Properties

Label 1386.2.a.p.1.2
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
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] = [1386,2,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, 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("1386.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.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: \(11.0672657201\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.46410 q^{10} -1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +5.46410 q^{19} +3.46410 q^{20} -1.00000 q^{22} -6.92820 q^{23} +7.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} +5.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +3.46410 q^{35} -4.92820 q^{37} +5.46410 q^{38} +3.46410 q^{40} -3.46410 q^{41} -10.9282 q^{43} -1.00000 q^{44} -6.92820 q^{46} -9.46410 q^{47} +1.00000 q^{49} +7.00000 q^{50} +2.00000 q^{52} +0.928203 q^{53} -3.46410 q^{55} +1.00000 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} +5.46410 q^{62} +1.00000 q^{64} +6.92820 q^{65} +14.9282 q^{67} -3.46410 q^{68} +3.46410 q^{70} +12.0000 q^{71} -0.535898 q^{73} -4.92820 q^{74} +5.46410 q^{76} -1.00000 q^{77} -10.9282 q^{79} +3.46410 q^{80} -3.46410 q^{82} -4.39230 q^{83} -12.0000 q^{85} -10.9282 q^{86} -1.00000 q^{88} +0.928203 q^{89} +2.00000 q^{91} -6.92820 q^{92} -9.46410 q^{94} +18.9282 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{19} - 2 q^{22} + 14 q^{25} + 4 q^{26} + 2 q^{28} + 12 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{37} + 4 q^{38} - 8 q^{43} - 2 q^{44} - 12 q^{47} + 2 q^{49} + 14 q^{50} + 4 q^{52} - 12 q^{53} + 2 q^{56} + 12 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} + 16 q^{67} + 24 q^{71} - 8 q^{73} + 4 q^{74} + 4 q^{76} - 2 q^{77} - 8 q^{79} + 12 q^{83} - 24 q^{85} - 8 q^{86} - 2 q^{88} - 12 q^{89} + 4 q^{91} - 12 q^{94} + 24 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.46410 1.09545
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) 3.46410 0.774597
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) 5.46410 0.886394
\(39\) 0 0
\(40\) 3.46410 0.547723
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −10.9282 −1.66654 −0.833268 0.552870i \(-0.813533\pi\)
−0.833268 + 0.552870i \(0.813533\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.92820 −1.02151
\(47\) −9.46410 −1.38048 −0.690241 0.723580i \(-0.742495\pi\)
−0.690241 + 0.723580i \(0.742495\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.00000 0.989949
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 0.928203 0.127499 0.0637493 0.997966i \(-0.479694\pi\)
0.0637493 + 0.997966i \(0.479694\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 5.46410 0.693942
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.92820 0.859338
\(66\) 0 0
\(67\) 14.9282 1.82377 0.911885 0.410445i \(-0.134627\pi\)
0.911885 + 0.410445i \(0.134627\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) 3.46410 0.414039
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −0.535898 −0.0627222 −0.0313611 0.999508i \(-0.509984\pi\)
−0.0313611 + 0.999508i \(0.509984\pi\)
\(74\) −4.92820 −0.572892
\(75\) 0 0
\(76\) 5.46410 0.626775
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −10.9282 −1.22952 −0.614759 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) 3.46410 0.387298
\(81\) 0 0
\(82\) −3.46410 −0.382546
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −10.9282 −1.17842
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) −9.46410 −0.976148
\(95\) 18.9282 1.94199
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) 19.8564 1.97579 0.987893 0.155136i \(-0.0495815\pi\)
0.987893 + 0.155136i \(0.0495815\pi\)
\(102\) 0 0
\(103\) −13.4641 −1.32666 −0.663329 0.748328i \(-0.730857\pi\)
−0.663329 + 0.748328i \(0.730857\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0.928203 0.0901551
