Properties

Label 3630.2.a.bk.1.1
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
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] = [3630,2,Mod(1,3630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3630, 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("3630.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.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: \(28.9856959337\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\chi\) \(=\) 3630.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.73205 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.73205 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -3.00000 q^{13} -1.73205 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.73205 q^{17} +1.00000 q^{18} +3.00000 q^{19} -1.00000 q^{20} +1.73205 q^{21} -8.19615 q^{23} -1.00000 q^{24} +1.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} -1.73205 q^{28} +6.46410 q^{29} +1.00000 q^{30} +6.19615 q^{31} +1.00000 q^{32} -1.73205 q^{34} +1.73205 q^{35} +1.00000 q^{36} -1.19615 q^{37} +3.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} +10.7321 q^{41} +1.73205 q^{42} +2.19615 q^{43} -1.00000 q^{45} -8.19615 q^{46} +6.00000 q^{47} -1.00000 q^{48} -4.00000 q^{49} +1.00000 q^{50} +1.73205 q^{51} -3.00000 q^{52} +2.19615 q^{53} -1.00000 q^{54} -1.73205 q^{56} -3.00000 q^{57} +6.46410 q^{58} +14.1962 q^{59} +1.00000 q^{60} -5.66025 q^{61} +6.19615 q^{62} -1.73205 q^{63} +1.00000 q^{64} +3.00000 q^{65} -6.39230 q^{67} -1.73205 q^{68} +8.19615 q^{69} +1.73205 q^{70} +3.00000 q^{71} +1.00000 q^{72} -1.26795 q^{73} -1.19615 q^{74} -1.00000 q^{75} +3.00000 q^{76} +3.00000 q^{78} +10.3923 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.7321 q^{82} -0.464102 q^{83} +1.73205 q^{84} +1.73205 q^{85} +2.19615 q^{86} -6.46410 q^{87} -16.3923 q^{89} -1.00000 q^{90} +5.19615 q^{91} -8.19615 q^{92} -6.19615 q^{93} +6.00000 q^{94} -3.00000 q^{95} -1.00000 q^{96} +10.1962 q^{97} -4.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 6 q^{13} + 2 q^{15} + 2 q^{16} + 2 q^{18} + 6 q^{19} - 2 q^{20} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 6 q^{26} - 2 q^{27} + 6 q^{29} + 2 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{36} + 8 q^{37} + 6 q^{38} + 6 q^{39} - 2 q^{40} + 18 q^{41} - 6 q^{43} - 2 q^{45} - 6 q^{46} + 12 q^{47} - 2 q^{48} - 8 q^{49} + 2 q^{50} - 6 q^{52} - 6 q^{53} - 2 q^{54} - 6 q^{57} + 6 q^{58} + 18 q^{59} + 2 q^{60} + 6 q^{61} + 2 q^{62} + 2 q^{64} + 6 q^{65} + 8 q^{67} + 6 q^{69} + 6 q^{71} + 2 q^{72} - 6 q^{73} + 8 q^{74} - 2 q^{75} + 6 q^{76} + 6 q^{78} - 2 q^{80} + 2 q^{81} + 18 q^{82} + 6 q^{83} - 6 q^{86} - 6 q^{87} - 12 q^{89} - 2 q^{90} - 6 q^{92} - 2 q^{93} + 12 q^{94} - 6 q^{95} - 2 q^{96} + 10 q^{97} - 8 q^{98}+O(q^{100}) \) 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
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −1.73205 −0.462910
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) −8.19615 −1.70902 −0.854508 0.519438i \(-0.826141\pi\)
−0.854508 + 0.519438i \(0.826141\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) −1.73205 −0.327327
\(29\) 6.46410 1.20035 0.600177 0.799867i \(-0.295097\pi\)
0.600177 + 0.799867i \(0.295097\pi\)
\(30\) 1.00000 0.182574
\(31\) 6.19615 1.11286 0.556431 0.830894i \(-0.312170\pi\)
0.556431 + 0.830894i \(0.312170\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.73205 −0.297044
\(35\) 1.73205 0.292770
\(36\) 1.00000 0.166667
\(37\) −1.19615 −0.196646 −0.0983231 0.995155i \(-0.531348\pi\)
−0.0983231 + 0.995155i \(0.531348\pi\)
\(38\) 3.00000 0.486664
\(39\) 3.00000 0.480384
\(40\) −1.00000 −0.158114
\(41\) 10.7321 1.67606 0.838032 0.545621i \(-0.183706\pi\)
0.838032 + 0.545621i \(0.183706\pi\)
\(42\) 1.73205 0.267261
\(43\) 2.19615 0.334910 0.167455 0.985880i \(-0.446445\pi\)
0.167455 + 0.985880i \(0.446445\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) −8.19615 −1.20846
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.00000 −0.571429
\(50\) 1.00000 0.141421
\(51\) 1.73205 0.242536
\(52\) −3.00000 −0.416025
\(53\) 2.19615 0.301665 0.150832 0.988559i \(-0.451805\pi\)
0.150832 + 0.988559i \(0.451805\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.73205 −0.231455
\(57\) −3.00000 −0.397360
\(58\) 6.46410 0.848778
\(59\) 14.1962 1.84818 0.924091 0.382173i \(-0.124824\pi\)
0.924091 + 0.382173i \(0.124824\pi\)
\(60\) 1.00000 0.129099
\(61\) −5.66025 −0.724721 −0.362361 0.932038i \(-0.618029\pi\)
−0.362361 + 0.932038i \(0.618029\pi\)
\(62\) 6.19615 0.786912
\(63\) −1.73205 −0.218218
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −6.39230 −0.780944 −0.390472 0.920615i \(-0.627688\pi\)
−0.390472 + 0.920615i \(0.627688\pi\)
\(68\) −1.73205 −0.210042
\(69\) 8.19615 0.986701
\(70\) 1.73205 0.207020
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.26795 −0.148402 −0.0742011 0.997243i \(-0.523641\pi\)
