Properties

Label 2310.2.a.z.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $2$
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] = [2310,2,Mod(1,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.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: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2310.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -3.46410 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -3.46410 q^{17} -1.00000 q^{18} +1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -1.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.92820 q^{29} -1.00000 q^{30} +10.9282 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.46410 q^{34} -1.00000 q^{35} +1.00000 q^{36} +3.46410 q^{37} -3.46410 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +1.00000 q^{44} +1.00000 q^{45} +1.46410 q^{46} +6.92820 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -3.46410 q^{51} -3.46410 q^{52} +6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -8.92820 q^{58} +8.00000 q^{59} +1.00000 q^{60} -4.92820 q^{61} -10.9282 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.46410 q^{65} -1.00000 q^{66} -5.46410 q^{67} -3.46410 q^{68} -1.46410 q^{69} +1.00000 q^{70} +1.07180 q^{71} -1.00000 q^{72} +15.8564 q^{73} -3.46410 q^{74} +1.00000 q^{75} -1.00000 q^{77} +3.46410 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +13.8564 q^{83} -1.00000 q^{84} -3.46410 q^{85} +8.92820 q^{87} -1.00000 q^{88} +8.53590 q^{89} -1.00000 q^{90} +3.46410 q^{91} -1.46410 q^{92} +10.9282 q^{93} -6.92820 q^{94} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 8 q^{31} - 2 q^{32} + 2 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{40} + 4 q^{41} + 2 q^{42} + 2 q^{44} + 2 q^{45} - 4 q^{46} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 4 q^{58} + 16 q^{59} + 2 q^{60} + 4 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{66} - 4 q^{67} + 4 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} + 4 q^{73} + 2 q^{75} - 2 q^{77} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} - 2 q^{84} + 4 q^{87} - 2 q^{88} + 24 q^{89} - 2 q^{90} + 4 q^{92} + 8 q^{93} - 2 q^{96} - 20 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −1.46410 −0.305286 −0.152643 0.988281i \(-0.548779\pi\)
−0.152643 + 0.988281i \(0.548779\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 3.46410 0.679366
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 8.92820 1.65793 0.828963 0.559304i \(-0.188931\pi\)
0.828963 + 0.559304i \(0.188931\pi\)
\(30\) −1.00000 −0.182574
\(31\) 10.9282 1.96276 0.981382 0.192068i \(-0.0615194\pi\)
0.981382 + 0.192068i \(0.0615194\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 3.46410 0.594089
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 3.46410 0.569495 0.284747 0.958603i \(-0.408090\pi\)
0.284747 + 0.958603i \(0.408090\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 1.46410 0.215870
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −3.46410 −0.485071
\(52\) −3.46410 −0.480384
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.92820 −1.17233
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 1.00000 0.129099
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −10.9282 −1.38788
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −3.46410 −0.429669
\(66\) −1.00000 −0.123091
\(67\) −5.46410 −0.667546 −0.333773 0.942653i \(-0.608322\pi\)
−0.333773 + 0.942653i \(0.608322\pi\)
\(68\) −3.46410 −0.420084
\(69\) −1.46410 −0.176257
\(70\) 1.00000 0.119523
\(71\) 1.07180 0.127199 0.0635994 0.997976i \(-0.479742\pi\)
0.0635994 + 0.997976i \(0.479742\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.8564 1.85585 0.927926 0.372764i \(-0.121590\pi\)
0.927926 + 0.372764i \(0.121590\pi\)
\(74\) −3.46410 −0.402694
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 3.46410 0.392232
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 13.8564 1.52094 0.760469 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) −1.00000 −0.109109
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 8.92820 0.957204
\(88\) −1.00000 −0.106600
\(89\) 8.53590 0.904803 0.452402 0.891814i \(-0.350567\pi\)
0.452402 + 0.891814i \(0.350567\pi\)
\(90\) −1.00000 −0.105409
\(91\) 3.46410 0.363137
\(92\) −1.46410 −0.152643
\(93\) 10.9282 1.13320
\(94\) −6.92820 −0.714590
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −7.85641 −0.781742 −0.390871 0.920446i \(-0.627826\pi\)
−0.390871 + 0.920446i \(0.627826\pi\)
\(102\) 3.46410 0.342997
\(103\) −2.92820 −0.288524 −0.144262 0.989539i \(-0.546081\pi\)