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) 15.8564 1.51877 0.759384 0.650643i \(-0.225500\pi\)
0.759384 + 0.650643i \(0.225500\pi\)
\(110\) −3.46410 −0.330289
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −7.85641 −0.739069 −0.369534 0.929217i \(-0.620483\pi\)
−0.369534 + 0.929217i \(0.620483\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −6.92820 −0.637793
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 5.46410 0.490691
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −10.9282 −0.969721 −0.484861 0.874591i \(-0.661130\pi\)
−0.484861 + 0.874591i \(0.661130\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 6.92820 0.607644
\(131\) 4.39230 0.383757 0.191879 0.981419i \(-0.438542\pi\)
0.191879 + 0.981419i \(0.438542\pi\)
\(132\) 0 0
\(133\) 5.46410 0.473798
\(134\) 14.9282 1.28960
\(135\) 0 0
\(136\) −3.46410 −0.297044
\(137\) −7.85641 −0.671218 −0.335609 0.942001i \(-0.608942\pi\)
−0.335609 + 0.942001i \(0.608942\pi\)
\(138\) 0 0
\(139\) −13.4641 −1.14201 −0.571005 0.820947i \(-0.693446\pi\)
−0.571005 + 0.820947i \(0.693446\pi\)
\(140\) 3.46410 0.292770
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 20.7846 1.72607
\(146\) −0.535898 −0.0443513
\(147\) 0 0
\(148\) −4.92820 −0.405096
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 5.46410 0.443197
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 18.9282 1.52035
\(156\) 0 0
\(157\) −2.39230 −0.190927 −0.0954634 0.995433i \(-0.530433\pi\)
−0.0954634 + 0.995433i \(0.530433\pi\)
\(158\) −10.9282 −0.869401
\(159\) 0 0
\(160\) 3.46410 0.273861
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) −4.39230 −0.340909
\(167\) −18.9282 −1.46471 −0.732354 0.680924i \(-0.761578\pi\)
−0.732354 + 0.680924i \(0.761578\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −12.0000 −0.920358
\(171\) 0 0
\(172\) −10.9282 −0.833268
\(173\) 0.928203 0.0705700 0.0352850 0.999377i \(-0.488766\pi\)
0.0352850 + 0.999377i \(0.488766\pi\)
\(174\) 0 0
\(175\) 7.00000 0.529150
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 0.928203 0.0695718
\(179\) 20.7846 1.55351 0.776757 0.629800i \(-0.216863\pi\)
0.776757 + 0.629800i \(0.216863\pi\)
\(180\) 0 0
\(181\) 6.39230 0.475136 0.237568 0.971371i \(-0.423650\pi\)
0.237568 + 0.971371i \(0.423650\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −6.92820 −0.510754
\(185\) −17.0718 −1.25514
\(186\) 0 0
\(187\) 3.46410 0.253320
\(188\) −9.46410 −0.690241
\(189\) 0 0
\(190\) 18.9282 1.37320
\(191\) −20.7846 −1.50392 −0.751961 0.659208i \(-0.770892\pi\)
−0.751961 + 0.659208i \(0.770892\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 0.392305 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(200\) 7.00000 0.494975
\(201\) 0 0
\(202\) 19.8564 1.39709
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) −13.4641 −0.938088
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −5.46410 −0.377960
\(210\) 0 0
\(211\) 16.7846 1.15550 0.577750 0.816214i \(-0.303931\pi\)
0.577750 + 0.816214i \(0.303931\pi\)
\(212\) 0.928203 0.0637493
\(213\) 0 0
\(214\) 6.92820 0.473602
\(215\) −37.8564 −2.58179
\(216\) 0 0
\(217\) 5.46410 0.370927
\(218\) 15.8564 1.07393
\(219\) 0 0
\(220\) −3.46410 −0.233550
\(221\) −6.92820 −0.466041
\(222\) 0 0
\(223\) −8.39230 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −7.85641 −0.522600
\(227\) 4.39230 0.291528 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(228\) 0 0
\(229\) −21.3205 −1.40890 −0.704449 0.709754i \(-0.748806\pi\)
−0.704449 + 0.709754i \(0.748806\pi\)
\(230\) −24.0000 −1.58251
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) 0 0
\(235\) −32.7846 −2.13863
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 9.60770 0.618886 0.309443 0.950918i \(-0.399857\pi\)
0.309443 + 0.950918i \(0.399857\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 3.46410 0.221313
\(246\) 0 0
\(247\) 10.9282 0.695345
\(248\) 5.46410 0.346971
\(249\) 0 0
