Properties

Label 1287.2.a.k.1.1
Level $1287$
Weight $2$
Character 1287.1
Self dual yes
Analytic conductor $10.277$
Analytic rank $0$
Dimension $4$
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] = [1287,2,Mod(1,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, 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("1287.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.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: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.06150\) of defining polynomial
Character \(\chi\) \(=\) 1287.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.57641 q^{2} +4.63791 q^{4} +4.12300 q^{5} +2.15845 q^{7} -6.79636 q^{8} +O(q^{10})\) \(q-2.57641 q^{2} +4.63791 q^{4} +4.12300 q^{5} +2.15845 q^{7} -6.79636 q^{8} -10.6226 q^{10} -1.00000 q^{11} -1.00000 q^{13} -5.56105 q^{14} +8.23442 q^{16} +3.09317 q^{17} -2.68783 q^{19} +19.1221 q^{20} +2.57641 q^{22} -3.31128 q^{23} +11.9991 q^{25} +2.57641 q^{26} +10.0107 q^{28} +3.34673 q^{29} +9.96928 q^{31} -7.62255 q^{32} -7.96928 q^{34} +8.89927 q^{35} +2.05966 q^{37} +6.92497 q^{38} -28.0214 q^{40} -9.49394 q^{41} +2.84717 q^{43} -4.63791 q^{44} +8.53122 q^{46} +9.46972 q^{47} -2.34111 q^{49} -30.9147 q^{50} -4.63791 q^{52} +5.34111 q^{53} -4.12300 q^{55} -14.6696 q^{56} -8.62255 q^{58} -4.62255 q^{59} -0.193895 q^{61} -25.6850 q^{62} +3.17003 q^{64} -4.12300 q^{65} -5.78662 q^{67} +14.3458 q^{68} -22.9282 q^{70} +3.78295 q^{71} +7.47534 q^{73} -5.30655 q^{74} -12.4659 q^{76} -2.15845 q^{77} -17.2049 q^{79} +33.9505 q^{80} +24.4603 q^{82} +0.218111 q^{83} +12.7531 q^{85} -7.33549 q^{86} +6.79636 q^{88} +1.52939 q^{89} -2.15845 q^{91} -15.3574 q^{92} -24.3979 q^{94} -11.0819 q^{95} +12.1462 q^{97} +6.03167 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 3 q^{4} + 6 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 3 q^{4} + 6 q^{7} - 9 q^{8} - 8 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} + 5 q^{16} - 6 q^{17} + 8 q^{19} + 24 q^{20} + 3 q^{22} + 4 q^{23} + 12 q^{25} + 3 q^{26} - q^{28} + 10 q^{29} + 2 q^{31} + 4 q^{32} + 6 q^{34} + 6 q^{35} + 12 q^{37} + 5 q^{38} - 30 q^{40} - 8 q^{41} + 26 q^{43} - 3 q^{44} + 6 q^{46} + 18 q^{47} + 6 q^{49} - 29 q^{50} - 3 q^{52} + 6 q^{53} - 33 q^{56} + 16 q^{59} - 12 q^{61} - 12 q^{62} + 5 q^{64} + 2 q^{67} + 18 q^{68} - 28 q^{70} + 14 q^{71} + 22 q^{73} - 28 q^{74} - 6 q^{76} - 6 q^{77} - 10 q^{79} + 26 q^{80} + 42 q^{82} + 2 q^{83} + 48 q^{85} - 2 q^{86} + 9 q^{88} - 10 q^{89} - 6 q^{91} - 17 q^{92} - 14 q^{94} - 2 q^{95} + 22 q^{97} + 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.57641 −1.82180 −0.910900 0.412627i \(-0.864611\pi\)
