Properties

Label 2310.2.a.bb.1.2
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +3.46410 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -1.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.46410 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +6.92820 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.46410 q^{34} -1.00000 q^{35} +1.00000 q^{36} +7.46410 q^{37} -3.46410 q^{39} -1.00000 q^{40} -8.92820 q^{41} -1.00000 q^{42} -2.92820 q^{43} +1.00000 q^{44} -1.00000 q^{45} -1.46410 q^{46} +6.92820 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.46410 q^{51} +3.46410 q^{52} +12.9282 q^{53} -1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +2.00000 q^{58} -2.92820 q^{59} +1.00000 q^{60} -4.92820 q^{61} +6.92820 q^{62} +1.00000 q^{63} +1.00000 q^{64} -3.46410 q^{65} -1.00000 q^{66} +9.46410 q^{67} -3.46410 q^{68} +1.46410 q^{69} -1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -0.928203 q^{73} +7.46410 q^{74} -1.00000 q^{75} +1.00000 q^{77} -3.46410 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.92820 q^{82} +6.92820 q^{83} -1.00000 q^{84} +3.46410 q^{85} -2.92820 q^{86} -2.00000 q^{87} +1.00000 q^{88} +3.46410 q^{89} -1.00000 q^{90} +3.46410 q^{91} -1.46410 q^{92} -6.92820 q^{93} +6.92820 q^{94} -1.00000 q^{96} -11.8564 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{18} - 2 q^{20} - 2 q^{21} + 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{28} + 4 q^{29} + 2 q^{30} + 2 q^{32} - 2 q^{33} - 2 q^{35} + 2 q^{36} + 8 q^{37} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 8 q^{43} + 2 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 12 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{58} + 8 q^{59} + 2 q^{60} + 4 q^{61} + 2 q^{63} + 2 q^{64} - 2 q^{66} + 12 q^{67} - 4 q^{69} - 2 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} + 8 q^{74} - 2 q^{75} + 2 q^{77} + 16 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} - 2 q^{84} + 8 q^{86} - 4 q^{87} + 2 q^{88} - 2 q^{90} + 4 q^{92} - 2 q^{96} + 4 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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 \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 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
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −3.46410 −0.594089
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 7.46410 1.22709 0.613545 0.789659i \(-0.289743\pi\)
0.613545 + 0.789659i \(0.289743\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) −1.00000 −0.158114
\(41\) −8.92820 −1.39435 −0.697176 0.716900i \(-0.745560\pi\)
−0.697176 + 0.716900i \(0.745560\pi\)
\(42\) −1.00000 −0.154303
\(43\) −2.92820 −0.446547 −0.223273 0.974756i \(-0.571674\pi\)
−0.223273 + 0.974756i \(0.571674\pi\)
\(44\) 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
\(53\) 12.9282 1.77583 0.887913 0.460012i \(-0.152155\pi\)
0.887913 + 0.460012i \(0.152155\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −2.92820 −0.381220 −0.190610 0.981666i \(-0.561047\pi\)
−0.190610 + 0.981666i \(0.561047\pi\)
\(60\) 1.00000 0.129099
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 6.92820 0.879883
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −3.46410 −0.429669
\(66\) −1.00000 −0.123091
\(67\) 9.46410 1.15622 0.578112 0.815957i \(-0.303790\pi\)
0.578112 + 0.815957i \(0.303790\pi\)
\(68\) −3.46410 −0.420084
\(69\) 1.46410 0.176257
\(70\) −1.00000 −0.119523
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) 7.46410 0.867684
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) −3.46410 −0.392232
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −8.92820 −0.985955
\(83\) 6.92820 0.760469 0.380235 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) −1.00000 −0.109109
\(85\) 3.46410 0.375735
\(86\) −2.92820 −0.315756
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) −1.00000 −0.105409
\(91\) 3.46410 0.363137
\(92\) −1.46410 −0.152643
\(93\) −6.92820 −0.718421
\(94\) 6.92820 0.714590
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −11.8564 −1.20384 −0.601918 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 11.8564 1.17976 0.589878 0.807492i \(-0.299176\pi\)
0.589878 + 0.807492i \(0.299176\pi\)
\(102\) 3.46410 0.342997
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 3.46410 0.339683
\(105\) 1.00000 0.0975900
\(106\) 12.9282 1.25570
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.3923 1.37853 0.689266 0.724508i \(-0.257933\pi\)
0.689266 + 0.724508i \(0.257933\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −7.46410 −0.708461
