) −0.228053 −0.0119697
\(364\) −3.16939 −0.166121
\(365\) 2.51101 0.131432
\(366\) 1.10651 0.0578384
\(367\) −3.89111 −0.203114 −0.101557 0.994830i \(-0.532382\pi\)
−0.101557 + 0.994830i \(0.532382\pi\)
\(368\) −0.875665 −0.0456472
\(369\) −23.0232 −1.19854
\(370\) 5.63729 0.293069
\(371\) 9.60308 0.498567
\(372\) 0.606772 0.0314597
\(373\) 23.0020 1.19100 0.595500 0.803356i \(-0.296954\pi\)
0.595500 + 0.803356i \(0.296954\pi\)
\(374\) 1.66554 0.0861231
\(375\) −0.228053 −0.0117766
\(376\) −13.2060 −0.681047
\(377\) −5.31732 −0.273856
\(378\) 3.82372 0.196671
\(379\) 9.10158 0.467517 0.233758 0.972295i \(-0.424898\pi\)
0.233758 + 0.972295i \(0.424898\pi\)
\(380\) 7.42051 0.380664
\(381\) 4.60474 0.235908
\(382\) 16.1843 0.828061
\(383\) 12.0564 0.616051 0.308025 0.951378i \(-0.400332\pi\)
0.308025 + 0.951378i \(0.400332\pi\)
\(384\) 0.228053 0.0116378
\(385\) −2.81890 −0.143665
\(386\) −25.5558 −1.30075
\(387\) 2.94799 0.149855
\(388\) −6.50507 −0.330245
\(389\) 27.1985 1.37902 0.689510 0.724277i \(-0.257826\pi\)
0.689510 + 0.724277i \(0.257826\pi\)
\(390\) 0.256408 0.0129837
\(391\) 1.45846 0.0737573
\(392\) −0.946220 −0.0477913
\(393\) 4.00582 0.202067
\(394\) −19.6077 −0.987821
\(395\) −10.7456 −0.540672
\(396\) −2.94799 −0.148142
\(397\) −17.7469 −0.890693 −0.445346 0.895358i \(-0.646919\pi\)
−0.445346 + 0.895358i \(0.646919\pi\)
\(398\) 21.6580 1.08562
\(399\) 4.77034 0.238816
\(400\) 1.00000 0.0500000
\(401\) 6.79422 0.339287 0.169644 0.985505i \(-0.445738\pi\)
0.169644 + 0.985505i \(0.445738\pi\)
\(402\) −0.512565 −0.0255644
\(403\) −2.99148 −0.149016
\(404\) 4.52576 0.225165
\(405\) 8.53463 0.424089
\(406\) −13.3315 −0.661629
\(407\) −5.63729 −0.279430
\(408\) −0.379832 −0.0188045
\(409\) −5.47349 −0.270647 −0.135323 0.990801i \(-0.543207\pi\)
−0.135323 + 0.990801i \(0.543207\pi\)
\(410\) −7.80977 −0.385697
\(411\) 2.41948 0.119344
\(412\) −2.63398 −0.129767
\(413\) −2.14053 −0.105328
\(414\) −2.58145 −0.126872
\(415\) −12.0367 −0.590860
\(416\) −1.12433 −0.0551250
\(417\) 0.288468 0.0141264
\(418\) −7.42051 −0.362949
\(419\) −1.27871 −0.0624689 −0.0312344 0.999512i \(-0.509944\pi\)
−0.0312344 + 0.999512i \(0.509944\pi\)
\(420\) 0.642859 0.0313683
\(421\) −11.6440 −0.567492 −0.283746 0.958899i \(-0.591577\pi\)
−0.283746 + 0.958899i \(0.591577\pi\)
\(422\) −12.3193 −0.599696
\(423\) −38.9312 −1.89290
\(424\) 3.40667 0.165443
\(425\) −1.66554 −0.0807906
\(426\) 1.69023 0.0818921
\(427\) −13.6773 −0.661893
\(428\) −9.33164 −0.451062
\(429\) −0.256408 −0.0123795
\(430\) 1.00000 0.0482243
\(431\) 26.0044 1.25259 0.626295 0.779586i \(-0.284571\pi\)
0.626295 + 0.779586i \(0.284571\pi\)
\(432\) 1.35646 0.0652626
\(433\) 1.06003 0.0509419 0.0254710 0.999676i \(-0.491891\pi\)
0.0254710 + 0.999676i \(0.491891\pi\)
\(434\) −7.50016 −0.360019
\(435\) 1.07853 0.0517117
\(436\) −4.97087 −0.238061
\(437\) −6.49788 −0.310836
\(438\) 0.572644 0.0273620
\(439\) 15.3574 0.732970 0.366485 0.930424i \(-0.380561\pi\)
0.366485 + 0.930424i \(0.380561\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −2.78945 −0.132831