−0.910900 + 0.412627i \(0.864611\pi\)
\(3\) 0 0
\(4\) 4.63791 2.31896
\(5\) 4.12300 1.84386 0.921930 0.387356i \(-0.126611\pi\)
0.921930 + 0.387356i \(0.126611\pi\)
\(6\) 0 0
\(7\) 2.15845 0.815816 0.407908 0.913023i \(-0.366258\pi\)
0.407908 + 0.913023i \(0.366258\pi\)
\(8\) −6.79636 −2.40288
\(9\) 0 0
\(10\) −10.6226 −3.35915
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −5.56105 −1.48625
\(15\) 0 0
\(16\) 8.23442 2.05860
\(17\) 3.09317 0.750203 0.375101 0.926984i \(-0.377608\pi\)
0.375101 + 0.926984i \(0.377608\pi\)
\(18\) 0 0
\(19\) −2.68783 −0.616631 −0.308316 0.951284i \(-0.599765\pi\)
−0.308316 + 0.951284i \(0.599765\pi\)
\(20\) 19.1221 4.27583
\(21\) 0 0
\(22\) 2.57641 0.549294
\(23\) −3.31128 −0.690449 −0.345224 0.938520i \(-0.612197\pi\)
−0.345224 + 0.938520i \(0.612197\pi\)
\(24\) 0 0
\(25\) 11.9991 2.39982
\(26\) 2.57641 0.505277
\(27\) 0 0
\(28\) 10.0107 1.89184
\(29\) 3.34673 0.621471 0.310736 0.950496i \(-0.399425\pi\)
0.310736 + 0.950496i \(0.399425\pi\)
\(30\) 0 0
\(31\) 9.96928 1.79054 0.895268 0.445529i \(-0.146984\pi\)
0.895268 + 0.445529i \(0.146984\pi\)
\(32\) −7.62255 −1.34749
\(33\) 0 0
\(34\) −7.96928 −1.36672
\(35\) 8.89927 1.50425
\(36\) 0 0
\(37\) 2.05966 0.338607 0.169303 0.985564i \(-0.445848\pi\)
0.169303 + 0.985564i \(0.445848\pi\)
\(38\) 6.92497 1.12338
\(39\) 0 0
\(40\) −28.0214 −4.43057
\(41\) −9.49394 −1.48270 −0.741352 0.671116i \(-0.765815\pi\)
−0.741352 + 0.671116i \(0.765815\pi\)
\(42\) 0 0
\(43\) 2.84717 0.434189 0.217095 0.976151i \(-0.430342\pi\)
0.217095 + 0.976151i \(0.430342\pi\)
\(44\) −4.63791 −0.699192
\(45\) 0 0
\(46\) 8.53122 1.25786
\(47\) 9.46972 1.38130 0.690651 0.723189i \(-0.257324\pi\)
0.690651 + 0.723189i \(0.257324\pi\)
\(48\) 0 0
\(49\) −2.34111 −0.334444
\(50\) −30.9147 −4.37200
\(51\) 0 0
\(52\) −4.63791 −0.643163
\(53\) 5.34111 0.733658 0.366829 0.930288i \(-0.380443\pi\)
0.366829 + 0.930288i \(0.380443\pi\)
\(54\) 0 0
\(55\) −4.12300 −0.555945
\(56\) −14.6696 −1.96031
\(57\) 0 0
\(58\) −8.62255 −1.13220
\(59\) −4.62255 −0.601805 −0.300903 0.953655i \(-0.597288\pi\)
−0.300903 + 0.953655i \(0.597288\pi\)
\(60\) 0 0
\(61\) −0.193895 −0.0248258 −0.0124129 0.999923i \(-0.503951\pi\)
−0.0124129 + 0.999923i \(0.503951\pi\)
\(62\) −25.6850 −3.26200
\(63\) 0 0
\(64\) 3.17003 0.396253
\(65\) −4.12300 −0.511395
\(66\) 0 0
\(67\) −5.78662 −0.706948 −0.353474 0.935444i \(-0.615000\pi\)