\(250\) 6.92820 0.438178
\(251\) −30.9282 −1.95217 −0.976085 0.217387i \(-0.930247\pi\)
−0.976085 + 0.217387i \(0.930247\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) −10.9282 −0.685696
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.9282 −0.806439 −0.403220 0.915103i \(-0.632109\pi\)
−0.403220 + 0.915103i \(0.632109\pi\)
\(258\) 0 0
\(259\) −4.92820 −0.306224
\(260\) 6.92820 0.429669
\(261\) 0 0
\(262\) 4.39230 0.271357
\(263\) 5.07180 0.312740 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(264\) 0 0
\(265\) 3.21539 0.197520
\(266\) 5.46410 0.335026
\(267\) 0 0
\(268\) 14.9282 0.911885
\(269\) −24.2487 −1.47847 −0.739235 0.673448i \(-0.764813\pi\)
−0.739235 + 0.673448i \(0.764813\pi\)
\(270\) 0 0
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) −3.46410 −0.210042
\(273\) 0 0
\(274\) −7.85641 −0.474623
\(275\) −7.00000 −0.422116
\(276\) 0 0
\(277\) −11.8564 −0.712382 −0.356191 0.934413i \(-0.615925\pi\)
−0.356191 + 0.934413i \(0.615925\pi\)
\(278\) −13.4641 −0.807523
\(279\) 0 0
\(280\) 3.46410 0.207020
\(281\) 7.85641 0.468674 0.234337 0.972155i \(-0.424708\pi\)
0.234337 + 0.972155i \(0.424708\pi\)
\(282\) 0 0
\(283\) −13.4641 −0.800358 −0.400179 0.916437i \(-0.631052\pi\)
−0.400179 + 0.916437i \(0.631052\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −3.46410 −0.204479
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 20.7846 1.22051
\(291\) 0 0
\(292\) −0.535898 −0.0313611
\(293\) 24.9282 1.45632 0.728161 0.685407i \(-0.240375\pi\)
0.728161 + 0.685407i \(0.240375\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) −4.92820 −0.286446
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −13.8564 −0.801337
\(300\) 0 0
\(301\) −10.9282 −0.629891
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 5.46410 0.313388
\(305\) 6.92820 0.396708
\(306\) 0 0
\(307\) 10.5359 0.601315 0.300658 0.953732i \(-0.402794\pi\)
0.300658 + 0.953732i \(0.402794\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 18.9282 1.07505
\(311\) −28.3923 −1.60998 −0.804990 0.593288i \(-0.797829\pi\)
−0.804990 + 0.593288i \(0.797829\pi\)
\(312\) 0 0
\(313\) −16.9282 −0.956839 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(314\) −2.39230 −0.135006
\(315\) 0 0
\(316\) −10.9282 −0.614759
\(317\) −12.9282 −0.726120 −0.363060 0.931766i \(-0.618268\pi\)
−0.363060 + 0.931766i \(0.618268\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 3.46410 0.193649
\(321\) 0 0
\(322\) −6.92820 −0.386094
\(323\) −18.9282 −1.05319
\(324\) 0 0
\(325\) 14.0000 0.776580
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −3.46410 −0.191273
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −9.07180 −0.498631 −0.249316 0.968422i \(-0.580206\pi\)
−0.249316 + 0.968422i \(0.580206\pi\)
\(332\) −4.39230 −0.241059
\(333\) 0 0
\(334\) −18.9282 −1.03571
\(335\) 51.7128 2.82537
\(336\) 0 0
\(337\) 20.9282 1.14003 0.570016 0.821634i \(-0.306937\pi\)
0.570016 + 0.821634i \(0.306937\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) −5.46410 −0.295898
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −10.9282 −0.589209
\(345\) 0 0
\(346\) 0.928203 0.0499005
\(347\) 20.7846 1.11578 0.557888 0.829916i \(-0.311612\pi\)
0.557888 + 0.829916i \(0.311612\pi\)
\(348\) 0 0
\(349\) 10.7846 0.577287 0.288643 0.957437i \(-0.406796\pi\)
0.288643 + 0.957437i \(0.406796\pi\)
\(350\) 7.00000 0.374166
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) 41.5692 2.20627
\(356\) 0.928203 0.0491947
\(357\) 0 0
\(358\) 20.7846 1.09850
\(359\) 5.07180 0.267679 0.133840 0.991003i \(-0.457269\pi\)
0.133840 + 0.991003i \(0.457269\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) 6.39230 0.335972
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −1.85641 −0.0971688
\(366\) 0 0
\(367\) 19.3205 1.00852 0.504261 0.863551i \(-0.331765\pi\)
0.504261 + 0.863551i \(0.331765\pi\)
\(368\) −6.92820 −0.361158
\(369\) 0 0