\(112\) 1.00000 0.0944911
\(113\) 11.4641 1.07845 0.539226 0.842161i \(-0.318717\pi\)
0.539226 + 0.842161i \(0.318717\pi\)
\(114\) 0 0
\(115\) 1.46410 0.136528
\(116\) 2.00000 0.185695
\(117\) 3.46410 0.320256
\(118\) −2.92820 −0.269563
\(119\) −3.46410 −0.317554
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −4.92820 −0.446179
\(123\) 8.92820 0.805029
\(124\) 6.92820 0.622171
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 10.9282 0.969721 0.484861 0.874591i \(-0.338870\pi\)
0.484861 + 0.874591i \(0.338870\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.92820 0.257814
\(130\) −3.46410 −0.303822
\(131\) −1.07180 −0.0936433 −0.0468217 0.998903i \(-0.514909\pi\)
−0.0468217 + 0.998903i \(0.514909\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 9.46410 0.817574
\(135\) 1.00000 0.0860663
\(136\) −3.46410 −0.297044
\(137\) −4.53590 −0.387528 −0.193764 0.981048i \(-0.562070\pi\)
−0.193764 + 0.981048i \(0.562070\pi\)
\(138\) 1.46410 0.124633
\(139\) −2.92820 −0.248367 −0.124183 0.992259i \(-0.539631\pi\)
−0.124183 + 0.992259i \(0.539631\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −6.92820 −0.583460
\(142\) −4.00000 −0.335673
\(143\) 3.46410 0.289683
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −0.928203 −0.0768186
\(147\) −1.00000 −0.0824786
\(148\) 7.46410 0.613545
\(149\) 12.9282 1.05912 0.529560 0.848273i \(-0.322357\pi\)
0.529560 + 0.848273i \(0.322357\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 1.00000 0.0805823
\(155\) −6.92820 −0.556487
\(156\) −3.46410 −0.277350
\(157\) −18.7846 −1.49918 −0.749588 0.661905i \(-0.769748\pi\)
−0.749588 + 0.661905i \(0.769748\pi\)
\(158\) 8.00000 0.636446
\(159\) −12.9282 −1.02527
\(160\) −1.00000 −0.0790569
\(161\) −1.46410 −0.115387
\(162\) 1.00000 0.0785674
\(163\) 9.46410 0.741286 0.370643 0.928775i \(-0.379137\pi\)
0.370643 + 0.928775i \(0.379137\pi\)
\(164\) −8.92820 −0.697176
\(165\) 1.00000 0.0778499
\(166\) 6.92820 0.537733
\(167\) −9.46410 −0.732354 −0.366177 0.930545i \(-0.619334\pi\)
−0.366177 + 0.930545i \(0.619334\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −1.00000 −0.0769231
\(170\) 3.46410 0.265684
\(171\) 0 0
\(172\) −2.92820 −0.223273
\(173\) −8.92820 −0.678799 −0.339399 0.940642i \(-0.610224\pi\)
−0.339399 + 0.940642i \(0.610224\pi\)
\(174\) −2.00000 −0.151620
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 2.92820 0.220097
\(178\) 3.46410 0.259645
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.5359 −1.22910 −0.614552 0.788876i \(-0.710663\pi\)
−0.614552 + 0.788876i \(0.710663\pi\)
\(182\) 3.46410 0.256776
\(183\) 4.92820 0.364303
\(184\) −1.46410 −0.107935
\(185\) −7.46410 −0.548772
\(186\) −6.92820 −0.508001
\(187\) −3.46410 −0.253320
\(188\) 6.92820 0.505291
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 14.9282 1.08017 0.540083 0.841611i \(-0.318393\pi\)
0.540083 + 0.841611i \(0.318393\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.0718 0.796965 0.398483 0.917176i \(-0.369537\pi\)
0.398483 + 0.917176i \(0.369537\pi\)
\(194\) −11.8564 −0.851240
\(195\) 3.46410 0.248069
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) −6.92820 −0.491127 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) 1.00000 0.0707107
\(201\) −9.46410 −0.667546
\(202\) 11.8564 0.834214
\(203\) 2.00000 0.140372
\(204\) 3.46410 0.242536
\(205\) 8.92820 0.623573
\(206\) 16.0000 1.11477
\(207\) −1.46410 −0.101762
\(208\) 3.46410 0.240192
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) −23.3205 −1.60545 −0.802725 0.596349i \(-0.796617\pi\)
−0.802725 + 0.596349i \(0.796617\pi\)
\(212\) 12.9282 0.887913
\(213\) 4.00000 0.274075
\(214\) −6.92820 −0.473602
\(215\) 2.92820 0.199702
\(216\) −1.00000 −0.0680414
\(217\) 6.92820 0.470317
\(218\) 14.3923 0.974770
\(219\) 0.928203 0.0627222
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) −7.46410 −0.500958
\(223\) 24.7846 1.65970 0.829850 0.557986i \(-0.188426\pi\)
0.829850 + 0.557986i \(0.188426\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 11.4641 0.762581
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) 2.39230 0.158088 0.0790440 0.996871i \(-0.474813\pi\)
0.0790440 + 0.996871i \(0.474813\pi\)
\(230\) 1.46410 0.0965400
\(231\) −1.00000 −0.0657952
\(232\) 2.00000 0.131306
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) 3.46410 0.226455
\(235\) −6.92820 −0.451946
\(236\) −2.92820 −0.190610
\(237\) −8.00000 −0.519656
\(238\) −3.46410 −0.224544