\(442\) 1.87263 0.0890717
\(443\) −13.0010 −0.617697 −0.308848 0.951111i \(-0.599944\pi\)
−0.308848 + 0.951111i \(0.599944\pi\)
\(444\) 1.28560 0.0610119
\(445\) −5.21429 −0.247181
\(446\) 20.5666 0.973855
\(447\) −2.18219 −0.103214
\(448\) −2.81890 −0.133181
\(449\) −22.9889 −1.08491 −0.542457 0.840084i \(-0.682506\pi\)
−0.542457 + 0.840084i \(0.682506\pi\)
\(450\) 2.94799 0.138970
\(451\) 7.80977 0.367748
\(452\) −8.23164 −0.387184
\(453\) −3.59339 −0.168832
\(454\) 24.9903 1.17285
\(455\) −3.16939 −0.148583
\(456\) 1.69227 0.0792477
\(457\) −17.3573 −0.811942 −0.405971 0.913886i \(-0.633067\pi\)
−0.405971 + 0.913886i \(0.633067\pi\)
\(458\) 23.3892 1.09291
\(459\) −2.25923 −0.105452
\(460\) −0.875665 −0.0408281
\(461\) 2.22579 0.103665 0.0518327 0.998656i \(-0.483494\pi\)
0.0518327 + 0.998656i \(0.483494\pi\)
\(462\) −0.642859 −0.0299085
\(463\) −12.5943 −0.585306 −0.292653 0.956219i \(-0.594538\pi\)
−0.292653 + 0.956219i \(0.594538\pi\)
\(464\) −4.72931 −0.219553
\(465\) 0.606772 0.0281384
\(466\) 7.48056 0.346530
\(467\) −15.4563 −0.715232 −0.357616 0.933869i \(-0.616410\pi\)
−0.357616 + 0.933869i \(0.616410\pi\)
\(468\) −3.31453 −0.153214
\(469\) 6.33569 0.292555
\(470\) −13.2060 −0.609147
\(471\) 1.02074 0.0470331
\(472\) −0.759347 −0.0349518
\(473\) −1.00000 −0.0459800
\(474\) −2.45057 −0.112559
\(475\) 7.42051 0.340476
\(476\) 4.69500 0.215195
\(477\) 10.0428 0.459830
\(478\) −10.7522 −0.491792
\(479\) 16.3881 0.748793 0.374397 0.927269i \(-0.377850\pi\)
0.374397 + 0.927269i \(0.377850\pi\)
\(480\) 0.228053 0.0104091
\(481\) −6.33820 −0.288997
\(482\) −10.8505 −0.494228
\(483\) −0.562930 −0.0256142
\(484\) 1.00000 0.0454545
\(485\) −6.50507 −0.295380
\(486\) 6.01572 0.272879
\(487\) −3.92891 −0.178036 −0.0890178 0.996030i \(-0.528373\pi\)
−0.0890178 + 0.996030i \(0.528373\pi\)
\(488\) −4.85201 −0.219640
\(489\) −3.55026 −0.160548
\(490\) −0.946220 −0.0427459
\(491\) 9.17767 0.414183 0.207091 0.978322i \(-0.433600\pi\)
0.207091 + 0.978322i \(0.433600\pi\)
\(492\) −1.78104 −0.0802956
\(493\) 7.87686 0.354756
\(494\) −8.34313 −0.375375
\(495\) −2.94799 −0.132502
\(496\) −2.66066 −0.119467
\(497\) −20.8925 −0.937159
\(498\) −2.74501 −0.123007
\(499\) −39.8502 −1.78394 −0.891969 0.452096i \(-0.850676\pi\)
−0.891969 + 0.452096i \(0.850676\pi\)
\(500\) 1.00000 0.0447214
\(501\) 1.37264 0.0613248
\(502\) 1.64877 0.0735881
\(503\) −25.6544 −1.14387 −0.571937 0.820297i \(-0.693808\pi\)
−0.571937 + 0.820297i \(0.693808\pi\)
\(504\) −8.31011 −0.370162
\(505\) 4.52576 0.201394
\(506\) 0.875665 0.0389281
\(507\) 2.67640 0.118863
\(508\) −20.1915 −0.895854
\(509\) 21.5400 0.954746 0.477373 0.878701i \(-0.341589\pi\)
0.477373 + 0.878701i \(0.341589\pi\)
\(510\) −0.379832 −0.0168192
\(511\) −7.07830 −0.313126
\(512\) −1.00000 −0.0441942
\(513\) 10.0656 0.444407
\(514\) 21.6789 0.956215
\(515\) −2.63398 −0.116067
\(516\) 0.228053 0.0100395
\(517\) 13.2060 0.580799
\(518\) −15.8910 −0.698210
\(519\) 0.912762 0.0400658
