Properties

Label 1672.2.a.i.1.2
Level $1672$
Weight $2$
Character 1672.1
Self dual yes
Analytic conductor $13.351$
Analytic rank $1$
Dimension $6$
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] = [1672,2,Mod(1,1672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1672, 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("1672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1672.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: \(13.3509872180\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.106392688.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.25613\) of defining polynomial
Character \(\chi\) \(=\) 1672.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} +O(q^{10})\) \(q-2.25613 q^{3} -4.20604 q^{5} +3.20604 q^{7} +2.09011 q^{9} -1.00000 q^{11} +0.723884 q^{13} +9.48935 q^{15} +0.655737 q^{17} +1.00000 q^{19} -7.23323 q^{21} -2.68543 q^{23} +12.6907 q^{25} +2.05282 q^{27} -2.64351 q^{29} +9.48935 q^{31} +2.25613 q^{33} -13.4847 q^{35} +4.20992 q^{37} -1.63317 q^{39} +5.04141 q^{41} -10.7480 q^{43} -8.79110 q^{45} -4.84428 q^{47} +3.27866 q^{49} -1.47943 q^{51} +2.80135 q^{53} +4.20604 q^{55} -2.25613 q^{57} -8.16817 q^{59} +1.66797 q^{61} +6.70098 q^{63} -3.04468 q^{65} -1.57845 q^{67} +6.05867 q^{69} -10.3445 q^{71} -3.35322 q^{73} -28.6319 q^{75} -3.20604 q^{77} -17.4366 q^{79} -10.9018 q^{81} +4.13518 q^{83} -2.75805 q^{85} +5.96409 q^{87} -15.8240 q^{89} +2.32080 q^{91} -21.4092 q^{93} -4.20604 q^{95} +5.57301 q^{97} -2.09011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.25613 −1.30258 −0.651288 0.758831i \(-0.725771\pi\)
−0.651288 + 0.758831i \(0.725771\pi\)
\(4\) 0 0
\(5\) −4.20604 −1.88100 −0.940498 0.339799i \(-0.889641\pi\)
−0.940498 + 0.339799i \(0.889641\pi\)
\(6\) 0 0
\(7\) 3.20604 1.21177 0.605884 0.795553i \(-0.292820\pi\)
0.605884 + 0.795553i \(0.292820\pi\)
\(8\) 0 0
\(9\) 2.09011 0.696705
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.723884 0.200769 0.100385 0.994949i \(-0.467993\pi\)
0.100385 + 0.994949i \(0.467993\pi\)
\(14\) 0 0
\(15\) 9.48935 2.45014
\(16\) 0 0
\(17\) 0.655737 0.159040 0.0795198 0.996833i \(-0.474661\pi\)
0.0795198 + 0.996833i \(0.474661\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −7.23323 −1.57842
\(22\) 0 0
\(23\) −2.68543 −0.559950 −0.279975 0.960007i \(-0.590326\pi\)
−0.279975 + 0.960007i \(0.590326\pi\)
\(24\) 0 0
\(25\) 12.6907 2.53815
\(26\) 0 0
\(27\) 2.05282 0.395065
\(28\) 0 0
\(29\) −2.64351 −0.490887 −0.245443 0.969411i \(-0.578934\pi\)
−0.245443 + 0.969411i \(0.578934\pi\)
\(30\) 0 0
\(31\) 9.48935 1.70434 0.852169 0.523266i \(-0.175287\pi\)
0.852169 + 0.523266i \(0.175287\pi\)
\(32\) 0 0
\(33\) 2.25613 0.392742
\(34\) 0 0
\(35\) −13.4847 −2.27933
\(36\) 0 0
\(37\) 4.20992 0.692107 0.346053 0.938215i \(-0.387522\pi\)
0.346053 + 0.938215i \(0.387522\pi\)
\(38\) 0 0
\(39\) −1.63317 −0.261517
\(40\) 0 0
\(41\) 5.04141 0.787336 0.393668 0.919253i \(-0.371206\pi\)
0.393668 + 0.919253i \(0.371206\pi\)
\(42\) 0 0
\(43\) −10.7480 −1.63905 −0.819526 0.573042i \(-0.805763\pi\)
−0.819526 + 0.573042i \(0.805763\pi\)
\(44\) 0 0
\(45\) −8.79110 −1.31050
\(46\) 0 0
\(47\) −4.84428 −0.706611 −0.353306 0.935508i \(-0.614942\pi\)
−0.353306 + 0.935508i \(0.614942\pi\)
\(48\) 0 0
\(49\) 3.27866 0.468380
\(50\) 0 0
\(51\) −1.47943 −0.207161
\(52\) 0 0
\(53\) 2.80135 0.384795 0.192397 0.981317i \(-0.438374\pi\)
0.192397 + 0.981317i \(0.438374\pi\)
\(54\) 0 0
\(55\) 4.20604 0.567142
\(56\) 0 0
\(57\) −2.25613 −0.298831
\(58\) 0 0
\(59\) −8.16817 −1.06341 −0.531703 0.846931i \(-0.678448\pi\)
−0.531703 + 0.846931i \(0.678448\pi\)
\(60\) 0 0
\(61\) 1.66797 0.213562 0.106781 0.994283i \(-0.465946\pi\)
0.106781 + 0.994283i \(0.465946\pi\)
\(62\) 0 0
\(63\) 6.70098 0.844244
\(64\) 0 0
\(65\) −3.04468 −0.377646
\(66\) 0 0
\(67\) −1.57845 −0.192839 −0.0964193 0.995341i \(-0.530739\pi\)
−0.0964193 + 0.995341i \(0.530739\pi\)
\(68\) 0 0
\(69\) 6.05867 0.729378
\(70\) 0 0
\(71\) −10.3445 −1.22766 −0.613832 0.789436i \(-0.710373\pi\)
−0.613832 + 0.789436i \(0.710373\pi\)
\(72\) 0 0
\(73\) −3.35322 −0.392465 −0.196232 0.980557i \(-0.562871\pi\)
