Properties

Label 6045.2.a.o.1.2
Level $6045$
Weight $2$
Character 6045.1
Self dual yes
Analytic conductor $48.270$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\chi\) \(=\) 6045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.73205 q^{10} -5.73205 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.46410 q^{14} +1.00000 q^{15} -5.00000 q^{16} +3.46410 q^{17} +1.73205 q^{18} +3.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -9.92820 q^{22} -3.00000 q^{23} +1.73205 q^{24} +1.00000 q^{25} +1.73205 q^{26} -1.00000 q^{27} +2.00000 q^{28} -1.73205 q^{29} +1.73205 q^{30} -1.00000 q^{31} -5.19615 q^{32} +5.73205 q^{33} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +4.92820 q^{37} +5.19615 q^{38} -1.00000 q^{39} +1.73205 q^{40} +1.00000 q^{41} -3.46410 q^{42} +4.53590 q^{43} -5.73205 q^{44} -1.00000 q^{45} -5.19615 q^{46} -1.73205 q^{47} +5.00000 q^{48} -3.00000 q^{49} +1.73205 q^{50} -3.46410 q^{51} +1.00000 q^{52} +7.46410 q^{53} -1.73205 q^{54} +5.73205 q^{55} -3.46410 q^{56} -3.00000 q^{57} -3.00000 q^{58} -0.535898 q^{59} +1.00000 q^{60} -5.46410 q^{61} -1.73205 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +9.92820 q^{66} +10.0000 q^{67} +3.46410 q^{68} +3.00000 q^{69} -3.46410 q^{70} -2.00000 q^{71} -1.73205 q^{72} +2.53590 q^{73} +8.53590 q^{74} -1.00000 q^{75} +3.00000 q^{76} -11.4641 q^{77} -1.73205 q^{78} +3.46410 q^{79} +5.00000 q^{80} +1.00000 q^{81} +1.73205 q^{82} +11.8564 q^{83} -2.00000 q^{84} -3.46410 q^{85} +7.85641 q^{86} +1.73205 q^{87} +9.92820 q^{88} +14.0000 q^{89} -1.73205 q^{90} +2.00000 q^{91} -3.00000 q^{92} +1.00000 q^{93} -3.00000 q^{94} -3.00000 q^{95} +5.19615 q^{96} -6.92820 q^{97} -5.19615 q^{98} -5.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 8 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{15} - 10 q^{16} + 6 q^{19} - 2 q^{20} - 4 q^{21} - 6 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{28} - 2 q^{31} + 8 q^{33} + 12 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{39} + 2 q^{41} + 16 q^{43} - 8 q^{44} - 2 q^{45} + 10 q^{48} - 6 q^{49} + 2 q^{52} + 8 q^{53} + 8 q^{55} - 6 q^{57} - 6 q^{58} - 8 q^{59} + 2 q^{60} - 4 q^{61} + 4 q^{63} + 2 q^{64} - 2 q^{65} + 6 q^{66} + 20 q^{67} + 6 q^{69} - 4 q^{71} + 12 q^{73} + 24 q^{74} - 2 q^{75} + 6 q^{76} - 16 q^{77} + 10 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{84} - 12 q^{86} + 6 q^{88} + 28 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{93} - 6 q^{94} - 6 q^{95} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.73205 −0.707107
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 0.333333
\(10\) −1.73205 −0.547723
\(11\) −5.73205 −1.72828 −0.864139 0.503253i \(-0.832136\pi\)
−0.864139 + 0.503253i \(0.832136\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 3.46410 0.925820
\(15\) 1.00000 0.258199
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 1.73205 0.408248
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −9.92820 −2.11670
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.73205 0.353553
\(25\) 1.00000 0.200000
\(26\) 1.73205 0.339683
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 1.73205 0.316228
\(31\) −1.00000 −0.179605
\(32\) −5.19615 −0.918559
\(33\) 5.73205 0.997822
\(34\) 6.00000 1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) 5.19615 0.842927
\(39\) −1.00000 −0.160128
\(40\) 1.73205 0.273861
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) −3.46410 −0.534522
\(43\) 4.53590 0.691718 0.345859 0.938286i \(-0.387588\pi\)
0.345859 + 0.938286i \(0.387588\pi\)
\(44\) −5.73205 −0.864139
\(45\) −1.00000 −0.149071
\(46\) −5.19615 −0.766131
\(47\) −1.73205 −0.252646 −0.126323 0.991989i \(-0.540318\pi\)
−0.126323 + 0.991989i \(0.540318\pi\)
\(48\) 5.00000 0.721688
\(49\) −3.00000 −0.428571
\(50\) 1.73205 0.244949
\(51\) −3.46410 −0.485071
\(52\) 1.00000 0.138675
\(53\) 7.46410 1.02527 0.512637 0.858606i \(-0.328669\pi\)
0.512637 + 0.858606i \(0.328669\pi\)
\(54\) −1.73205 −0.235702
\(55\) 5.73205 0.772910
\(56\) −3.46410 −0.462910
\(57\) −3.00000 −0.397360
\(58\) −3.00000 −0.393919
\(59\) −0.535898 −0.0697680 −0.0348840 0.999391i \(-0.511106\pi\)
−0.0348840 + 0.999391i \(0.511106\pi\)
\(60\) 1.00000 0.129099
\(61\) −5.46410 −0.699607 −0.349803 0.936823i \(-0.613752\pi\)
−0.349803 + 0.936823i \(0.613752\pi\)
\(62\) −1.73205 −0.219971
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 9.92820 1.22208
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 3.46410 0.420084
\(69\) 3.00000 0.361158
\(70\) −3.46410 −0.414039
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.73205 −0.204124
\(73\) 2.53590 0.296804 0.148402 0.988927i \(-0.452587\pi\)
0.148402 + 0.988927i \(0.452587\pi\)
\(74\) 8.53590 0.992278
\(75\) −1.00000 −0.115470
\(76\) 3.00000 0.344124
\(77\) −11.4641 −1.30646
\(78\) −1.73205 −0.196116
\(79\) 3.46410 0.389742 0.194871 0.980829i \(-0.437571\pi\)
0.194871 + 0.980829i \(0.437571\pi\)