−0.144262 + 0.989539i \(0.546081\pi\)
\(104\) 3.46410 0.339683
\(105\) −1.00000 −0.0975900
\(106\) −6.00000 −0.582772
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.46410 −0.331801 −0.165900 0.986143i \(-0.553053\pi\)
−0.165900 + 0.986143i \(0.553053\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 3.46410 0.328798
\(112\) −1.00000 −0.0944911
\(113\) −2.39230 −0.225049 −0.112525 0.993649i \(-0.535894\pi\)
−0.112525 + 0.993649i \(0.535894\pi\)
\(114\) 0 0
\(115\) −1.46410 −0.136528
\(116\) 8.92820 0.828963
\(117\) −3.46410 −0.320256
\(118\) −8.00000 −0.736460
\(119\) 3.46410 0.317554
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 4.92820 0.446179
\(123\) 2.00000 0.180334
\(124\) 10.9282 0.981382
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) −10.9282 −0.969721 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.46410 0.303822
\(131\) 17.8564 1.56012 0.780061 0.625704i \(-0.215188\pi\)
0.780061 + 0.625704i \(0.215188\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 5.46410 0.472026
\(135\) 1.00000 0.0860663
\(136\) 3.46410 0.297044
\(137\) 3.46410 0.295958 0.147979 0.988990i \(-0.452723\pi\)
0.147979 + 0.988990i \(0.452723\pi\)
\(138\) 1.46410 0.124633
\(139\) −10.9282 −0.926918 −0.463459 0.886118i \(-0.653392\pi\)
−0.463459 + 0.886118i \(0.653392\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 6.92820 0.583460
\(142\) −1.07180 −0.0899432
\(143\) −3.46410 −0.289683
\(144\) 1.00000 0.0833333
\(145\) 8.92820 0.741447
\(146\) −15.8564 −1.31229
\(147\) 1.00000 0.0824786
\(148\) 3.46410 0.284747
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 1.00000 0.0805823
\(155\) 10.9282 0.877774
\(156\) −3.46410 −0.277350
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 4.00000 0.318223
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 1.46410 0.115387
\(162\) −1.00000 −0.0785674
\(163\) −19.3205 −1.51330 −0.756649 0.653821i \(-0.773165\pi\)
−0.756649 + 0.653821i \(0.773165\pi\)
\(164\) 2.00000 0.156174
\(165\) 1.00000 0.0778499
\(166\) −13.8564 −1.07547
\(167\) −17.4641 −1.35141 −0.675706 0.737171i \(-0.736161\pi\)
−0.675706 + 0.737171i \(0.736161\pi\)
\(168\) 1.00000 0.0771517
\(169\) −1.00000 −0.0769231
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −8.92820 −0.676845
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 8.00000 0.601317
\(178\) −8.53590 −0.639793
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 1.00000 0.0745356
\(181\) 3.46410 0.257485 0.128742 0.991678i \(-0.458906\pi\)
0.128742 + 0.991678i \(0.458906\pi\)
\(182\) −3.46410 −0.256776
\(183\) −4.92820 −0.364303
\(184\) 1.46410 0.107935
\(185\) 3.46410 0.254686
\(186\) −10.9282 −0.801295
\(187\) −3.46410 −0.253320
\(188\) 6.92820 0.505291
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −9.85641 −0.713185 −0.356592 0.934260i \(-0.616061\pi\)
−0.356592 + 0.934260i \(0.616061\pi\)
\(192\) 1.00000 0.0721688
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) 10.0000 0.717958
\(195\) −3.46410 −0.248069
\(196\) 1.00000 0.0714286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 18.9282 1.34178 0.670892 0.741555i \(-0.265911\pi\)
0.670892 + 0.741555i \(0.265911\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −5.46410 −0.385408
\(202\) 7.85641 0.552775
\(203\) −8.92820 −0.626637
\(204\) −3.46410 −0.242536
\(205\) 2.00000 0.139686
\(206\) 2.92820 0.204018
\(207\) −1.46410 −0.101762
\(208\) −3.46410 −0.240192
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 4.39230 0.302379 0.151189 0.988505i \(-0.451690\pi\)
0.151189 + 0.988505i \(0.451690\pi\)
\(212\) 6.00000 0.412082
\(213\) 1.07180 0.0734383
\(214\) −6.92820 −0.473602
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −10.9282 −0.741855
\(218\) 3.46410 0.234619
\(219\) 15.8564 1.07148
\(220\) 1.00000 0.0674200
\(221\) 12.0000 0.807207
\(222\) −3.46410 −0.232495
\(223\) 27.7128 1.85579 0.927894 0.372845i \(-0.121618\pi\)
0.927894 + 0.372845i \(0.121618\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.39230 0.159134
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 16.5359 1.09272 0.546361 0.837549i \(-0.316012\pi\)
0.546361 + 0.837549i \(0.316012\pi\)
\(230\) 1.46410 0.0965400
\(231\) −1.00000 −0.0657952
\(232\) −8.92820 −0.586165
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 3.46410 0.226455
\(235\) 6.92820 0.451946
\(236\) 8.00000 0.520756
\(237\) −4.00000 −0.259828
\(238\) −3.46410 −0.224544
\(239\) 21.4641 1.38840 0.694199 0.719783i \(-0.255759\pi\)