\(520\) −1.12433 −0.0493053
\(521\) −11.6344 −0.509713 −0.254857 0.966979i \(-0.582028\pi\)
−0.254857 + 0.966979i \(0.582028\pi\)
\(522\) −13.9420 −0.610223
\(523\) −1.08302 −0.0473570 −0.0236785 0.999720i \(-0.507538\pi\)
−0.0236785 + 0.999720i \(0.507538\pi\)
\(524\) −17.5653 −0.767345
\(525\) 0.642859 0.0280567
\(526\) 15.0436 0.655934
\(527\) 4.43145 0.193037
\(528\) −0.228053 −0.00992473
\(529\) −22.2332 −0.966661
\(530\) 3.40667 0.147976
\(531\) −2.23855 −0.0971447
\(532\) −20.9177 −0.906897
\(533\) 8.78080 0.380339
\(534\) −1.18914 −0.0514589
\(535\) −9.33164 −0.403442
\(536\) 2.24757 0.0970803
\(537\) 0.0317454 0.00136991
\(538\) −17.5715 −0.757562
\(539\) 0.946220 0.0407566
\(540\) 1.35646 0.0583726
\(541\) −16.5158 −0.710072 −0.355036 0.934853i \(-0.615531\pi\)
−0.355036 + 0.934853i \(0.615531\pi\)
\(542\) 5.85055 0.251303
\(543\) −1.38178 −0.0592979
\(544\) 1.66554 0.0714095
\(545\) −4.97087 −0.212929
\(546\) −0.722789 −0.0309325
\(547\) −20.5525 −0.878759 −0.439380 0.898301i \(-0.644802\pi\)
−0.439380 + 0.898301i \(0.644802\pi\)
\(548\) −10.6093 −0.453206
\(549\) −14.3037 −0.610466
\(550\) −1.00000 −0.0426401
\(551\) −35.0939 −1.49505
\(552\) −0.199698 −0.00849971
\(553\) 30.2909 1.28810
\(554\) −5.20593 −0.221179
\(555\) 1.28560 0.0545707
\(556\) −1.26492 −0.0536445
\(557\) 18.4175 0.780377 0.390188 0.920735i \(-0.372410\pi\)
0.390188 + 0.920735i \(0.372410\pi\)
\(558\) −7.84362 −0.332047
\(559\) −1.12433 −0.0475543
\(560\) −2.81890 −0.119120
\(561\) 0.379832 0.0160365
\(562\) −0.709992 −0.0299492
\(563\) −26.4824 −1.11610 −0.558049 0.829808i \(-0.688450\pi\)
−0.558049 + 0.829808i \(0.688450\pi\)
\(564\) −3.01167 −0.126814
\(565\) −8.23164 −0.346308
\(566\) 12.9171 0.542947
\(567\) −24.0583 −1.01035
\(568\) −7.41158 −0.310983
\(569\) 28.7858 1.20676 0.603381 0.797453i \(-0.293820\pi\)
0.603381 + 0.797453i \(0.293820\pi\)
\(570\) 1.69227 0.0708813
\(571\) 42.4795 1.77771 0.888855 0.458188i \(-0.151501\pi\)
0.888855 + 0.458188i \(0.151501\pi\)
\(572\) 1.12433 0.0470108
\(573\) 3.69088 0.154189
\(574\) 22.0150 0.918889
\(575\) −0.875665 −0.0365178
\(576\) −2.94799 −0.122833
\(577\) 31.9934 1.33190 0.665952 0.745995i \(-0.268026\pi\)
0.665952 + 0.745995i \(0.268026\pi\)
\(578\) 14.2260 0.591722
\(579\) −5.82807 −0.242206
\(580\) −4.72931 −0.196374
\(581\) 33.9304 1.40767
\(582\) −1.48350 −0.0614931
\(583\) −3.40667 −0.141090
\(584\) −2.51101 −0.103906
\(585\) −3.31453 −0.137039
\(586\) −31.7028 −1.30963
\(587\) 33.9644 1.40186 0.700930 0.713230i \(-0.252769\pi\)
0.700930 + 0.713230i \(0.252769\pi\)
\(588\) −0.215788 −0.00889896
\(589\) −19.7435 −0.813516
\(590\) −0.759347 −0.0312618
\(591\) −4.47159 −0.183937
\(592\) −5.63729 −0.231691
\(593\) 4.34754 0.178532 0.0892660 0.996008i \(-0.471548\pi\)
0.0892660 + 0.996008i \(0.471548\pi\)
\(594\) −1.35646 −0.0556561
\(595\) 4.69500 0.192476
\(596\) 9.56881 0.391954
\(597\) 4.93917 0.202147
\(598\) 0.984541 0.0402609
\(599\) −46.5405 −1.90159 −0.950796 0.309819i \(-0.899732\pi\)
−0.950796 + 0.309819i \(0.899732\pi\)