\(80\) 5.00000 0.559017
\(81\) 1.00000 0.111111
\(82\) 1.73205 0.191273
\(83\) 11.8564 1.30141 0.650705 0.759331i \(-0.274474\pi\)
0.650705 + 0.759331i \(0.274474\pi\)
\(84\) −2.00000 −0.218218
\(85\) −3.46410 −0.375735
\(86\) 7.85641 0.847178
\(87\) 1.73205 0.185695
\(88\) 9.92820 1.05835
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −1.73205 −0.182574
\(91\) 2.00000 0.209657
\(92\) −3.00000 −0.312772
\(93\) 1.00000 0.103695
\(94\) −3.00000 −0.309426
\(95\) −3.00000 −0.307794
\(96\) 5.19615 0.530330
\(97\) −6.92820 −0.703452 −0.351726 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) −5.19615 −0.524891
\(99\) −5.73205 −0.576093
\(100\) 1.00000 0.100000
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) −6.00000 −0.594089
\(103\) 5.73205 0.564796 0.282398 0.959297i \(-0.408870\pi\)
0.282398 + 0.959297i \(0.408870\pi\)
\(104\) −1.73205 −0.169842
\(105\) 2.00000 0.195180
\(106\) 12.9282 1.25570
\(107\) 17.3205 1.67444 0.837218 0.546869i \(-0.184180\pi\)
0.837218 + 0.546869i \(0.184180\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.9282 1.04673 0.523366 0.852108i \(-0.324676\pi\)
0.523366 + 0.852108i \(0.324676\pi\)
\(110\) 9.92820 0.946617
\(111\) −4.92820 −0.467764
\(112\) −10.0000 −0.944911
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −5.19615 −0.486664
\(115\) 3.00000 0.279751
\(116\) −1.73205 −0.160817
\(117\) 1.00000 0.0924500
\(118\) −0.928203 −0.0854480
\(119\) 6.92820 0.635107
\(120\) −1.73205 −0.158114
\(121\) 21.8564 1.98695
\(122\) −9.46410 −0.856840
\(123\) −1.00000 −0.0901670
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 3.46410 0.308607
\(127\) 11.4641 1.01727 0.508637 0.860981i \(-0.330149\pi\)
0.508637 + 0.860981i \(0.330149\pi\)
\(128\) 12.1244 1.07165
\(129\) −4.53590 −0.399364
\(130\) −1.73205 −0.151911
\(131\) 11.0000 0.961074 0.480537 0.876974i \(-0.340442\pi\)
0.480537 + 0.876974i \(0.340442\pi\)
\(132\) 5.73205 0.498911
\(133\) 6.00000 0.520266
\(134\) 17.3205 1.49626
\(135\) 1.00000 0.0860663
\(136\) −6.00000 −0.514496
\(137\) −1.92820 −0.164738 −0.0823688 0.996602i \(-0.526249\pi\)
−0.0823688 + 0.996602i \(0.526249\pi\)
\(138\) 5.19615 0.442326
\(139\) −0.660254 −0.0560020 −0.0280010 0.999608i \(-0.508914\pi\)
−0.0280010 + 0.999608i \(0.508914\pi\)
\(140\) −2.00000 −0.169031
\(141\) 1.73205 0.145865
\(142\) −3.46410 −0.290701
\(143\) −5.73205 −0.479338
\(144\) −5.00000 −0.416667
\(145\) 1.73205 0.143839
\(146\) 4.39230 0.363510
\(147\) 3.00000 0.247436
\(148\) 4.92820 0.405096
\(149\) −17.9282 −1.46874 −0.734368 0.678752i \(-0.762521\pi\)
−0.734368 + 0.678752i \(0.762521\pi\)
\(150\) −1.73205 −0.141421
\(151\) 2.39230 0.194683 0.0973415 0.995251i \(-0.468966\pi\)
0.0973415 + 0.995251i \(0.468966\pi\)
\(152\) −5.19615 −0.421464
\(153\) 3.46410 0.280056
\(154\) −19.8564 −1.60007
\(155\) 1.00000 0.0803219
\(156\) −1.00000 −0.0800641
\(157\) −16.5359 −1.31971 −0.659854 0.751394i \(-0.729382\pi\)
−0.659854 + 0.751394i \(0.729382\pi\)
\(158\) 6.00000 0.477334
\(159\) −7.46410 −0.591942
\(160\) 5.19615 0.410792
\(161\) −6.00000 −0.472866
\(162\) 1.73205 0.136083
\(163\) 19.4641 1.52455 0.762273 0.647256i \(-0.224083\pi\)
0.762273 + 0.647256i \(0.224083\pi\)
\(164\) 1.00000 0.0780869
\(165\) −5.73205 −0.446240
\(166\) 20.5359 1.59389
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 3.46410 0.267261
\(169\) 1.00000 0.0769231
\(170\) −6.00000 −0.460179
\(171\) 3.00000 0.229416
\(172\) 4.53590 0.345859
\(173\) 1.19615 0.0909418 0.0454709 0.998966i \(-0.485521\pi\)
0.0454709 + 0.998966i \(0.485521\pi\)
\(174\) 3.00000 0.227429
\(175\) 2.00000 0.151186
\(176\) 28.6603 2.16035
\(177\) 0.535898 0.0402806
\(178\) 24.2487 1.81752
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 12.9282 0.960946 0.480473 0.877010i \(-0.340465\pi\)
0.480473 + 0.877010i \(0.340465\pi\)
\(182\) 3.46410 0.256776
\(183\) 5.46410 0.403918
\(184\) 5.19615 0.383065
\(185\) −4.92820 −0.362329
\(186\) 1.73205 0.127000
\(187\) −19.8564 −1.45204
\(188\) −1.73205 −0.126323
\(189\) −2.00000 −0.145479
\(190\) −5.19615 −0.376969
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0.124356 0.00895132 0.00447566 0.999990i \(-0.498575\pi\)
0.00447566 + 0.999990i \(0.498575\pi\)
\(194\) −12.0000 −0.861550
\(195\) 1.00000 0.0716115
\(196\) −3.00000 −0.214286
\(197\) 11.9282 0.849849 0.424925 0.905229i \(-0.360301\pi\)
0.424925 + 0.905229i \(0.360301\pi\)
\(198\) −9.92820 −0.705567
\(199\) 21.1962 1.50256 0.751278 0.659986i \(-0.229438\pi\)
0.751278 + 0.659986i \(0.229438\pi\)
\(200\) −1.73205 −0.122474
\(201\) −10.0000 −0.705346
\(202\) −24.0000 −1.68863
\(203\) −3.46410 −0.243132
\(204\) −3.46410 −0.242536
\(205\) −1.00000 −0.0698430
\(206\) 9.92820 0.691731
\(207\) −3.00000 −0.208514
\(208\) −5.00000 −0.346688
\(209\) −17.1962 −1.18948
\(210\) 3.46410 0.239046
\(211\) 14.5359 1.00069 0.500346 0.865825i \(-0.333206\pi\)
0.500346 + 0.865825i \(0.333206\pi\)