0.694199 + 0.719783i \(0.255759\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.7846 −1.21002 −0.605012 0.796217i \(-0.706832\pi\)
−0.605012 + 0.796217i \(0.706832\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −4.92820 −0.315496
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −10.9282 −0.693942
\(249\) 13.8564 0.878114
\(250\) −1.00000 −0.0632456
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −1.46410 −0.0920473
\(254\) 10.9282 0.685696
\(255\) −3.46410 −0.216930
\(256\) 1.00000 0.0625000
\(257\) 3.07180 0.191613 0.0958067 0.995400i \(-0.469457\pi\)
0.0958067 + 0.995400i \(0.469457\pi\)
\(258\) 0 0
\(259\) −3.46410 −0.215249
\(260\) −3.46410 −0.214834
\(261\) 8.92820 0.552642
\(262\) −17.8564 −1.10317
\(263\) 5.07180 0.312740 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 8.53590 0.522388
\(268\) −5.46410 −0.333773
\(269\) 16.9282 1.03213 0.516065 0.856549i \(-0.327396\pi\)
0.516065 + 0.856549i \(0.327396\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −3.46410 −0.210042
\(273\) 3.46410 0.209657
\(274\) −3.46410 −0.209274
\(275\) 1.00000 0.0603023
\(276\) −1.46410 −0.0881286
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 10.9282 0.655430
\(279\) 10.9282 0.654254
\(280\) 1.00000 0.0597614
\(281\) 21.3205 1.27187 0.635937 0.771741i \(-0.280614\pi\)
0.635937 + 0.771741i \(0.280614\pi\)
\(282\) −6.92820 −0.412568
\(283\) −24.3923 −1.44997 −0.724986 0.688764i \(-0.758154\pi\)
−0.724986 + 0.688764i \(0.758154\pi\)
\(284\) 1.07180 0.0635994
\(285\) 0 0
\(286\) 3.46410 0.204837
\(287\) −2.00000 −0.118056
\(288\) −1.00000 −0.0589256
\(289\) −5.00000 −0.294118
\(290\) −8.92820 −0.524282
\(291\) −10.0000 −0.586210
\(292\) 15.8564 0.927926
\(293\) −18.7846 −1.09741 −0.548704 0.836016i \(-0.684879\pi\)
−0.548704 + 0.836016i \(0.684879\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −3.46410 −0.201347
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) 5.07180 0.293310
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) −7.85641 −0.451339
\(304\) 0 0
\(305\) −4.92820 −0.282188
\(306\) 3.46410 0.198030
\(307\) −24.3923 −1.39214 −0.696071 0.717973i \(-0.745070\pi\)
−0.696071 + 0.717973i \(0.745070\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −2.92820 −0.166580
\(310\) −10.9282 −0.620680
\(311\) −21.4641 −1.21712 −0.608559 0.793509i \(-0.708252\pi\)
−0.608559 + 0.793509i \(0.708252\pi\)
\(312\) 3.46410 0.196116
\(313\) 27.8564 1.57454 0.787269 0.616610i \(-0.211495\pi\)
0.787269 + 0.616610i \(0.211495\pi\)
\(314\) 6.00000 0.338600
\(315\) −1.00000 −0.0563436
\(316\) −4.00000 −0.225018
\(317\) 11.8564 0.665922 0.332961 0.942941i \(-0.391952\pi\)
0.332961 + 0.942941i \(0.391952\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.92820 0.499883
\(320\) 1.00000 0.0559017
\(321\) 6.92820 0.386695
\(322\) −1.46410 −0.0815912
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −3.46410 −0.192154
\(326\) 19.3205 1.07006
\(327\) −3.46410 −0.191565
\(328\) −2.00000 −0.110432
\(329\) −6.92820 −0.381964
\(330\) −1.00000 −0.0550482
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 13.8564 0.760469
\(333\) 3.46410 0.189832
\(334\) 17.4641 0.955593
\(335\) −5.46410 −0.298536
\(336\) −1.00000 −0.0545545
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 1.00000 0.0543928
\(339\) −2.39230 −0.129932
\(340\) −3.46410 −0.187867
\(341\) 10.9282 0.591795
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.46410 −0.0788246
\(346\) −6.00000 −0.322562
\(347\) −14.9282 −0.801388 −0.400694 0.916212i \(-0.631231\pi\)
−0.400694 + 0.916212i \(0.631231\pi\)
\(348\) 8.92820 0.478602
\(349\) −7.07180 −0.378545 −0.189272 0.981925i \(-0.560613\pi\)
−0.189272 + 0.981925i \(0.560613\pi\)
\(350\) 1.00000 0.0534522
\(351\) −3.46410 −0.184900
\(352\) −1.00000 −0.0533002
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) −8.00000 −0.425195
\(355\) 1.07180 0.0568851
\(356\) 8.53590 0.452402
\(357\) 3.46410 0.183340
\(358\) 6.92820 0.366167
\(359\) −5.46410 −0.288384 −0.144192 0.989550i \(-0.546058\pi\)
−0.144192 + 0.989550i \(0.546058\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −3.46410 −0.182069
\(363\) 1.00000 0.0524864
\(364\) 3.46410 0.181568
\(365\) 15.8564 0.829962
\(366\) 4.92820 0.257601
\(367\) 16.7846 0.876149 0.438075 0.898939i \(-0.355661\pi\)
0.438075 + 0.898939i \(0.355661\pi\)
\(368\) −1.46410 −0.0763216
\(369\) 2.00000 0.104116
\(370\) −3.46410 −0.180090
\(371\) −6.00000 −0.311504
\(372\) 10.9282 0.566601
\(373\) 34.7846 1.80108 0.900539 0.434774i \(-0.143172\pi\)
0.900539 + 0.434774i \(0.143172\pi\)
\(374\) 3.46410 0.179124