\(600\) 0.228053 0.00931022
\(601\) 6.17544 0.251902 0.125951 0.992036i \(-0.459802\pi\)
0.125951 + 0.992036i \(0.459802\pi\)
\(602\) −2.81890 −0.114890
\(603\) 6.62583 0.269824
\(604\) 15.7568 0.641136
\(605\) 1.00000 0.0406558
\(606\) 1.03211 0.0419267
\(607\) −11.5532 −0.468929 −0.234464 0.972125i \(-0.575334\pi\)
−0.234464 + 0.972125i \(0.575334\pi\)
\(608\) −7.42051 −0.300941
\(609\) −3.04028 −0.123198
\(610\) −4.85201 −0.196452
\(611\) 14.8480 0.600684
\(612\) 4.91000 0.198475
\(613\) −9.44772 −0.381590 −0.190795 0.981630i \(-0.561107\pi\)
−0.190795 + 0.981630i \(0.561107\pi\)
\(614\) 12.5516 0.506543
\(615\) −1.78104 −0.0718185
\(616\) 2.81890 0.113577
\(617\) −1.03392 −0.0416239 −0.0208120 0.999783i \(-0.506625\pi\)
−0.0208120 + 0.999783i \(0.506625\pi\)
\(618\) −0.600688 −0.0241632
\(619\) −25.0261 −1.00588 −0.502941 0.864321i \(-0.667749\pi\)
−0.502941 + 0.864321i \(0.667749\pi\)
\(620\) −2.66066 −0.106855
\(621\) −1.18780 −0.0476649
\(622\) 31.6930 1.27077
\(623\) 14.6986 0.588887
\(624\) −0.256408 −0.0102645
\(625\) 1.00000 0.0400000
\(626\) 29.8594 1.19342
\(627\) −1.69227 −0.0675827
\(628\) −4.47588 −0.178607
\(629\) 9.38914 0.374370
\(630\) −8.31011 −0.331083
\(631\) −12.3126 −0.490158 −0.245079 0.969503i \(-0.578814\pi\)
−0.245079 + 0.969503i \(0.578814\pi\)
\(632\) 10.7456 0.427439
\(633\) −2.80946 −0.111666
\(634\) −21.6465 −0.859693
\(635\) −20.1915 −0.801276
\(636\) 0.776902 0.0308062
\(637\) 1.06387 0.0421520
\(638\) 4.72931 0.187235
\(639\) −21.8493 −0.864345
\(640\) −1.00000 −0.0395285
\(641\) 9.17973 0.362578 0.181289 0.983430i \(-0.441973\pi\)
0.181289 + 0.983430i \(0.441973\pi\)
\(642\) −2.12811 −0.0839897
\(643\) 0.391841 0.0154527 0.00772635 0.999970i \(-0.497541\pi\)
0.00772635 + 0.999970i \(0.497541\pi\)
\(644\) 2.46842 0.0972693
\(645\) 0.228053 0.00897957
\(646\) 12.3592 0.486265
\(647\) −30.6471 −1.20486 −0.602431 0.798171i \(-0.705801\pi\)
−0.602431 + 0.798171i \(0.705801\pi\)
\(648\) −8.53463 −0.335272
\(649\) 0.759347 0.0298070
\(650\) −1.12433 −0.0441000
\(651\) −1.71043 −0.0670371
\(652\) 15.5677 0.609679
\(653\) 22.1546 0.866977 0.433488 0.901159i \(-0.357283\pi\)
0.433488 + 0.901159i \(0.357283\pi\)
\(654\) −1.13362 −0.0443281
\(655\) −17.5653 −0.686334
\(656\) 7.80977 0.304920
\(657\) −7.40245 −0.288797
\(658\) 37.2264 1.45124
\(659\) 32.6360 1.27132 0.635659 0.771970i \(-0.280729\pi\)
0.635659 + 0.771970i \(0.280729\pi\)
\(660\) −0.228053 −0.00887695
\(661\) −13.2055 −0.513633 −0.256816 0.966460i \(-0.582674\pi\)
−0.256816 + 0.966460i \(0.582674\pi\)
\(662\) 30.4783 1.18457
\(663\) 0.427058 0.0165855
\(664\) 12.0367 0.467116
\(665\) −20.9177 −0.811153
\(666\) −16.6187 −0.643961
\(667\) 4.14129 0.160351
\(668\) −6.01893 −0.232879
\(669\) 4.69027 0.181336
\(670\) 2.24757 0.0868313
\(671\) 4.85201 0.187310
\(672\) −0.642859 −0.0247988
\(673\) 10.7752 0.415353 0.207677 0.978198i \(-0.433410\pi\)
0.207677 + 0.978198i \(0.433410\pi\)
\(674\) 23.7603 0.915214
\(675\) 1.35646 0.0522100
\(676\) −11.7359 −0.451380