−0.0742011 + 0.997243i \(0.523641\pi\)
\(74\) −1.19615 −0.139050
\(75\) −1.00000 −0.115470
\(76\) 3.00000 0.344124
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 10.7321 1.18516
\(83\) −0.464102 −0.0509418 −0.0254709 0.999676i \(-0.508109\pi\)
−0.0254709 + 0.999676i \(0.508109\pi\)
\(84\) 1.73205 0.188982
\(85\) 1.73205 0.187867
\(86\) 2.19615 0.236817
\(87\) −6.46410 −0.693024
\(88\) 0 0
\(89\) −16.3923 −1.73758 −0.868790 0.495180i \(-0.835102\pi\)
−0.868790 + 0.495180i \(0.835102\pi\)
\(90\) −1.00000 −0.105409
\(91\) 5.19615 0.544705
\(92\) −8.19615 −0.854508
\(93\) −6.19615 −0.642511
\(94\) 6.00000 0.618853
\(95\) −3.00000 −0.307794
\(96\) −1.00000 −0.102062
\(97\) 10.1962 1.03526 0.517631 0.855604i \(-0.326814\pi\)
0.517631 + 0.855604i \(0.326814\pi\)
\(98\) −4.00000 −0.404061
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.07180 0.206151 0.103076 0.994674i \(-0.467132\pi\)
0.103076 + 0.994674i \(0.467132\pi\)
\(102\) 1.73205 0.171499
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) −3.00000 −0.294174
\(105\) −1.73205 −0.169031
\(106\) 2.19615 0.213309
\(107\) −3.46410 −0.334887 −0.167444 0.985882i \(-0.553551\pi\)
−0.167444 + 0.985882i \(0.553551\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 9.46410 0.906497 0.453248 0.891384i \(-0.350265\pi\)
0.453248 + 0.891384i \(0.350265\pi\)
\(110\) 0 0
\(111\) 1.19615 0.113534
\(112\) −1.73205 −0.163663
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −3.00000 −0.280976
\(115\) 8.19615 0.764295
\(116\) 6.46410 0.600177
\(117\) −3.00000 −0.277350
\(118\) 14.1962 1.30686
\(119\) 3.00000 0.275010
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −5.66025 −0.512455
\(123\) −10.7321 −0.967676
\(124\) 6.19615 0.556431
\(125\) −1.00000 −0.0894427
\(126\) −1.73205 −0.154303
\(127\) 9.46410 0.839803 0.419902 0.907570i \(-0.362065\pi\)
0.419902 + 0.907570i \(0.362065\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.19615 −0.193360
\(130\) 3.00000 0.263117
\(131\) 10.7321 0.937664 0.468832 0.883287i \(-0.344675\pi\)
0.468832 + 0.883287i \(0.344675\pi\)
\(132\) 0 0
\(133\) −5.19615 −0.450564
\(134\) −6.39230 −0.552211
\(135\) 1.00000 0.0860663
\(136\) −1.73205 −0.148522
\(137\) 7.39230 0.631567 0.315784 0.948831i \(-0.397733\pi\)
0.315784 + 0.948831i \(0.397733\pi\)
\(138\) 8.19615 0.697703
\(139\) −10.8564 −0.920828 −0.460414 0.887704i \(-0.652299\pi\)
−0.460414 + 0.887704i \(0.652299\pi\)
\(140\) 1.73205 0.146385
\(141\) −6.00000 −0.505291
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.46410 −0.536814
\(146\) −1.26795 −0.104936
\(147\) 4.00000 0.329914
\(148\) −1.19615 −0.0983231
\(149\) 14.5359 1.19083 0.595414 0.803419i \(-0.296988\pi\)
0.595414 + 0.803419i \(0.296988\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 11.6603 0.948898 0.474449 0.880283i \(-0.342647\pi\)
0.474449 + 0.880283i \(0.342647\pi\)
\(152\) 3.00000 0.243332
\(153\) −1.73205 −0.140028
\(154\) 0 0
\(155\) −6.19615 −0.497687
\(156\) 3.00000 0.240192
\(157\) 19.5885 1.56333 0.781665 0.623699i \(-0.214371\pi\)
0.781665 + 0.623699i \(0.214371\pi\)
\(158\) 10.3923 0.826767
\(159\) −2.19615 −0.174166
\(160\) −1.00000 −0.0790569
\(161\) 14.1962 1.11881
\(162\) 1.00000 0.0785674
\(163\) 10.1962 0.798624 0.399312 0.916815i \(-0.369249\pi\)
0.399312 + 0.916815i \(0.369249\pi\)
\(164\) 10.7321 0.838032
\(165\) 0 0
\(166\) −0.464102 −0.0360213
\(167\) 23.3205 1.80460 0.902298 0.431114i \(-0.141879\pi\)
0.902298 + 0.431114i \(0.141879\pi\)
\(168\) 1.73205 0.133631
\(169\) −4.00000 −0.307692
\(170\) 1.73205 0.132842
\(171\) 3.00000 0.229416
\(172\) 2.19615 0.167455
\(173\) 3.80385 0.289201 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(174\) −6.46410 −0.490042
\(175\) −1.73205 −0.130931
\(176\) 0 0
\(177\) −14.1962 −1.06705
\(178\) −16.3923 −1.22866
\(179\) −2.19615 −0.164148 −0.0820741 0.996626i \(-0.526154\pi\)
−0.0820741 + 0.996626i \(0.526154\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.5885 −1.67899 −0.839493 0.543370i \(-0.817148\pi\)
−0.839493 + 0.543370i \(0.817148\pi\)
\(182\) 5.19615 0.385164
\(183\) 5.66025 0.418418
\(184\) −8.19615 −0.604228
\(185\) 1.19615 0.0879429
\(186\) −6.19615 −0.454324
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 1.73205 0.125988
\(190\) −3.00000 −0.217643
\(191\) −13.3923 −0.969033 −0.484517 0.874782i \(-0.661004\pi\)
−0.484517 + 0.874782i \(0.661004\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.46410 0.681241 0.340620 0.940201i \(-0.389363\pi\)
0.340620 + 0.940201i \(0.389363\pi\)
\(194\) 10.1962 0.732041
\(195\) −3.00000 −0.214834
\(196\) −4.00000 −0.285714
\(197\) 2.53590 0.180675 0.0903376 0.995911i \(-0.471205\pi\)
0.0903376 + 0.995911i \(0.471205\pi\)
\(198\) 0 0
\(199\) 8.39230 0.594915 0.297457 0.954735i \(-0.403861\pi\)
0.297457 + 0.954735i \(0.403861\pi\)