\(375\) 1.00000 0.0516398
\(376\) −6.92820 −0.357295
\(377\) −30.9282 −1.59288
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −10.9282 −0.559869
\(382\) 9.85641 0.504298
\(383\) −1.07180 −0.0547663 −0.0273831 0.999625i \(-0.508717\pi\)
−0.0273831 + 0.999625i \(0.508717\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.00000 −0.0509647
\(386\) −15.8564 −0.807070
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 28.9282 1.46672 0.733359 0.679842i \(-0.237951\pi\)
0.733359 + 0.679842i \(0.237951\pi\)
\(390\) 3.46410 0.175412
\(391\) 5.07180 0.256492
\(392\) −1.00000 −0.0505076
\(393\) 17.8564 0.900737
\(394\) −2.00000 −0.100759
\(395\) −4.00000 −0.201262
\(396\) 1.00000 0.0502519
\(397\) 34.7846 1.74579 0.872895 0.487909i \(-0.162240\pi\)
0.872895 + 0.487909i \(0.162240\pi\)
\(398\) −18.9282 −0.948785
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −19.8564 −0.991582 −0.495791 0.868442i \(-0.665122\pi\)
−0.495791 + 0.868442i \(0.665122\pi\)
\(402\) 5.46410 0.272525
\(403\) −37.8564 −1.88576
\(404\) −7.85641 −0.390871
\(405\) 1.00000 0.0496904
\(406\) 8.92820 0.443099
\(407\) 3.46410 0.171709
\(408\) 3.46410 0.171499
\(409\) −7.07180 −0.349678 −0.174839 0.984597i \(-0.555940\pi\)
−0.174839 + 0.984597i \(0.555940\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 3.46410 0.170872
\(412\) −2.92820 −0.144262
\(413\) −8.00000 −0.393654
\(414\) 1.46410 0.0719567
\(415\) 13.8564 0.680184
\(416\) 3.46410 0.169842
\(417\) −10.9282 −0.535156
\(418\) 0 0
\(419\) −2.92820 −0.143052 −0.0715260 0.997439i \(-0.522787\pi\)
−0.0715260 + 0.997439i \(0.522787\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −31.8564 −1.55259 −0.776293 0.630372i \(-0.782902\pi\)
−0.776293 + 0.630372i \(0.782902\pi\)
\(422\) −4.39230 −0.213814
\(423\) 6.92820 0.336861
\(424\) −6.00000 −0.291386
\(425\) −3.46410 −0.168034
\(426\) −1.07180 −0.0519287
\(427\) 4.92820 0.238492
\(428\) 6.92820 0.334887
\(429\) −3.46410 −0.167248
\(430\) 0 0
\(431\) −32.3923 −1.56028 −0.780141 0.625603i \(-0.784853\pi\)
−0.780141 + 0.625603i \(0.784853\pi\)
\(432\) 1.00000 0.0481125
\(433\) −23.8564 −1.14647 −0.573233 0.819393i \(-0.694311\pi\)
−0.573233 + 0.819393i \(0.694311\pi\)
\(434\) 10.9282 0.524571
\(435\) 8.92820 0.428075
\(436\) −3.46410 −0.165900
\(437\) 0 0
\(438\) −15.8564 −0.757648
\(439\) 34.9282 1.66703 0.833516 0.552495i \(-0.186324\pi\)
0.833516 + 0.552495i \(0.186324\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −6.14359 −0.291891 −0.145945 0.989293i \(-0.546622\pi\)
−0.145945 + 0.989293i \(0.546622\pi\)
\(444\) 3.46410 0.164399
\(445\) 8.53590 0.404640
\(446\) −27.7128 −1.31224
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −11.8564 −0.559538 −0.279769 0.960067i \(-0.590258\pi\)
−0.279769 + 0.960067i \(0.590258\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 2.00000 0.0941763
\(452\) −2.39230 −0.112525
\(453\) −20.0000 −0.939682
\(454\) 8.00000 0.375459
\(455\) 3.46410 0.162400
\(456\) 0 0
\(457\) −11.8564 −0.554619 −0.277310 0.960781i \(-0.589443\pi\)
−0.277310 + 0.960781i \(0.589443\pi\)
\(458\) −16.5359 −0.772672
\(459\) −3.46410 −0.161690
\(460\) −1.46410 −0.0682641
\(461\) 25.7128 1.19757 0.598783 0.800912i \(-0.295651\pi\)
0.598783 + 0.800912i \(0.295651\pi\)
\(462\) 1.00000 0.0465242
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 8.92820 0.414481
\(465\) 10.9282 0.506783
\(466\) −19.8564 −0.919830
\(467\) −20.7846 −0.961797 −0.480899 0.876776i \(-0.659689\pi\)
−0.480899 + 0.876776i \(0.659689\pi\)
\(468\) −3.46410 −0.160128
\(469\) 5.46410 0.252309
\(470\) −6.92820 −0.319574
\(471\) −6.00000 −0.276465
\(472\) −8.00000 −0.368230
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 3.46410 0.158777
\(477\) 6.00000 0.274721
\(478\) −21.4641 −0.981745
\(479\) 21.8564 0.998645 0.499322 0.866416i \(-0.333582\pi\)
0.499322 + 0.866416i \(0.333582\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −12.0000 −0.547153
\(482\) 18.7846 0.855616
\(483\) 1.46410 0.0666189
\(484\) 1.00000 0.0454545
\(485\) −10.0000 −0.454077
\(486\) −1.00000 −0.0453609
\(487\) −41.8564 −1.89669 −0.948347 0.317234i \(-0.897246\pi\)
−0.948347 + 0.317234i \(0.897246\pi\)
\(488\) 4.92820 0.223089
\(489\) −19.3205 −0.873704
\(490\) −1.00000 −0.0451754
\(491\) 1.85641 0.0837785 0.0418892 0.999122i \(-0.486662\pi\)
0.0418892 + 0.999122i \(0.486662\pi\)
\(492\) 2.00000 0.0901670
\(493\) −30.9282 −1.39294
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) 10.9282 0.490691
\(497\) −1.07180 −0.0480767
\(498\) −13.8564 −0.620920