\(370\) −17.0718 −0.887520
\(371\) 0.928203 0.0481899
\(372\) 0 0
\(373\) 29.7128 1.53847 0.769236 0.638965i \(-0.220637\pi\)
0.769236 + 0.638965i \(0.220637\pi\)
\(374\) 3.46410 0.179124
\(375\) 0 0
\(376\) −9.46410 −0.488074
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) 18.9282 0.970996
\(381\) 0 0
\(382\) −20.7846 −1.06343
\(383\) −14.5359 −0.742750 −0.371375 0.928483i \(-0.621114\pi\)
−0.371375 + 0.928483i \(0.621114\pi\)
\(384\) 0 0
\(385\) −3.46410 −0.176547
\(386\) 26.0000 1.32337
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −37.8564 −1.90476
\(396\) 0 0
\(397\) −2.39230 −0.120066 −0.0600332 0.998196i \(-0.519121\pi\)
−0.0600332 + 0.998196i \(0.519121\pi\)
\(398\) 0.392305 0.0196645
\(399\) 0 0
\(400\) 7.00000 0.350000
\(401\) −4.14359 −0.206921 −0.103461 0.994634i \(-0.532992\pi\)
−0.103461 + 0.994634i \(0.532992\pi\)
\(402\) 0 0
\(403\) 10.9282 0.544373
\(404\) 19.8564 0.987893
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 4.92820 0.244282
\(408\) 0 0
\(409\) −9.32051 −0.460869 −0.230435 0.973088i \(-0.574015\pi\)
−0.230435 + 0.973088i \(0.574015\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) −13.4641 −0.663329
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) −15.2154 −0.746894
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −5.46410 −0.267258
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −37.7128 −1.83801 −0.919005 0.394246i \(-0.871006\pi\)
−0.919005 + 0.394246i \(0.871006\pi\)
\(422\) 16.7846 0.817062
\(423\) 0 0
\(424\) 0.928203 0.0450775
\(425\) −24.2487 −1.17624
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) −37.8564 −1.82560
\(431\) 18.9282 0.911739 0.455870 0.890047i \(-0.349328\pi\)
0.455870 + 0.890047i \(0.349328\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 5.46410 0.262285
\(435\) 0 0
\(436\) 15.8564 0.759384
\(437\) −37.8564 −1.81092
\(438\) 0 0
\(439\) −24.7846 −1.18290 −0.591452 0.806340i \(-0.701445\pi\)
−0.591452 + 0.806340i \(0.701445\pi\)
\(440\) −3.46410 −0.165145
\(441\) 0 0
\(442\) −6.92820 −0.329541
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 3.21539 0.152424
\(446\) −8.39230 −0.397387
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 19.8564 0.937082 0.468541 0.883442i \(-0.344780\pi\)
0.468541 + 0.883442i \(0.344780\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) −7.85641 −0.369534
\(453\) 0 0
\(454\) 4.39230 0.206141
\(455\) 6.92820 0.324799
\(456\) 0 0
\(457\) 15.8564 0.741731 0.370866 0.928687i \(-0.379061\pi\)
0.370866 + 0.928687i \(0.379061\pi\)
\(458\) −21.3205 −0.996242
\(459\) 0 0
\(460\) −24.0000 −1.11901
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 7.85641 0.363941
\(467\) 15.7128 0.727102 0.363551 0.931574i \(-0.381564\pi\)
0.363551 + 0.931574i \(0.381564\pi\)
\(468\) 0 0
\(469\) 14.9282 0.689320
\(470\) −32.7846 −1.51224
\(471\) 0 0
\(472\) −6.92820 −0.318896
\(473\) 10.9282 0.502479
\(474\) 0 0
\(475\) 38.2487 1.75497
\(476\) −3.46410 −0.158777
\(477\) 0 0
\(478\) 24.0000 1.09773
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) 0 0
\(481\) −9.85641 −0.449413
\(482\) 9.60770 0.437619
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 6.92820 0.314594
\(486\) 0 0
\(487\) 40.7846 1.84813 0.924064 0.382239i \(-0.124847\pi\)
0.924064 + 0.382239i \(0.124847\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 3.46410 0.156492
\(491\) 1.85641 0.0837785 0.0418892 0.999122i \(-0.486662\pi\)
0.0418892 + 0.999122i \(0.486662\pi\)
\(492\) 0 0
\(493\) −20.7846 −0.936092
\(494\) 10.9282 0.491683
\(495\) 0 0
\(496\) 5.46410 0.245345
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −9.07180 −0.406109 −0.203055 0.979167i \(-0.565087\pi\)
−0.203055 + 0.979167i \(0.565087\pi\)
\(500\) 6.92820 0.309839
\(501\) 0 0
\(502\) −30.9282 −1.38039
\(503\) −18.9282 −0.843967 −0.421983 0.906604i \(-0.638666\pi\)
−0.421983 + 0.906604i \(0.638666\pi\)
\(504\) 0 0