\(239\) 4.39230 0.284115 0.142057 0.989858i \(-0.454628\pi\)
0.142057 + 0.989858i \(0.454628\pi\)
\(240\) 1.00000 0.0645497
\(241\) −7.07180 −0.455534 −0.227767 0.973716i \(-0.573143\pi\)
−0.227767 + 0.973716i \(0.573143\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −4.92820 −0.315496
\(245\) −1.00000 −0.0638877
\(246\) 8.92820 0.569241
\(247\) 0 0
\(248\) 6.92820 0.439941
\(249\) −6.92820 −0.439057
\(250\) −1.00000 −0.0632456
\(251\) −5.07180 −0.320129 −0.160064 0.987107i \(-0.551170\pi\)
−0.160064 + 0.987107i \(0.551170\pi\)
\(252\) 1.00000 0.0629941
\(253\) −1.46410 −0.0920473
\(254\) 10.9282 0.685696
\(255\) −3.46410 −0.216930
\(256\) 1.00000 0.0625000
\(257\) −16.9282 −1.05595 −0.527976 0.849259i \(-0.677049\pi\)
−0.527976 + 0.849259i \(0.677049\pi\)
\(258\) 2.92820 0.182302
\(259\) 7.46410 0.463797
\(260\) −3.46410 −0.214834
\(261\) 2.00000 0.123797
\(262\) −1.07180 −0.0662158
\(263\) −16.7846 −1.03498 −0.517492 0.855688i \(-0.673134\pi\)
−0.517492 + 0.855688i \(0.673134\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −12.9282 −0.794173
\(266\) 0 0
\(267\) −3.46410 −0.212000
\(268\) 9.46410 0.578112
\(269\) −16.9282 −1.03213 −0.516065 0.856549i \(-0.672604\pi\)
−0.516065 + 0.856549i \(0.672604\pi\)
\(270\) 1.00000 0.0608581
\(271\) −5.85641 −0.355751 −0.177876 0.984053i \(-0.556922\pi\)
−0.177876 + 0.984053i \(0.556922\pi\)
\(272\) −3.46410 −0.210042
\(273\) −3.46410 −0.209657
\(274\) −4.53590 −0.274024
\(275\) 1.00000 0.0603023
\(276\) 1.46410 0.0881286
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −2.92820 −0.175622
\(279\) 6.92820 0.414781
\(280\) −1.00000 −0.0597614
\(281\) −14.3923 −0.858573 −0.429286 0.903168i \(-0.641235\pi\)
−0.429286 + 0.903168i \(0.641235\pi\)
\(282\) −6.92820 −0.412568
\(283\) 1.46410 0.0870318 0.0435159 0.999053i \(-0.486144\pi\)
0.0435159 + 0.999053i \(0.486144\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 3.46410 0.204837
\(287\) −8.92820 −0.527015
\(288\) 1.00000 0.0589256
\(289\) −5.00000 −0.294118
\(290\) −2.00000 −0.117444
\(291\) 11.8564 0.695035
\(292\) −0.928203 −0.0543190
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 2.92820 0.170487
\(296\) 7.46410 0.433842
\(297\) −1.00000 −0.0580259
\(298\) 12.9282 0.748911
\(299\) −5.07180 −0.293310
\(300\) −1.00000 −0.0577350
\(301\) −2.92820 −0.168779
\(302\) 8.00000 0.460348
\(303\) −11.8564 −0.681133
\(304\) 0 0
\(305\) 4.92820 0.282188
\(306\) −3.46410 −0.198030
\(307\) 7.32051 0.417803 0.208902 0.977937i \(-0.433011\pi\)
0.208902 + 0.977937i \(0.433011\pi\)
\(308\) 1.00000 0.0569803
\(309\) −16.0000 −0.910208
\(310\) −6.92820 −0.393496
\(311\) 6.53590 0.370617 0.185308 0.982680i \(-0.440672\pi\)
0.185308 + 0.982680i \(0.440672\pi\)
\(312\) −3.46410 −0.196116
\(313\) 21.7128 1.22728 0.613640 0.789586i \(-0.289704\pi\)
0.613640 + 0.789586i \(0.289704\pi\)
\(314\) −18.7846 −1.06008
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) −3.07180 −0.172529 −0.0862646 0.996272i \(-0.527493\pi\)
−0.0862646 + 0.996272i \(0.527493\pi\)
\(318\) −12.9282 −0.724978
\(319\) 2.00000 0.111979
\(320\) −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
\(326\) 9.46410 0.524168
\(327\) −14.3923 −0.795896
\(328\) −8.92820 −0.492978
\(329\) 6.92820 0.381964
\(330\) 1.00000 0.0550482
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 6.92820 0.380235
\(333\) 7.46410 0.409030
\(334\) −9.46410 −0.517853
\(335\) −9.46410 −0.517079
\(336\) −1.00000 −0.0545545
\(337\) 22.7846 1.24116 0.620578 0.784144i \(-0.286898\pi\)
0.620578 + 0.784144i \(0.286898\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −11.4641 −0.622645
\(340\) 3.46410 0.187867
\(341\) 6.92820 0.375183
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.92820 −0.157878
\(345\) −1.46410 −0.0788246
\(346\) −8.92820 −0.479983
\(347\) −1.07180 −0.0575371 −0.0287685 0.999586i \(-0.509159\pi\)
−0.0287685 + 0.999586i \(0.509159\pi\)
\(348\) −2.00000 −0.107211
\(349\) −28.9282 −1.54849 −0.774246 0.632885i \(-0.781870\pi\)
−0.774246 + 0.632885i \(0.781870\pi\)
\(350\) 1.00000 0.0534522
\(351\) −3.46410 −0.184900
\(352\) 1.00000 0.0533002
\(353\) 20.9282 1.11390 0.556948 0.830547i \(-0.311972\pi\)
0.556948 + 0.830547i \(0.311972\pi\)
\(354\) 2.92820 0.155632
\(355\) 4.00000 0.212298
\(356\) 3.46410 0.183597
\(357\) 3.46410 0.183340
\(358\) −20.7846 −1.09850
\(359\) 9.46410 0.499496 0.249748 0.968311i \(-0.419652\pi\)
0.249748 + 0.968311i \(0.419652\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −16.5359 −0.869108
\(363\) −1.00000 −0.0524864