\(200\) 1.00000 0.0707107
\(201\) 6.39230 0.450878
\(202\) 2.07180 0.145771
\(203\) −11.1962 −0.785816
\(204\) 1.73205 0.121268
\(205\) −10.7321 −0.749559
\(206\) 5.00000 0.348367
\(207\) −8.19615 −0.569672
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) −1.73205 −0.119523
\(211\) −11.5359 −0.794164 −0.397082 0.917783i \(-0.629977\pi\)
−0.397082 + 0.917783i \(0.629977\pi\)
\(212\) 2.19615 0.150832
\(213\) −3.00000 −0.205557
\(214\) −3.46410 −0.236801
\(215\) −2.19615 −0.149776
\(216\) −1.00000 −0.0680414
\(217\) −10.7321 −0.728539
\(218\) 9.46410 0.640990
\(219\) 1.26795 0.0856801
\(220\) 0 0
\(221\) 5.19615 0.349531
\(222\) 1.19615 0.0802805
\(223\) 13.7846 0.923086 0.461543 0.887118i \(-0.347296\pi\)
0.461543 + 0.887118i \(0.347296\pi\)
\(224\) −1.73205 −0.115728
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.3205 −1.14960 −0.574801 0.818293i \(-0.694921\pi\)
−0.574801 + 0.818293i \(0.694921\pi\)
\(228\) −3.00000 −0.198680
\(229\) −16.7846 −1.10916 −0.554579 0.832131i \(-0.687121\pi\)
−0.554579 + 0.832131i \(0.687121\pi\)
\(230\) 8.19615 0.540438
\(231\) 0 0
\(232\) 6.46410 0.424389
\(233\) 6.92820 0.453882 0.226941 0.973909i \(-0.427128\pi\)
0.226941 + 0.973909i \(0.427128\pi\)
\(234\) −3.00000 −0.196116
\(235\) −6.00000 −0.391397
\(236\) 14.1962 0.924091
\(237\) −10.3923 −0.675053
\(238\) 3.00000 0.194461
\(239\) 29.1962 1.88854 0.944271 0.329169i \(-0.106769\pi\)
0.944271 + 0.329169i \(0.106769\pi\)
\(240\) 1.00000 0.0645497
\(241\) 25.0526 1.61378 0.806889 0.590704i \(-0.201150\pi\)
0.806889 + 0.590704i \(0.201150\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −5.66025 −0.362361
\(245\) 4.00000 0.255551
\(246\) −10.7321 −0.684251
\(247\) −9.00000 −0.572656
\(248\) 6.19615 0.393456
\(249\) 0.464102 0.0294112
\(250\) −1.00000 −0.0632456
\(251\) 4.39230 0.277240 0.138620 0.990346i \(-0.455733\pi\)
0.138620 + 0.990346i \(0.455733\pi\)
\(252\) −1.73205 −0.109109
\(253\) 0 0
\(254\) 9.46410 0.593831
\(255\) −1.73205 −0.108465
\(256\) 1.00000 0.0625000
\(257\) −11.7846 −0.735104 −0.367552 0.930003i \(-0.619804\pi\)
−0.367552 + 0.930003i \(0.619804\pi\)
\(258\) −2.19615 −0.136726
\(259\) 2.07180 0.128735
\(260\) 3.00000 0.186052
\(261\) 6.46410 0.400118
\(262\) 10.7321 0.663028
\(263\) −8.19615 −0.505396 −0.252698 0.967545i \(-0.581318\pi\)
−0.252698 + 0.967545i \(0.581318\pi\)
\(264\) 0 0
\(265\) −2.19615 −0.134909
\(266\) −5.19615 −0.318597
\(267\) 16.3923 1.00319
\(268\) −6.39230 −0.390472
\(269\) 11.1962 0.682641 0.341321 0.939947i \(-0.389126\pi\)
0.341321 + 0.939947i \(0.389126\pi\)
\(270\) 1.00000 0.0608581
\(271\) −26.7846 −1.62705 −0.813525 0.581531i \(-0.802454\pi\)
−0.813525 + 0.581531i \(0.802454\pi\)
\(272\) −1.73205 −0.105021
\(273\) −5.19615 −0.314485
\(274\) 7.39230 0.446585
\(275\) 0 0
\(276\) 8.19615 0.493350
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −10.8564 −0.651124
\(279\) 6.19615 0.370954
\(280\) 1.73205 0.103510
\(281\) −18.5885 −1.10889 −0.554447 0.832219i \(-0.687070\pi\)
−0.554447 + 0.832219i \(0.687070\pi\)
\(282\) −6.00000 −0.357295
\(283\) 7.60770 0.452231 0.226115 0.974101i \(-0.427397\pi\)
0.226115 + 0.974101i \(0.427397\pi\)
\(284\) 3.00000 0.178017
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −18.5885 −1.09724
\(288\) 1.00000 0.0589256
\(289\) −14.0000 −0.823529
\(290\) −6.46410 −0.379585
\(291\) −10.1962 −0.597709
\(292\) −1.26795 −0.0742011
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 4.00000 0.233285
\(295\) −14.1962 −0.826532
\(296\) −1.19615 −0.0695249
\(297\) 0 0
\(298\) 14.5359 0.842042
\(299\) 24.5885 1.42199
\(300\) −1.00000 −0.0577350
\(301\) −3.80385 −0.219250
\(302\) 11.6603 0.670972
\(303\) −2.07180 −0.119022
\(304\) 3.00000 0.172062
\(305\) 5.66025 0.324105
\(306\) −1.73205 −0.0990148
\(307\) 19.2679 1.09968 0.549840 0.835270i \(-0.314689\pi\)
0.549840 + 0.835270i \(0.314689\pi\)
\(308\) 0 0
\(309\) −5.00000 −0.284440
\(310\) −6.19615 −0.351918
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 3.00000 0.169842
\(313\) −22.1962 −1.25460 −0.627300 0.778777i \(-0.715840\pi\)
−0.627300 + 0.778777i \(0.715840\pi\)
\(314\) 19.5885 1.10544
\(315\) 1.73205 0.0975900
\(316\) 10.3923 0.584613
\(317\) 22.9808 1.29073 0.645364 0.763875i \(-0.276706\pi\)
0.645364 + 0.763875i \(0.276706\pi\)
\(318\) −2.19615 −0.123154
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 3.46410 0.193347
\(322\) 14.1962 0.791121
\(323\) −5.19615 −0.289122
\(324\) 1.00000 0.0555556
\(325\) −3.00000 −0.166410
\(326\) 10.1962 0.564713
\(327\) −9.46410 −0.523366
\(328\) 10.7321 0.592578
\(329\) −10.3923 −0.572946
\(330\) 0 0
\(331\) −25.5885 −1.40647 −0.703234 0.710958i \(-0.748262\pi\)
−0.703234 + 0.710958i \(0.748262\pi\)
\(332\) −0.464102 −0.0254709
\(333\) −1.19615 −0.0655487
\(334\) 23.3205 1.27604
\(335\) 6.39230 0.349249