\(212\) 7.46410 0.512637
\(213\) 2.00000 0.137038
\(214\) 30.0000 2.05076
\(215\) −4.53590 −0.309346
\(216\) 1.73205 0.117851
\(217\) −2.00000 −0.135769
\(218\) 18.9282 1.28198
\(219\) −2.53590 −0.171360
\(220\) 5.73205 0.386455
\(221\) 3.46410 0.233021
\(222\) −8.53590 −0.572892
\(223\) 26.7128 1.78882 0.894411 0.447246i \(-0.147595\pi\)
0.894411 + 0.447246i \(0.147595\pi\)
\(224\) −10.3923 −0.694365
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 22.5167 1.49448 0.747242 0.664552i \(-0.231378\pi\)
0.747242 + 0.664552i \(0.231378\pi\)
\(228\) −3.00000 −0.198680
\(229\) 4.66025 0.307958 0.153979 0.988074i \(-0.450791\pi\)
0.153979 + 0.988074i \(0.450791\pi\)
\(230\) 5.19615 0.342624
\(231\) 11.4641 0.754283
\(232\) 3.00000 0.196960
\(233\) 1.73205 0.113470 0.0567352 0.998389i \(-0.481931\pi\)
0.0567352 + 0.998389i \(0.481931\pi\)
\(234\) 1.73205 0.113228
\(235\) 1.73205 0.112987
\(236\) −0.535898 −0.0348840
\(237\) −3.46410 −0.225018
\(238\) 12.0000 0.777844
\(239\) −5.32051 −0.344155 −0.172078 0.985083i \(-0.555048\pi\)
−0.172078 + 0.985083i \(0.555048\pi\)
\(240\) −5.00000 −0.322749
\(241\) −18.5167 −1.19276 −0.596381 0.802701i \(-0.703395\pi\)
−0.596381 + 0.802701i \(0.703395\pi\)
\(242\) 37.8564 2.43350
\(243\) −1.00000 −0.0641500
\(244\) −5.46410 −0.349803
\(245\) 3.00000 0.191663
\(246\) −1.73205 −0.110432
\(247\) 3.00000 0.190885
\(248\) 1.73205 0.109985
\(249\) −11.8564 −0.751369
\(250\) −1.73205 −0.109545
\(251\) 16.3923 1.03467 0.517337 0.855782i \(-0.326924\pi\)
0.517337 + 0.855782i \(0.326924\pi\)
\(252\) 2.00000 0.125988
\(253\) 17.1962 1.08111
\(254\) 19.8564 1.24590
\(255\) 3.46410 0.216930
\(256\) 19.0000 1.18750
\(257\) 17.7321 1.10609 0.553047 0.833150i \(-0.313465\pi\)
0.553047 + 0.833150i \(0.313465\pi\)
\(258\) −7.85641 −0.489119
\(259\) 9.85641 0.612447
\(260\) −1.00000 −0.0620174
\(261\) −1.73205 −0.107211
\(262\) 19.0526 1.17707
\(263\) −20.8564 −1.28606 −0.643031 0.765841i \(-0.722323\pi\)
−0.643031 + 0.765841i \(0.722323\pi\)
\(264\) −9.92820 −0.611039
\(265\) −7.46410 −0.458516
\(266\) 10.3923 0.637193
\(267\) −14.0000 −0.856786
\(268\) 10.0000 0.610847
\(269\) −18.9282 −1.15407 −0.577036 0.816718i \(-0.695791\pi\)
−0.577036 + 0.816718i \(0.695791\pi\)
\(270\) 1.73205 0.105409
\(271\) 11.4641 0.696395 0.348197 0.937421i \(-0.386794\pi\)
0.348197 + 0.937421i \(0.386794\pi\)
\(272\) −17.3205 −1.05021
\(273\) −2.00000 −0.121046
\(274\) −3.33975 −0.201761
\(275\) −5.73205 −0.345656
\(276\) 3.00000 0.180579
\(277\) 16.0718 0.965661 0.482830 0.875714i \(-0.339609\pi\)
0.482830 + 0.875714i \(0.339609\pi\)
\(278\) −1.14359 −0.0685882
\(279\) −1.00000 −0.0598684
\(280\) 3.46410 0.207020
\(281\) −11.7846 −0.703011 −0.351505 0.936186i \(-0.614330\pi\)
−0.351505 + 0.936186i \(0.614330\pi\)
\(282\) 3.00000 0.178647
\(283\) 19.5885 1.16441 0.582206 0.813041i \(-0.302190\pi\)
0.582206 + 0.813041i \(0.302190\pi\)
\(284\) −2.00000 −0.118678
\(285\) 3.00000 0.177705
\(286\) −9.92820 −0.587067
\(287\) 2.00000 0.118056
\(288\) −5.19615 −0.306186
\(289\) −5.00000 −0.294118
\(290\) 3.00000 0.176166
\(291\) 6.92820 0.406138
\(292\) 2.53590 0.148402
\(293\) −3.46410 −0.202375 −0.101187 0.994867i \(-0.532264\pi\)
−0.101187 + 0.994867i \(0.532264\pi\)
\(294\) 5.19615 0.303046
\(295\) 0.535898 0.0312012
\(296\) −8.53590 −0.496139
\(297\) 5.73205 0.332607
\(298\) −31.0526 −1.79883
\(299\) −3.00000 −0.173494
\(300\) −1.00000 −0.0577350
\(301\) 9.07180 0.522890
\(302\) 4.14359 0.238437
\(303\) 13.8564 0.796030
\(304\) −15.0000 −0.860309
\(305\) 5.46410 0.312874
\(306\) 6.00000 0.342997
\(307\) −21.8564 −1.24741 −0.623706 0.781659i \(-0.714374\pi\)
−0.623706 + 0.781659i \(0.714374\pi\)
\(308\) −11.4641 −0.653228
\(309\) −5.73205 −0.326085
\(310\) 1.73205 0.0983739
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 1.73205 0.0980581
\(313\) 10.8564 0.613640 0.306820 0.951767i \(-0.400735\pi\)
0.306820 + 0.951767i \(0.400735\pi\)
\(314\) −28.6410 −1.61631
\(315\) −2.00000 −0.112687
\(316\) 3.46410 0.194871
\(317\) −4.53590 −0.254761 −0.127381 0.991854i \(-0.540657\pi\)
−0.127381 + 0.991854i \(0.540657\pi\)
\(318\) −12.9282 −0.724978
\(319\) 9.92820 0.555873
\(320\) −1.00000 −0.0559017
\(321\) −17.3205 −0.966736
\(322\) −10.3923 −0.579141
\(323\) 10.3923 0.578243
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 33.7128 1.86718
\(327\) −10.9282 −0.604331
\(328\) −1.73205 −0.0956365
\(329\) −3.46410 −0.190982
\(330\) −9.92820 −0.546530
\(331\) −3.32051 −0.182512 −0.0912558 0.995827i \(-0.529088\pi\)
−0.0912558 + 0.995827i \(0.529088\pi\)
\(332\) 11.8564 0.650705
\(333\) 4.92820 0.270064
\(334\) −31.1769 −1.70592
\(335\) −10.0000 −0.546358
\(336\) 10.0000 0.545545
\(337\) 16.0718 0.875487 0.437743 0.899100i \(-0.355778\pi\)
0.437743 + 0.899100i \(0.355778\pi\)
\(338\) 1.73205 0.0942111
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) 5.73205 0.310408
\(342\) 5.19615 0.280976