−0.353474 + 0.935444i \(0.615000\pi\)
\(68\) 14.3458 1.73969
\(69\) 0 0
\(70\) −22.9282 −2.74045
\(71\) 3.78295 0.448953 0.224477 0.974479i \(-0.427933\pi\)
0.224477 + 0.974479i \(0.427933\pi\)
\(72\) 0 0
\(73\) 7.47534 0.874922 0.437461 0.899237i \(-0.355878\pi\)
0.437461 + 0.899237i \(0.355878\pi\)
\(74\) −5.30655 −0.616874
\(75\) 0 0
\(76\) −12.4659 −1.42994
\(77\) −2.15845 −0.245978
\(78\) 0 0
\(79\) −17.2049 −1.93571 −0.967853 0.251517i \(-0.919071\pi\)
−0.967853 + 0.251517i \(0.919071\pi\)
\(80\) 33.9505 3.79578
\(81\) 0 0
\(82\) 24.4603 2.70119
\(83\) 0.218111 0.0239408 0.0119704 0.999928i \(-0.496190\pi\)
0.0119704 + 0.999928i \(0.496190\pi\)
\(84\) 0 0
\(85\) 12.7531 1.38327
\(86\) −7.33549 −0.791006
\(87\) 0 0
\(88\) 6.79636 0.724494
\(89\) 1.52939 0.162115 0.0810574 0.996709i \(-0.474170\pi\)
0.0810574 + 0.996709i \(0.474170\pi\)
\(90\) 0 0
\(91\) −2.15845 −0.226267
\(92\) −15.3574 −1.60112
\(93\) 0 0
\(94\) −24.3979 −2.51645
\(95\) −11.0819 −1.13698
\(96\) 0 0
\(97\) 12.1462 1.23326 0.616628 0.787255i \(-0.288498\pi\)
0.616628 + 0.787255i \(0.288498\pi\)
\(98\) 6.03167 0.609290
\(99\) 0 0
\(100\) 55.6508 5.56508
\(101\) 13.2162 1.31506 0.657529 0.753429i \(-0.271602\pi\)
0.657529 + 0.753429i \(0.271602\pi\)
\(102\) 0 0
\(103\) −4.61783 −0.455008 −0.227504 0.973777i \(-0.573056\pi\)
−0.227504 + 0.973777i \(0.573056\pi\)
\(104\) 6.79636 0.666438
\(105\) 0 0
\(106\) −13.7609 −1.33658
\(107\) −10.3458 −1.00017 −0.500085 0.865976i \(-0.666698\pi\)
−0.500085 + 0.865976i \(0.666698\pi\)
\(108\) 0 0
\(109\) 6.83032 0.654226 0.327113 0.944985i \(-0.393924\pi\)
0.327113 + 0.944985i \(0.393924\pi\)
\(110\) 10.6226 1.01282
\(111\) 0 0
\(112\) 17.7735 1.67944
\(113\) 7.40077 0.696206 0.348103 0.937456i \(-0.386826\pi\)
0.348103 + 0.937456i \(0.386826\pi\)
\(114\) 0 0
\(115\) −13.6524 −1.27309
\(116\) 15.5218 1.44117
\(117\) 0 0
\(118\) 11.9096 1.09637
\(119\) 6.67643 0.612028
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.499555 0.0452276
\(123\) 0 0
\(124\) 46.2367 4.15217
\(125\) 28.8573 2.58108
\(126\) 0 0
\(127\) 8.40550 0.745867 0.372934 0.927858i \(-0.378352\pi\)
0.372934 + 0.927858i \(0.378352\pi\)
\(128\) 7.07780 0.625595
\(129\) 0 0
\(130\) 10.6226 0.931660
\(131\) 0.630117 0.0550536 0.0275268 0.999621i \(-0.491237\pi\)
0.0275268 + 0.999621i \(0.491237\pi\)
\(132\) 0 0
\(133\) −5.80155 −0.503058
\(134\) 14.9087 1.28792
\(135\) 0 0
\(136\) −21.0223 −1.80264
\(137\) −4.81645 −0.411497 −0.205748 0.978605i \(-0.565963\pi\)
−0.205748 + 0.978605i \(0.565963\pi\)