\(505\) 68.7846 3.06087
\(506\) 6.92820 0.307996
\(507\) 0 0
\(508\) −10.9282 −0.484861
\(509\) 41.3205 1.83150 0.915750 0.401749i \(-0.131598\pi\)
0.915750 + 0.401749i \(0.131598\pi\)
\(510\) 0 0
\(511\) −0.535898 −0.0237067
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −12.9282 −0.570239
\(515\) −46.6410 −2.05525
\(516\) 0 0
\(517\) 9.46410 0.416231
\(518\) −4.92820 −0.216533
\(519\) 0 0
\(520\) 6.92820 0.303822
\(521\) −36.9282 −1.61785 −0.808927 0.587909i \(-0.799951\pi\)
−0.808927 + 0.587909i \(0.799951\pi\)
\(522\) 0 0
\(523\) 24.3923 1.06660 0.533301 0.845926i \(-0.320952\pi\)
0.533301 + 0.845926i \(0.320952\pi\)
\(524\) 4.39230 0.191879
\(525\) 0 0
\(526\) 5.07180 0.221141
\(527\) −18.9282 −0.824525
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 3.21539 0.139668
\(531\) 0 0
\(532\) 5.46410 0.236899
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 14.9282 0.644800
\(537\) 0 0
\(538\) −24.2487 −1.04544
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −25.7128 −1.10548 −0.552740 0.833354i \(-0.686418\pi\)
−0.552740 + 0.833354i \(0.686418\pi\)
\(542\) −24.7846 −1.06459
\(543\) 0 0
\(544\) −3.46410 −0.148522
\(545\) 54.9282 2.35287
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −7.85641 −0.335609
\(549\) 0 0
\(550\) −7.00000 −0.298481
\(551\) 32.7846 1.39667
\(552\) 0 0
\(553\) −10.9282 −0.464714
\(554\) −11.8564 −0.503730
\(555\) 0 0
\(556\) −13.4641 −0.571005
\(557\) −21.7128 −0.920001 −0.460001 0.887919i \(-0.652151\pi\)
−0.460001 + 0.887919i \(0.652151\pi\)
\(558\) 0 0
\(559\) −21.8564 −0.924427
\(560\) 3.46410 0.146385
\(561\) 0 0
\(562\) 7.85641 0.331403
\(563\) 18.2487 0.769091 0.384546 0.923106i \(-0.374358\pi\)
0.384546 + 0.923106i \(0.374358\pi\)
\(564\) 0 0
\(565\) −27.2154 −1.14496
\(566\) −13.4641 −0.565938
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 31.8564 1.33549 0.667745 0.744390i \(-0.267260\pi\)
0.667745 + 0.744390i \(0.267260\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −3.46410 −0.144589
\(575\) −48.4974 −2.02248
\(576\) 0 0
\(577\) −30.7846 −1.28158 −0.640790 0.767716i \(-0.721393\pi\)
−0.640790 + 0.767716i \(0.721393\pi\)
\(578\) −5.00000 −0.207973
\(579\) 0 0
\(580\) 20.7846 0.863034
\(581\) −4.39230 −0.182224
\(582\) 0 0
\(583\) −0.928203 −0.0384422
\(584\) −0.535898 −0.0221756
\(585\) 0 0
\(586\) 24.9282 1.02977
\(587\) 20.7846 0.857873 0.428936 0.903335i \(-0.358888\pi\)
0.428936 + 0.903335i \(0.358888\pi\)
\(588\) 0 0
\(589\) 29.8564 1.23021
\(590\) −24.0000 −0.988064
\(591\) 0 0
\(592\) −4.92820 −0.202548
\(593\) 20.5359 0.843308 0.421654 0.906757i \(-0.361450\pi\)
0.421654 + 0.906757i \(0.361450\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −13.8564 −0.566631
\(599\) −1.85641 −0.0758507 −0.0379254 0.999281i \(-0.512075\pi\)
−0.0379254 + 0.999281i \(0.512075\pi\)
\(600\) 0 0
\(601\) 18.3923 0.750238 0.375119 0.926977i \(-0.377602\pi\)
0.375119 + 0.926977i \(0.377602\pi\)
\(602\) −10.9282 −0.445400
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) −19.7128 −0.800118 −0.400059 0.916489i \(-0.631010\pi\)
−0.400059 + 0.916489i \(0.631010\pi\)
\(608\) 5.46410 0.221599
\(609\) 0 0
\(610\) 6.92820 0.280515
\(611\) −18.9282 −0.765753
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 10.5359 0.425194
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 19.8564 0.799389 0.399694 0.916648i \(-0.369116\pi\)
0.399694 + 0.916648i \(0.369116\pi\)
\(618\) 0 0
\(619\) −31.7128 −1.27465 −0.637323 0.770597i \(-0.719958\pi\)
−0.637323 + 0.770597i \(0.719958\pi\)
\(620\) 18.9282 0.760175
\(621\) 0 0
\(622\) −28.3923 −1.13843
\(623\) 0.928203 0.0371877
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −16.9282 −0.676587
\(627\) 0 0
\(628\) −2.39230 −0.0954634
\(629\) 17.0718 0.680697
\(630\) 0 0
\(631\) −0.784610 −0.0312348 −0.0156174 0.999878i \(-0.504971\pi\)
−0.0156174 + 0.999878i \(0.504971\pi\)