\(343\) −20.0000 −1.07990
\(344\) −7.85641 −0.423589
\(345\) −3.00000 −0.161515
\(346\) 2.07180 0.111380
\(347\) 8.07180 0.433317 0.216658 0.976247i \(-0.430484\pi\)
0.216658 + 0.976247i \(0.430484\pi\)
\(348\) 1.73205 0.0928477
\(349\) −20.3923 −1.09158 −0.545788 0.837924i \(-0.683769\pi\)
−0.545788 + 0.837924i \(0.683769\pi\)
\(350\) 3.46410 0.185164
\(351\) −1.00000 −0.0533761
\(352\) 29.7846 1.58753
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0.928203 0.0493334
\(355\) 2.00000 0.106149
\(356\) 14.0000 0.741999
\(357\) −6.92820 −0.366679
\(358\) 13.8564 0.732334
\(359\) 0.143594 0.00757858 0.00378929 0.999993i \(-0.498794\pi\)
0.00378929 + 0.999993i \(0.498794\pi\)
\(360\) 1.73205 0.0912871
\(361\) −10.0000 −0.526316
\(362\) 22.3923 1.17691
\(363\) −21.8564 −1.14716
\(364\) 2.00000 0.104828
\(365\) −2.53590 −0.132735
\(366\) 9.46410 0.494697
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 15.0000 0.781929
\(369\) 1.00000 0.0520579
\(370\) −8.53590 −0.443760
\(371\) 14.9282 0.775034
\(372\) 1.00000 0.0518476
\(373\) 3.07180 0.159052 0.0795258 0.996833i \(-0.474659\pi\)
0.0795258 + 0.996833i \(0.474659\pi\)
\(374\) −34.3923 −1.77838
\(375\) 1.00000 0.0516398
\(376\) 3.00000 0.154713
\(377\) −1.73205 −0.0892052
\(378\) −3.46410 −0.178174
\(379\) 34.8564 1.79045 0.895227 0.445611i \(-0.147014\pi\)
0.895227 + 0.445611i \(0.147014\pi\)
\(380\) −3.00000 −0.153897
\(381\) −11.4641 −0.587324
\(382\) −36.3731 −1.86101
\(383\) −20.5359 −1.04934 −0.524668 0.851307i \(-0.675810\pi\)
−0.524668 + 0.851307i \(0.675810\pi\)
\(384\) −12.1244 −0.618718
\(385\) 11.4641 0.584265
\(386\) 0.215390 0.0109631
\(387\) 4.53590 0.230573
\(388\) −6.92820 −0.351726
\(389\) −2.92820 −0.148466 −0.0742329 0.997241i \(-0.523651\pi\)
−0.0742329 + 0.997241i \(0.523651\pi\)
\(390\) 1.73205 0.0877058
\(391\) −10.3923 −0.525561
\(392\) 5.19615 0.262445
\(393\) −11.0000 −0.554877
\(394\) 20.6603 1.04085
\(395\) −3.46410 −0.174298
\(396\) −5.73205 −0.288046
\(397\) −27.0526 −1.35773 −0.678865 0.734264i \(-0.737528\pi\)
−0.678865 + 0.734264i \(0.737528\pi\)
\(398\) 36.7128 1.84025
\(399\) −6.00000 −0.300376
\(400\) −5.00000 −0.250000
\(401\) 34.6410 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(402\) −17.3205 −0.863868
\(403\) −1.00000 −0.0498135
\(404\) −13.8564 −0.689382
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) −28.2487 −1.40024
\(408\) 6.00000 0.297044
\(409\) −26.5167 −1.31116 −0.655582 0.755124i \(-0.727577\pi\)
−0.655582 + 0.755124i \(0.727577\pi\)
\(410\) −1.73205 −0.0855399
\(411\) 1.92820 0.0951113
\(412\) 5.73205 0.282398
\(413\) −1.07180 −0.0527397
\(414\) −5.19615 −0.255377
\(415\) −11.8564 −0.582008
\(416\) −5.19615 −0.254762
\(417\) 0.660254 0.0323328
\(418\) −29.7846 −1.45681
\(419\) −0.856406 −0.0418382 −0.0209191 0.999781i \(-0.506659\pi\)
−0.0209191 + 0.999781i \(0.506659\pi\)
\(420\) 2.00000 0.0975900
\(421\) −38.6410 −1.88325 −0.941624 0.336667i \(-0.890701\pi\)
−0.941624 + 0.336667i \(0.890701\pi\)
\(422\) 25.1769 1.22559
\(423\) −1.73205 −0.0842152
\(424\) −12.9282 −0.627849
\(425\) 3.46410 0.168034
\(426\) 3.46410 0.167836
\(427\) −10.9282 −0.528853
\(428\) 17.3205 0.837218
\(429\) 5.73205 0.276746
\(430\) −7.85641 −0.378870
\(431\) −6.39230 −0.307906 −0.153953 0.988078i \(-0.549201\pi\)
−0.153953 + 0.988078i \(0.549201\pi\)
\(432\) 5.00000 0.240563
\(433\) −21.9282 −1.05380 −0.526901 0.849927i \(-0.676646\pi\)
−0.526901 + 0.849927i \(0.676646\pi\)
\(434\) −3.46410 −0.166282
\(435\) −1.73205 −0.0830455
\(436\) 10.9282 0.523366
\(437\) −9.00000 −0.430528
\(438\) −4.39230 −0.209872
\(439\) 1.07180 0.0511541 0.0255770 0.999673i \(-0.491858\pi\)
0.0255770 + 0.999673i \(0.491858\pi\)
\(440\) −9.92820 −0.473309
\(441\) −3.00000 −0.142857
\(442\) 6.00000 0.285391
\(443\) −28.3923 −1.34896 −0.674480 0.738294i \(-0.735632\pi\)
−0.674480 + 0.738294i \(0.735632\pi\)
\(444\) −4.92820 −0.233882
\(445\) −14.0000 −0.663664
\(446\) 46.2679 2.19085
\(447\) 17.9282 0.847975
\(448\) 2.00000 0.0944911
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 1.73205 0.0816497
\(451\) −5.73205 −0.269912
\(452\) 0 0
\(453\) −2.39230 −0.112400
\(454\) 39.0000 1.83036
\(455\) −2.00000 −0.0937614
\(456\) 5.19615 0.243332
\(457\) −19.4641 −0.910492 −0.455246 0.890366i \(-0.650449\pi\)
−0.455246 + 0.890366i \(0.650449\pi\)
\(458\) 8.07180 0.377170
\(459\) −3.46410 −0.161690
\(460\) 3.00000 0.139876
\(461\) 23.0718 1.07456 0.537280 0.843404i \(-0.319452\pi\)
0.537280 + 0.843404i \(0.319452\pi\)
\(462\) 19.8564 0.923804
\(463\) 2.92820 0.136085 0.0680426 0.997682i \(-0.478325\pi\)
0.0680426 + 0.997682i \(0.478325\pi\)
\(464\) 8.66025 0.402042
\(465\) −1.00000 −0.0463739
\(466\) 3.00000 0.138972
\(467\) 26.6410 1.23280 0.616400 0.787434i \(-0.288591\pi\)
0.616400 + 0.787434i \(0.288591\pi\)
\(468\) 1.00000 0.0462250
\(469\) 20.0000 0.923514
\(470\) 3.00000 0.138380
\(471\) 16.5359 0.761934