−0.196232 + 0.980557i \(0.562871\pi\)
\(74\) 0 0
\(75\) −28.6319 −3.30613
\(76\) 0 0
\(77\) −3.20604 −0.365362
\(78\) 0 0
\(79\) −17.4366 −1.96177 −0.980884 0.194591i \(-0.937662\pi\)
−0.980884 + 0.194591i \(0.937662\pi\)
\(80\) 0 0
\(81\) −10.9018 −1.21131
\(82\) 0 0
\(83\) 4.13518 0.453895 0.226948 0.973907i \(-0.427125\pi\)
0.226948 + 0.973907i \(0.427125\pi\)
\(84\) 0 0
\(85\) −2.75805 −0.299153
\(86\) 0 0
\(87\) 5.96409 0.639417
\(88\) 0 0
\(89\) −15.8240 −1.67734 −0.838668 0.544643i \(-0.816665\pi\)
−0.838668 + 0.544643i \(0.816665\pi\)
\(90\) 0 0
\(91\) 2.32080 0.243286
\(92\) 0 0
\(93\) −21.4092 −2.22003
\(94\) 0 0
\(95\) −4.20604 −0.431530
\(96\) 0 0
\(97\) 5.57301 0.565853 0.282926 0.959142i \(-0.408695\pi\)
0.282926 + 0.959142i \(0.408695\pi\)
\(98\) 0 0
\(99\) −2.09011 −0.210064
\(100\) 0 0
\(101\) 7.98882 0.794917 0.397458 0.917620i \(-0.369892\pi\)
0.397458 + 0.917620i \(0.369892\pi\)
\(102\) 0 0
\(103\) −14.7152 −1.44993 −0.724964 0.688787i \(-0.758144\pi\)
−0.724964 + 0.688787i \(0.758144\pi\)
\(104\) 0 0
\(105\) 30.4232 2.96900
\(106\) 0 0
\(107\) −1.11242 −0.107542 −0.0537710 0.998553i \(-0.517124\pi\)
−0.0537710 + 0.998553i \(0.517124\pi\)
\(108\) 0 0
\(109\) −8.03175 −0.769302 −0.384651 0.923062i \(-0.625678\pi\)
−0.384651 + 0.923062i \(0.625678\pi\)
\(110\) 0 0
\(111\) −9.49812 −0.901522
\(112\) 0 0
\(113\) 10.1703 0.956742 0.478371 0.878158i \(-0.341227\pi\)
0.478371 + 0.878158i \(0.341227\pi\)
\(114\) 0 0
\(115\) 11.2950 1.05326
\(116\) 0 0
\(117\) 1.51300 0.139877
\(118\) 0 0
\(119\) 2.10232 0.192719
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −11.3741 −1.02556
\(124\) 0 0
\(125\) −32.3475 −2.89325
\(126\) 0 0
\(127\) −11.6874 −1.03709 −0.518544 0.855051i \(-0.673526\pi\)
−0.518544 + 0.855051i \(0.673526\pi\)
\(128\) 0 0
\(129\) 24.2488 2.13499
\(130\) 0 0
\(131\) 15.2777 1.33482 0.667409 0.744691i \(-0.267403\pi\)
0.667409 + 0.744691i \(0.267403\pi\)
\(132\) 0 0
\(133\) 3.20604 0.277999
\(134\) 0 0
\(135\) −8.63422 −0.743115
\(136\) 0 0
\(137\) −12.8730 −1.09982 −0.549909 0.835225i \(-0.685338\pi\)
−0.549909 + 0.835225i \(0.685338\pi\)
\(138\) 0 0
\(139\) −13.1802 −1.11793 −0.558965 0.829191i \(-0.688801\pi\)
−0.558965 + 0.829191i \(0.688801\pi\)
\(140\) 0 0
\(141\) 10.9293 0.920415
\(142\) 0 0
\(143\) −0.723884 −0.0605342
\(144\) 0 0
\(145\) 11.1187 0.923356
\(146\) 0 0
\(147\) −7.39708 −0.610101
\(148\) 0 0
\(149\) 16.5422 1.35519 0.677594 0.735436i \(-0.263023\pi\)
0.677594 + 0.735436i \(0.263023\pi\)
\(150\) 0 0
\(151\) 16.0851 1.30899 0.654495 0.756067i \(-0.272881\pi\)
0.654495 + 0.756067i \(0.272881\pi\)
\(152\) 0 0
\(153\) 1.37057 0.110804
\(154\) 0 0
\(155\) −39.9126 −3.20585
\(156\) 0 0
\(157\) 17.3684 1.38615 0.693074 0.720866i \(-0.256256\pi\)
0.693074 + 0.720866i \(0.256256\pi\)
\(158\) 0 0
\(159\) −6.32020 −0.501224
\(160\) 0 0
\(161\) −8.60958 −0.678530
\(162\) 0 0
\(163\) −18.3725 −1.43904 −0.719521 0.694471i \(-0.755638\pi\)
−0.719521 + 0.694471i \(0.755638\pi\)
\(164\) 0 0
\(165\) −9.48935 −0.738745
\(166\) 0 0
\(167\) −2.62676 −0.203265 −0.101632 0.994822i \(-0.532407\pi\)
−0.101632 + 0.994822i \(0.532407\pi\)
\(168\) 0 0
\(169\) −12.4760 −0.959692
\(170\) 0 0
\(171\) 2.09011 0.159835
\(172\) 0 0
\(173\) −6.23475 −0.474020 −0.237010 0.971507i \(-0.576167\pi\)
−0.237010 + 0.971507i \(0.576167\pi\)
\(174\) 0 0
\(175\) 40.6869 3.07564
\(176\) 0 0
\(177\) 18.4284 1.38517
\(178\) 0 0
\(179\) 4.29357 0.320917 0.160458 0.987043i \(-0.448703\pi\)
0.160458 + 0.987043i \(0.448703\pi\)
\(180\) 0 0
\(181\) 4.52254 0.336158 0.168079 0.985774i \(-0.446244\pi\)
0.168079 + 0.985774i \(0.446244\pi\)
\(182\) 0 0
\(183\) −3.76316 −0.278181
\(184\) 0 0
\(185\) −17.7071 −1.30185
\(186\) 0 0
\(187\) −0.655737 −0.0479523
\(188\) 0 0
\(189\) 6.58140 0.478727
\(190\) 0 0
\(191\) 25.0370 1.81162 0.905808 0.423689i \(-0.139265\pi\)
0.905808 + 0.423689i \(0.139265\pi\)
\(192\) 0 0
\(193\) 4.60176 0.331242 0.165621 0.986189i \(-0.447037\pi\)
0.165621 + 0.986189i \(0.447037\pi\)
\(194\) 0 0
\(195\) 6.86919 0.491913
\(196\) 0 0
\(197\) 5.51765 0.393117 0.196558 0.980492i \(-0.437024\pi\)