\(364\) 3.46410 0.181568
\(365\) 0.928203 0.0485844
\(366\) 4.92820 0.257601
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) −1.46410 −0.0763216
\(369\) −8.92820 −0.464784
\(370\) −7.46410 −0.388040
\(371\) 12.9282 0.671199
\(372\) −6.92820 −0.359211
\(373\) −4.92820 −0.255173 −0.127586 0.991827i \(-0.540723\pi\)
−0.127586 + 0.991827i \(0.540723\pi\)
\(374\) −3.46410 −0.179124
\(375\) 1.00000 0.0516398
\(376\) 6.92820 0.357295
\(377\) 6.92820 0.356821
\(378\) −1.00000 −0.0514344
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −10.9282 −0.559869
\(382\) 14.9282 0.763793
\(383\) −12.7846 −0.653263 −0.326632 0.945152i \(-0.605914\pi\)
−0.326632 + 0.945152i \(0.605914\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.00000 −0.0509647
\(386\) 11.0718 0.563540
\(387\) −2.92820 −0.148849
\(388\) −11.8564 −0.601918
\(389\) 21.7128 1.10088 0.550442 0.834874i \(-0.314459\pi\)
0.550442 + 0.834874i \(0.314459\pi\)
\(390\) 3.46410 0.175412
\(391\) 5.07180 0.256492
\(392\) 1.00000 0.0505076
\(393\) 1.07180 0.0540650
\(394\) −18.0000 −0.906827
\(395\) −8.00000 −0.402524
\(396\) 1.00000 0.0502519
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −6.92820 −0.347279
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −25.7128 −1.28404 −0.642018 0.766689i \(-0.721903\pi\)
−0.642018 + 0.766689i \(0.721903\pi\)
\(402\) −9.46410 −0.472026
\(403\) 24.0000 1.19553
\(404\) 11.8564 0.589878
\(405\) −1.00000 −0.0496904
\(406\) 2.00000 0.0992583
\(407\) 7.46410 0.369982
\(408\) 3.46410 0.171499
\(409\) −12.9282 −0.639259 −0.319629 0.947543i \(-0.603558\pi\)
−0.319629 + 0.947543i \(0.603558\pi\)
\(410\) 8.92820 0.440933
\(411\) 4.53590 0.223739
\(412\) 16.0000 0.788263
\(413\) −2.92820 −0.144087
\(414\) −1.46410 −0.0719567
\(415\) −6.92820 −0.340092
\(416\) 3.46410 0.169842
\(417\) 2.92820 0.143395
\(418\) 0 0
\(419\) 13.8564 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(420\) 1.00000 0.0487950
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −23.3205 −1.13522
\(423\) 6.92820 0.336861
\(424\) 12.9282 0.627849
\(425\) −3.46410 −0.168034
\(426\) 4.00000 0.193801
\(427\) −4.92820 −0.238492
\(428\) −6.92820 −0.334887
\(429\) −3.46410 −0.167248
\(430\) 2.92820 0.141210
\(431\) −23.3205 −1.12331 −0.561655 0.827372i \(-0.689835\pi\)
−0.561655 + 0.827372i \(0.689835\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −17.7128 −0.851223 −0.425612 0.904906i \(-0.639941\pi\)
−0.425612 + 0.904906i \(0.639941\pi\)
\(434\) 6.92820 0.332564
\(435\) 2.00000 0.0958927
\(436\) 14.3923 0.689266
\(437\) 0 0
\(438\) 0.928203 0.0443513
\(439\) −16.7846 −0.801086 −0.400543 0.916278i \(-0.631178\pi\)
−0.400543 + 0.916278i \(0.631178\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 26.9282 1.27940 0.639699 0.768626i \(-0.279059\pi\)
0.639699 + 0.768626i \(0.279059\pi\)
\(444\) −7.46410 −0.354231
\(445\) −3.46410 −0.164214
\(446\) 24.7846 1.17359
\(447\) −12.9282 −0.611483
\(448\) 1.00000 0.0472456
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.92820 −0.420413
\(452\) 11.4641 0.539226
\(453\) −8.00000 −0.375873
\(454\) 6.92820 0.325157
\(455\) −3.46410 −0.162400
\(456\) 0 0
\(457\) 19.0718 0.892141 0.446071 0.894998i \(-0.352823\pi\)
0.446071 + 0.894998i \(0.352823\pi\)
\(458\) 2.39230 0.111785
\(459\) 3.46410 0.161690
\(460\) 1.46410 0.0682641
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 20.7846 0.965943 0.482971 0.875636i \(-0.339558\pi\)
0.482971 + 0.875636i \(0.339558\pi\)
\(464\) 2.00000 0.0928477
\(465\) 6.92820 0.321288
\(466\) 2.00000 0.0926482
\(467\) −30.9282 −1.43119 −0.715593 0.698517i \(-0.753844\pi\)
−0.715593 + 0.698517i \(0.753844\pi\)
\(468\) 3.46410 0.160128
\(469\) 9.46410 0.437012
\(470\) −6.92820 −0.319574
\(471\) 18.7846 0.865549
\(472\) −2.92820 −0.134781
\(473\) −2.92820 −0.134639
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −3.46410 −0.158777
\(477\) 12.9282 0.591942
\(478\) 4.39230 0.200899
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) 1.00000 0.0456435
\(481\) 25.8564 1.17895
\(482\) −7.07180 −0.322112
\(483\) 1.46410 0.0666189
\(484\) 1.00000 0.0454545
\(485\) 11.8564 0.538372
\(486\) −1.00000 −0.0453609
\(487\) 1.07180 0.0485677 0.0242839 0.999705i \(-0.492269\pi\)
0.0242839 + 0.999705i \(0.492269\pi\)
\(488\) −4.92820 −0.223089
\(489\) −9.46410 −0.427981
\(490\) −1.00000 −0.0451754
\(491\) 1.85641 0.0837785 0.0418892 0.999122i \(-0.486662\pi\)
0.0418892 + 0.999122i \(0.486662\pi\)
\(492\) 8.92820 0.402514
\(493\) −6.92820 −0.312031