\(499\) −12.7846 −0.572318 −0.286159 0.958182i \(-0.592379\pi\)
−0.286159 + 0.958182i \(0.592379\pi\)
\(500\) 1.00000 0.0447214
\(501\) −17.4641 −0.780239
\(502\) 8.00000 0.357057
\(503\) −19.6077 −0.874264 −0.437132 0.899397i \(-0.644006\pi\)
−0.437132 + 0.899397i \(0.644006\pi\)
\(504\) 1.00000 0.0445435
\(505\) −7.85641 −0.349605
\(506\) 1.46410 0.0650873
\(507\) −1.00000 −0.0444116
\(508\) −10.9282 −0.484861
\(509\) 0.928203 0.0411419 0.0205709 0.999788i \(-0.493452\pi\)
0.0205709 + 0.999788i \(0.493452\pi\)
\(510\) 3.46410 0.153393
\(511\) −15.8564 −0.701446
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.07180 −0.135491
\(515\) −2.92820 −0.129032
\(516\) 0 0
\(517\) 6.92820 0.304702
\(518\) 3.46410 0.152204
\(519\) 6.00000 0.263371
\(520\) 3.46410 0.151911
\(521\) 34.1051 1.49417 0.747086 0.664727i \(-0.231452\pi\)
0.747086 + 0.664727i \(0.231452\pi\)
\(522\) −8.92820 −0.390777
\(523\) −19.3205 −0.844827 −0.422413 0.906403i \(-0.638817\pi\)
−0.422413 + 0.906403i \(0.638817\pi\)
\(524\) 17.8564 0.780061
\(525\) −1.00000 −0.0436436
\(526\) −5.07180 −0.221141
\(527\) −37.8564 −1.64905
\(528\) 1.00000 0.0435194
\(529\) −20.8564 −0.906800
\(530\) −6.00000 −0.260623
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −6.92820 −0.300094
\(534\) −8.53590 −0.369384
\(535\) 6.92820 0.299532
\(536\) 5.46410 0.236013
\(537\) −6.92820 −0.298974
\(538\) −16.9282 −0.729827
\(539\) 1.00000 0.0430730
\(540\) 1.00000 0.0430331
\(541\) −0.535898 −0.0230401 −0.0115200 0.999934i \(-0.503667\pi\)
−0.0115200 + 0.999934i \(0.503667\pi\)
\(542\) −8.00000 −0.343629
\(543\) 3.46410 0.148659
\(544\) 3.46410 0.148522
\(545\) −3.46410 −0.148386
\(546\) −3.46410 −0.148250
\(547\) 38.6410 1.65217 0.826085 0.563545i \(-0.190563\pi\)
0.826085 + 0.563545i \(0.190563\pi\)
\(548\) 3.46410 0.147979
\(549\) −4.92820 −0.210331
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 1.46410 0.0623163
\(553\) 4.00000 0.170097
\(554\) 14.0000 0.594803
\(555\) 3.46410 0.147043
\(556\) −10.9282 −0.463459
\(557\) −16.1436 −0.684026 −0.342013 0.939695i \(-0.611109\pi\)
−0.342013 + 0.939695i \(0.611109\pi\)
\(558\) −10.9282 −0.462628
\(559\) 0 0
\(560\) −1.00000 −0.0422577
\(561\) −3.46410 −0.146254
\(562\) −21.3205 −0.899351
\(563\) −21.0718 −0.888070 −0.444035 0.896009i \(-0.646453\pi\)
−0.444035 + 0.896009i \(0.646453\pi\)
\(564\) 6.92820 0.291730
\(565\) −2.39230 −0.100645
\(566\) 24.3923 1.02529
\(567\) −1.00000 −0.0419961
\(568\) −1.07180 −0.0449716
\(569\) 6.67949 0.280019 0.140009 0.990150i \(-0.455287\pi\)
0.140009 + 0.990150i \(0.455287\pi\)
\(570\) 0 0
\(571\) 21.1769 0.886226 0.443113 0.896466i \(-0.353874\pi\)
0.443113 + 0.896466i \(0.353874\pi\)
\(572\) −3.46410 −0.144841
\(573\) −9.85641 −0.411757
\(574\) 2.00000 0.0834784
\(575\) −1.46410 −0.0610573
\(576\) 1.00000 0.0416667
\(577\) −15.8564 −0.660111 −0.330055 0.943962i \(-0.607067\pi\)
−0.330055 + 0.943962i \(0.607067\pi\)
\(578\) 5.00000 0.207973
\(579\) 15.8564 0.658970
\(580\) 8.92820 0.370723
\(581\) −13.8564 −0.574861
\(582\) 10.0000 0.414513
\(583\) 6.00000 0.248495
\(584\) −15.8564 −0.656143
\(585\) −3.46410 −0.143223
\(586\) 18.7846 0.775985
\(587\) −22.9282 −0.946348 −0.473174 0.880969i \(-0.656892\pi\)
−0.473174 + 0.880969i \(0.656892\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) 2.00000 0.0822690
\(592\) 3.46410 0.142374
\(593\) −12.2487 −0.502994 −0.251497 0.967858i \(-0.580923\pi\)
−0.251497 + 0.967858i \(0.580923\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 3.46410 0.142014
\(596\) 6.00000 0.245770
\(597\) 18.9282 0.774680
\(598\) −5.07180 −0.207401
\(599\) 23.7128 0.968879 0.484440 0.874825i \(-0.339024\pi\)
0.484440 + 0.874825i \(0.339024\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 3.07180 0.125301 0.0626506 0.998036i \(-0.480045\pi\)
0.0626506 + 0.998036i \(0.480045\pi\)
\(602\) 0 0
\(603\) −5.46410 −0.222515
\(604\) −20.0000 −0.813788
\(605\) 1.00000 0.0406558
\(606\) 7.85641 0.319145
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −8.92820 −0.361789
\(610\) 4.92820 0.199537
\(611\) −24.0000 −0.970936
\(612\) −3.46410 −0.140028
\(613\) −43.8564 −1.77134 −0.885672 0.464312i \(-0.846302\pi\)
−0.885672 + 0.464312i \(0.846302\pi\)
\(614\) 24.3923 0.984393
\(615\) 2.00000 0.0806478
\(616\) 1.00000 0.0402911
\(617\) 3.46410 0.139459 0.0697297 0.997566i \(-0.477786\pi\)
0.0697297 + 0.997566i \(0.477786\pi\)
\(618\) 2.92820 0.117790
\(619\) −30.5359 −1.22734 −0.613671 0.789562i \(-0.710308\pi\)
−0.613671 + 0.789562i \(0.710308\pi\)
\(620\) 10.9282 0.438887