0.196558 + 0.980492i \(0.437024\pi\)
\(198\) 0 0
\(199\) −9.84534 −0.697917 −0.348959 0.937138i \(-0.613465\pi\)
−0.348959 + 0.937138i \(0.613465\pi\)
\(200\) 0 0
\(201\) 3.56119 0.251187
\(202\) 0 0
\(203\) −8.47517 −0.594841
\(204\) 0 0
\(205\) −21.2043 −1.48098
\(206\) 0 0
\(207\) −5.61285 −0.390120
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −20.3536 −1.40120 −0.700601 0.713553i \(-0.747085\pi\)
−0.700601 + 0.713553i \(0.747085\pi\)
\(212\) 0 0
\(213\) 23.3385 1.59913
\(214\) 0 0
\(215\) 45.2064 3.08305
\(216\) 0 0
\(217\) 30.4232 2.06526
\(218\) 0 0
\(219\) 7.56530 0.511215
\(220\) 0 0
\(221\) 0.474677 0.0319303
\(222\) 0 0
\(223\) 5.69489 0.381358 0.190679 0.981652i \(-0.438931\pi\)
0.190679 + 0.981652i \(0.438931\pi\)
\(224\) 0 0
\(225\) 26.5251 1.76834
\(226\) 0 0
\(227\) −16.0945 −1.06823 −0.534116 0.845411i \(-0.679356\pi\)
−0.534116 + 0.845411i \(0.679356\pi\)
\(228\) 0 0
\(229\) 10.1886 0.673280 0.336640 0.941633i \(-0.390709\pi\)
0.336640 + 0.941633i \(0.390709\pi\)
\(230\) 0 0
\(231\) 7.23323 0.475911
\(232\) 0 0
\(233\) 16.5112 1.08168 0.540841 0.841125i \(-0.318106\pi\)
0.540841 + 0.841125i \(0.318106\pi\)
\(234\) 0 0
\(235\) 20.3752 1.32913
\(236\) 0 0
\(237\) 39.3392 2.55535
\(238\) 0 0
\(239\) −28.1261 −1.81933 −0.909664 0.415345i \(-0.863661\pi\)
−0.909664 + 0.415345i \(0.863661\pi\)
\(240\) 0 0
\(241\) 18.8092 1.21160 0.605802 0.795615i \(-0.292852\pi\)
0.605802 + 0.795615i \(0.292852\pi\)
\(242\) 0 0
\(243\) 18.4373 1.18276
\(244\) 0 0
\(245\) −13.7902 −0.881021
\(246\) 0 0
\(247\) 0.723884 0.0460596
\(248\) 0 0
\(249\) −9.32950 −0.591233
\(250\) 0 0
\(251\) 19.2974 1.21804 0.609020 0.793155i \(-0.291563\pi\)
0.609020 + 0.793155i \(0.291563\pi\)
\(252\) 0 0
\(253\) 2.68543 0.168831
\(254\) 0 0
\(255\) 6.22252 0.389669
\(256\) 0 0
\(257\) −21.9480 −1.36908 −0.684540 0.728976i \(-0.739997\pi\)
−0.684540 + 0.728976i \(0.739997\pi\)
\(258\) 0 0
\(259\) 13.4972 0.838672
\(260\) 0 0
\(261\) −5.52523 −0.342003
\(262\) 0 0
\(263\) −9.31030 −0.574098 −0.287049 0.957916i \(-0.592674\pi\)
−0.287049 + 0.957916i \(0.592674\pi\)
\(264\) 0 0
\(265\) −11.7826 −0.723797
\(266\) 0 0
\(267\) 35.7009 2.18486
\(268\) 0 0
\(269\) −18.1406 −1.10605 −0.553026 0.833164i \(-0.686527\pi\)
−0.553026 + 0.833164i \(0.686527\pi\)
\(270\) 0 0
\(271\) 25.8372 1.56950 0.784750 0.619813i \(-0.212792\pi\)
0.784750 + 0.619813i \(0.212792\pi\)
\(272\) 0 0
\(273\) −5.23602 −0.316898
\(274\) 0 0
\(275\) −12.6907 −0.765280
\(276\) 0 0
\(277\) −14.9438 −0.897886 −0.448943 0.893560i \(-0.648199\pi\)
−0.448943 + 0.893560i \(0.648199\pi\)
\(278\) 0 0
\(279\) 19.8338 1.18742
\(280\) 0 0
\(281\) −24.4663 −1.45954 −0.729769 0.683694i \(-0.760372\pi\)
−0.729769 + 0.683694i \(0.760372\pi\)
\(282\) 0 0
\(283\) 8.68159 0.516067 0.258034 0.966136i \(-0.416925\pi\)
0.258034 + 0.966136i \(0.416925\pi\)
\(284\) 0 0
\(285\) 9.48935 0.562101
\(286\) 0 0
\(287\) 16.1629 0.954068
\(288\) 0 0
\(289\) −16.5700 −0.974706
\(290\) 0 0
\(291\) −12.5734 −0.737067
\(292\) 0 0
\(293\) −18.2517 −1.06628 −0.533138 0.846028i \(-0.678987\pi\)
−0.533138 + 0.846028i \(0.678987\pi\)
\(294\) 0 0
\(295\) 34.3556 2.00026
\(296\) 0 0
\(297\) −2.05282 −0.119117
\(298\) 0 0
\(299\) −1.94394 −0.112421
\(300\) 0 0
\(301\) −34.4584 −1.98615
\(302\) 0 0
\(303\) −18.0238 −1.03544
\(304\) 0 0
\(305\) −7.01555 −0.401709
\(306\) 0 0
\(307\) 5.22644 0.298289 0.149144 0.988815i \(-0.452348\pi\)
0.149144 + 0.988815i \(0.452348\pi\)
\(308\) 0 0
\(309\) 33.1993 1.88864
\(310\) 0 0
\(311\) −5.53926 −0.314103 −0.157051 0.987590i \(-0.550199\pi\)
−0.157051 + 0.987590i \(0.550199\pi\)
\(312\) 0 0
\(313\) −12.7911 −0.722998 −0.361499 0.932372i \(-0.617735\pi\)
−0.361499 + 0.932372i \(0.617735\pi\)
\(314\) 0 0
\(315\) −28.1846 −1.58802
\(316\) 0 0
\(317\) −29.2257 −1.64148 −0.820740 0.571301i \(-0.806439\pi\)
−0.820740 + 0.571301i \(0.806439\pi\)
\(318\) 0 0
\(319\) 2.64351 0.148008
\(320\) 0 0
\(321\) 2.50977 0.140082
\(322\) 0 0
\(323\) 0.655737 0.0364862
\(324\) 0 0
\(325\) 9.18661 0.509582
\(326\) 0 0
\(327\) 18.1207 1.00207
\(328\) 0 0