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 6.92820 0.311086
\(497\) −4.00000 −0.179425
\(498\) −6.92820 −0.310460
\(499\) 17.0718 0.764239 0.382119 0.924113i \(-0.375194\pi\)
0.382119 + 0.924113i \(0.375194\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 9.46410 0.422825
\(502\) −5.07180 −0.226365
\(503\) 12.3923 0.552546 0.276273 0.961079i \(-0.410901\pi\)
0.276273 + 0.961079i \(0.410901\pi\)
\(504\) 1.00000 0.0445435
\(505\) −11.8564 −0.527603
\(506\) −1.46410 −0.0650873
\(507\) 1.00000 0.0444116
\(508\) 10.9282 0.484861
\(509\) −32.9282 −1.45952 −0.729758 0.683705i \(-0.760367\pi\)
−0.729758 + 0.683705i \(0.760367\pi\)
\(510\) −3.46410 −0.153393
\(511\) −0.928203 −0.0410613
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −16.9282 −0.746671
\(515\) −16.0000 −0.705044
\(516\) 2.92820 0.128907
\(517\) 6.92820 0.304702
\(518\) 7.46410 0.327954
\(519\) 8.92820 0.391905
\(520\) −3.46410 −0.151911
\(521\) 27.4641 1.20322 0.601612 0.798788i \(-0.294525\pi\)
0.601612 + 0.798788i \(0.294525\pi\)
\(522\) 2.00000 0.0875376
\(523\) 44.3923 1.94114 0.970570 0.240819i \(-0.0774161\pi\)
0.970570 + 0.240819i \(0.0774161\pi\)
\(524\) −1.07180 −0.0468217
\(525\) −1.00000 −0.0436436
\(526\) −16.7846 −0.731844
\(527\) −24.0000 −1.04546
\(528\) −1.00000 −0.0435194
\(529\) −20.8564 −0.906800
\(530\) −12.9282 −0.561565
\(531\) −2.92820 −0.127073
\(532\) 0 0
\(533\) −30.9282 −1.33965
\(534\) −3.46410 −0.149906
\(535\) 6.92820 0.299532
\(536\) 9.46410 0.408787
\(537\) 20.7846 0.896922
\(538\) −16.9282 −0.729827
\(539\) 1.00000 0.0430730
\(540\) 1.00000 0.0430331
\(541\) 9.32051 0.400720 0.200360 0.979722i \(-0.435789\pi\)
0.200360 + 0.979722i \(0.435789\pi\)
\(542\) −5.85641 −0.251554
\(543\) 16.5359 0.709623
\(544\) −3.46410 −0.148522
\(545\) −14.3923 −0.616499
\(546\) −3.46410 −0.148250
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −4.53590 −0.193764
\(549\) −4.92820 −0.210331
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) 1.46410 0.0623163
\(553\) 8.00000 0.340195
\(554\) −10.0000 −0.424859
\(555\) 7.46410 0.316833
\(556\) −2.92820 −0.124183
\(557\) −31.8564 −1.34980 −0.674900 0.737910i \(-0.735813\pi\)
−0.674900 + 0.737910i \(0.735813\pi\)
\(558\) 6.92820 0.293294
\(559\) −10.1436 −0.429028
\(560\) −1.00000 −0.0422577
\(561\) 3.46410 0.146254
\(562\) −14.3923 −0.607103
\(563\) −31.7128 −1.33654 −0.668268 0.743921i \(-0.732964\pi\)
−0.668268 + 0.743921i \(0.732964\pi\)
\(564\) −6.92820 −0.291730
\(565\) −11.4641 −0.482298
\(566\) 1.46410 0.0615408
\(567\) 1.00000 0.0419961
\(568\) −4.00000 −0.167836
\(569\) −39.1769 −1.64238 −0.821191 0.570654i \(-0.806690\pi\)
−0.821191 + 0.570654i \(0.806690\pi\)
\(570\) 0 0
\(571\) 23.3205 0.975933 0.487966 0.872862i \(-0.337739\pi\)
0.487966 + 0.872862i \(0.337739\pi\)
\(572\) 3.46410 0.144841
\(573\) −14.9282 −0.623635
\(574\) −8.92820 −0.372656
\(575\) −1.46410 −0.0610573
\(576\) 1.00000 0.0416667
\(577\) 45.7128 1.90305 0.951525 0.307572i \(-0.0995167\pi\)
0.951525 + 0.307572i \(0.0995167\pi\)
\(578\) −5.00000 −0.207973
\(579\) −11.0718 −0.460128
\(580\) −2.00000 −0.0830455
\(581\) 6.92820 0.287430
\(582\) 11.8564 0.491464
\(583\) 12.9282 0.535431
\(584\) −0.928203 −0.0384093
\(585\) −3.46410 −0.143223
\(586\) −14.0000 −0.578335
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) 2.92820 0.120552
\(591\) 18.0000 0.740421
\(592\) 7.46410 0.306773
\(593\) 27.1769 1.11602 0.558011 0.829834i \(-0.311565\pi\)
0.558011 + 0.829834i \(0.311565\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 3.46410 0.142014
\(596\) 12.9282 0.529560
\(597\) 6.92820 0.283552
\(598\) −5.07180 −0.207401
\(599\) −3.21539 −0.131377 −0.0656886 0.997840i \(-0.520924\pi\)
−0.0656886 + 0.997840i \(0.520924\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −23.0718 −0.941118 −0.470559 0.882368i \(-0.655948\pi\)
−0.470559 + 0.882368i \(0.655948\pi\)
\(602\) −2.92820 −0.119345
\(603\) 9.46410 0.385408
\(604\) 8.00000 0.325515
\(605\) −1.00000 −0.0406558
\(606\) −11.8564 −0.481634
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 4.92820 0.199537
\(611\) 24.0000 0.970936
\(612\) −3.46410 −0.140028
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 7.32051 0.295432
\(615\) −8.92820 −0.360020
\(616\) 1.00000 0.0402911
\(617\) −2.39230 −0.0963106 −0.0481553 0.998840i \(-0.515334\pi\)
−0.0481553 + 0.998840i \(0.515334\pi\)
\(618\) −16.0000 −0.643614
\(619\) 15.3205 0.615783 0.307892 0.951421i \(-0.400377\pi\)
0.307892 + 0.951421i \(0.400377\pi\)
\(620\) −6.92820 −0.278243