\(621\) −1.46410 −0.0587524
\(622\) 21.4641 0.860632
\(623\) −8.53590 −0.341984
\(624\) −3.46410 −0.138675
\(625\) 1.00000 0.0400000
\(626\) −27.8564 −1.11337
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −12.0000 −0.478471
\(630\) 1.00000 0.0398410
\(631\) −2.92820 −0.116570 −0.0582850 0.998300i \(-0.518563\pi\)
−0.0582850 + 0.998300i \(0.518563\pi\)
\(632\) 4.00000 0.159111
\(633\) 4.39230 0.174578
\(634\) −11.8564 −0.470878
\(635\) −10.9282 −0.433673
\(636\) 6.00000 0.237915
\(637\) −3.46410 −0.137253
\(638\) −8.92820 −0.353471
\(639\) 1.07180 0.0423996
\(640\) −1.00000 −0.0395285
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) −6.92820 −0.273434
\(643\) −33.8564 −1.33517 −0.667583 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(644\) 1.46410 0.0576937
\(645\) 0 0
\(646\) 0 0
\(647\) −30.9282 −1.21591 −0.607957 0.793970i \(-0.708011\pi\)
−0.607957 + 0.793970i \(0.708011\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.00000 0.314027
\(650\) 3.46410 0.135873
\(651\) −10.9282 −0.428310
\(652\) −19.3205 −0.756649
\(653\) 16.9282 0.662452 0.331226 0.943551i \(-0.392538\pi\)
0.331226 + 0.943551i \(0.392538\pi\)
\(654\) 3.46410 0.135457
\(655\) 17.8564 0.697708
\(656\) 2.00000 0.0780869
\(657\) 15.8564 0.618617
\(658\) 6.92820 0.270089
\(659\) −9.85641 −0.383951 −0.191976 0.981400i \(-0.561489\pi\)
−0.191976 + 0.981400i \(0.561489\pi\)
\(660\) 1.00000 0.0389249
\(661\) 43.4641 1.69056 0.845279 0.534325i \(-0.179434\pi\)
0.845279 + 0.534325i \(0.179434\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) −13.8564 −0.537733
\(665\) 0 0
\(666\) −3.46410 −0.134231
\(667\) −13.0718 −0.506142
\(668\) −17.4641 −0.675706
\(669\) 27.7128 1.07144
\(670\) 5.46410 0.211097
\(671\) −4.92820 −0.190251
\(672\) 1.00000 0.0385758
\(673\) −35.8564 −1.38216 −0.691081 0.722777i \(-0.742865\pi\)
−0.691081 + 0.722777i \(0.742865\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 1.00000 0.0384900
\(676\) −1.00000 −0.0384615
\(677\) 19.8564 0.763144 0.381572 0.924339i \(-0.375383\pi\)
0.381572 + 0.924339i \(0.375383\pi\)
\(678\) 2.39230 0.0918759
\(679\) 10.0000 0.383765
\(680\) 3.46410 0.132842
\(681\) −8.00000 −0.306561
\(682\) −10.9282 −0.418463
\(683\) 37.5692 1.43755 0.718773 0.695245i \(-0.244704\pi\)
0.718773 + 0.695245i \(0.244704\pi\)
\(684\) 0 0
\(685\) 3.46410 0.132357
\(686\) 1.00000 0.0381802
\(687\) 16.5359 0.630884
\(688\) 0 0
\(689\) −20.7846 −0.791831
\(690\) 1.46410 0.0557374
\(691\) −14.5359 −0.552972 −0.276486 0.961018i \(-0.589170\pi\)
−0.276486 + 0.961018i \(0.589170\pi\)
\(692\) 6.00000 0.228086
\(693\) −1.00000 −0.0379869
\(694\) 14.9282 0.566667
\(695\) −10.9282 −0.414530
\(696\) −8.92820 −0.338423
\(697\) −6.92820 −0.262424
\(698\) 7.07180 0.267671
\(699\) 19.8564 0.751038
\(700\) −1.00000 −0.0377964
\(701\) −10.7846 −0.407329 −0.203665 0.979041i \(-0.565285\pi\)
−0.203665 + 0.979041i \(0.565285\pi\)
\(702\) 3.46410 0.130744
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 6.92820 0.260931
\(706\) 12.9282 0.486559
\(707\) 7.85641 0.295471
\(708\) 8.00000 0.300658
\(709\) 9.71281 0.364772 0.182386 0.983227i \(-0.441618\pi\)
0.182386 + 0.983227i \(0.441618\pi\)
\(710\) −1.07180 −0.0402238
\(711\) −4.00000 −0.150012
\(712\) −8.53590 −0.319896
\(713\) −16.0000 −0.599205
\(714\) −3.46410 −0.129641
\(715\) −3.46410 −0.129550
\(716\) −6.92820 −0.258919
\(717\) 21.4641 0.801592
\(718\) 5.46410 0.203918
\(719\) 29.4641 1.09883 0.549413 0.835551i \(-0.314851\pi\)
0.549413 + 0.835551i \(0.314851\pi\)
\(720\) 1.00000 0.0372678
\(721\) 2.92820 0.109052
\(722\) 19.0000 0.707107
\(723\) −18.7846 −0.698607
\(724\) 3.46410 0.128742
\(725\) 8.92820 0.331585
\(726\) −1.00000 −0.0371135
\(727\) 18.1436 0.672909 0.336454 0.941700i \(-0.390772\pi\)
0.336454 + 0.941700i \(0.390772\pi\)
\(728\) −3.46410 −0.128388
\(729\) 1.00000 0.0370370
\(730\) −15.8564 −0.586872
\(731\) 0 0
\(732\) −4.92820 −0.182152
\(733\) 36.5359 1.34948 0.674742 0.738054i \(-0.264255\pi\)
0.674742 + 0.738054i \(0.264255\pi\)
\(734\) −16.7846 −0.619531
\(735\) 1.00000 0.0368856
\(736\) 1.46410 0.0539675
\(737\) −5.46410 −0.201273
\(738\) −2.00000 −0.0736210
\(739\) −14.5359 −0.534712 −0.267356 0.963598i \(-0.586150\pi\)
−0.267356 + 0.963598i \(0.586150\pi\)
\(740\) 3.46410 0.127343
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −10.9282 −0.400647
\(745\) 6.00000 0.219823
\(746\) −34.7846 −1.27356
\(747\) 13.8564 0.506979
\(748\) −3.46410 −0.126660
\(749\) −6.92820 −0.253151
\(750\) −1.00000 −0.0365148