\(472\) 0.928203 0.0427240
\(473\) −26.0000 −1.19548
\(474\) −6.00000 −0.275589
\(475\) 3.00000 0.137649
\(476\) 6.92820 0.317554
\(477\) 7.46410 0.341758
\(478\) −9.21539 −0.421502
\(479\) 21.6077 0.987281 0.493640 0.869666i \(-0.335666\pi\)
0.493640 + 0.869666i \(0.335666\pi\)
\(480\) −5.19615 −0.237171
\(481\) 4.92820 0.224707
\(482\) −32.0718 −1.46083
\(483\) 6.00000 0.273009
\(484\) 21.8564 0.993473
\(485\) 6.92820 0.314594
\(486\) −1.73205 −0.0785674
\(487\) −1.78461 −0.0808684 −0.0404342 0.999182i \(-0.512874\pi\)
−0.0404342 + 0.999182i \(0.512874\pi\)
\(488\) 9.46410 0.428420
\(489\) −19.4641 −0.880197
\(490\) 5.19615 0.234738
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) −1.00000 −0.0450835
\(493\) −6.00000 −0.270226
\(494\) 5.19615 0.233786
\(495\) 5.73205 0.257637
\(496\) 5.00000 0.224507
\(497\) −4.00000 −0.179425
\(498\) −20.5359 −0.920236
\(499\) 15.6077 0.698696 0.349348 0.936993i \(-0.386403\pi\)
0.349348 + 0.936993i \(0.386403\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 18.0000 0.804181
\(502\) 28.3923 1.26721
\(503\) −2.00000 −0.0891756 −0.0445878 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) −3.46410 −0.154303
\(505\) 13.8564 0.616602
\(506\) 29.7846 1.32409
\(507\) −1.00000 −0.0444116
\(508\) 11.4641 0.508637
\(509\) 6.39230 0.283334 0.141667 0.989914i \(-0.454754\pi\)
0.141667 + 0.989914i \(0.454754\pi\)
\(510\) 6.00000 0.265684
\(511\) 5.07180 0.224363
\(512\) 8.66025 0.382733
\(513\) −3.00000 −0.132453
\(514\) 30.7128 1.35468
\(515\) −5.73205 −0.252584
\(516\) −4.53590 −0.199682
\(517\) 9.92820 0.436642
\(518\) 17.0718 0.750092
\(519\) −1.19615 −0.0525053
\(520\) 1.73205 0.0759555
\(521\) 7.85641 0.344195 0.172098 0.985080i \(-0.444946\pi\)
0.172098 + 0.985080i \(0.444946\pi\)
\(522\) −3.00000 −0.131306
\(523\) −12.9282 −0.565311 −0.282655 0.959222i \(-0.591215\pi\)
−0.282655 + 0.959222i \(0.591215\pi\)
\(524\) 11.0000 0.480537
\(525\) −2.00000 −0.0872872
\(526\) −36.1244 −1.57510
\(527\) −3.46410 −0.150899
\(528\) −28.6603 −1.24728
\(529\) −14.0000 −0.608696
\(530\) −12.9282 −0.561565
\(531\) −0.535898 −0.0232560
\(532\) 6.00000 0.260133
\(533\) 1.00000 0.0433148
\(534\) −24.2487 −1.04934
\(535\) −17.3205 −0.748831
\(536\) −17.3205 −0.748132
\(537\) −8.00000 −0.345225
\(538\) −32.7846 −1.41344
\(539\) 17.1962 0.740691
\(540\) 1.00000 0.0430331
\(541\) 25.3205 1.08861 0.544307 0.838886i \(-0.316793\pi\)
0.544307 + 0.838886i \(0.316793\pi\)
\(542\) 19.8564 0.852906
\(543\) −12.9282 −0.554802
\(544\) −18.0000 −0.771744
\(545\) −10.9282 −0.468113
\(546\) −3.46410 −0.148250
\(547\) 27.0526 1.15668 0.578342 0.815794i \(-0.303700\pi\)
0.578342 + 0.815794i \(0.303700\pi\)
\(548\) −1.92820 −0.0823688
\(549\) −5.46410 −0.233202
\(550\) −9.92820 −0.423340
\(551\) −5.19615 −0.221364
\(552\) −5.19615 −0.221163
\(553\) 6.92820 0.294617
\(554\) 27.8372 1.18269
\(555\) 4.92820 0.209191
\(556\) −0.660254 −0.0280010
\(557\) 29.7846 1.26201 0.631007 0.775777i \(-0.282642\pi\)
0.631007 + 0.775777i \(0.282642\pi\)
\(558\) −1.73205 −0.0733236
\(559\) 4.53590 0.191848
\(560\) 10.0000 0.422577
\(561\) 19.8564 0.838338
\(562\) −20.4115 −0.861009
\(563\) −34.2487 −1.44341 −0.721706 0.692200i \(-0.756642\pi\)
−0.721706 + 0.692200i \(0.756642\pi\)
\(564\) 1.73205 0.0729325
\(565\) 0 0
\(566\) 33.9282 1.42611
\(567\) 2.00000 0.0839921
\(568\) 3.46410 0.145350
\(569\) 19.7128 0.826404 0.413202 0.910639i \(-0.364410\pi\)
0.413202 + 0.910639i \(0.364410\pi\)
\(570\) 5.19615 0.217643
\(571\) −23.0526 −0.964720 −0.482360 0.875973i \(-0.660220\pi\)
−0.482360 + 0.875973i \(0.660220\pi\)
\(572\) −5.73205 −0.239669
\(573\) 21.0000 0.877288
\(574\) 3.46410 0.144589
\(575\) −3.00000 −0.125109
\(576\) 1.00000 0.0416667
\(577\) −43.3013 −1.80266 −0.901328 0.433138i \(-0.857406\pi\)
−0.901328 + 0.433138i \(0.857406\pi\)
\(578\) −8.66025 −0.360219
\(579\) −0.124356 −0.00516804
\(580\) 1.73205 0.0719195
\(581\) 23.7128 0.983773
\(582\) 12.0000 0.497416
\(583\) −42.7846 −1.77196
\(584\) −4.39230 −0.181755
\(585\) −1.00000 −0.0413449
\(586\) −6.00000 −0.247858
\(587\) 24.3923 1.00678 0.503389 0.864060i \(-0.332086\pi\)
0.503389 + 0.864060i \(0.332086\pi\)
\(588\) 3.00000 0.123718
\(589\) −3.00000 −0.123613
\(590\) 0.928203 0.0382135
\(591\) −11.9282 −0.490661
\(592\) −24.6410 −1.01274
\(593\) 23.3205 0.957658 0.478829 0.877908i \(-0.341061\pi\)
0.478829 + 0.877908i \(0.341061\pi\)
\(594\) 9.92820 0.407359
\(595\) −6.92820 −0.284029
\(596\) −17.9282 −0.734368
\(597\) −21.1962 −0.867501
\(598\) −5.19615 −0.212486
\(599\) −10.8564 −0.443581 −0.221790 0.975094i \(-0.571190\pi\)
−0.221790 + 0.975094i \(0.571190\pi\)
\(600\) 1.73205 0.0707107
\(601\) 11.6077 0.473488 0.236744 0.971572i \(-0.423920\pi\)
0.236744 + 0.971572i \(0.423920\pi\)
\(602\) 15.7128 0.640406
\(603\) 10.0000 0.407231
\(604\) 2.39230 0.0973415
\(605\) −21.8564 −0.888589
\(606\) 24.0000 0.974933