\(621\) 1.46410 0.0587524
\(622\) 6.53590 0.262066
\(623\) 3.46410 0.138786
\(624\) −3.46410 −0.138675
\(625\) 1.00000 0.0400000
\(626\) 21.7128 0.867819
\(627\) 0 0
\(628\) −18.7846 −0.749588
\(629\) −25.8564 −1.03096
\(630\) −1.00000 −0.0398410
\(631\) 16.7846 0.668185 0.334092 0.942540i \(-0.391570\pi\)
0.334092 + 0.942540i \(0.391570\pi\)
\(632\) 8.00000 0.318223
\(633\) 23.3205 0.926907
\(634\) −3.07180 −0.121997
\(635\) −10.9282 −0.433673
\(636\) −12.9282 −0.512637
\(637\) 3.46410 0.137253
\(638\) 2.00000 0.0791808
\(639\) −4.00000 −0.158238
\(640\) −1.00000 −0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 6.92820 0.273434
\(643\) −23.7128 −0.935142 −0.467571 0.883956i \(-0.654871\pi\)
−0.467571 + 0.883956i \(0.654871\pi\)
\(644\) −1.46410 −0.0576937
\(645\) −2.92820 −0.115298
\(646\) 0 0
\(647\) −22.9282 −0.901401 −0.450700 0.892675i \(-0.648826\pi\)
−0.450700 + 0.892675i \(0.648826\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.92820 −0.114942
\(650\) 3.46410 0.135873
\(651\) −6.92820 −0.271538
\(652\) 9.46410 0.370643
\(653\) 31.8564 1.24664 0.623319 0.781968i \(-0.285784\pi\)
0.623319 + 0.781968i \(0.285784\pi\)
\(654\) −14.3923 −0.562784
\(655\) 1.07180 0.0418786
\(656\) −8.92820 −0.348588
\(657\) −0.928203 −0.0362127
\(658\) 6.92820 0.270089
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) 1.00000 0.0389249
\(661\) 1.60770 0.0625321 0.0312660 0.999511i \(-0.490046\pi\)
0.0312660 + 0.999511i \(0.490046\pi\)
\(662\) −4.00000 −0.155464
\(663\) 12.0000 0.466041
\(664\) 6.92820 0.268866
\(665\) 0 0
\(666\) 7.46410 0.289228
\(667\) −2.92820 −0.113380
\(668\) −9.46410 −0.366177
\(669\) −24.7846 −0.958228
\(670\) −9.46410 −0.365630
\(671\) −4.92820 −0.190251
\(672\) −1.00000 −0.0385758
\(673\) −15.0718 −0.580975 −0.290488 0.956879i \(-0.593817\pi\)
−0.290488 + 0.956879i \(0.593817\pi\)
\(674\) 22.7846 0.877630
\(675\) −1.00000 −0.0384900
\(676\) −1.00000 −0.0384615
\(677\) −40.9282 −1.57300 −0.786499 0.617591i \(-0.788109\pi\)
−0.786499 + 0.617591i \(0.788109\pi\)
\(678\) −11.4641 −0.440276
\(679\) −11.8564 −0.455007
\(680\) 3.46410 0.132842
\(681\) −6.92820 −0.265489
\(682\) 6.92820 0.265295
\(683\) 26.9282 1.03038 0.515190 0.857076i \(-0.327722\pi\)
0.515190 + 0.857076i \(0.327722\pi\)
\(684\) 0 0
\(685\) 4.53590 0.173308
\(686\) 1.00000 0.0381802
\(687\) −2.39230 −0.0912721
\(688\) −2.92820 −0.111637
\(689\) 44.7846 1.70616
\(690\) −1.46410 −0.0557374
\(691\) 3.60770 0.137243 0.0686216 0.997643i \(-0.478140\pi\)
0.0686216 + 0.997643i \(0.478140\pi\)
\(692\) −8.92820 −0.339399
\(693\) 1.00000 0.0379869
\(694\) −1.07180 −0.0406848
\(695\) 2.92820 0.111073
\(696\) −2.00000 −0.0758098
\(697\) 30.9282 1.17149
\(698\) −28.9282 −1.09495
\(699\) −2.00000 −0.0756469
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −3.46410 −0.130744
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 6.92820 0.260931
\(706\) 20.9282 0.787643
\(707\) 11.8564 0.445906
\(708\) 2.92820 0.110049
\(709\) −7.85641 −0.295054 −0.147527 0.989058i \(-0.547131\pi\)
−0.147527 + 0.989058i \(0.547131\pi\)
\(710\) 4.00000 0.150117
\(711\) 8.00000 0.300023
\(712\) 3.46410 0.129823
\(713\) −10.1436 −0.379881
\(714\) 3.46410 0.129641
\(715\) −3.46410 −0.129550
\(716\) −20.7846 −0.776757
\(717\) −4.39230 −0.164034
\(718\) 9.46410 0.353197
\(719\) −4.39230 −0.163805 −0.0819027 0.996640i \(-0.526100\pi\)
−0.0819027 + 0.996640i \(0.526100\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 16.0000 0.595871
\(722\) −19.0000 −0.707107
\(723\) 7.07180 0.263003
\(724\) −16.5359 −0.614552
\(725\) 2.00000 0.0742781
\(726\) −1.00000 −0.0371135
\(727\) 13.0718 0.484806 0.242403 0.970176i \(-0.422064\pi\)
0.242403 + 0.970176i \(0.422064\pi\)
\(728\) 3.46410 0.128388
\(729\) 1.00000 0.0370370
\(730\) 0.928203 0.0343543
\(731\) 10.1436 0.375174
\(732\) 4.92820 0.182152
\(733\) −24.2487 −0.895647 −0.447823 0.894122i \(-0.647801\pi\)
−0.447823 + 0.894122i \(0.647801\pi\)
\(734\) 16.0000 0.590571
\(735\) 1.00000 0.0368856
\(736\) −1.46410 −0.0539675
\(737\) 9.46410 0.348615
\(738\) −8.92820 −0.328652
\(739\) 15.3205 0.563574 0.281787 0.959477i \(-0.409073\pi\)
0.281787 + 0.959477i \(0.409073\pi\)
\(740\) −7.46410 −0.274386
\(741\) 0 0
\(742\) 12.9282 0.474609
\(743\) 13.8564 0.508342 0.254171 0.967159i \(-0.418197\pi\)
0.254171 + 0.967159i \(0.418197\pi\)
\(744\) −6.92820 −0.254000
\(745\) −12.9282 −0.473653
\(746\) −4.92820 −0.180434
\(747\) 6.92820 0.253490
\(748\) −3.46410 −0.126660
\(749\) −6.92820 −0.253151
\(750\) 1.00000 0.0365148