\(336\) 1.73205 0.0944911
\(337\) −27.1244 −1.47756 −0.738779 0.673948i \(-0.764597\pi\)
−0.738779 + 0.673948i \(0.764597\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) 1.73205 0.0939336
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) 19.0526 1.02874
\(344\) 2.19615 0.118409
\(345\) −8.19615 −0.441266
\(346\) 3.80385 0.204496
\(347\) 15.2487 0.818594 0.409297 0.912401i \(-0.365774\pi\)
0.409297 + 0.912401i \(0.365774\pi\)
\(348\) −6.46410 −0.346512
\(349\) 26.7846 1.43375 0.716874 0.697203i \(-0.245572\pi\)
0.716874 + 0.697203i \(0.245572\pi\)
\(350\) −1.73205 −0.0925820
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) −14.1962 −0.754517
\(355\) −3.00000 −0.159223
\(356\) −16.3923 −0.868790
\(357\) −3.00000 −0.158777
\(358\) −2.19615 −0.116070
\(359\) 27.4641 1.44950 0.724750 0.689012i \(-0.241955\pi\)
0.724750 + 0.689012i \(0.241955\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −10.0000 −0.526316
\(362\) −22.5885 −1.18722
\(363\) 0 0
\(364\) 5.19615 0.272352
\(365\) 1.26795 0.0663675
\(366\) 5.66025 0.295866
\(367\) −32.1769 −1.67962 −0.839811 0.542879i \(-0.817334\pi\)
−0.839811 + 0.542879i \(0.817334\pi\)
\(368\) −8.19615 −0.427254
\(369\) 10.7321 0.558688
\(370\) 1.19615 0.0621850
\(371\) −3.80385 −0.197486
\(372\) −6.19615 −0.321256
\(373\) −27.9282 −1.44607 −0.723034 0.690813i \(-0.757253\pi\)
−0.723034 + 0.690813i \(0.757253\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 6.00000 0.309426
\(377\) −19.3923 −0.998755
\(378\) 1.73205 0.0890871
\(379\) −35.9808 −1.84821 −0.924104 0.382142i \(-0.875187\pi\)
−0.924104 + 0.382142i \(0.875187\pi\)
\(380\) −3.00000 −0.153897
\(381\) −9.46410 −0.484861
\(382\) −13.3923 −0.685210
\(383\) 14.7846 0.755458 0.377729 0.925916i \(-0.376705\pi\)
0.377729 + 0.925916i \(0.376705\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 9.46410 0.481710
\(387\) 2.19615 0.111637
\(388\) 10.1962 0.517631
\(389\) 8.78461 0.445397 0.222699 0.974887i \(-0.428513\pi\)
0.222699 + 0.974887i \(0.428513\pi\)
\(390\) −3.00000 −0.151911
\(391\) 14.1962 0.717930
\(392\) −4.00000 −0.202031
\(393\) −10.7321 −0.541360
\(394\) 2.53590 0.127757
\(395\) −10.3923 −0.522894
\(396\) 0 0
\(397\) 23.5885 1.18387 0.591935 0.805985i \(-0.298364\pi\)
0.591935 + 0.805985i \(0.298364\pi\)
\(398\) 8.39230 0.420668
\(399\) 5.19615 0.260133
\(400\) 1.00000 0.0500000
\(401\) 2.19615 0.109671 0.0548353 0.998495i \(-0.482537\pi\)
0.0548353 + 0.998495i \(0.482537\pi\)
\(402\) 6.39230 0.318819
\(403\) −18.5885 −0.925957
\(404\) 2.07180 0.103076
\(405\) −1.00000 −0.0496904
\(406\) −11.1962 −0.555656
\(407\) 0 0
\(408\) 1.73205 0.0857493
\(409\) 12.9282 0.639259 0.319629 0.947543i \(-0.396442\pi\)
0.319629 + 0.947543i \(0.396442\pi\)
\(410\) −10.7321 −0.530018
\(411\) −7.39230 −0.364636
\(412\) 5.00000 0.246332
\(413\) −24.5885 −1.20992
\(414\) −8.19615 −0.402819
\(415\) 0.464102 0.0227819
\(416\) −3.00000 −0.147087
\(417\) 10.8564 0.531641
\(418\) 0 0
\(419\) 22.3923 1.09394 0.546968 0.837154i \(-0.315782\pi\)
0.546968 + 0.837154i \(0.315782\pi\)
\(420\) −1.73205 −0.0845154
\(421\) −40.1962 −1.95904 −0.979520 0.201345i \(-0.935469\pi\)
−0.979520 + 0.201345i \(0.935469\pi\)
\(422\) −11.5359 −0.561559
\(423\) 6.00000 0.291730
\(424\) 2.19615 0.106655
\(425\) −1.73205 −0.0840168
\(426\) −3.00000 −0.145350
\(427\) 9.80385 0.474441
\(428\) −3.46410 −0.167444
\(429\) 0 0
\(430\) −2.19615 −0.105908
\(431\) −1.05256 −0.0507000 −0.0253500 0.999679i \(-0.508070\pi\)
−0.0253500 + 0.999679i \(0.508070\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 4.78461 0.229934 0.114967 0.993369i \(-0.463324\pi\)
0.114967 + 0.993369i \(0.463324\pi\)
\(434\) −10.7321 −0.515155
\(435\) 6.46410 0.309930
\(436\) 9.46410 0.453248
\(437\) −24.5885 −1.17623
\(438\) 1.26795 0.0605850
\(439\) −14.7846 −0.705631 −0.352815 0.935693i \(-0.614776\pi\)
−0.352815 + 0.935693i \(0.614776\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 5.19615 0.247156
\(443\) −25.9808 −1.23438 −0.617192 0.786813i \(-0.711730\pi\)
−0.617192 + 0.786813i \(0.711730\pi\)
\(444\) 1.19615 0.0567669
\(445\) 16.3923 0.777070
\(446\) 13.7846 0.652720
\(447\) −14.5359 −0.687524
\(448\) −1.73205 −0.0818317
\(449\) −13.6077 −0.642187 −0.321093 0.947048i \(-0.604050\pi\)
−0.321093 + 0.947048i \(0.604050\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 0 0
\(453\) −11.6603 −0.547847
\(454\) −17.3205 −0.812892
\(455\) −5.19615 −0.243599
\(456\) −3.00000 −0.140488
\(457\) −19.2679 −0.901317 −0.450658 0.892697i \(-0.648811\pi\)
−0.450658 + 0.892697i \(0.648811\pi\)
\(458\) −16.7846 −0.784293
\(459\) 1.73205 0.0808452
\(460\) 8.19615 0.382148
\(461\) −8.07180 −0.375941 −0.187971 0.982175i \(-0.560191\pi\)
−0.187971 + 0.982175i \(0.560191\pi\)
\(462\) 0 0
\(463\) 6.39230 0.297076 0.148538 0.988907i \(-0.452543\pi\)