\(677\) −16.0165 −0.615564 −0.307782 0.951457i \(-0.599587\pi\)
−0.307782 + 0.951457i \(0.599587\pi\)
\(678\) −1.87725 −0.0720953
\(679\) 18.3372 0.703716
\(680\) 1.66554 0.0638706
\(681\) 5.69910 0.218390
\(682\) 2.66066 0.101882
\(683\) −2.47119 −0.0945576 −0.0472788 0.998882i \(-0.515055\pi\)
−0.0472788 + 0.998882i \(0.515055\pi\)
\(684\) −21.8756 −0.836434
\(685\) −10.6093 −0.405360
\(686\) −17.0650 −0.651546
\(687\) 5.33398 0.203504
\(688\) −1.00000 −0.0381246
\(689\) −3.83024 −0.145921
\(690\) −0.199698 −0.00760238
\(691\) −34.7962 −1.32371 −0.661855 0.749632i \(-0.730231\pi\)
−0.661855 + 0.749632i \(0.730231\pi\)
\(692\) −4.00241 −0.152149
\(693\) 8.31011 0.315675
\(694\) 14.2525 0.541016
\(695\) −1.26492 −0.0479811
\(696\) −1.07853 −0.0408817
\(697\) −13.0075 −0.492694
\(698\) 5.27164 0.199534
\(699\) 1.70596 0.0645255
\(700\) −2.81890 −0.106545
\(701\) 22.8672 0.863683 0.431841 0.901950i \(-0.357864\pi\)
0.431841 + 0.901950i \(0.357864\pi\)
\(702\) −1.52511 −0.0575616
\(703\) −41.8316 −1.57771
\(704\) 1.00000 0.0376889
\(705\) −3.01167 −0.113426
\(706\) 21.3297 0.802753
\(707\) −12.7577 −0.479802
\(708\) −0.173171 −0.00650818
\(709\) 13.4279 0.504294 0.252147 0.967689i \(-0.418863\pi\)
0.252147 + 0.967689i \(0.418863\pi\)
\(710\) −7.41158 −0.278152
\(711\) 31.6781 1.18802
\(712\) 5.21429 0.195414
\(713\) 2.32985 0.0872536
\(714\) 1.07071 0.0400703
\(715\) 1.12433 0.0420477
\(716\) −0.139202 −0.00520221
\(717\) −2.45206 −0.0915739
\(718\) −4.53953 −0.169414
\(719\) 3.01743 0.112531 0.0562656 0.998416i \(-0.482081\pi\)
0.0562656 + 0.998416i \(0.482081\pi\)
\(720\) −2.94799 −0.109865
\(721\) 7.42495 0.276520
\(722\) −36.0639 −1.34216
\(723\) −2.47449 −0.0920274
\(724\) 6.05904 0.225182
\(725\) −4.72931 −0.175642
\(726\) 0.228053 0.00846384
\(727\) −40.5835 −1.50516 −0.752579 0.658502i \(-0.771190\pi\)
−0.752579 + 0.658502i \(0.771190\pi\)
\(728\) 3.16939 0.117465
\(729\) −24.2320 −0.897481
\(730\) −2.51101 −0.0929368
\(731\) 1.66554 0.0616023
\(732\) −1.10651 −0.0408979
\(733\) 48.4636 1.79004 0.895022 0.446021i \(-0.147160\pi\)
0.895022 + 0.446021i \(0.147160\pi\)
\(734\) 3.89111 0.143623
\(735\) −0.215788 −0.00795947
\(736\) 0.875665 0.0322775
\(737\) −2.24757 −0.0827904
\(738\) 23.0232 0.847494
\(739\) 34.9368 1.28517 0.642586 0.766213i \(-0.277861\pi\)
0.642586 + 0.766213i \(0.277861\pi\)
\(740\) −5.63729 −0.207231
\(741\) −1.90268 −0.0698965
\(742\) −9.60308 −0.352540
\(743\) −41.3831 −1.51820 −0.759100 0.650974i \(-0.774361\pi\)
−0.759100 + 0.650974i \(0.774361\pi\)
\(744\) −0.606772 −0.0222454
\(745\) 9.56881 0.350574
\(746\) −23.0020 −0.842164
\(747\) 35.4842 1.29830
\(748\) −1.66554 −0.0608982
\(749\) 26.3050 0.961164
\(750\) 0.228053 0.00832732
\(751\) 24.2343 0.884322 0.442161 0.896936i \(-0.354212\pi\)
0.442161 + 0.896936i \(0.354212\pi\)
\(752\) 13.2060 0.481573
\(753\) 0.376006 0.0137024
\(754\) 5.31732 0.193646
\(755\) 15.7568 0.573449
\(756\) −3.82372 −0.139067
\(757\) −34.1443 −1.24100 −0.620498 0.784208i \(-0.713069\pi\)
−0.620498 + 0.784208i \(0.713069\pi\)