\(329\) −15.5309 −0.856249
\(330\) 0 0
\(331\) −15.8740 −0.872513 −0.436256 0.899822i \(-0.643696\pi\)
−0.436256 + 0.899822i \(0.643696\pi\)
\(332\) 0 0
\(333\) 8.79922 0.482194
\(334\) 0 0
\(335\) 6.63903 0.362729
\(336\) 0 0
\(337\) 2.93726 0.160003 0.0800014 0.996795i \(-0.474508\pi\)
0.0800014 + 0.996795i \(0.474508\pi\)
\(338\) 0 0
\(339\) −22.9455 −1.24623
\(340\) 0 0
\(341\) −9.48935 −0.513877
\(342\) 0 0
\(343\) −11.9307 −0.644200
\(344\) 0 0
\(345\) −25.4830 −1.37196
\(346\) 0 0
\(347\) −8.74992 −0.469721 −0.234860 0.972029i \(-0.575463\pi\)
−0.234860 + 0.972029i \(0.575463\pi\)
\(348\) 0 0
\(349\) −18.4164 −0.985807 −0.492904 0.870084i \(-0.664064\pi\)
−0.492904 + 0.870084i \(0.664064\pi\)
\(350\) 0 0
\(351\) 1.48600 0.0793169
\(352\) 0 0
\(353\) −19.9919 −1.06406 −0.532031 0.846725i \(-0.678571\pi\)
−0.532031 + 0.846725i \(0.678571\pi\)
\(354\) 0 0
\(355\) 43.5093 2.30923
\(356\) 0 0
\(357\) −4.74310 −0.251031
\(358\) 0 0
\(359\) 3.29606 0.173960 0.0869798 0.996210i \(-0.472278\pi\)
0.0869798 + 0.996210i \(0.472278\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.25613 −0.118416
\(364\) 0 0
\(365\) 14.1038 0.738225
\(366\) 0 0
\(367\) −25.2683 −1.31900 −0.659498 0.751707i \(-0.729231\pi\)
−0.659498 + 0.751707i \(0.729231\pi\)
\(368\) 0 0
\(369\) 10.5371 0.548541
\(370\) 0 0
\(371\) 8.98122 0.466282
\(372\) 0 0
\(373\) −5.25095 −0.271884 −0.135942 0.990717i \(-0.543406\pi\)
−0.135942 + 0.990717i \(0.543406\pi\)
\(374\) 0 0
\(375\) 72.9801 3.76868
\(376\) 0 0
\(377\) −1.91359 −0.0985550
\(378\) 0 0
\(379\) −24.3964 −1.25316 −0.626580 0.779357i \(-0.715546\pi\)
−0.626580 + 0.779357i \(0.715546\pi\)
\(380\) 0 0
\(381\) 26.3682 1.35089
\(382\) 0 0
\(383\) 29.5585 1.51037 0.755184 0.655512i \(-0.227547\pi\)
0.755184 + 0.655512i \(0.227547\pi\)
\(384\) 0 0
\(385\) 13.4847 0.687244
\(386\) 0 0
\(387\) −22.4645 −1.14194
\(388\) 0 0
\(389\) 33.0958 1.67803 0.839013 0.544112i \(-0.183133\pi\)
0.839013 + 0.544112i \(0.183133\pi\)
\(390\) 0 0
\(391\) −1.76093 −0.0890543
\(392\) 0 0
\(393\) −34.4684 −1.73870
\(394\) 0 0
\(395\) 73.3389 3.69008
\(396\) 0 0
\(397\) −1.55966 −0.0782769 −0.0391385 0.999234i \(-0.512461\pi\)
−0.0391385 + 0.999234i \(0.512461\pi\)
\(398\) 0 0
\(399\) −7.23323 −0.362114
\(400\) 0 0
\(401\) 17.9881 0.898282 0.449141 0.893461i \(-0.351730\pi\)
0.449141 + 0.893461i \(0.351730\pi\)
\(402\) 0 0
\(403\) 6.86919 0.342179
\(404\) 0 0
\(405\) 45.8532 2.27846
\(406\) 0 0
\(407\) −4.20992 −0.208678
\(408\) 0 0
\(409\) 31.8247 1.57363 0.786816 0.617188i \(-0.211728\pi\)
0.786816 + 0.617188i \(0.211728\pi\)
\(410\) 0 0
\(411\) 29.0432 1.43260
\(412\) 0 0
\(413\) −26.1874 −1.28860
\(414\) 0 0
\(415\) −17.3927 −0.853775
\(416\) 0 0
\(417\) 29.7362 1.45619
\(418\) 0 0
\(419\) −22.0175 −1.07563 −0.537814 0.843064i \(-0.680750\pi\)
−0.537814 + 0.843064i \(0.680750\pi\)
\(420\) 0 0
\(421\) −17.5466 −0.855171 −0.427585 0.903975i \(-0.640636\pi\)
−0.427585 + 0.903975i \(0.640636\pi\)
\(422\) 0 0
\(423\) −10.1251 −0.492300
\(424\) 0 0
\(425\) 8.32178 0.403666
\(426\) 0 0
\(427\) 5.34758 0.258788
\(428\) 0 0
\(429\) 1.63317 0.0788504
\(430\) 0 0
\(431\) 0.854799 0.0411742 0.0205871 0.999788i \(-0.493446\pi\)
0.0205871 + 0.999788i \(0.493446\pi\)
\(432\) 0 0
\(433\) −17.0259 −0.818214 −0.409107 0.912486i \(-0.634160\pi\)
−0.409107 + 0.912486i \(0.634160\pi\)
\(434\) 0 0
\(435\) −25.0852 −1.20274
\(436\) 0 0
\(437\) −2.68543 −0.128461
\(438\) 0 0
\(439\) 2.81936 0.134561 0.0672804 0.997734i \(-0.478568\pi\)
0.0672804 + 0.997734i \(0.478568\pi\)
\(440\) 0 0
\(441\) 6.85278 0.326323
\(442\) 0 0
\(443\) −19.4356 −0.923414 −0.461707 0.887033i \(-0.652763\pi\)
−0.461707 + 0.887033i \(0.652763\pi\)
\(444\) 0 0
\(445\) 66.5561 3.15506
\(446\) 0 0
\(447\) −37.3213 −1.76524
\(448\) 0 0
\(449\) 13.7032 0.646693 0.323347 0.946281i \(-0.395192\pi\)
0.323347 + 0.946281i \(0.395192\pi\)
\(450\) 0 0
\(451\) −5.04141 −0.237391
\(452\) 0 0
\(453\) −36.2901 −1.70506
\(454\) 0 0
\(455\) −9.76135 −0.457619
\(456\) 0 0
\(457\) −11.2586 −0.526653 −0.263326 0.964707i \(-0.584820\pi\)
−0.263326 + 0.964707i \(0.584820\pi\)
\(458\) 0 0