0.148538 + 0.988907i \(0.452543\pi\)
\(464\) 6.46410 0.300088
\(465\) 6.19615 0.287340
\(466\) 6.92820 0.320943
\(467\) 30.3731 1.40550 0.702749 0.711438i \(-0.251956\pi\)
0.702749 + 0.711438i \(0.251956\pi\)
\(468\) −3.00000 −0.138675
\(469\) 11.0718 0.511248
\(470\) −6.00000 −0.276759
\(471\) −19.5885 −0.902588
\(472\) 14.1962 0.653431
\(473\) 0 0
\(474\) −10.3923 −0.477334
\(475\) 3.00000 0.137649
\(476\) 3.00000 0.137505
\(477\) 2.19615 0.100555
\(478\) 29.1962 1.33540
\(479\) −7.73205 −0.353286 −0.176643 0.984275i \(-0.556524\pi\)
−0.176643 + 0.984275i \(0.556524\pi\)
\(480\) 1.00000 0.0456435
\(481\) 3.58846 0.163620
\(482\) 25.0526 1.14111
\(483\) −14.1962 −0.645947
\(484\) 0 0
\(485\) −10.1962 −0.462983
\(486\) −1.00000 −0.0453609
\(487\) −11.0000 −0.498458 −0.249229 0.968445i \(-0.580177\pi\)
−0.249229 + 0.968445i \(0.580177\pi\)
\(488\) −5.66025 −0.256228
\(489\) −10.1962 −0.461086
\(490\) 4.00000 0.180702
\(491\) 33.1244 1.49488 0.747441 0.664329i \(-0.231282\pi\)
0.747441 + 0.664329i \(0.231282\pi\)
\(492\) −10.7321 −0.483838
\(493\) −11.1962 −0.504249
\(494\) −9.00000 −0.404929
\(495\) 0 0
\(496\) 6.19615 0.278215
\(497\) −5.19615 −0.233079
\(498\) 0.464102 0.0207969
\(499\) 35.9808 1.61072 0.805360 0.592786i \(-0.201972\pi\)
0.805360 + 0.592786i \(0.201972\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −23.3205 −1.04188
\(502\) 4.39230 0.196038
\(503\) 4.73205 0.210992 0.105496 0.994420i \(-0.466357\pi\)
0.105496 + 0.994420i \(0.466357\pi\)
\(504\) −1.73205 −0.0771517
\(505\) −2.07180 −0.0921937
\(506\) 0 0
\(507\) 4.00000 0.177646
\(508\) 9.46410 0.419902
\(509\) −2.78461 −0.123426 −0.0617128 0.998094i \(-0.519656\pi\)
−0.0617128 + 0.998094i \(0.519656\pi\)
\(510\) −1.73205 −0.0766965
\(511\) 2.19615 0.0971521
\(512\) 1.00000 0.0441942
\(513\) −3.00000 −0.132453
\(514\) −11.7846 −0.519797
\(515\) −5.00000 −0.220326
\(516\) −2.19615 −0.0966802
\(517\) 0 0
\(518\) 2.07180 0.0910295
\(519\) −3.80385 −0.166970
\(520\) 3.00000 0.131559
\(521\) −32.1962 −1.41054 −0.705270 0.708939i \(-0.749174\pi\)
−0.705270 + 0.708939i \(0.749174\pi\)
\(522\) 6.46410 0.282926
\(523\) 15.1244 0.661342 0.330671 0.943746i \(-0.392725\pi\)
0.330671 + 0.943746i \(0.392725\pi\)
\(524\) 10.7321 0.468832
\(525\) 1.73205 0.0755929
\(526\) −8.19615 −0.357369
\(527\) −10.7321 −0.467495
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) −2.19615 −0.0953948
\(531\) 14.1962 0.616061
\(532\) −5.19615 −0.225282
\(533\) −32.1962 −1.39457
\(534\) 16.3923 0.709364
\(535\) 3.46410 0.149766
\(536\) −6.39230 −0.276106
\(537\) 2.19615 0.0947710
\(538\) 11.1962 0.482700
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −42.5885 −1.83102 −0.915510 0.402294i \(-0.868213\pi\)
−0.915510 + 0.402294i \(0.868213\pi\)
\(542\) −26.7846 −1.15050
\(543\) 22.5885 0.969363
\(544\) −1.73205 −0.0742611
\(545\) −9.46410 −0.405398
\(546\) −5.19615 −0.222375
\(547\) −42.0000 −1.79579 −0.897895 0.440209i \(-0.854904\pi\)
−0.897895 + 0.440209i \(0.854904\pi\)
\(548\) 7.39230 0.315784
\(549\) −5.66025 −0.241574
\(550\) 0 0
\(551\) 19.3923 0.826140
\(552\) 8.19615 0.348851
\(553\) −18.0000 −0.765438
\(554\) 0 0
\(555\) −1.19615 −0.0507738
\(556\) −10.8564 −0.460414
\(557\) −6.67949 −0.283019 −0.141510 0.989937i \(-0.545196\pi\)
−0.141510 + 0.989937i \(0.545196\pi\)
\(558\) 6.19615 0.262304
\(559\) −6.58846 −0.278662
\(560\) 1.73205 0.0731925
\(561\) 0 0
\(562\) −18.5885 −0.784107
\(563\) 32.3205 1.36215 0.681074 0.732215i \(-0.261513\pi\)
0.681074 + 0.732215i \(0.261513\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) 7.60770 0.319775
\(567\) −1.73205 −0.0727393
\(568\) 3.00000 0.125877
\(569\) 20.1962 0.846667 0.423333 0.905974i \(-0.360860\pi\)
0.423333 + 0.905974i \(0.360860\pi\)
\(570\) 3.00000 0.125656
\(571\) −39.4641 −1.65152 −0.825761 0.564021i \(-0.809254\pi\)
−0.825761 + 0.564021i \(0.809254\pi\)
\(572\) 0 0
\(573\) 13.3923 0.559472
\(574\) −18.5885 −0.775867
\(575\) −8.19615 −0.341803
\(576\) 1.00000 0.0416667
\(577\) −9.60770 −0.399974 −0.199987 0.979799i \(-0.564090\pi\)
−0.199987 + 0.979799i \(0.564090\pi\)
\(578\) −14.0000 −0.582323
\(579\) −9.46410 −0.393315
\(580\) −6.46410 −0.268407
\(581\) 0.803848 0.0333492
\(582\) −10.1962 −0.422644
\(583\) 0 0
\(584\) −1.26795 −0.0524681
\(585\) 3.00000 0.124035
\(586\) 24.0000 0.991431
\(587\) 35.1962 1.45270 0.726350 0.687325i \(-0.241215\pi\)
0.726350 + 0.687325i \(0.241215\pi\)
\(588\) 4.00000 0.164957
\(589\) 18.5885 0.765924
\(590\) −14.1962 −0.584446
\(591\) −2.53590 −0.104313
\(592\) −1.19615 −0.0491616
\(593\) 47.5692 1.95343 0.976717 0.214532i \(-0.0688228\pi\)
0.976717 + 0.214532i \(0.0688228\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 14.5359 0.595414
\(597\) −8.39230 −0.343474
\(598\) 24.5885 1.00550
\(599\) 13.6077 0.555995 0.277998 0.960582i \(-0.410329\pi\)