\(758\) −9.10158 −0.330584
\(759\) 0.199698 0.00724858
\(760\) −7.42051 −0.269170
\(761\) 15.2944 0.554421 0.277211 0.960809i \(-0.410590\pi\)
0.277211 + 0.960809i \(0.410590\pi\)
\(762\) −4.60474 −0.166812
\(763\) 14.0124 0.507283
\(764\) −16.1843 −0.585528
\(765\) 4.91000 0.177521
\(766\) −12.0564 −0.435614
\(767\) 0.853760 0.0308275
\(768\) −0.228053 −0.00822915
\(769\) −5.86852 −0.211624 −0.105812 0.994386i \(-0.533744\pi\)
−0.105812 + 0.994386i \(0.533744\pi\)
\(770\) 2.81890 0.101586
\(771\) 4.94394 0.178052
\(772\) 25.5558 0.919772
\(773\) −1.03649 −0.0372801 −0.0186401 0.999826i \(-0.505934\pi\)
−0.0186401 + 0.999826i \(0.505934\pi\)
\(774\) −2.94799 −0.105963
\(775\) −2.66066 −0.0955739
\(776\) 6.50507 0.233519
\(777\) −3.62399 −0.130010
\(778\) −27.1985 −0.975114
\(779\) 57.9525 2.07636
\(780\) −0.256408 −0.00918087
\(781\) 7.41158 0.265207
\(782\) −1.45846 −0.0521543
\(783\) −6.41510 −0.229257
\(784\) 0.946220 0.0337936
\(785\) −4.47588 −0.159751
\(786\) −4.00582 −0.142883
\(787\) −0.328194 −0.0116989 −0.00584943 0.999983i \(-0.501862\pi\)
−0.00584943 + 0.999983i \(0.501862\pi\)
\(788\) 19.6077 0.698495
\(789\) 3.43075 0.122138
\(790\) 10.7456 0.382313
\(791\) 23.2042 0.825046
\(792\) 2.94799 0.104752
\(793\) 5.45528 0.193723
\(794\) 17.7469 0.629815
\(795\) 0.776902 0.0275539
\(796\) −21.6580 −0.767647
\(797\) 17.5447 0.621466 0.310733 0.950497i \(-0.399425\pi\)
0.310733 + 0.950497i \(0.399425\pi\)
\(798\) −4.77034 −0.168868
\(799\) −21.9951 −0.778132
\(800\) −1.00000 −0.0353553
\(801\) 15.3717 0.543132
\(802\) −6.79422 −0.239912
\(803\) 2.51101 0.0886117
\(804\) 0.512565 0.0180768
\(805\) 2.46842 0.0870003
\(806\) 2.99148 0.105370
\(807\) −4.00723 −0.141061
\(808\) −4.52576 −0.159216
\(809\) 25.6981 0.903495 0.451748 0.892146i \(-0.350801\pi\)
0.451748 + 0.892146i \(0.350801\pi\)
\(810\) −8.53463 −0.299876
\(811\) −33.9530 −1.19225 −0.596125 0.802892i \(-0.703294\pi\)
−0.596125 + 0.802892i \(0.703294\pi\)
\(812\) 13.3315 0.467843
\(813\) 1.33424 0.0467937
\(814\) 5.63729 0.197587
\(815\) 15.5677 0.545313
\(816\) 0.379832 0.0132968
\(817\) −7.42051 −0.259611
\(818\) 5.47349 0.191376
\(819\) 9.34334 0.326483
\(820\) 7.80977 0.272729
\(821\) −9.62751 −0.336003 −0.168001 0.985787i \(-0.553731\pi\)
−0.168001 + 0.985787i \(0.553731\pi\)
\(822\) −2.41948 −0.0843890
\(823\) 16.0071 0.557971 0.278986 0.960295i \(-0.410002\pi\)
0.278986 + 0.960295i \(0.410002\pi\)
\(824\) 2.63398 0.0917592
\(825\) −0.228053 −0.00793978
\(826\) 2.14053 0.0744784
\(827\) 43.8703 1.52552 0.762759 0.646682i \(-0.223844\pi\)
0.762759 + 0.646682i \(0.223844\pi\)
\(828\) 2.58145 0.0897117
\(829\) −51.3496 −1.78344 −0.891722 0.452583i \(-0.850503\pi\)
−0.891722 + 0.452583i \(0.850503\pi\)
\(830\) 12.0367 0.417801
\(831\) −1.18723 −0.0411845
\(832\) 1.12433 0.0389793
\(833\) −1.57597 −0.0546041
\(834\) −0.288468 −0.00998885
\(835\) −6.01893 −0.208294
\(836\) 7.42051 0.256644
\(837\) −3.60908 −0.124748
\(838\) 1.27871 0.0441722
\(839\) 42.5718 1.46974 0.734871 0.678207i \(-0.237243\pi\)