\(459\) 1.34611 0.0628310
\(460\) 0 0
\(461\) 17.5115 0.815594 0.407797 0.913073i \(-0.366297\pi\)
0.407797 + 0.913073i \(0.366297\pi\)
\(462\) 0 0
\(463\) −20.4636 −0.951022 −0.475511 0.879710i \(-0.657737\pi\)
−0.475511 + 0.879710i \(0.657737\pi\)
\(464\) 0 0
\(465\) 90.0479 4.17587
\(466\) 0 0
\(467\) −11.1897 −0.517796 −0.258898 0.965905i \(-0.583359\pi\)
−0.258898 + 0.965905i \(0.583359\pi\)
\(468\) 0 0
\(469\) −5.06057 −0.233676
\(470\) 0 0
\(471\) −39.1853 −1.80556
\(472\) 0 0
\(473\) 10.7480 0.494193
\(474\) 0 0
\(475\) 12.6907 0.582291
\(476\) 0 0
\(477\) 5.85514 0.268088
\(478\) 0 0
\(479\) 38.8664 1.77585 0.887926 0.459986i \(-0.152145\pi\)
0.887926 + 0.459986i \(0.152145\pi\)
\(480\) 0 0
\(481\) 3.04749 0.138954
\(482\) 0 0
\(483\) 19.4243 0.883837
\(484\) 0 0
\(485\) −23.4403 −1.06437
\(486\) 0 0
\(487\) −23.6592 −1.07210 −0.536050 0.844186i \(-0.680084\pi\)
−0.536050 + 0.844186i \(0.680084\pi\)
\(488\) 0 0
\(489\) 41.4506 1.87446
\(490\) 0 0
\(491\) 5.74101 0.259088 0.129544 0.991574i \(-0.458649\pi\)
0.129544 + 0.991574i \(0.458649\pi\)
\(492\) 0 0
\(493\) −1.73345 −0.0780704
\(494\) 0 0
\(495\) 8.79110 0.395130
\(496\) 0 0
\(497\) −33.1648 −1.48764
\(498\) 0 0
\(499\) −4.85986 −0.217557 −0.108779 0.994066i \(-0.534694\pi\)
−0.108779 + 0.994066i \(0.534694\pi\)
\(500\) 0 0
\(501\) 5.92630 0.264768
\(502\) 0 0
\(503\) 5.08324 0.226650 0.113325 0.993558i \(-0.463850\pi\)
0.113325 + 0.993558i \(0.463850\pi\)
\(504\) 0 0
\(505\) −33.6012 −1.49524
\(506\) 0 0
\(507\) 28.1474 1.25007
\(508\) 0 0
\(509\) 35.2272 1.56142 0.780708 0.624896i \(-0.214859\pi\)
0.780708 + 0.624896i \(0.214859\pi\)
\(510\) 0 0
\(511\) −10.7505 −0.475576
\(512\) 0 0
\(513\) 2.05282 0.0906341
\(514\) 0 0
\(515\) 61.8925 2.72731
\(516\) 0 0
\(517\) 4.84428 0.213051
\(518\) 0 0
\(519\) 14.0664 0.617447
\(520\) 0 0
\(521\) −13.2922 −0.582344 −0.291172 0.956671i \(-0.594045\pi\)
−0.291172 + 0.956671i \(0.594045\pi\)
\(522\) 0 0
\(523\) −29.0749 −1.27136 −0.635679 0.771954i \(-0.719280\pi\)
−0.635679 + 0.771954i \(0.719280\pi\)
\(524\) 0 0
\(525\) −91.7949 −4.00626
\(526\) 0 0
\(527\) 6.22252 0.271057
\(528\) 0 0
\(529\) −15.7885 −0.686456
\(530\) 0 0
\(531\) −17.0724 −0.740880
\(532\) 0 0
\(533\) 3.64939 0.158073
\(534\) 0 0
\(535\) 4.67889 0.202286
\(536\) 0 0
\(537\) −9.68685 −0.418018
\(538\) 0 0
\(539\) −3.27866 −0.141222
\(540\) 0 0
\(541\) 25.8688 1.11219 0.556093 0.831120i \(-0.312300\pi\)
0.556093 + 0.831120i \(0.312300\pi\)
\(542\) 0 0
\(543\) −10.2034 −0.437871
\(544\) 0 0
\(545\) 33.7818 1.44705
\(546\) 0 0
\(547\) 10.5966 0.453076 0.226538 0.974002i \(-0.427259\pi\)
0.226538 + 0.974002i \(0.427259\pi\)
\(548\) 0 0
\(549\) 3.48626 0.148790
\(550\) 0 0
\(551\) −2.64351 −0.112617
\(552\) 0 0
\(553\) −55.9023 −2.37721
\(554\) 0 0
\(555\) 39.9494 1.69576
\(556\) 0 0
\(557\) 46.0565 1.95148 0.975738 0.218939i \(-0.0702597\pi\)
0.975738 + 0.218939i \(0.0702597\pi\)
\(558\) 0 0
\(559\) −7.78029 −0.329071
\(560\) 0 0
\(561\) 1.47943 0.0624615
\(562\) 0 0
\(563\) −37.4009 −1.57626 −0.788129 0.615510i \(-0.788950\pi\)
−0.788129 + 0.615510i \(0.788950\pi\)
\(564\) 0 0
\(565\) −42.7767 −1.79963
\(566\) 0 0
\(567\) −34.9514 −1.46782
\(568\) 0 0
\(569\) 6.67373 0.279777 0.139889 0.990167i \(-0.455326\pi\)
0.139889 + 0.990167i \(0.455326\pi\)
\(570\) 0 0
\(571\) −28.1302 −1.17721 −0.588606 0.808420i \(-0.700323\pi\)
−0.588606 + 0.808420i \(0.700323\pi\)
\(572\) 0 0
\(573\) −56.4867 −2.35977
\(574\) 0 0
\(575\) −34.0800 −1.42124
\(576\) 0 0
\(577\) 21.1279 0.879567 0.439784 0.898104i \(-0.355055\pi\)
0.439784 + 0.898104i \(0.355055\pi\)
\(578\) 0 0
\(579\) −10.3822 −0.431468
\(580\) 0 0
\(581\) 13.2575 0.550015
\(582\) 0 0
\(583\) −2.80135 −0.116020
\(584\) 0 0
\(585\) −6.36373 −0.263108
\(586\) 0 0
\(587\) 20.6634 0.852869 0.426435 0.904518i \(-0.359769\pi\)
0.426435 + 0.904518i \(0.359769\pi\)
\(588\) 0 0
\(589\) 9.48935 0.391002
\(590\) 0 0
\(591\) −12.4485 −0.512064
\(592\) 0 0
\(593\) −7.59734 −0.311985 −0.155993 0.987758i \(-0.549858\pi\)
−0.155993 + 0.987758i \(0.549858\pi\)
\(594\) 0 0
\(595\) −8.84242 −0.362504
\(596\) 0 0