0.277998 + 0.960582i \(0.410329\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −12.9282 −0.527352 −0.263676 0.964611i \(-0.584935\pi\)
−0.263676 + 0.964611i \(0.584935\pi\)
\(602\) −3.80385 −0.155033
\(603\) −6.39230 −0.260315
\(604\) 11.6603 0.474449
\(605\) 0 0
\(606\) −2.07180 −0.0841610
\(607\) 35.4449 1.43866 0.719331 0.694667i \(-0.244448\pi\)
0.719331 + 0.694667i \(0.244448\pi\)
\(608\) 3.00000 0.121666
\(609\) 11.1962 0.453691
\(610\) 5.66025 0.229177
\(611\) −18.0000 −0.728202
\(612\) −1.73205 −0.0700140
\(613\) 34.8564 1.40784 0.703918 0.710281i \(-0.251432\pi\)
0.703918 + 0.710281i \(0.251432\pi\)
\(614\) 19.2679 0.777591
\(615\) 10.7321 0.432758
\(616\) 0 0
\(617\) −40.1769 −1.61746 −0.808731 0.588179i \(-0.799845\pi\)
−0.808731 + 0.588179i \(0.799845\pi\)
\(618\) −5.00000 −0.201129
\(619\) 26.8038 1.07734 0.538669 0.842518i \(-0.318927\pi\)
0.538669 + 0.842518i \(0.318927\pi\)
\(620\) −6.19615 −0.248843
\(621\) 8.19615 0.328900
\(622\) 0 0
\(623\) 28.3923 1.13751
\(624\) 3.00000 0.120096
\(625\) 1.00000 0.0400000
\(626\) −22.1962 −0.887137
\(627\) 0 0
\(628\) 19.5885 0.781665
\(629\) 2.07180 0.0826079
\(630\) 1.73205 0.0690066
\(631\) 25.3731 1.01009 0.505043 0.863094i \(-0.331477\pi\)
0.505043 + 0.863094i \(0.331477\pi\)
\(632\) 10.3923 0.413384
\(633\) 11.5359 0.458511
\(634\) 22.9808 0.912683
\(635\) −9.46410 −0.375571
\(636\) −2.19615 −0.0870831
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 3.00000 0.118678
\(640\) −1.00000 −0.0395285
\(641\) −38.7846 −1.53190 −0.765950 0.642900i \(-0.777731\pi\)
−0.765950 + 0.642900i \(0.777731\pi\)
\(642\) 3.46410 0.136717
\(643\) −24.7846 −0.977410 −0.488705 0.872449i \(-0.662530\pi\)
−0.488705 + 0.872449i \(0.662530\pi\)
\(644\) 14.1962 0.559407
\(645\) 2.19615 0.0864734
\(646\) −5.19615 −0.204440
\(647\) 40.3923 1.58799 0.793993 0.607927i \(-0.207999\pi\)
0.793993 + 0.607927i \(0.207999\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −3.00000 −0.117670
\(651\) 10.7321 0.420622
\(652\) 10.1962 0.399312
\(653\) 8.19615 0.320740 0.160370 0.987057i \(-0.448731\pi\)
0.160370 + 0.987057i \(0.448731\pi\)
\(654\) −9.46410 −0.370076
\(655\) −10.7321 −0.419336
\(656\) 10.7321 0.419016
\(657\) −1.26795 −0.0494674
\(658\) −10.3923 −0.405134
\(659\) −1.60770 −0.0626269 −0.0313135 0.999510i \(-0.509969\pi\)
−0.0313135 + 0.999510i \(0.509969\pi\)
\(660\) 0 0
\(661\) −12.9808 −0.504893 −0.252447 0.967611i \(-0.581235\pi\)
−0.252447 + 0.967611i \(0.581235\pi\)
\(662\) −25.5885 −0.994524
\(663\) −5.19615 −0.201802
\(664\) −0.464102 −0.0180106
\(665\) 5.19615 0.201498
\(666\) −1.19615 −0.0463500
\(667\) −52.9808 −2.05142
\(668\) 23.3205 0.902298
\(669\) −13.7846 −0.532944
\(670\) 6.39230 0.246956
\(671\) 0 0
\(672\) 1.73205 0.0668153
\(673\) −41.5692 −1.60238 −0.801188 0.598413i \(-0.795798\pi\)
−0.801188 + 0.598413i \(0.795798\pi\)
\(674\) −27.1244 −1.04479
\(675\) −1.00000 −0.0384900
\(676\) −4.00000 −0.153846
\(677\) −38.1962 −1.46800 −0.733999 0.679151i \(-0.762348\pi\)
−0.733999 + 0.679151i \(0.762348\pi\)
\(678\) 0 0
\(679\) −17.6603 −0.677738
\(680\) 1.73205 0.0664211
\(681\) 17.3205 0.663723
\(682\) 0 0
\(683\) 31.9808 1.22371 0.611855 0.790970i \(-0.290424\pi\)
0.611855 + 0.790970i \(0.290424\pi\)
\(684\) 3.00000 0.114708
\(685\) −7.39230 −0.282445
\(686\) 19.0526 0.727430
\(687\) 16.7846 0.640373
\(688\) 2.19615 0.0837275
\(689\) −6.58846 −0.251000
\(690\) −8.19615 −0.312022
\(691\) −25.5885 −0.973431 −0.486715 0.873561i \(-0.661805\pi\)
−0.486715 + 0.873561i \(0.661805\pi\)
\(692\) 3.80385 0.144601
\(693\) 0 0
\(694\) 15.2487 0.578833
\(695\) 10.8564 0.411807
\(696\) −6.46410 −0.245021
\(697\) −18.5885 −0.704088
\(698\) 26.7846 1.01381
\(699\) −6.92820 −0.262049
\(700\) −1.73205 −0.0654654
\(701\) 24.7128 0.933390 0.466695 0.884418i \(-0.345445\pi\)
0.466695 + 0.884418i \(0.345445\pi\)
\(702\) 3.00000 0.113228
\(703\) −3.58846 −0.135341
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) −36.0000 −1.35488
\(707\) −3.58846 −0.134958
\(708\) −14.1962 −0.533524
\(709\) 31.8038 1.19442 0.597209 0.802085i \(-0.296276\pi\)
0.597209 + 0.802085i \(0.296276\pi\)
\(710\) −3.00000 −0.112588
\(711\) 10.3923 0.389742
\(712\) −16.3923 −0.614328
\(713\) −50.7846 −1.90190
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) −2.19615 −0.0820741
\(717\) −29.1962 −1.09035
\(718\) 27.4641 1.02495
\(719\) 22.3923 0.835092 0.417546 0.908656i \(-0.362890\pi\)
0.417546 + 0.908656i \(0.362890\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.66025 −0.322525
\(722\) −10.0000 −0.372161
\(723\) −25.0526 −0.931715
\(724\) −22.5885 −0.839493
\(725\) 6.46410 0.240071
\(726\) 0 0
\(727\) −12.6077 −0.467594 −0.233797 0.972285i \(-0.575115\pi\)
−0.233797 + 0.972285i \(0.575115\pi\)
\(728\) 5.19615 0.192582
\(729\) 1.00000 0.0370370
\(730\) 1.26795 0.0469289
\(731\) −3.80385 −0.140690