0.734871 + 0.678207i \(0.237243\pi\)
\(840\) −0.642859 −0.0221808
\(841\) −6.63365 −0.228747
\(842\) 11.6440 0.401278
\(843\) −0.161916 −0.00557668
\(844\) 12.3193 0.424049
\(845\) −11.7359 −0.403726
\(846\) 38.9312 1.33848
\(847\) −2.81890 −0.0968587
\(848\) −3.40667 −0.116986
\(849\) 2.94579 0.101099
\(850\) 1.66554 0.0571276
\(851\) 4.93638 0.169217
\(852\) −1.69023 −0.0579064
\(853\) 10.7587 0.368372 0.184186 0.982891i \(-0.441035\pi\)
0.184186 + 0.982891i \(0.441035\pi\)
\(854\) 13.6773 0.468029
\(855\) −21.8756 −0.748129
\(856\) 9.33164 0.318949
\(857\) −43.8362 −1.49741 −0.748707 0.662901i \(-0.769325\pi\)
−0.748707 + 0.662901i \(0.769325\pi\)
\(858\) 0.256408 0.00875362
\(859\) 16.8246 0.574047 0.287023 0.957924i \(-0.407334\pi\)
0.287023 + 0.957924i \(0.407334\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 5.02059 0.171101
\(862\) −26.0044 −0.885715
\(863\) 17.2444 0.587008 0.293504 0.955958i \(-0.405179\pi\)
0.293504 + 0.955958i \(0.405179\pi\)
\(864\) −1.35646 −0.0461476
\(865\) −4.00241 −0.136086
\(866\) −1.06003 −0.0360214
\(867\) 3.24427 0.110181
\(868\) 7.50016 0.254572
\(869\) −10.7456 −0.364521
\(870\) −1.07853 −0.0365657
\(871\) −2.52702 −0.0856249
\(872\) 4.97087 0.168335
\(873\) 19.1769 0.649040
\(874\) 6.49788 0.219794
\(875\) −2.81890 −0.0952964
\(876\) −0.572644 −0.0193478
\(877\) −3.05479 −0.103153 −0.0515765 0.998669i \(-0.516425\pi\)
−0.0515765 + 0.998669i \(0.516425\pi\)
\(878\) −15.3574 −0.518288
\(879\) −7.22992 −0.243859
\(880\) 1.00000 0.0337100
\(881\) 20.4644 0.689464 0.344732 0.938701i \(-0.387970\pi\)
0.344732 + 0.938701i \(0.387970\pi\)
\(882\) 2.78945 0.0939256
\(883\) −13.9109 −0.468139 −0.234070 0.972220i \(-0.575204\pi\)
−0.234070 + 0.972220i \(0.575204\pi\)
\(884\) −1.87263 −0.0629832
\(885\) −0.173171 −0.00582109
\(886\) 13.0010 0.436778
\(887\) 1.06369 0.0357152 0.0178576 0.999841i \(-0.494315\pi\)
0.0178576 + 0.999841i \(0.494315\pi\)
\(888\) −1.28560 −0.0431420
\(889\) 56.9180 1.90897
\(890\) 5.21429 0.174784
\(891\) 8.53463 0.285921
\(892\) −20.5666 −0.688620
\(893\) 97.9952 3.27928
\(894\) 2.18219 0.0729835
\(895\) −0.139202 −0.00465300
\(896\) 2.81890 0.0941730
\(897\) 0.224527 0.00749675
\(898\) 22.9889 0.767150
\(899\) 12.5831 0.419670
\(900\) −2.94799 −0.0982664
\(901\) 5.67395 0.189027
\(902\) −7.80977 −0.260037
\(903\) −0.642859 −0.0213930
\(904\) 8.23164 0.273780
\(905\) 6.05904 0.201409
\(906\) 3.59339 0.119382
\(907\) −30.2393 −1.00408 −0.502040 0.864844i \(-0.667417\pi\)
−0.502040 + 0.864844i \(0.667417\pi\)
\(908\) −24.9903 −0.829331
\(909\) −13.3419 −0.442523
\(910\) 3.16939 0.105064
\(911\) 40.3214 1.33591 0.667954 0.744202i \(-0.267170\pi\)
0.667954 + 0.744202i \(0.267170\pi\)
\(912\) −1.69227 −0.0560366
\(913\) −12.0367 −0.398358
\(914\) 17.3573 0.574130
\(915\) −1.10651 −0.0365802
\(916\) −23.3892 −0.772801
\(917\) 49.5150 1.63513
\(918\) 2.25923 0.0745659
\(919\) −35.9397 −1.18554 −0.592770 0.805372i \(-0.701966\pi\)
−0.592770 + 0.805372i \(0.701966\pi\)
\(920\) 0.875665 0.0288698