\(597\) 22.2123 0.909091
\(598\) 0 0
\(599\) 33.9894 1.38877 0.694385 0.719604i \(-0.255677\pi\)
0.694385 + 0.719604i \(0.255677\pi\)
\(600\) 0 0
\(601\) 22.0050 0.897602 0.448801 0.893632i \(-0.351851\pi\)
0.448801 + 0.893632i \(0.351851\pi\)
\(602\) 0 0
\(603\) −3.29915 −0.134352
\(604\) 0 0
\(605\) −4.20604 −0.171000
\(606\) 0 0
\(607\) −11.9767 −0.486120 −0.243060 0.970011i \(-0.578151\pi\)
−0.243060 + 0.970011i \(0.578151\pi\)
\(608\) 0 0
\(609\) 19.1211 0.774825
\(610\) 0 0
\(611\) −3.50670 −0.141866
\(612\) 0 0
\(613\) 19.7909 0.799345 0.399673 0.916658i \(-0.369124\pi\)
0.399673 + 0.916658i \(0.369124\pi\)
\(614\) 0 0
\(615\) 47.8397 1.92908
\(616\) 0 0
\(617\) −31.1026 −1.25214 −0.626072 0.779765i \(-0.715338\pi\)
−0.626072 + 0.779765i \(0.715338\pi\)
\(618\) 0 0
\(619\) −14.4469 −0.580671 −0.290335 0.956925i \(-0.593767\pi\)
−0.290335 + 0.956925i \(0.593767\pi\)
\(620\) 0 0
\(621\) −5.51269 −0.221217
\(622\) 0 0
\(623\) −50.7322 −2.03254
\(624\) 0 0
\(625\) 72.6010 2.90404
\(626\) 0 0
\(627\) 2.25613 0.0901011
\(628\) 0 0
\(629\) 2.76060 0.110072
\(630\) 0 0
\(631\) −4.05580 −0.161459 −0.0807295 0.996736i \(-0.525725\pi\)
−0.0807295 + 0.996736i \(0.525725\pi\)
\(632\) 0 0
\(633\) 45.9204 1.82517
\(634\) 0 0
\(635\) 49.1575 1.95076
\(636\) 0 0
\(637\) 2.37337 0.0940363
\(638\) 0 0
\(639\) −21.6212 −0.855320
\(640\) 0 0
\(641\) −32.8261 −1.29655 −0.648277 0.761405i \(-0.724510\pi\)
−0.648277 + 0.761405i \(0.724510\pi\)
\(642\) 0 0
\(643\) 22.0138 0.868139 0.434070 0.900879i \(-0.357077\pi\)
0.434070 + 0.900879i \(0.357077\pi\)
\(644\) 0 0
\(645\) −101.991 −4.01591
\(646\) 0 0
\(647\) 42.5626 1.67331 0.836654 0.547731i \(-0.184508\pi\)
0.836654 + 0.547731i \(0.184508\pi\)
\(648\) 0 0
\(649\) 8.16817 0.320629
\(650\) 0 0
\(651\) −68.6387 −2.69016
\(652\) 0 0
\(653\) −37.0860 −1.45129 −0.725645 0.688070i \(-0.758458\pi\)
−0.725645 + 0.688070i \(0.758458\pi\)
\(654\) 0 0
\(655\) −64.2585 −2.51079
\(656\) 0 0
\(657\) −7.00862 −0.273432
\(658\) 0 0
\(659\) −33.0772 −1.28850 −0.644252 0.764814i \(-0.722831\pi\)
−0.644252 + 0.764814i \(0.722831\pi\)
\(660\) 0 0
\(661\) 37.6789 1.46554 0.732770 0.680477i \(-0.238227\pi\)
0.732770 + 0.680477i \(0.238227\pi\)
\(662\) 0 0
\(663\) −1.07093 −0.0415916
\(664\) 0 0
\(665\) −13.4847 −0.522914
\(666\) 0 0
\(667\) 7.09894 0.274872
\(668\) 0 0
\(669\) −12.8484 −0.496748
\(670\) 0 0
\(671\) −1.66797 −0.0643914
\(672\) 0 0
\(673\) −37.0065 −1.42650 −0.713248 0.700912i \(-0.752777\pi\)
−0.713248 + 0.700912i \(0.752777\pi\)
\(674\) 0 0
\(675\) 26.0518 1.00273
\(676\) 0 0
\(677\) −23.1551 −0.889923 −0.444961 0.895550i \(-0.646783\pi\)
−0.444961 + 0.895550i \(0.646783\pi\)
\(678\) 0 0
\(679\) 17.8672 0.685682
\(680\) 0 0
\(681\) 36.3114 1.39145
\(682\) 0 0
\(683\) −7.10245 −0.271768 −0.135884 0.990725i \(-0.543387\pi\)
−0.135884 + 0.990725i \(0.543387\pi\)
\(684\) 0 0
\(685\) 54.1445 2.06875
\(686\) 0 0
\(687\) −22.9867 −0.876999
\(688\) 0 0
\(689\) 2.02785 0.0772549
\(690\) 0 0
\(691\) 15.3153 0.582622 0.291311 0.956628i \(-0.405909\pi\)
0.291311 + 0.956628i \(0.405909\pi\)
\(692\) 0 0
\(693\) −6.70098 −0.254549
\(694\) 0 0
\(695\) 55.4364 2.10282
\(696\) 0 0
\(697\) 3.30584 0.125218
\(698\) 0 0
\(699\) −37.2513 −1.40897
\(700\) 0 0
\(701\) 32.0180 1.20930 0.604651 0.796490i \(-0.293312\pi\)
0.604651 + 0.796490i \(0.293312\pi\)
\(702\) 0 0
\(703\) 4.20992 0.158780
\(704\) 0 0
\(705\) −45.9691 −1.73130
\(706\) 0 0
\(707\) 25.6124 0.963254
\(708\) 0 0
\(709\) −9.61140 −0.360964 −0.180482 0.983578i \(-0.557766\pi\)
−0.180482 + 0.983578i \(0.557766\pi\)
\(710\) 0 0
\(711\) −36.4445 −1.36677
\(712\) 0 0
\(713\) −25.4830 −0.954345
\(714\) 0 0
\(715\) 3.04468 0.113865
\(716\) 0 0
\(717\) 63.4562 2.36981
\(718\) 0 0
\(719\) −18.3795 −0.685440 −0.342720 0.939438i \(-0.611348\pi\)
−0.342720 + 0.939438i \(0.611348\pi\)
\(720\) 0 0
\(721\) −47.1773 −1.75698
\(722\) 0 0
\(723\) −42.4359 −1.57821
\(724\) 0 0
\(725\) −33.5480 −1.24594
\(726\) 0 0
\(727\) −11.6194 −0.430941 −0.215471 0.976510i \(-0.569129\pi\)
−0.215471 + 0.976510i \(0.569129\pi\)
\(728\) 0 0
\(729\) −8.89168 −0.329322
\(730\) 0 0