\(751\) −0.784610 −0.0286308 −0.0143154 0.999898i \(-0.504557\pi\)
−0.0143154 + 0.999898i \(0.504557\pi\)
\(752\) 6.92820 0.252646
\(753\) 5.07180 0.184827
\(754\) 6.92820 0.252310
\(755\) −8.00000 −0.291150
\(756\) −1.00000 −0.0363696
\(757\) −50.1051 −1.82110 −0.910551 0.413397i \(-0.864342\pi\)
−0.910551 + 0.413397i \(0.864342\pi\)
\(758\) −28.0000 −1.01701
\(759\) 1.46410 0.0531435
\(760\) 0 0
\(761\) −38.7846 −1.40594 −0.702971 0.711219i \(-0.748143\pi\)
−0.702971 + 0.711219i \(0.748143\pi\)
\(762\) −10.9282 −0.395887
\(763\) 14.3923 0.521036
\(764\) 14.9282 0.540083
\(765\) 3.46410 0.125245
\(766\) −12.7846 −0.461927
\(767\) −10.1436 −0.366264
\(768\) −1.00000 −0.0360844
\(769\) 38.7846 1.39861 0.699304 0.714824i \(-0.253493\pi\)
0.699304 + 0.714824i \(0.253493\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 16.9282 0.609654
\(772\) 11.0718 0.398483
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −2.92820 −0.105252
\(775\) 6.92820 0.248868
\(776\) −11.8564 −0.425620
\(777\) −7.46410 −0.267773
\(778\) 21.7128 0.778442
\(779\) 0 0
\(780\) 3.46410 0.124035
\(781\) −4.00000 −0.143131
\(782\) 5.07180 0.181367
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 18.7846 0.670451
\(786\) 1.07180 0.0382297
\(787\) 41.4641 1.47804 0.739018 0.673686i \(-0.235290\pi\)
0.739018 + 0.673686i \(0.235290\pi\)
\(788\) −18.0000 −0.641223
\(789\) 16.7846 0.597548
\(790\) −8.00000 −0.284627
\(791\) 11.4641 0.407617
\(792\) 1.00000 0.0355335
\(793\) −17.0718 −0.606237
\(794\) −10.0000 −0.354887
\(795\) 12.9282 0.458516
\(796\) −6.92820 −0.245564
\(797\) 31.8564 1.12841 0.564206 0.825634i \(-0.309182\pi\)
0.564206 + 0.825634i \(0.309182\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 3.46410 0.122398
\(802\) −25.7128 −0.907951
\(803\) −0.928203 −0.0327556
\(804\) −9.46410 −0.333773
\(805\) 1.46410 0.0516028
\(806\) 24.0000 0.845364
\(807\) 16.9282 0.595901
\(808\) 11.8564 0.417107
\(809\) −11.4641 −0.403056 −0.201528 0.979483i \(-0.564591\pi\)
−0.201528 + 0.979483i \(0.564591\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 13.8564 0.486564 0.243282 0.969956i \(-0.421776\pi\)
0.243282 + 0.969956i \(0.421776\pi\)
\(812\) 2.00000 0.0701862
\(813\) 5.85641 0.205393
\(814\) 7.46410 0.261617
\(815\) −9.46410 −0.331513
\(816\) 3.46410 0.121268
\(817\) 0 0
\(818\) −12.9282 −0.452024
\(819\) 3.46410 0.121046
\(820\) 8.92820 0.311786
\(821\) 12.1436 0.423814 0.211907 0.977290i \(-0.432033\pi\)
0.211907 + 0.977290i \(0.432033\pi\)
\(822\) 4.53590 0.158208
\(823\) 49.8564 1.73789 0.868943 0.494913i \(-0.164800\pi\)
0.868943 + 0.494913i \(0.164800\pi\)
\(824\) 16.0000 0.557386
\(825\) −1.00000 −0.0348155
\(826\) −2.92820 −0.101885
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) −1.46410 −0.0508810
\(829\) 51.1769 1.77745 0.888724 0.458443i \(-0.151593\pi\)
0.888724 + 0.458443i \(0.151593\pi\)
\(830\) −6.92820 −0.240481
\(831\) 10.0000 0.346896
\(832\) 3.46410 0.120096
\(833\) −3.46410 −0.120024
\(834\) 2.92820 0.101395
\(835\) 9.46410 0.327519
\(836\) 0 0
\(837\) −6.92820 −0.239474
\(838\) 13.8564 0.478662
\(839\) 30.5359 1.05422 0.527108 0.849798i \(-0.323276\pi\)
0.527108 + 0.849798i \(0.323276\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) 14.3923 0.495697
\(844\) −23.3205 −0.802725
\(845\) 1.00000 0.0344010
\(846\) 6.92820 0.238197
\(847\) 1.00000 0.0343604
\(848\) 12.9282 0.443956
\(849\) −1.46410 −0.0502478
\(850\) −3.46410 −0.118818
\(851\) −10.9282 −0.374614
\(852\) 4.00000 0.137038
\(853\) 15.1769 0.519648 0.259824 0.965656i \(-0.416336\pi\)
0.259824 + 0.965656i \(0.416336\pi\)
\(854\) −4.92820 −0.168640
\(855\) 0 0
\(856\) −6.92820 −0.236801
\(857\) −0.535898 −0.0183059 −0.00915297 0.999958i \(-0.502914\pi\)
−0.00915297 + 0.999958i \(0.502914\pi\)
\(858\) −3.46410 −0.118262
\(859\) 20.3923 0.695776 0.347888 0.937536i \(-0.386899\pi\)
0.347888 + 0.937536i \(0.386899\pi\)
\(860\) 2.92820 0.0998509
\(861\) 8.92820 0.304272
\(862\) −23.3205 −0.794300
\(863\) 42.2487 1.43816 0.719081 0.694926i \(-0.244563\pi\)
0.719081 + 0.694926i \(0.244563\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 8.92820 0.303568
\(866\) −17.7128 −0.601906
\(867\) 5.00000 0.169809
\(868\) 6.92820 0.235159
\(869\) 8.00000 0.271381
\(870\) 2.00000 0.0678064
\(871\) 32.7846 1.11086
\(872\) 14.3923 0.487385
\(873\) −11.8564 −0.401279
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0.928203 0.0313611
\(877\) −7.85641 −0.265292 −0.132646 0.991163i \(-0.542347\pi\)
−0.132646 + 0.991163i \(0.542347\pi\)