\(607\) 32.2487 1.30893 0.654467 0.756090i \(-0.272893\pi\)
0.654467 + 0.756090i \(0.272893\pi\)
\(608\) −15.5885 −0.632195
\(609\) 3.46410 0.140372
\(610\) 9.46410 0.383190
\(611\) −1.73205 −0.0700713
\(612\) 3.46410 0.140028
\(613\) 30.7846 1.24338 0.621689 0.783264i \(-0.286447\pi\)
0.621689 + 0.783264i \(0.286447\pi\)
\(614\) −37.8564 −1.52776
\(615\) 1.00000 0.0403239
\(616\) 19.8564 0.800037
\(617\) 25.3205 1.01937 0.509683 0.860362i \(-0.329763\pi\)
0.509683 + 0.860362i \(0.329763\pi\)
\(618\) −9.92820 −0.399371
\(619\) 4.67949 0.188085 0.0940423 0.995568i \(-0.470021\pi\)
0.0940423 + 0.995568i \(0.470021\pi\)
\(620\) 1.00000 0.0401610
\(621\) 3.00000 0.120386
\(622\) −34.6410 −1.38898
\(623\) 28.0000 1.12180
\(624\) 5.00000 0.200160
\(625\) 1.00000 0.0400000
\(626\) 18.8038 0.751553
\(627\) 17.1962 0.686748
\(628\) −16.5359 −0.659854
\(629\) 17.0718 0.680697
\(630\) −3.46410 −0.138013
\(631\) 17.7128 0.705136 0.352568 0.935786i \(-0.385309\pi\)
0.352568 + 0.935786i \(0.385309\pi\)
\(632\) −6.00000 −0.238667
\(633\) −14.5359 −0.577750
\(634\) −7.85641 −0.312018
\(635\) −11.4641 −0.454939
\(636\) −7.46410 −0.295971
\(637\) −3.00000 −0.118864
\(638\) 17.1962 0.680802
\(639\) −2.00000 −0.0791188
\(640\) −12.1244 −0.479257
\(641\) 6.80385 0.268736 0.134368 0.990932i \(-0.457100\pi\)
0.134368 + 0.990932i \(0.457100\pi\)
\(642\) −30.0000 −1.18401
\(643\) 3.21539 0.126803 0.0634013 0.997988i \(-0.479805\pi\)
0.0634013 + 0.997988i \(0.479805\pi\)
\(644\) −6.00000 −0.236433
\(645\) 4.53590 0.178601
\(646\) 18.0000 0.708201
\(647\) −10.0718 −0.395963 −0.197982 0.980206i \(-0.563439\pi\)
−0.197982 + 0.980206i \(0.563439\pi\)
\(648\) −1.73205 −0.0680414
\(649\) 3.07180 0.120579
\(650\) 1.73205 0.0679366
\(651\) 2.00000 0.0783862
\(652\) 19.4641 0.762273
\(653\) 18.8038 0.735851 0.367926 0.929855i \(-0.380068\pi\)
0.367926 + 0.929855i \(0.380068\pi\)
\(654\) −18.9282 −0.740151
\(655\) −11.0000 −0.429806
\(656\) −5.00000 −0.195217
\(657\) 2.53590 0.0989348
\(658\) −6.00000 −0.233904
\(659\) −48.4974 −1.88919 −0.944596 0.328236i \(-0.893546\pi\)
−0.944596 + 0.328236i \(0.893546\pi\)
\(660\) −5.73205 −0.223120
\(661\) 20.3923 0.793169 0.396584 0.917998i \(-0.370195\pi\)
0.396584 + 0.917998i \(0.370195\pi\)
\(662\) −5.75129 −0.223530
\(663\) −3.46410 −0.134535
\(664\) −20.5359 −0.796947
\(665\) −6.00000 −0.232670
\(666\) 8.53590 0.330759
\(667\) 5.19615 0.201196
\(668\) −18.0000 −0.696441
\(669\) −26.7128 −1.03278
\(670\) −17.3205 −0.669150
\(671\) 31.3205 1.20911
\(672\) 10.3923 0.400892
\(673\) −40.6410 −1.56660 −0.783298 0.621646i \(-0.786464\pi\)
−0.783298 + 0.621646i \(0.786464\pi\)
\(674\) 27.8372 1.07225
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) 5.07180 0.194925 0.0974625 0.995239i \(-0.468927\pi\)
0.0974625 + 0.995239i \(0.468927\pi\)
\(678\) 0 0
\(679\) −13.8564 −0.531760
\(680\) 6.00000 0.230089
\(681\) −22.5167 −0.862840
\(682\) 9.92820 0.380171
\(683\) −34.2679 −1.31123 −0.655613 0.755097i \(-0.727590\pi\)
−0.655613 + 0.755097i \(0.727590\pi\)
\(684\) 3.00000 0.114708
\(685\) 1.92820 0.0736729
\(686\) −34.6410 −1.32260
\(687\) −4.66025 −0.177800
\(688\) −22.6795 −0.864648
\(689\) 7.46410 0.284360
\(690\) −5.19615 −0.197814
\(691\) 6.14359 0.233713 0.116857 0.993149i \(-0.462718\pi\)
0.116857 + 0.993149i \(0.462718\pi\)
\(692\) 1.19615 0.0454709
\(693\) −11.4641 −0.435485
\(694\) 13.9808 0.530702
\(695\) 0.660254 0.0250449
\(696\) −3.00000 −0.113715
\(697\) 3.46410 0.131212
\(698\) −35.3205 −1.33690
\(699\) −1.73205 −0.0655122
\(700\) 2.00000 0.0755929
\(701\) −48.3923 −1.82775 −0.913876 0.405993i \(-0.866926\pi\)
−0.913876 + 0.405993i \(0.866926\pi\)
\(702\) −1.73205 −0.0653720
\(703\) 14.7846 0.557612
\(704\) −5.73205 −0.216035
\(705\) −1.73205 −0.0652328
\(706\) 24.2487 0.912612
\(707\) −27.7128 −1.04225
\(708\) 0.535898 0.0201403
\(709\) 28.6603 1.07636 0.538179 0.842830i \(-0.319112\pi\)
0.538179 + 0.842830i \(0.319112\pi\)
\(710\) 3.46410 0.130005
\(711\) 3.46410 0.129914
\(712\) −24.2487 −0.908759
\(713\) 3.00000 0.112351
\(714\) −12.0000 −0.449089
\(715\) 5.73205 0.214367
\(716\) 8.00000 0.298974
\(717\) 5.32051 0.198698
\(718\) 0.248711 0.00928182
\(719\) −47.5692 −1.77403 −0.887016 0.461738i \(-0.847226\pi\)
−0.887016 + 0.461738i \(0.847226\pi\)
\(720\) 5.00000 0.186339
\(721\) 11.4641 0.426945
\(722\) −17.3205 −0.644603
\(723\) 18.5167 0.688642
\(724\) 12.9282 0.480473
\(725\) −1.73205 −0.0643268
\(726\) −37.8564 −1.40498
\(727\) −49.3205 −1.82920 −0.914598 0.404364i \(-0.867493\pi\)
−0.914598 + 0.404364i \(0.867493\pi\)
\(728\) −3.46410 −0.128388
\(729\) 1.00000 0.0370370
\(730\) −4.39230 −0.162566
\(731\) 15.7128 0.581159
\(732\) 5.46410 0.201959
\(733\) −13.7321 −0.507205 −0.253602 0.967309i \(-0.581615\pi\)
−0.253602 + 0.967309i \(0.581615\pi\)
\(734\) 55.4256 2.04580
\(735\) −3.00000 −0.110657
\(736\) 15.5885 0.574598
\(737\) −57.3205 −2.11143
\(738\) 1.73205 0.0637577