\(751\) −26.9282 −0.982624 −0.491312 0.870984i \(-0.663483\pi\)
−0.491312 + 0.870984i \(0.663483\pi\)
\(752\) 6.92820 0.252646
\(753\) −8.00000 −0.291536
\(754\) 30.9282 1.12634
\(755\) −20.0000 −0.727875
\(756\) −1.00000 −0.0363696
\(757\) −4.53590 −0.164860 −0.0824300 0.996597i \(-0.526268\pi\)
−0.0824300 + 0.996597i \(0.526268\pi\)
\(758\) 4.00000 0.145287
\(759\) −1.46410 −0.0531435
\(760\) 0 0
\(761\) 21.7128 0.787089 0.393544 0.919306i \(-0.371249\pi\)
0.393544 + 0.919306i \(0.371249\pi\)
\(762\) 10.9282 0.395887
\(763\) 3.46410 0.125409
\(764\) −9.85641 −0.356592
\(765\) −3.46410 −0.125245
\(766\) 1.07180 0.0387256
\(767\) −27.7128 −1.00065
\(768\) 1.00000 0.0360844
\(769\) 28.6410 1.03282 0.516411 0.856341i \(-0.327268\pi\)
0.516411 + 0.856341i \(0.327268\pi\)
\(770\) 1.00000 0.0360375
\(771\) 3.07180 0.110628
\(772\) 15.8564 0.570685
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 10.9282 0.392553
\(776\) 10.0000 0.358979
\(777\) −3.46410 −0.124274
\(778\) −28.9282 −1.03713
\(779\) 0 0
\(780\) −3.46410 −0.124035
\(781\) 1.07180 0.0383519
\(782\) −5.07180 −0.181367
\(783\) 8.92820 0.319068
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) −17.8564 −0.636917
\(787\) 21.4641 0.765113 0.382556 0.923932i \(-0.375044\pi\)
0.382556 + 0.923932i \(0.375044\pi\)
\(788\) 2.00000 0.0712470
\(789\) 5.07180 0.180561
\(790\) 4.00000 0.142314
\(791\) 2.39230 0.0850606
\(792\) −1.00000 −0.0355335
\(793\) 17.0718 0.606237
\(794\) −34.7846 −1.23446
\(795\) 6.00000 0.212798
\(796\) 18.9282 0.670892
\(797\) 33.7128 1.19417 0.597085 0.802178i \(-0.296326\pi\)
0.597085 + 0.802178i \(0.296326\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) 8.53590 0.301601
\(802\) 19.8564 0.701154
\(803\) 15.8564 0.559560
\(804\) −5.46410 −0.192704
\(805\) 1.46410 0.0516028
\(806\) 37.8564 1.33344
\(807\) 16.9282 0.595901
\(808\) 7.85641 0.276387
\(809\) −7.17691 −0.252327 −0.126163 0.992009i \(-0.540266\pi\)
−0.126163 + 0.992009i \(0.540266\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −35.7128 −1.25405 −0.627023 0.779001i \(-0.715727\pi\)
−0.627023 + 0.779001i \(0.715727\pi\)
\(812\) −8.92820 −0.313319
\(813\) 8.00000 0.280572
\(814\) −3.46410 −0.121417
\(815\) −19.3205 −0.676768
\(816\) −3.46410 −0.121268
\(817\) 0 0
\(818\) 7.07180 0.247260
\(819\) 3.46410 0.121046
\(820\) 2.00000 0.0698430
\(821\) −42.7846 −1.49319 −0.746597 0.665277i \(-0.768313\pi\)
−0.746597 + 0.665277i \(0.768313\pi\)
\(822\) −3.46410 −0.120824
\(823\) 28.7846 1.00337 0.501684 0.865051i \(-0.332714\pi\)
0.501684 + 0.865051i \(0.332714\pi\)
\(824\) 2.92820 0.102009
\(825\) 1.00000 0.0348155
\(826\) 8.00000 0.278356
\(827\) −30.9282 −1.07548 −0.537740 0.843111i \(-0.680722\pi\)
−0.537740 + 0.843111i \(0.680722\pi\)
\(828\) −1.46410 −0.0508810
\(829\) −16.2487 −0.564341 −0.282171 0.959364i \(-0.591054\pi\)
−0.282171 + 0.959364i \(0.591054\pi\)
\(830\) −13.8564 −0.480963
\(831\) −14.0000 −0.485655
\(832\) −3.46410 −0.120096
\(833\) −3.46410 −0.120024
\(834\) 10.9282 0.378413
\(835\) −17.4641 −0.604370
\(836\) 0 0
\(837\) 10.9282 0.377734
\(838\) 2.92820 0.101153
\(839\) −27.3205 −0.943209 −0.471604 0.881810i \(-0.656325\pi\)
−0.471604 + 0.881810i \(0.656325\pi\)
\(840\) 1.00000 0.0345033
\(841\) 50.7128 1.74872
\(842\) 31.8564 1.09784
\(843\) 21.3205 0.734317
\(844\) 4.39230 0.151189
\(845\) −1.00000 −0.0344010
\(846\) −6.92820 −0.238197
\(847\) −1.00000 −0.0343604
\(848\) 6.00000 0.206041
\(849\) −24.3923 −0.837142
\(850\) 3.46410 0.118818
\(851\) −5.07180 −0.173859
\(852\) 1.07180 0.0367192
\(853\) 26.3923 0.903655 0.451828 0.892105i \(-0.350772\pi\)
0.451828 + 0.892105i \(0.350772\pi\)
\(854\) −4.92820 −0.168640
\(855\) 0 0
\(856\) −6.92820 −0.236801
\(857\) −8.53590 −0.291581 −0.145790 0.989316i \(-0.546573\pi\)
−0.145790 + 0.989316i \(0.546573\pi\)
\(858\) 3.46410 0.118262
\(859\) 36.3923 1.24169 0.620845 0.783934i \(-0.286790\pi\)
0.620845 + 0.783934i \(0.286790\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 32.3923 1.10329
\(863\) 24.1051 0.820548 0.410274 0.911962i \(-0.365433\pi\)
0.410274 + 0.911962i \(0.365433\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 23.8564 0.810674
\(867\) −5.00000 −0.169809
\(868\) −10.9282 −0.370927
\(869\) −4.00000 −0.135691
\(870\) −8.92820 −0.302694
\(871\) 18.9282 0.641358
\(872\) 3.46410 0.117309
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 15.8564 0.535738
\(877\) 20.1436 0.680201 0.340100 0.940389i \(-0.389539\pi\)
0.340100 + 0.940389i \(0.389539\pi\)
\(878\) −34.9282 −1.17877
\(879\) −18.7846 −0.633589