\(921\) 2.86244 0.0943205
\(922\) −2.22579 −0.0733026
\(923\) 8.33310 0.274287
\(924\) 0.642859 0.0211485
\(925\) −5.63729 −0.185353
\(926\) 12.5943 0.413874
\(927\) 7.76496 0.255035
\(928\) 4.72931 0.155247
\(929\) 48.4403 1.58928 0.794638 0.607084i \(-0.207661\pi\)
0.794638 + 0.607084i \(0.207661\pi\)
\(930\) −0.606772 −0.0198968
\(931\) 7.02143 0.230118
\(932\) −7.48056 −0.245034
\(933\) 7.22769 0.236624
\(934\) 15.4563 0.505745
\(935\) −1.66554 −0.0544690
\(936\) 3.31453 0.108339
\(937\) 57.6303 1.88270 0.941350 0.337431i \(-0.109558\pi\)
0.941350 + 0.337431i \(0.109558\pi\)
\(938\) −6.33569 −0.206868
\(939\) 6.80953 0.222221
\(940\) 13.2060 0.430732
\(941\) 24.7222 0.805920 0.402960 0.915218i \(-0.367981\pi\)
0.402960 + 0.915218i \(0.367981\pi\)
\(942\) −1.02074 −0.0332574
\(943\) −6.83875 −0.222700
\(944\) 0.759347 0.0247146
\(945\) −3.82372 −0.124386
\(946\) 1.00000 0.0325128
\(947\) 20.3291 0.660608 0.330304 0.943875i \(-0.392849\pi\)
0.330304 + 0.943875i \(0.392849\pi\)
\(948\) 2.45057 0.0795910
\(949\) 2.82322 0.0916456
\(950\) −7.42051 −0.240753
\(951\) −4.93655 −0.160079
\(952\) −4.69500 −0.152166
\(953\) −23.9699 −0.776461 −0.388231 0.921562i \(-0.626914\pi\)
−0.388231 + 0.921562i \(0.626914\pi\)
\(954\) −10.0428 −0.325149
\(955\) −16.1843 −0.523712
\(956\) 10.7522 0.347750
\(957\) 1.07853 0.0348640
\(958\) −16.3881 −0.529477
\(959\) 29.9066 0.965733
\(960\) −0.228053 −0.00736038
\(961\) −23.9209 −0.771641
\(962\) 6.33820 0.204352
\(963\) 27.5096 0.886484
\(964\) 10.8505 0.349472
\(965\) 25.5558 0.822670
\(966\) 0.562930 0.0181120
\(967\) 37.5049 1.20608 0.603038 0.797712i \(-0.293957\pi\)
0.603038 + 0.797712i \(0.293957\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 2.81854 0.0905446
\(970\) 6.50507 0.208865
\(971\) 53.6069 1.72033 0.860163 0.510019i \(-0.170362\pi\)
0.860163 + 0.510019i \(0.170362\pi\)
\(972\) −6.01572 −0.192954
\(973\) 3.56569 0.114311
\(974\) 3.92891 0.125890
\(975\) −0.256408 −0.00821162
\(976\) 4.85201 0.155309
\(977\) −16.9339 −0.541763 −0.270881 0.962613i \(-0.587315\pi\)
−0.270881 + 0.962613i \(0.587315\pi\)
\(978\) 3.55026 0.113525
\(979\) −5.21429 −0.166650
\(980\) 0.946220 0.0302259
\(981\) 14.6541 0.467869
\(982\) −9.17767 −0.292871
\(983\) 20.9921 0.669545 0.334772 0.942299i \(-0.391341\pi\)
0.334772 + 0.942299i \(0.391341\pi\)
\(984\) 1.78104 0.0567775
\(985\) 19.6077 0.624753
\(986\) −7.87686 −0.250850
\(987\) 8.48960 0.270227
\(988\) 8.34313 0.265430
\(989\) 0.875665 0.0278445
\(990\) 2.94799 0.0936933
\(991\) −44.9992 −1.42945 −0.714723 0.699408i \(-0.753447\pi\)
−0.714723 + 0.699408i \(0.753447\pi\)
\(992\) 2.66066 0.0844762
\(993\) 6.95068 0.220573
\(994\) 20.8925 0.662671
\(995\) −21.6580 −0.686605
\(996\) 2.74501 0.0869790
\(997\) −19.5085 −0.617841 −0.308921 0.951088i \(-0.599968\pi\)
−0.308921 + 0.951088i \(0.599968\pi\)
\(998\) 39.8502 1.26143
\(999\) −7.64675 −0.241932
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 4730.2.a.w.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.w.1.6 8 1.1 even 1 trivial