\(739\) −24.7846 −0.911717 −0.455858 0.890052i \(-0.650668\pi\)
−0.455858 + 0.890052i \(0.650668\pi\)
\(740\) −4.92820 −0.181164
\(741\) −3.00000 −0.110208
\(742\) 25.8564 0.949219
\(743\) 9.07180 0.332812 0.166406 0.986057i \(-0.446784\pi\)
0.166406 + 0.986057i \(0.446784\pi\)
\(744\) −1.73205 −0.0635001
\(745\) 17.9282 0.656839
\(746\) 5.32051 0.194798
\(747\) 11.8564 0.433803
\(748\) −19.8564 −0.726022
\(749\) 34.6410 1.26576
\(750\) 1.73205 0.0632456
\(751\) 7.32051 0.267129 0.133565 0.991040i \(-0.457358\pi\)
0.133565 + 0.991040i \(0.457358\pi\)
\(752\) 8.66025 0.315807
\(753\) −16.3923 −0.597369
\(754\) −3.00000 −0.109254
\(755\) −2.39230 −0.0870649
\(756\) −2.00000 −0.0727393
\(757\) −39.0000 −1.41748 −0.708740 0.705470i \(-0.750736\pi\)
−0.708740 + 0.705470i \(0.750736\pi\)
\(758\) 60.3731 2.19285
\(759\) −17.1962 −0.624181
\(760\) 5.19615 0.188484
\(761\) 16.7846 0.608442 0.304221 0.952602i \(-0.401604\pi\)
0.304221 + 0.952602i \(0.401604\pi\)
\(762\) −19.8564 −0.719322
\(763\) 21.8564 0.791255
\(764\) −21.0000 −0.759753
\(765\) −3.46410 −0.125245
\(766\) −35.5692 −1.28517
\(767\) −0.535898 −0.0193502
\(768\) −19.0000 −0.685603
\(769\) 22.3923 0.807487 0.403744 0.914872i \(-0.367709\pi\)
0.403744 + 0.914872i \(0.367709\pi\)
\(770\) 19.8564 0.715575
\(771\) −17.7321 −0.638604
\(772\) 0.124356 0.00447566
\(773\) −4.85641 −0.174673 −0.0873364 0.996179i \(-0.527836\pi\)
−0.0873364 + 0.996179i \(0.527836\pi\)
\(774\) 7.85641 0.282393
\(775\) −1.00000 −0.0359211
\(776\) 12.0000 0.430775
\(777\) −9.85641 −0.353597
\(778\) −5.07180 −0.181833
\(779\) 3.00000 0.107486
\(780\) 1.00000 0.0358057
\(781\) 11.4641 0.410218
\(782\) −18.0000 −0.643679
\(783\) 1.73205 0.0618984
\(784\) 15.0000 0.535714
\(785\) 16.5359 0.590192
\(786\) −19.0526 −0.679582
\(787\) −31.6410 −1.12788 −0.563940 0.825816i \(-0.690715\pi\)
−0.563940 + 0.825816i \(0.690715\pi\)
\(788\) 11.9282 0.424925
\(789\) 20.8564 0.742508
\(790\) −6.00000 −0.213470
\(791\) 0 0
\(792\) 9.92820 0.352783
\(793\) −5.46410 −0.194036
\(794\) −46.8564 −1.66287
\(795\) 7.46410 0.264724
\(796\) 21.1962 0.751278
\(797\) −16.6410 −0.589455 −0.294728 0.955581i \(-0.595229\pi\)
−0.294728 + 0.955581i \(0.595229\pi\)
\(798\) −10.3923 −0.367884
\(799\) −6.00000 −0.212265
\(800\) −5.19615 −0.183712
\(801\) 14.0000 0.494666
\(802\) 60.0000 2.11867
\(803\) −14.5359 −0.512961
\(804\) −10.0000 −0.352673
\(805\) 6.00000 0.211472
\(806\) −1.73205 −0.0610089
\(807\) 18.9282 0.666304
\(808\) 24.0000 0.844317
\(809\) −20.7846 −0.730748 −0.365374 0.930861i \(-0.619059\pi\)
−0.365374 + 0.930861i \(0.619059\pi\)
\(810\) −1.73205 −0.0608581
\(811\) 46.9282 1.64787 0.823936 0.566683i \(-0.191773\pi\)
0.823936 + 0.566683i \(0.191773\pi\)
\(812\) −3.46410 −0.121566
\(813\) −11.4641 −0.402064
\(814\) −48.9282 −1.71493
\(815\) −19.4641 −0.681798
\(816\) 17.3205 0.606339
\(817\) 13.6077 0.476073
\(818\) −45.9282 −1.60584
\(819\) 2.00000 0.0698857
\(820\) −1.00000 −0.0349215
\(821\) −0.784610 −0.0273831 −0.0136915 0.999906i \(-0.504358\pi\)
−0.0136915 + 0.999906i \(0.504358\pi\)
\(822\) 3.33975 0.116487
\(823\) −29.6077 −1.03206 −0.516030 0.856571i \(-0.672591\pi\)
−0.516030 + 0.856571i \(0.672591\pi\)
\(824\) −9.92820 −0.345865
\(825\) 5.73205 0.199564
\(826\) −1.85641 −0.0645926
\(827\) −24.2487 −0.843210 −0.421605 0.906780i \(-0.638533\pi\)
−0.421605 + 0.906780i \(0.638533\pi\)
\(828\) −3.00000 −0.104257
\(829\) −24.5359 −0.852167 −0.426083 0.904684i \(-0.640107\pi\)
−0.426083 + 0.904684i \(0.640107\pi\)
\(830\) −20.5359 −0.712811
\(831\) −16.0718 −0.557524
\(832\) 1.00000 0.0346688
\(833\) −10.3923 −0.360072
\(834\) 1.14359 0.0395994
\(835\) 18.0000 0.622916
\(836\) −17.1962 −0.594741
\(837\) 1.00000 0.0345651
\(838\) −1.48334 −0.0512411
\(839\) 20.9282 0.722522 0.361261 0.932465i \(-0.382346\pi\)
0.361261 + 0.932465i \(0.382346\pi\)
\(840\) −3.46410 −0.119523
\(841\) −26.0000 −0.896552
\(842\) −66.9282 −2.30650
\(843\) 11.7846 0.405884
\(844\) 14.5359 0.500346
\(845\) −1.00000 −0.0344010
\(846\) −3.00000 −0.103142
\(847\) 43.7128 1.50199
\(848\) −37.3205 −1.28159
\(849\) −19.5885 −0.672274
\(850\) 6.00000 0.205798
\(851\) −14.7846 −0.506810
\(852\) 2.00000 0.0685189
\(853\) −19.3397 −0.662180 −0.331090 0.943599i \(-0.607416\pi\)
−0.331090 + 0.943599i \(0.607416\pi\)
\(854\) −18.9282 −0.647710
\(855\) −3.00000 −0.102598
\(856\) −30.0000 −1.02538
\(857\) −14.9282 −0.509938 −0.254969 0.966949i \(-0.582065\pi\)
−0.254969 + 0.966949i \(0.582065\pi\)
\(858\) 9.92820 0.338943
\(859\) −35.1769 −1.20022 −0.600110 0.799917i \(-0.704877\pi\)
−0.600110 + 0.799917i \(0.704877\pi\)
\(860\) −4.53590 −0.154673
\(861\) −2.00000 −0.0681598
\(862\) −11.0718 −0.377107
\(863\) −6.78461 −0.230951 −0.115475 0.993310i \(-0.536839\pi\)
−0.115475 + 0.993310i \(0.536839\pi\)
\(864\) 5.19615 0.176777
\(865\) −1.19615 −0.0406704
\(866\) −37.9808 −1.29064
\(867\) 5.00000 0.169809