\(632\) −10.9282 −0.434701
\(633\) 0 0
\(634\) −12.9282 −0.513445
\(635\) −37.8564 −1.50229
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 3.46410 0.136931
\(641\) −31.8564 −1.25825 −0.629126 0.777303i \(-0.716587\pi\)
−0.629126 + 0.777303i \(0.716587\pi\)
\(642\) 0 0
\(643\) 14.9282 0.588711 0.294355 0.955696i \(-0.404895\pi\)
0.294355 + 0.955696i \(0.404895\pi\)
\(644\) −6.92820 −0.273009
\(645\) 0 0
\(646\) −18.9282 −0.744720
\(647\) −0.679492 −0.0267136 −0.0133568 0.999911i \(-0.504252\pi\)
−0.0133568 + 0.999911i \(0.504252\pi\)
\(648\) 0 0
\(649\) 6.92820 0.271956
\(650\) 14.0000 0.549125
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 33.7128 1.31928 0.659642 0.751580i \(-0.270708\pi\)
0.659642 + 0.751580i \(0.270708\pi\)
\(654\) 0 0
\(655\) 15.2154 0.594514
\(656\) −3.46410 −0.135250
\(657\) 0 0
\(658\) −9.46410 −0.368949
\(659\) 17.0718 0.665023 0.332511 0.943099i \(-0.392104\pi\)
0.332511 + 0.943099i \(0.392104\pi\)
\(660\) 0 0
\(661\) 44.2487 1.72108 0.860538 0.509387i \(-0.170128\pi\)
0.860538 + 0.509387i \(0.170128\pi\)
\(662\) −9.07180 −0.352585
\(663\) 0 0
\(664\) −4.39230 −0.170454
\(665\) 18.9282 0.734004
\(666\) 0 0
\(667\) −41.5692 −1.60957
\(668\) −18.9282 −0.732354
\(669\) 0 0
\(670\) 51.7128 1.99784
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 20.9282 0.806723 0.403361 0.915041i \(-0.367842\pi\)
0.403361 + 0.915041i \(0.367842\pi\)
\(674\) 20.9282 0.806124
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −2.78461 −0.107021 −0.0535106 0.998567i \(-0.517041\pi\)
−0.0535106 + 0.998567i \(0.517041\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −12.0000 −0.460179
\(681\) 0 0
\(682\) −5.46410 −0.209231
\(683\) −3.21539 −0.123033 −0.0615167 0.998106i \(-0.519594\pi\)
−0.0615167 + 0.998106i \(0.519594\pi\)
\(684\) 0 0
\(685\) −27.2154 −1.03985
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −10.9282 −0.416634
\(689\) 1.85641 0.0707235
\(690\) 0 0
\(691\) −33.0718 −1.25811 −0.629055 0.777361i \(-0.716558\pi\)
−0.629055 + 0.777361i \(0.716558\pi\)
\(692\) 0.928203 0.0352850
\(693\) 0 0
\(694\) 20.7846 0.788973
\(695\) −46.6410 −1.76919
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 10.7846 0.408203
\(699\) 0 0
\(700\) 7.00000 0.264575
\(701\) −45.7128 −1.72655 −0.863275 0.504735i \(-0.831590\pi\)
−0.863275 + 0.504735i \(0.831590\pi\)
\(702\) 0 0
\(703\) −26.9282 −1.01562
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −12.9282 −0.486559
\(707\) 19.8564 0.746777
\(708\) 0 0
\(709\) 46.7846 1.75703 0.878516 0.477712i \(-0.158534\pi\)
0.878516 + 0.477712i \(0.158534\pi\)
\(710\) 41.5692 1.56007
\(711\) 0 0
\(712\) 0.928203 0.0347859
\(713\) −37.8564 −1.41773
\(714\) 0 0
\(715\) −6.92820 −0.259100
\(716\) 20.7846 0.776757
\(717\) 0 0
\(718\) 5.07180 0.189278
\(719\) 37.1769 1.38646 0.693232 0.720714i \(-0.256186\pi\)
0.693232 + 0.720714i \(0.256186\pi\)
\(720\) 0 0
\(721\) −13.4641 −0.501429
\(722\) 10.8564 0.404034
\(723\) 0 0
\(724\) 6.39230 0.237568
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) 33.1769 1.23046 0.615232 0.788346i \(-0.289062\pi\)
0.615232 + 0.788346i \(0.289062\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) −1.85641 −0.0687087
\(731\) 37.8564 1.40017
\(732\) 0 0
\(733\) 12.1436 0.448534 0.224267 0.974528i \(-0.428001\pi\)
0.224267 + 0.974528i \(0.428001\pi\)
\(734\) 19.3205 0.713133
\(735\) 0 0
\(736\) −6.92820 −0.255377
\(737\) −14.9282 −0.549887
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) −17.0718 −0.627572
\(741\) 0 0
\(742\) 0.928203 0.0340754
\(743\) 41.5692 1.52503 0.762513 0.646972i \(-0.223965\pi\)
0.762513 + 0.646972i \(0.223965\pi\)
\(744\) 0 0
\(745\) 20.7846 0.761489
\(746\) 29.7128 1.08786
\(747\) 0 0
\(748\) 3.46410 0.126660
\(749\) 6.92820 0.253151
\(750\) 0 0
\(751\) −24.7846 −0.904403 −0.452202 0.891916i \(-0.649361\pi\)
−0.452202 + 0.891916i \(0.649361\pi\)