\(732\) 5.66025 0.209209
\(733\) 25.8564 0.955028 0.477514 0.878624i \(-0.341538\pi\)
0.477514 + 0.878624i \(0.341538\pi\)
\(734\) −32.1769 −1.18767
\(735\) −4.00000 −0.147542
\(736\) −8.19615 −0.302114
\(737\) 0 0
\(738\) 10.7321 0.395052
\(739\) 17.7846 0.654217 0.327109 0.944987i \(-0.393926\pi\)
0.327109 + 0.944987i \(0.393926\pi\)
\(740\) 1.19615 0.0439714
\(741\) 9.00000 0.330623
\(742\) −3.80385 −0.139644
\(743\) −34.9808 −1.28332 −0.641660 0.766989i \(-0.721754\pi\)
−0.641660 + 0.766989i \(0.721754\pi\)
\(744\) −6.19615 −0.227162
\(745\) −14.5359 −0.532554
\(746\) −27.9282 −1.02252
\(747\) −0.464102 −0.0169806
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 1.00000 0.0365148
\(751\) −18.9808 −0.692618 −0.346309 0.938121i \(-0.612565\pi\)
−0.346309 + 0.938121i \(0.612565\pi\)
\(752\) 6.00000 0.218797
\(753\) −4.39230 −0.160064
\(754\) −19.3923 −0.706226
\(755\) −11.6603 −0.424360
\(756\) 1.73205 0.0629941
\(757\) −42.3923 −1.54077 −0.770387 0.637576i \(-0.779937\pi\)
−0.770387 + 0.637576i \(0.779937\pi\)
\(758\) −35.9808 −1.30688
\(759\) 0 0
\(760\) −3.00000 −0.108821
\(761\) 0.679492 0.0246316 0.0123158 0.999924i \(-0.496080\pi\)
0.0123158 + 0.999924i \(0.496080\pi\)
\(762\) −9.46410 −0.342848
\(763\) −16.3923 −0.593441
\(764\) −13.3923 −0.484517
\(765\) 1.73205 0.0626224
\(766\) 14.7846 0.534190
\(767\) −42.5885 −1.53778
\(768\) −1.00000 −0.0360844
\(769\) −44.9090 −1.61946 −0.809729 0.586804i \(-0.800386\pi\)
−0.809729 + 0.586804i \(0.800386\pi\)
\(770\) 0 0
\(771\) 11.7846 0.424412
\(772\) 9.46410 0.340620
\(773\) −30.5885 −1.10019 −0.550095 0.835102i \(-0.685409\pi\)
−0.550095 + 0.835102i \(0.685409\pi\)
\(774\) 2.19615 0.0789391
\(775\) 6.19615 0.222572
\(776\) 10.1962 0.366021
\(777\) −2.07180 −0.0743253
\(778\) 8.78461 0.314944
\(779\) 32.1962 1.15355
\(780\) −3.00000 −0.107417
\(781\) 0 0
\(782\) 14.1962 0.507653
\(783\) −6.46410 −0.231008
\(784\) −4.00000 −0.142857
\(785\) −19.5885 −0.699142
\(786\) −10.7321 −0.382800
\(787\) −27.1244 −0.966879 −0.483439 0.875378i \(-0.660613\pi\)
−0.483439 + 0.875378i \(0.660613\pi\)
\(788\) 2.53590 0.0903376
\(789\) 8.19615 0.291791
\(790\) −10.3923 −0.369742
\(791\) 0 0
\(792\) 0 0
\(793\) 16.9808 0.603005
\(794\) 23.5885 0.837123
\(795\) 2.19615 0.0778895
\(796\) 8.39230 0.297457
\(797\) 42.5885 1.50856 0.754280 0.656553i \(-0.227986\pi\)
0.754280 + 0.656553i \(0.227986\pi\)
\(798\) 5.19615 0.183942
\(799\) −10.3923 −0.367653
\(800\) 1.00000 0.0353553
\(801\) −16.3923 −0.579194
\(802\) 2.19615 0.0775488
\(803\) 0 0
\(804\) 6.39230 0.225439
\(805\) −14.1962 −0.500349
\(806\) −18.5885 −0.654750
\(807\) −11.1962 −0.394123
\(808\) 2.07180 0.0728856
\(809\) −34.0526 −1.19722 −0.598612 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 23.5359 0.826457 0.413229 0.910627i \(-0.364401\pi\)
0.413229 + 0.910627i \(0.364401\pi\)
\(812\) −11.1962 −0.392908
\(813\) 26.7846 0.939377
\(814\) 0 0
\(815\) −10.1962 −0.357156
\(816\) 1.73205 0.0606339
\(817\) 6.58846 0.230501
\(818\) 12.9282 0.452024
\(819\) 5.19615 0.181568
\(820\) −10.7321 −0.374779
\(821\) −4.39230 −0.153292 −0.0766462 0.997058i \(-0.524421\pi\)
−0.0766462 + 0.997058i \(0.524421\pi\)
\(822\) −7.39230 −0.257836
\(823\) −15.7846 −0.550217 −0.275108 0.961413i \(-0.588714\pi\)
−0.275108 + 0.961413i \(0.588714\pi\)
\(824\) 5.00000 0.174183
\(825\) 0 0
\(826\) −24.5885 −0.855542
\(827\) −23.7846 −0.827072 −0.413536 0.910488i \(-0.635706\pi\)
−0.413536 + 0.910488i \(0.635706\pi\)
\(828\) −8.19615 −0.284836
\(829\) 14.5885 0.506678 0.253339 0.967378i \(-0.418471\pi\)
0.253339 + 0.967378i \(0.418471\pi\)
\(830\) 0.464102 0.0161092
\(831\) 0 0
\(832\) −3.00000 −0.104006
\(833\) 6.92820 0.240048
\(834\) 10.8564 0.375927
\(835\) −23.3205 −0.807039
\(836\) 0 0
\(837\) −6.19615 −0.214170
\(838\) 22.3923 0.773529
\(839\) 6.21539 0.214579 0.107290 0.994228i \(-0.465783\pi\)
0.107290 + 0.994228i \(0.465783\pi\)
\(840\) −1.73205 −0.0597614
\(841\) 12.7846 0.440849
\(842\) −40.1962 −1.38525
\(843\) 18.5885 0.640220
\(844\) −11.5359 −0.397082
\(845\) 4.00000 0.137604
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 2.19615 0.0754162
\(849\) −7.60770 −0.261095
\(850\) −1.73205 −0.0594089
\(851\) 9.80385 0.336072
\(852\) −3.00000 −0.102778
\(853\) 47.1051 1.61285 0.806424 0.591337i \(-0.201400\pi\)
0.806424 + 0.591337i \(0.201400\pi\)
\(854\) 9.80385 0.335481
\(855\) −3.00000 −0.102598
\(856\) −3.46410 −0.118401
\(857\) 7.98076 0.272618 0.136309 0.990666i \(-0.456476\pi\)
0.136309 + 0.990666i \(0.456476\pi\)
\(858\) 0 0
\(859\) 27.1769 0.927264 0.463632 0.886028i \(-0.346546\pi\)
0.463632 + 0.886028i \(0.346546\pi\)
\(860\) −2.19615 −0.0748882
\(861\) 18.5885 0.633493
\(862\) −1.05256 −0.0358503
\(863\) −22.3923 −0.762243 −0.381121 0.924525i \(-0.624462\pi\)