\(731\) −7.04785 −0.260674
\(732\) 0 0
\(733\) −23.4420 −0.865850 −0.432925 0.901430i \(-0.642519\pi\)
−0.432925 + 0.901430i \(0.642519\pi\)
\(734\) 0 0
\(735\) 31.1124 1.14760
\(736\) 0 0
\(737\) 1.57845 0.0581430
\(738\) 0 0
\(739\) 14.0970 0.518567 0.259284 0.965801i \(-0.416514\pi\)
0.259284 + 0.965801i \(0.416514\pi\)
\(740\) 0 0
\(741\) −1.63317 −0.0599962
\(742\) 0 0
\(743\) −39.6727 −1.45545 −0.727725 0.685869i \(-0.759422\pi\)
−0.727725 + 0.685869i \(0.759422\pi\)
\(744\) 0 0
\(745\) −69.5770 −2.54910
\(746\) 0 0
\(747\) 8.64300 0.316231
\(748\) 0 0
\(749\) −3.56647 −0.130316
\(750\) 0 0
\(751\) −5.16229 −0.188375 −0.0941874 0.995554i \(-0.530025\pi\)
−0.0941874 + 0.995554i \(0.530025\pi\)
\(752\) 0 0
\(753\) −43.5374 −1.58659
\(754\) 0 0
\(755\) −67.6546 −2.46220
\(756\) 0 0
\(757\) 28.3285 1.02962 0.514809 0.857305i \(-0.327863\pi\)
0.514809 + 0.857305i \(0.327863\pi\)
\(758\) 0 0
\(759\) −6.05867 −0.219916
\(760\) 0 0
\(761\) −31.0318 −1.12490 −0.562451 0.826831i \(-0.690141\pi\)
−0.562451 + 0.826831i \(0.690141\pi\)
\(762\) 0 0
\(763\) −25.7501 −0.932215
\(764\) 0 0
\(765\) −5.76465 −0.208421
\(766\) 0 0
\(767\) −5.91281 −0.213499
\(768\) 0 0
\(769\) 40.6414 1.46557 0.732783 0.680462i \(-0.238221\pi\)
0.732783 + 0.680462i \(0.238221\pi\)
\(770\) 0 0
\(771\) 49.5175 1.78333
\(772\) 0 0
\(773\) −0.817958 −0.0294199 −0.0147099 0.999892i \(-0.504682\pi\)
−0.0147099 + 0.999892i \(0.504682\pi\)
\(774\) 0 0
\(775\) 120.427 4.32586
\(776\) 0 0
\(777\) −30.4513 −1.09243
\(778\) 0 0
\(779\) 5.04141 0.180627
\(780\) 0 0
\(781\) 10.3445 0.370155
\(782\) 0 0
\(783\) −5.42663 −0.193932
\(784\) 0 0
\(785\) −73.0521 −2.60734
\(786\) 0 0
\(787\) −11.2343 −0.400459 −0.200229 0.979749i \(-0.564169\pi\)
−0.200229 + 0.979749i \(0.564169\pi\)
\(788\) 0 0
\(789\) 21.0052 0.747806
\(790\) 0 0
\(791\) 32.6063 1.15935
\(792\) 0 0
\(793\) 1.20742 0.0428767
\(794\) 0 0
\(795\) 26.5830 0.942801
\(796\) 0 0
\(797\) 4.03810 0.143037 0.0715184 0.997439i \(-0.477216\pi\)
0.0715184 + 0.997439i \(0.477216\pi\)
\(798\) 0 0
\(799\) −3.17658 −0.112379
\(800\) 0 0
\(801\) −33.0739 −1.16861
\(802\) 0 0
\(803\) 3.35322 0.118333
\(804\) 0 0
\(805\) 36.2122 1.27631
\(806\) 0 0
\(807\) 40.9275 1.44072
\(808\) 0 0
\(809\) 47.1270 1.65690 0.828448 0.560066i \(-0.189224\pi\)
0.828448 + 0.560066i \(0.189224\pi\)
\(810\) 0 0
\(811\) −37.6251 −1.32119 −0.660597 0.750740i \(-0.729697\pi\)
−0.660597 + 0.750740i \(0.729697\pi\)
\(812\) 0 0
\(813\) −58.2921 −2.04439
\(814\) 0 0
\(815\) 77.2752 2.70683
\(816\) 0 0
\(817\) −10.7480 −0.376024
\(818\) 0 0
\(819\) 4.85073 0.169498
\(820\) 0 0
\(821\) −54.5948 −1.90537 −0.952685 0.303959i \(-0.901691\pi\)
−0.952685 + 0.303959i \(0.901691\pi\)
\(822\) 0 0
\(823\) −5.10600 −0.177984 −0.0889920 0.996032i \(-0.528365\pi\)
−0.0889920 + 0.996032i \(0.528365\pi\)
\(824\) 0 0
\(825\) 28.6319 0.996835
\(826\) 0 0
\(827\) −21.9650 −0.763798 −0.381899 0.924204i \(-0.624730\pi\)
−0.381899 + 0.924204i \(0.624730\pi\)
\(828\) 0 0
\(829\) 30.3940 1.05563 0.527813 0.849360i \(-0.323012\pi\)
0.527813 + 0.849360i \(0.323012\pi\)
\(830\) 0 0
\(831\) 33.7151 1.16957
\(832\) 0 0
\(833\) 2.14994 0.0744910
\(834\) 0 0
\(835\) 11.0482 0.382340
\(836\) 0 0
\(837\) 19.4799 0.673324
\(838\) 0 0
\(839\) −5.38580 −0.185938 −0.0929692 0.995669i \(-0.529636\pi\)
−0.0929692 + 0.995669i \(0.529636\pi\)
\(840\) 0 0
\(841\) −22.0119 −0.759030
\(842\) 0 0
\(843\) 55.1991 1.90116
\(844\) 0 0
\(845\) 52.4745 1.80518
\(846\) 0 0
\(847\) 3.20604 0.110161
\(848\) 0 0
\(849\) −19.5868 −0.672217
\(850\) 0 0
\(851\) −11.3054 −0.387545
\(852\) 0 0
\(853\) 36.1015 1.23609 0.618046 0.786142i \(-0.287925\pi\)
0.618046 + 0.786142i \(0.287925\pi\)
\(854\) 0 0
\(855\) −8.79110 −0.300649
\(856\) 0 0
\(857\) −36.5886 −1.24984 −0.624922 0.780687i \(-0.714869\pi\)
−0.624922 + 0.780687i \(0.714869\pi\)
\(858\) 0 0
\(859\) −21.7813 −0.743168 −0.371584 0.928399i \(-0.621185\pi\)
−0.371584 + 0.928399i \(0.621185\pi\)
\(860\) 0 0
\(861\) −36.4656 −1.24275
\(862\) 0 0
\(863\) 46.9178 1.59710 0.798550 0.601929i \(-0.205601\pi\)
0.798550 + 0.601929i \(0.205601\pi\)