\(878\) −16.7846 −0.566453
\(879\) 14.0000 0.472208
\(880\) −1.00000 −0.0337100
\(881\) 17.3205 0.583543 0.291771 0.956488i \(-0.405755\pi\)
0.291771 + 0.956488i \(0.405755\pi\)
\(882\) 1.00000 0.0336718
\(883\) −48.1051 −1.61887 −0.809433 0.587212i \(-0.800225\pi\)
−0.809433 + 0.587212i \(0.800225\pi\)
\(884\) −12.0000 −0.403604
\(885\) −2.92820 −0.0984305
\(886\) 26.9282 0.904671
\(887\) −16.6795 −0.560043 −0.280021 0.959994i \(-0.590342\pi\)
−0.280021 + 0.959994i \(0.590342\pi\)
\(888\) −7.46410 −0.250479
\(889\) 10.9282 0.366520
\(890\) −3.46410 −0.116117
\(891\) 1.00000 0.0335013
\(892\) 24.7846 0.829850
\(893\) 0 0
\(894\) −12.9282 −0.432384
\(895\) 20.7846 0.694753
\(896\) 1.00000 0.0334077
\(897\) 5.07180 0.169342
\(898\) −22.0000 −0.734150
\(899\) 13.8564 0.462137
\(900\) 1.00000 0.0333333
\(901\) −44.7846 −1.49199
\(902\) −8.92820 −0.297277
\(903\) 2.92820 0.0974445
\(904\) 11.4641 0.381290
\(905\) 16.5359 0.549672
\(906\) −8.00000 −0.265782
\(907\) −21.1769 −0.703168 −0.351584 0.936156i \(-0.614357\pi\)
−0.351584 + 0.936156i \(0.614357\pi\)
\(908\) 6.92820 0.229920
\(909\) 11.8564 0.393252
\(910\) −3.46410 −0.114834
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 0 0
\(913\) 6.92820 0.229290
\(914\) 19.0718 0.630839
\(915\) −4.92820 −0.162921
\(916\) 2.39230 0.0790440
\(917\) −1.07180 −0.0353938
\(918\) 3.46410 0.114332
\(919\) 45.0718 1.48678 0.743391 0.668857i \(-0.233216\pi\)
0.743391 + 0.668857i \(0.233216\pi\)
\(920\) 1.46410 0.0482700
\(921\) −7.32051 −0.241219
\(922\) −18.0000 −0.592798
\(923\) −13.8564 −0.456089
\(924\) −1.00000 −0.0328976
\(925\) 7.46410 0.245418
\(926\) 20.7846 0.683025
\(927\) 16.0000 0.525509
\(928\) 2.00000 0.0656532
\(929\) −25.6077 −0.840161 −0.420081 0.907487i \(-0.637998\pi\)
−0.420081 + 0.907487i \(0.637998\pi\)
\(930\) 6.92820 0.227185
\(931\) 0 0
\(932\) 2.00000 0.0655122
\(933\) −6.53590 −0.213976
\(934\) −30.9282 −1.01200
\(935\) 3.46410 0.113288
\(936\) 3.46410 0.113228
\(937\) −3.85641 −0.125983 −0.0629917 0.998014i \(-0.520064\pi\)
−0.0629917 + 0.998014i \(0.520064\pi\)
\(938\) 9.46410 0.309014
\(939\) −21.7128 −0.708571
\(940\) −6.92820 −0.225973
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 18.7846 0.612036
\(943\) 13.0718 0.425676
\(944\) −2.92820 −0.0953049
\(945\) 1.00000 0.0325300
\(946\) −2.92820 −0.0952041
\(947\) −10.1436 −0.329622 −0.164811 0.986325i \(-0.552701\pi\)
−0.164811 + 0.986325i \(0.552701\pi\)
\(948\) −8.00000 −0.259828
\(949\) −3.21539 −0.104376
\(950\) 0 0
\(951\) 3.07180 0.0996098
\(952\) −3.46410 −0.112272
\(953\) −28.6410 −0.927774 −0.463887 0.885895i \(-0.653546\pi\)
−0.463887 + 0.885895i \(0.653546\pi\)
\(954\) 12.9282 0.418566
\(955\) −14.9282 −0.483065
\(956\) 4.39230 0.142057
\(957\) −2.00000 −0.0646508
\(958\) 13.8564 0.447680
\(959\) −4.53590 −0.146472
\(960\) 1.00000 0.0322749
\(961\) 17.0000 0.548387
\(962\) 25.8564 0.833644
\(963\) −6.92820 −0.223258
\(964\) −7.07180 −0.227767
\(965\) −11.0718 −0.356414
\(966\) 1.46410 0.0471067
\(967\) −26.9282 −0.865953 −0.432976 0.901405i \(-0.642537\pi\)
−0.432976 + 0.901405i \(0.642537\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 11.8564 0.380686
\(971\) 35.7128 1.14608 0.573039 0.819528i \(-0.305764\pi\)
0.573039 + 0.819528i \(0.305764\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.92820 −0.0938739
\(974\) 1.07180 0.0343426
\(975\) −3.46410 −0.110940
\(976\) −4.92820 −0.157748
\(977\) −37.3205 −1.19399 −0.596994 0.802245i \(-0.703639\pi\)
−0.596994 + 0.802245i \(0.703639\pi\)
\(978\) −9.46410 −0.302629
\(979\) 3.46410 0.110713
\(980\) −1.00000 −0.0319438
\(981\) 14.3923 0.459511
\(982\) 1.85641 0.0592403
\(983\) 3.21539 0.102555 0.0512775 0.998684i \(-0.483671\pi\)
0.0512775 + 0.998684i \(0.483671\pi\)
\(984\) 8.92820 0.284621
\(985\) 18.0000 0.573528
\(986\) −6.92820 −0.220639
\(987\) −6.92820 −0.220527
\(988\) 0 0
\(989\) 4.28719 0.136325
\(990\) −1.00000 −0.0317821
\(991\) −40.7846 −1.29557 −0.647783 0.761825i \(-0.724304\pi\)
−0.647783 + 0.761825i \(0.724304\pi\)
\(992\) 6.92820 0.219971
\(993\) 4.00000 0.126936
\(994\) −4.00000 −0.126872
\(995\) 6.92820 0.219639
\(996\) −6.92820 −0.219529
\(997\) 18.6795 0.591585 0.295793 0.955252i \(-0.404416\pi\)
0.295793 + 0.955252i \(0.404416\pi\)
\(998\) 17.0718 0.540398
\(999\) −7.46410 −0.236154
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.bb.1.2 2
3.2 odd 2 6930.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.bb.1.2 2 1.1 even 1 trivial
6930.2.a.bt.1.2 2 3.2 odd 2