−0.381121 + 0.924525i \(0.624462\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −3.80385 −0.129335
\(866\) 4.78461 0.162588
\(867\) 14.0000 0.475465
\(868\) −10.7321 −0.364270
\(869\) 0 0
\(870\) 6.46410 0.219154
\(871\) 19.1769 0.649785
\(872\) 9.46410 0.320495
\(873\) 10.1962 0.345087
\(874\) −24.5885 −0.831717
\(875\) 1.73205 0.0585540
\(876\) 1.26795 0.0428400
\(877\) −13.1436 −0.443828 −0.221914 0.975066i \(-0.571230\pi\)
−0.221914 + 0.975066i \(0.571230\pi\)
\(878\) −14.7846 −0.498956
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) 46.9808 1.58282 0.791411 0.611284i \(-0.209347\pi\)
0.791411 + 0.611284i \(0.209347\pi\)
\(882\) −4.00000 −0.134687
\(883\) −16.1962 −0.545044 −0.272522 0.962150i \(-0.587858\pi\)
−0.272522 + 0.962150i \(0.587858\pi\)
\(884\) 5.19615 0.174766
\(885\) 14.1962 0.477198
\(886\) −25.9808 −0.872841
\(887\) −17.0718 −0.573215 −0.286607 0.958048i \(-0.592528\pi\)
−0.286607 + 0.958048i \(0.592528\pi\)
\(888\) 1.19615 0.0401402
\(889\) −16.3923 −0.549780
\(890\) 16.3923 0.549471
\(891\) 0 0
\(892\) 13.7846 0.461543
\(893\) 18.0000 0.602347
\(894\) −14.5359 −0.486153
\(895\) 2.19615 0.0734093
\(896\) −1.73205 −0.0578638
\(897\) −24.5885 −0.820985
\(898\) −13.6077 −0.454095
\(899\) 40.0526 1.33583
\(900\) 1.00000 0.0333333
\(901\) −3.80385 −0.126725
\(902\) 0 0
\(903\) 3.80385 0.126584
\(904\) 0 0
\(905\) 22.5885 0.750866
\(906\) −11.6603 −0.387386
\(907\) 31.5692 1.04824 0.524119 0.851645i \(-0.324395\pi\)
0.524119 + 0.851645i \(0.324395\pi\)
\(908\) −17.3205 −0.574801
\(909\) 2.07180 0.0687172
\(910\) −5.19615 −0.172251
\(911\) 33.0000 1.09334 0.546669 0.837349i \(-0.315895\pi\)
0.546669 + 0.837349i \(0.315895\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 0 0
\(914\) −19.2679 −0.637327
\(915\) −5.66025 −0.187122
\(916\) −16.7846 −0.554579
\(917\) −18.5885 −0.613845
\(918\) 1.73205 0.0571662
\(919\) −9.80385 −0.323399 −0.161700 0.986840i \(-0.551698\pi\)
−0.161700 + 0.986840i \(0.551698\pi\)
\(920\) 8.19615 0.270219
\(921\) −19.2679 −0.634901
\(922\) −8.07180 −0.265830
\(923\) −9.00000 −0.296239
\(924\) 0 0
\(925\) −1.19615 −0.0393292
\(926\) 6.39230 0.210064
\(927\) 5.00000 0.164222
\(928\) 6.46410 0.212195
\(929\) −28.9808 −0.950828 −0.475414 0.879762i \(-0.657702\pi\)
−0.475414 + 0.879762i \(0.657702\pi\)
\(930\) 6.19615 0.203180
\(931\) −12.0000 −0.393284
\(932\) 6.92820 0.226941
\(933\) 0 0
\(934\) 30.3731 0.993837
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 4.14359 0.135365 0.0676827 0.997707i \(-0.478439\pi\)
0.0676827 + 0.997707i \(0.478439\pi\)
\(938\) 11.0718 0.361507
\(939\) 22.1962 0.724344
\(940\) −6.00000 −0.195698
\(941\) 5.53590 0.180465 0.0902326 0.995921i \(-0.471239\pi\)
0.0902326 + 0.995921i \(0.471239\pi\)
\(942\) −19.5885 −0.638226
\(943\) −87.9615 −2.86442
\(944\) 14.1962 0.462045
\(945\) −1.73205 −0.0563436
\(946\) 0 0
\(947\) 30.8038 1.00099 0.500495 0.865739i \(-0.333151\pi\)
0.500495 + 0.865739i \(0.333151\pi\)
\(948\) −10.3923 −0.337526
\(949\) 3.80385 0.123478
\(950\) 3.00000 0.0973329
\(951\) −22.9808 −0.745202
\(952\) 3.00000 0.0972306
\(953\) 0.928203 0.0300675 0.0150337 0.999887i \(-0.495214\pi\)
0.0150337 + 0.999887i \(0.495214\pi\)
\(954\) 2.19615 0.0711031
\(955\) 13.3923 0.433365
\(956\) 29.1962 0.944271
\(957\) 0 0
\(958\) −7.73205 −0.249811
\(959\) −12.8038 −0.413458
\(960\) 1.00000 0.0322749
\(961\) 7.39230 0.238461
\(962\) 3.58846 0.115697
\(963\) −3.46410 −0.111629
\(964\) 25.0526 0.806889
\(965\) −9.46410 −0.304660
\(966\) −14.1962 −0.456754
\(967\) 27.4641 0.883186 0.441593 0.897215i \(-0.354414\pi\)
0.441593 + 0.897215i \(0.354414\pi\)
\(968\) 0 0
\(969\) 5.19615 0.166924
\(970\) −10.1962 −0.327379
\(971\) 28.3923 0.911152 0.455576 0.890197i \(-0.349433\pi\)
0.455576 + 0.890197i \(0.349433\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 18.8038 0.602824
\(974\) −11.0000 −0.352463
\(975\) 3.00000 0.0960769
\(976\) −5.66025 −0.181180
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) −10.1962 −0.326037
\(979\) 0 0
\(980\) 4.00000 0.127775
\(981\) 9.46410 0.302166
\(982\) 33.1244 1.05704
\(983\) 58.9808 1.88119 0.940597 0.339525i \(-0.110266\pi\)
0.940597 + 0.339525i \(0.110266\pi\)
\(984\) −10.7321 −0.342125
\(985\) −2.53590 −0.0808004
\(986\) −11.1962 −0.356558
\(987\) 10.3923 0.330791
\(988\) −9.00000 −0.286328
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) 6.19615 0.196728
\(993\) 25.5885 0.812025
\(994\) −5.19615 −0.164812
\(995\) −8.39230 −0.266054
\(996\) 0.464102 0.0147056
\(997\) 29.1051 0.921768 0.460884 0.887460i \(-0.347532\pi\)
0.460884 + 0.887460i \(0.347532\pi\)
\(998\) 35.9808 1.13895
\(999\) 1.19615 0.0378446
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 3630.2.a.bk.1.1 yes 2
11.10 odd 2 3630.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3630.2.a.ba.1.2 2 11.10 odd 2
3630.2.a.bk.1.1 yes 2 1.1 even 1 trivial