\(864\) 0 0
\(865\) 26.2236 0.891629
\(866\) 0 0
\(867\) 37.3841 1.26963
\(868\) 0 0
\(869\) 17.4366 0.591496
\(870\) 0 0
\(871\) −1.14262 −0.0387161
\(872\) 0 0
\(873\) 11.6482 0.394233
\(874\) 0 0
\(875\) −103.707 −3.50594
\(876\) 0 0
\(877\) −41.9565 −1.41677 −0.708385 0.705827i \(-0.750576\pi\)
−0.708385 + 0.705827i \(0.750576\pi\)
\(878\) 0 0
\(879\) 41.1782 1.38891
\(880\) 0 0
\(881\) −22.6747 −0.763931 −0.381965 0.924177i \(-0.624753\pi\)
−0.381965 + 0.924177i \(0.624753\pi\)
\(882\) 0 0
\(883\) 52.7680 1.77578 0.887892 0.460052i \(-0.152169\pi\)
0.887892 + 0.460052i \(0.152169\pi\)
\(884\) 0 0
\(885\) −77.5107 −2.60549
\(886\) 0 0
\(887\) −46.4618 −1.56004 −0.780018 0.625758i \(-0.784790\pi\)
−0.780018 + 0.625758i \(0.784790\pi\)
\(888\) 0 0
\(889\) −37.4701 −1.25671
\(890\) 0 0
\(891\) 10.9018 0.365223
\(892\) 0 0
\(893\) −4.84428 −0.162108
\(894\) 0 0
\(895\) −18.0589 −0.603643
\(896\) 0 0
\(897\) 4.38577 0.146437
\(898\) 0 0
\(899\) −25.0852 −0.836637
\(900\) 0 0
\(901\) 1.83695 0.0611976
\(902\) 0 0
\(903\) 77.7426 2.58711
\(904\) 0 0
\(905\) −19.0220 −0.632312
\(906\) 0 0
\(907\) −40.1762 −1.33403 −0.667014 0.745045i \(-0.732428\pi\)
−0.667014 + 0.745045i \(0.732428\pi\)
\(908\) 0 0
\(909\) 16.6975 0.553823
\(910\) 0 0
\(911\) −34.8397 −1.15429 −0.577145 0.816642i \(-0.695833\pi\)
−0.577145 + 0.816642i \(0.695833\pi\)
\(912\) 0 0
\(913\) −4.13518 −0.136855
\(914\) 0 0
\(915\) 15.8280 0.523257
\(916\) 0 0
\(917\) 48.9808 1.61749
\(918\) 0 0
\(919\) 38.3172 1.26397 0.631983 0.774982i \(-0.282241\pi\)
0.631983 + 0.774982i \(0.282241\pi\)
\(920\) 0 0
\(921\) −11.7915 −0.388544
\(922\) 0 0
\(923\) −7.48821 −0.246477
\(924\) 0 0
\(925\) 53.4270 1.75667
\(926\) 0 0
\(927\) −30.7564 −1.01017
\(928\) 0 0
\(929\) 3.20133 0.105032 0.0525161 0.998620i \(-0.483276\pi\)
0.0525161 + 0.998620i \(0.483276\pi\)
\(930\) 0 0
\(931\) 3.27866 0.107454
\(932\) 0 0
\(933\) 12.4973 0.409143
\(934\) 0 0
\(935\) 2.75805 0.0901980
\(936\) 0 0
\(937\) −38.8123 −1.26794 −0.633972 0.773356i \(-0.718576\pi\)
−0.633972 + 0.773356i \(0.718576\pi\)
\(938\) 0 0
\(939\) 28.8585 0.941761
\(940\) 0 0
\(941\) 46.2516 1.50776 0.753879 0.657013i \(-0.228180\pi\)
0.753879 + 0.657013i \(0.228180\pi\)
\(942\) 0 0
\(943\) −13.5383 −0.440869
\(944\) 0 0
\(945\) −27.6816 −0.900483
\(946\) 0 0
\(947\) −7.01420 −0.227931 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(948\) 0 0
\(949\) −2.42734 −0.0787949
\(950\) 0 0
\(951\) 65.9370 2.13815
\(952\) 0 0
\(953\) 13.8853 0.449788 0.224894 0.974383i \(-0.427796\pi\)
0.224894 + 0.974383i \(0.427796\pi\)
\(954\) 0 0
\(955\) −105.307 −3.40764
\(956\) 0 0
\(957\) −5.96409 −0.192792
\(958\) 0 0
\(959\) −41.2714 −1.33272
\(960\) 0 0
\(961\) 59.0479 1.90477
\(962\) 0 0
\(963\) −2.32509 −0.0749250
\(964\) 0 0
\(965\) −19.3552 −0.623065
\(966\) 0 0
\(967\) −34.4880 −1.10906 −0.554529 0.832164i \(-0.687102\pi\)
−0.554529 + 0.832164i \(0.687102\pi\)
\(968\) 0 0
\(969\) −1.47943 −0.0475260
\(970\) 0 0
\(971\) 8.98878 0.288464 0.144232 0.989544i \(-0.453929\pi\)
0.144232 + 0.989544i \(0.453929\pi\)
\(972\) 0 0
\(973\) −42.2562 −1.35467
\(974\) 0 0
\(975\) −20.7262 −0.663769
\(976\) 0 0
\(977\) −49.8473 −1.59476 −0.797378 0.603480i \(-0.793780\pi\)
−0.797378 + 0.603480i \(0.793780\pi\)
\(978\) 0 0
\(979\) 15.8240 0.505736
\(980\) 0 0
\(981\) −16.7873 −0.535977
\(982\) 0 0
\(983\) −38.4740 −1.22713 −0.613565 0.789644i \(-0.710265\pi\)
−0.613565 + 0.789644i \(0.710265\pi\)
\(984\) 0 0
\(985\) −23.2074 −0.739451
\(986\) 0 0
\(987\) 35.0398 1.11533
\(988\) 0 0
\(989\) 28.8629 0.917788
\(990\) 0 0
\(991\) 26.0741 0.828271 0.414135 0.910215i \(-0.364084\pi\)
0.414135 + 0.910215i \(0.364084\pi\)
\(992\) 0 0
\(993\) 35.8137 1.13651
\(994\) 0 0
\(995\) 41.4098 1.31278
\(996\) 0 0
\(997\) 51.2926 1.62445 0.812227 0.583341i \(-0.198255\pi\)
0.812227 + 0.583341i \(0.198255\pi\)
\(998\) 0 0
\(999\) 8.64220 0.273427
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 1672.2.a.i.1.2 6
4.3 odd 2 3344.2.a.z.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.i.1.2 6 1.1 even 1 trivial
3344.2.a.z.1.5 6 4.3 odd 2