\(880\) 1.00000 0.0337100
\(881\) 0.535898 0.0180549 0.00902744 0.999959i \(-0.497126\pi\)
0.00902744 + 0.999959i \(0.497126\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 8.39230 0.282424 0.141212 0.989979i \(-0.454900\pi\)
0.141212 + 0.989979i \(0.454900\pi\)
\(884\) 12.0000 0.403604
\(885\) 8.00000 0.268917
\(886\) 6.14359 0.206398
\(887\) −6.53590 −0.219454 −0.109727 0.993962i \(-0.534998\pi\)
−0.109727 + 0.993962i \(0.534998\pi\)
\(888\) −3.46410 −0.116248
\(889\) 10.9282 0.366520
\(890\) −8.53590 −0.286124
\(891\) 1.00000 0.0335013
\(892\) 27.7128 0.927894
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) −6.92820 −0.231584
\(896\) 1.00000 0.0334077
\(897\) 5.07180 0.169342
\(898\) 11.8564 0.395653
\(899\) 97.5692 3.25412
\(900\) 1.00000 0.0333333
\(901\) −20.7846 −0.692436
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 2.39230 0.0795669
\(905\) 3.46410 0.115151
\(906\) 20.0000 0.664455
\(907\) −26.5359 −0.881110 −0.440555 0.897726i \(-0.645218\pi\)
−0.440555 + 0.897726i \(0.645218\pi\)
\(908\) −8.00000 −0.265489
\(909\) −7.85641 −0.260581
\(910\) −3.46410 −0.114834
\(911\) −42.6410 −1.41276 −0.706380 0.707833i \(-0.749673\pi\)
−0.706380 + 0.707833i \(0.749673\pi\)
\(912\) 0 0
\(913\) 13.8564 0.458580
\(914\) 11.8564 0.392175
\(915\) −4.92820 −0.162921
\(916\) 16.5359 0.546361
\(917\) −17.8564 −0.589670
\(918\) 3.46410 0.114332
\(919\) 12.7846 0.421725 0.210863 0.977516i \(-0.432373\pi\)
0.210863 + 0.977516i \(0.432373\pi\)
\(920\) 1.46410 0.0482700
\(921\) −24.3923 −0.803754
\(922\) −25.7128 −0.846806
\(923\) −3.71281 −0.122209
\(924\) −1.00000 −0.0328976
\(925\) 3.46410 0.113899
\(926\) −4.00000 −0.131448
\(927\) −2.92820 −0.0961748
\(928\) −8.92820 −0.293083
\(929\) 27.4641 0.901068 0.450534 0.892759i \(-0.351234\pi\)
0.450534 + 0.892759i \(0.351234\pi\)
\(930\) −10.9282 −0.358350
\(931\) 0 0
\(932\) 19.8564 0.650418
\(933\) −21.4641 −0.702703
\(934\) 20.7846 0.680093
\(935\) −3.46410 −0.113288
\(936\) 3.46410 0.113228
\(937\) −14.7846 −0.482992 −0.241496 0.970402i \(-0.577638\pi\)
−0.241496 + 0.970402i \(0.577638\pi\)
\(938\) −5.46410 −0.178409
\(939\) 27.8564 0.909059
\(940\) 6.92820 0.225973
\(941\) −21.7128 −0.707817 −0.353909 0.935280i \(-0.615148\pi\)
−0.353909 + 0.935280i \(0.615148\pi\)
\(942\) 6.00000 0.195491
\(943\) −2.92820 −0.0953554
\(944\) 8.00000 0.260378
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −44.7846 −1.45530 −0.727652 0.685946i \(-0.759388\pi\)
−0.727652 + 0.685946i \(0.759388\pi\)
\(948\) −4.00000 −0.129914
\(949\) −54.9282 −1.78304
\(950\) 0 0
\(951\) 11.8564 0.384470
\(952\) −3.46410 −0.112272
\(953\) −23.0718 −0.747369 −0.373684 0.927556i \(-0.621906\pi\)
−0.373684 + 0.927556i \(0.621906\pi\)
\(954\) −6.00000 −0.194257
\(955\) −9.85641 −0.318946
\(956\) 21.4641 0.694199
\(957\) 8.92820 0.288608
\(958\) −21.8564 −0.706148
\(959\) −3.46410 −0.111862
\(960\) 1.00000 0.0322749
\(961\) 88.4256 2.85244
\(962\) 12.0000 0.386896
\(963\) 6.92820 0.223258
\(964\) −18.7846 −0.605012
\(965\) 15.8564 0.510436
\(966\) −1.46410 −0.0471067
\(967\) −8.78461 −0.282494 −0.141247 0.989974i \(-0.545111\pi\)
−0.141247 + 0.989974i \(0.545111\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −58.9282 −1.89110 −0.945548 0.325483i \(-0.894473\pi\)
−0.945548 + 0.325483i \(0.894473\pi\)
\(972\) 1.00000 0.0320750
\(973\) 10.9282 0.350342
\(974\) 41.8564 1.34117
\(975\) −3.46410 −0.110940
\(976\) −4.92820 −0.157748
\(977\) −1.60770 −0.0514347 −0.0257174 0.999669i \(-0.508187\pi\)
−0.0257174 + 0.999669i \(0.508187\pi\)
\(978\) 19.3205 0.617802
\(979\) 8.53590 0.272808
\(980\) 1.00000 0.0319438
\(981\) −3.46410 −0.110600
\(982\) −1.85641 −0.0592403
\(983\) −26.6410 −0.849716 −0.424858 0.905260i \(-0.639676\pi\)
−0.424858 + 0.905260i \(0.639676\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 2.00000 0.0637253
\(986\) 30.9282 0.984955
\(987\) −6.92820 −0.220527
\(988\) 0 0
\(989\) 0 0
\(990\) −1.00000 −0.0317821
\(991\) −10.9282 −0.347146 −0.173573 0.984821i \(-0.555531\pi\)
−0.173573 + 0.984821i \(0.555531\pi\)
\(992\) −10.9282 −0.346971
\(993\) 4.00000 0.126936
\(994\) 1.07180 0.0339953
\(995\) 18.9282 0.600064
\(996\) 13.8564 0.439057
\(997\) 1.60770 0.0509162 0.0254581 0.999676i \(-0.491896\pi\)
0.0254581 + 0.999676i \(0.491896\pi\)
\(998\) 12.7846 0.404690
\(999\) 3.46410 0.109599
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 2310.2.a.z.1.1 2
3.2 odd 2 6930.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.z.1.1 2 1.1 even 1 trivial
6930.2.a.bw.1.1 2 3.2 odd 2