\(752\) −9.46410 −0.345120
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −55.4256 −2.01715
\(756\) 0 0
\(757\) 22.7846 0.828121 0.414060 0.910249i \(-0.364110\pi\)
0.414060 + 0.910249i \(0.364110\pi\)
\(758\) −17.8564 −0.648574
\(759\) 0 0
\(760\) 18.9282 0.686598
\(761\) 25.6077 0.928278 0.464139 0.885762i \(-0.346364\pi\)
0.464139 + 0.885762i \(0.346364\pi\)
\(762\) 0 0
\(763\) 15.8564 0.574040
\(764\) −20.7846 −0.751961
\(765\) 0 0
\(766\) −14.5359 −0.525203
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) 28.5359 1.02903 0.514515 0.857481i \(-0.327972\pi\)
0.514515 + 0.857481i \(0.327972\pi\)
\(770\) −3.46410 −0.124838
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) 32.5359 1.17023 0.585117 0.810949i \(-0.301048\pi\)
0.585117 + 0.810949i \(0.301048\pi\)
\(774\) 0 0
\(775\) 38.2487 1.37393
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −18.9282 −0.678173
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −8.28719 −0.295782
\(786\) 0 0
\(787\) 33.1769 1.18263 0.591315 0.806441i \(-0.298609\pi\)
0.591315 + 0.806441i \(0.298609\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −37.8564 −1.34687
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −2.39230 −0.0848997
\(795\) 0 0
\(796\) 0.392305 0.0139049
\(797\) 32.5359 1.15248 0.576240 0.817280i \(-0.304519\pi\)
0.576240 + 0.817280i \(0.304519\pi\)
\(798\) 0 0
\(799\) 32.7846 1.15984
\(800\) 7.00000 0.247487
\(801\) 0 0
\(802\) −4.14359 −0.146315
\(803\) 0.535898 0.0189114
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) 10.9282 0.384930
\(807\) 0 0
\(808\) 19.8564 0.698546
\(809\) −11.0718 −0.389264 −0.194632 0.980876i \(-0.562351\pi\)
−0.194632 + 0.980876i \(0.562351\pi\)
\(810\) 0 0
\(811\) −17.1769 −0.603163 −0.301582 0.953440i \(-0.597515\pi\)
−0.301582 + 0.953440i \(0.597515\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 4.92820 0.172733
\(815\) −13.8564 −0.485369
\(816\) 0 0
\(817\) −59.7128 −2.08909
\(818\) −9.32051 −0.325884
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 40.7846 1.42166 0.710831 0.703363i \(-0.248319\pi\)
0.710831 + 0.703363i \(0.248319\pi\)
\(824\) −13.4641 −0.469044
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 44.2487 1.53682 0.768411 0.639957i \(-0.221048\pi\)
0.768411 + 0.639957i \(0.221048\pi\)
\(830\) −15.2154 −0.528134
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) −65.5692 −2.26912
\(836\) −5.46410 −0.188980
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 33.4641 1.15531 0.577655 0.816281i \(-0.303968\pi\)
0.577655 + 0.816281i \(0.303968\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −37.7128 −1.29967
\(843\) 0 0
\(844\) 16.7846 0.577750
\(845\) −31.1769 −1.07252
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0.928203 0.0318746
\(849\) 0 0
\(850\) −24.2487 −0.831724
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) −49.7128 −1.70213 −0.851067 0.525057i \(-0.824044\pi\)
−0.851067 + 0.525057i \(0.824044\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 6.92820 0.236801
\(857\) 5.32051 0.181745 0.0908725 0.995863i \(-0.471034\pi\)
0.0908725 + 0.995863i \(0.471034\pi\)
\(858\) 0 0
\(859\) 52.7846 1.80099 0.900494 0.434869i \(-0.143205\pi\)
0.900494 + 0.434869i \(0.143205\pi\)
\(860\) −37.8564 −1.29089
\(861\) 0 0
\(862\) 18.9282 0.644697
\(863\) −44.7846 −1.52449 −0.762243 0.647291i \(-0.775902\pi\)
−0.762243 + 0.647291i \(0.775902\pi\)
\(864\) 0 0
\(865\) 3.21539 0.109327
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 5.46410 0.185464
\(869\) 10.9282 0.370714
\(870\) 0 0
\(871\) 29.8564 1.01165
\(872\) 15.8564 0.536966
\(873\) 0 0
\(874\) −37.8564 −1.28051
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −24.7846 −0.836440
\(879\) 0 0
\(880\) −3.46410 −0.116775
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) −41.8564 −1.40858 −0.704290 0.709912i \(-0.748735\pi\)
−0.704290 + 0.709912i \(0.748735\pi\)