\(868\) −2.00000 −0.0678844
\(869\) −19.8564 −0.673582
\(870\) −3.00000 −0.101710
\(871\) 10.0000 0.338837
\(872\) −18.9282 −0.640990
\(873\) −6.92820 −0.234484
\(874\) −15.5885 −0.527287
\(875\) −2.00000 −0.0676123
\(876\) −2.53590 −0.0856801
\(877\) 20.4115 0.689249 0.344624 0.938741i \(-0.388006\pi\)
0.344624 + 0.938741i \(0.388006\pi\)
\(878\) 1.85641 0.0626507
\(879\) 3.46410 0.116841
\(880\) −28.6603 −0.966137
\(881\) 3.33975 0.112519 0.0562595 0.998416i \(-0.482083\pi\)
0.0562595 + 0.998416i \(0.482083\pi\)
\(882\) −5.19615 −0.174964
\(883\) 47.0333 1.58280 0.791399 0.611300i \(-0.209353\pi\)
0.791399 + 0.611300i \(0.209353\pi\)
\(884\) 3.46410 0.116510
\(885\) −0.535898 −0.0180140
\(886\) −49.1769 −1.65213
\(887\) 0.287187 0.00964280 0.00482140 0.999988i \(-0.498465\pi\)
0.00482140 + 0.999988i \(0.498465\pi\)
\(888\) 8.53590 0.286446
\(889\) 22.9282 0.768987
\(890\) −24.2487 −0.812819
\(891\) −5.73205 −0.192031
\(892\) 26.7128 0.894411
\(893\) −5.19615 −0.173883
\(894\) 31.0526 1.03855
\(895\) −8.00000 −0.267411
\(896\) 24.2487 0.810093
\(897\) 3.00000 0.100167
\(898\) −17.3205 −0.577993
\(899\) 1.73205 0.0577671
\(900\) 1.00000 0.0333333
\(901\) 25.8564 0.861402
\(902\) −9.92820 −0.330573
\(903\) −9.07180 −0.301890
\(904\) 0 0
\(905\) −12.9282 −0.429748
\(906\) −4.14359 −0.137662
\(907\) 47.1769 1.56648 0.783242 0.621717i \(-0.213565\pi\)
0.783242 + 0.621717i \(0.213565\pi\)
\(908\) 22.5167 0.747242
\(909\) −13.8564 −0.459588
\(910\) −3.46410 −0.114834
\(911\) 37.1769 1.23173 0.615863 0.787853i \(-0.288807\pi\)
0.615863 + 0.787853i \(0.288807\pi\)
\(912\) 15.0000 0.496700
\(913\) −67.9615 −2.24920
\(914\) −33.7128 −1.11512
\(915\) −5.46410 −0.180638
\(916\) 4.66025 0.153979
\(917\) 22.0000 0.726504
\(918\) −6.00000 −0.198030
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) −5.19615 −0.171312
\(921\) 21.8564 0.720193
\(922\) 39.9615 1.31606
\(923\) −2.00000 −0.0658308
\(924\) 11.4641 0.377141
\(925\) 4.92820 0.162038
\(926\) 5.07180 0.166670
\(927\) 5.73205 0.188265
\(928\) 9.00000 0.295439
\(929\) −24.1051 −0.790863 −0.395432 0.918495i \(-0.629405\pi\)
−0.395432 + 0.918495i \(0.629405\pi\)
\(930\) −1.73205 −0.0567962
\(931\) −9.00000 −0.294963
\(932\) 1.73205 0.0567352
\(933\) 20.0000 0.654771
\(934\) 46.1436 1.50986
\(935\) 19.8564 0.649374
\(936\) −1.73205 −0.0566139
\(937\) −19.8564 −0.648681 −0.324340 0.945940i \(-0.605142\pi\)
−0.324340 + 0.945940i \(0.605142\pi\)
\(938\) 34.6410 1.13107
\(939\) −10.8564 −0.354285
\(940\) 1.73205 0.0564933
\(941\) −35.3205 −1.15142 −0.575708 0.817655i \(-0.695273\pi\)
−0.575708 + 0.817655i \(0.695273\pi\)
\(942\) 28.6410 0.933175
\(943\) −3.00000 −0.0976934
\(944\) 2.67949 0.0872100
\(945\) 2.00000 0.0650600
\(946\) −45.0333 −1.46416
\(947\) −31.0333 −1.00845 −0.504224 0.863573i \(-0.668221\pi\)
−0.504224 + 0.863573i \(0.668221\pi\)
\(948\) −3.46410 −0.112509
\(949\) 2.53590 0.0823187
\(950\) 5.19615 0.168585
\(951\) 4.53590 0.147087
\(952\) −12.0000 −0.388922
\(953\) 9.07180 0.293864 0.146932 0.989147i \(-0.453060\pi\)
0.146932 + 0.989147i \(0.453060\pi\)
\(954\) 12.9282 0.418566
\(955\) 21.0000 0.679544
\(956\) −5.32051 −0.172078
\(957\) −9.92820 −0.320933
\(958\) 37.4256 1.20917
\(959\) −3.85641 −0.124530
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) 8.53590 0.275208
\(963\) 17.3205 0.558146
\(964\) −18.5167 −0.596381
\(965\) −0.124356 −0.00400315
\(966\) 10.3923 0.334367
\(967\) −25.7846 −0.829177 −0.414589 0.910009i \(-0.636074\pi\)
−0.414589 + 0.910009i \(0.636074\pi\)
\(968\) −37.8564 −1.21675
\(969\) −10.3923 −0.333849
\(970\) 12.0000 0.385297
\(971\) −3.21539 −0.103187 −0.0515934 0.998668i \(-0.516430\pi\)
−0.0515934 + 0.998668i \(0.516430\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −1.32051 −0.0423335
\(974\) −3.09103 −0.0990431
\(975\) −1.00000 −0.0320256
\(976\) 27.3205 0.874508
\(977\) 19.4641 0.622712 0.311356 0.950293i \(-0.399217\pi\)
0.311356 + 0.950293i \(0.399217\pi\)
\(978\) −33.7128 −1.07802
\(979\) −80.2487 −2.56476
\(980\) 3.00000 0.0958315
\(981\) 10.9282 0.348911
\(982\) 10.3923 0.331632
\(983\) 44.1051 1.40673 0.703367 0.710826i \(-0.251679\pi\)
0.703367 + 0.710826i \(0.251679\pi\)
\(984\) 1.73205 0.0552158
\(985\) −11.9282 −0.380064
\(986\) −10.3923 −0.330958
\(987\) 3.46410 0.110264
\(988\) 3.00000 0.0954427
\(989\) −13.6077 −0.432700
\(990\) 9.92820 0.315539
\(991\) −26.5167 −0.842329 −0.421165 0.906984i \(-0.638379\pi\)
−0.421165 + 0.906984i \(0.638379\pi\)
\(992\) 5.19615 0.164978
\(993\) 3.32051 0.105373
\(994\) −6.92820 −0.219749
\(995\) −21.1962 −0.671963
\(996\) −11.8564 −0.375685
\(997\) −2.67949 −0.0848604 −0.0424302 0.999099i \(-0.513510\pi\)
−0.0424302 + 0.999099i \(0.513510\pi\)
\(998\) 27.0333 0.855725
\(999\) −4.92820 −0.155921
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 6045.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.o.1.2 2 1.1 even 1 trivial