\(884\) −6.92820 −0.233021
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) −10.1436 −0.340589 −0.170294 0.985393i \(-0.554472\pi\)
−0.170294 + 0.985393i \(0.554472\pi\)
\(888\) 0 0
\(889\) −10.9282 −0.366520
\(890\) 3.21539 0.107780
\(891\) 0 0
\(892\) −8.39230 −0.280995
\(893\) −51.7128 −1.73050
\(894\) 0 0
\(895\) 72.0000 2.40669
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 19.8564 0.662617
\(899\) 32.7846 1.09343
\(900\) 0 0
\(901\) −3.21539 −0.107120
\(902\) 3.46410 0.115342
\(903\) 0 0
\(904\) −7.85641 −0.261300
\(905\) 22.1436 0.736078
\(906\) 0 0
\(907\) 14.9282 0.495683 0.247841 0.968801i \(-0.420279\pi\)
0.247841 + 0.968801i \(0.420279\pi\)
\(908\) 4.39230 0.145764
\(909\) 0 0
\(910\) 6.92820 0.229668
\(911\) 48.4974 1.60679 0.803396 0.595446i \(-0.203024\pi\)
0.803396 + 0.595446i \(0.203024\pi\)
\(912\) 0 0
\(913\) 4.39230 0.145364
\(914\) 15.8564 0.524483
\(915\) 0 0
\(916\) −21.3205 −0.704449
\(917\) 4.39230 0.145047
\(918\) 0 0
\(919\) 21.8564 0.720976 0.360488 0.932764i \(-0.382610\pi\)
0.360488 + 0.932764i \(0.382610\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −34.4974 −1.13427
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −40.6410 −1.33339 −0.666694 0.745331i \(-0.732291\pi\)
−0.666694 + 0.745331i \(0.732291\pi\)
\(930\) 0 0
\(931\) 5.46410 0.179079
\(932\) 7.85641 0.257345
\(933\) 0 0
\(934\) 15.7128 0.514139
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −14.3923 −0.470176 −0.235088 0.971974i \(-0.575538\pi\)
−0.235088 + 0.971974i \(0.575538\pi\)
\(938\) 14.9282 0.487423
\(939\) 0 0
\(940\) −32.7846 −1.06932
\(941\) 11.0718 0.360930 0.180465 0.983581i \(-0.442240\pi\)
0.180465 + 0.983581i \(0.442240\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) −6.92820 −0.225494
\(945\) 0 0
\(946\) 10.9282 0.355307
\(947\) −49.8564 −1.62012 −0.810058 0.586350i \(-0.800564\pi\)
−0.810058 + 0.586350i \(0.800564\pi\)
\(948\) 0 0
\(949\) −1.07180 −0.0347920
\(950\) 38.2487 1.24095
\(951\) 0 0
\(952\) −3.46410 −0.112272
\(953\) 26.7846 0.867639 0.433819 0.901000i \(-0.357166\pi\)
0.433819 + 0.901000i \(0.357166\pi\)
\(954\) 0 0
\(955\) −72.0000 −2.32987
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 13.8564 0.447680
\(959\) −7.85641 −0.253697
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) −9.85641 −0.317783
\(963\) 0 0
\(964\) 9.60770 0.309443
\(965\) 90.0666 2.89935
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 6.92820 0.222451
\(971\) 15.7128 0.504248 0.252124 0.967695i \(-0.418871\pi\)
0.252124 + 0.967695i \(0.418871\pi\)
\(972\) 0 0
\(973\) −13.4641 −0.431639
\(974\) 40.7846 1.30682
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 47.5692 1.52187 0.760937 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(978\) 0 0
\(979\) −0.928203 −0.0296655
\(980\) 3.46410 0.110657
\(981\) 0 0
\(982\) 1.85641 0.0592403
\(983\) −23.3205 −0.743809 −0.371904 0.928271i \(-0.621295\pi\)
−0.371904 + 0.928271i \(0.621295\pi\)
\(984\) 0 0
\(985\) −62.3538 −1.98676
\(986\) −20.7846 −0.661917
\(987\) 0 0
\(988\) 10.9282 0.347672
\(989\) 75.7128 2.40753
\(990\) 0 0
\(991\) −2.14359 −0.0680935 −0.0340467 0.999420i \(-0.510840\pi\)
−0.0340467 + 0.999420i \(0.510840\pi\)
\(992\) 5.46410 0.173485
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) 1.35898 0.0430827
\(996\) 0 0
\(997\) 24.6410 0.780389 0.390194 0.920732i \(-0.372408\pi\)
0.390194 + 0.920732i \(0.372408\pi\)
\(998\) −9.07180 −0.287163
\(999\) 0 0
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 1386.2.a.p.1.2 2
3.2 odd 2 462.2.a.h.1.1 2
7.6 odd 2 9702.2.a.dd.1.1 2
12.11 even 2 3696.2.a.bc.1.1 2
21.20 even 2 3234.2.a.x.1.2 2
33.32 even 2 5082.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.h.1.1 2 3.2 odd 2
1386.2.a.p.1.2 2 1.1 even 1 trivial
3234.2.a.x.1.2 2 21.20 even 2
3696.2.a.bc.1.1 2 12.11 even 2
5082.2.a.bu.1.1 2 33.32 even 2
9702.2.a.dd.1.1 2 7.6 odd 2