\(138\) 0 0
\(139\) −10.5592 −0.895621 −0.447811 0.894128i \(-0.647796\pi\)
−0.447811 + 0.894128i \(0.647796\pi\)
\(140\) 41.2740 3.48829
\(141\) 0 0
\(142\) −9.74644 −0.817903
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) 13.7985 1.14591
\(146\) −19.2596 −1.59393
\(147\) 0 0
\(148\) 9.55254 0.785214
\(149\) 2.80049 0.229425 0.114712 0.993399i \(-0.463405\pi\)
0.114712 + 0.993399i \(0.463405\pi\)
\(150\) 0 0
\(151\) −8.24600 −0.671050 −0.335525 0.942031i \(-0.608914\pi\)
−0.335525 + 0.942031i \(0.608914\pi\)
\(152\) 18.2675 1.48169
\(153\) 0 0
\(154\) 5.56105 0.448122
\(155\) 41.1033 3.30150
\(156\) 0 0
\(157\) −13.2918 −1.06080 −0.530400 0.847747i \(-0.677958\pi\)
−0.530400 + 0.847747i \(0.677958\pi\)
\(158\) 44.3270 3.52647
\(159\) 0 0
\(160\) −31.4278 −2.48458
\(161\) −7.14721 −0.563279
\(162\) 0 0
\(163\) −2.85084 −0.223295 −0.111647 0.993748i \(-0.535613\pi\)
−0.111647 + 0.993748i \(0.535613\pi\)
\(164\) −44.0321 −3.43833
\(165\) 0 0
\(166\) −0.561944 −0.0436153
\(167\) 1.30093 0.100669 0.0503346 0.998732i \(-0.483971\pi\)
0.0503346 + 0.998732i \(0.483971\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −32.8573 −2.52004
\(171\) 0 0
\(172\) 13.2049 1.00687
\(173\) −8.52850 −0.648410 −0.324205 0.945987i \(-0.605097\pi\)
−0.324205 + 0.945987i \(0.605097\pi\)
\(174\) 0 0
\(175\) 25.8994 1.95781
\(176\) −8.23442 −0.620692
\(177\) 0 0
\(178\) −3.94034 −0.295341
\(179\) −19.3048 −1.44291 −0.721453 0.692463i \(-0.756525\pi\)
−0.721453 + 0.692463i \(0.756525\pi\)
\(180\) 0 0
\(181\) −26.3402 −1.95785 −0.978927 0.204213i \(-0.934537\pi\)
−0.978927 + 0.204213i \(0.934537\pi\)
\(182\) 5.56105 0.412213
\(183\) 0 0
\(184\) 22.5046 1.65906
\(185\) 8.49199 0.624344
\(186\) 0 0
\(187\) −3.09317 −0.226195
\(188\) 43.9198 3.20318
\(189\) 0 0
\(190\) 28.5517 2.07135
\(191\) 1.86015 0.134596 0.0672979 0.997733i \(-0.478562\pi\)
0.0672979 + 0.997733i \(0.478562\pi\)
\(192\) 0 0
\(193\) 16.4239 1.18222 0.591110 0.806591i \(-0.298690\pi\)
0.591110 + 0.806591i \(0.298690\pi\)
\(194\) −31.2935 −2.24674
\(195\) 0 0
\(196\) −10.8579 −0.775561
\(197\) 17.5929 1.25344 0.626721 0.779244i \(-0.284397\pi\)
0.626721 + 0.779244i \(0.284397\pi\)
\(198\) 0 0
\(199\) 0.434274 0.0307849 0.0153924 0.999882i \(-0.495100\pi\)
0.0153924 + 0.999882i \(0.495100\pi\)
\(200\) −81.5503 −5.76648
\(201\) 0 0
\(202\) −34.0503 −2.39577
\(203\) 7.22373 0.507006
\(204\) 0 0
\(205\) −39.1435 −2.73390
\(206\) 11.8974 0.828934
\(207\) 0 0
\(208\) −8.23442 −0.570954
\(209\) 2.68783 0.185921
\(210\) 0 0
\(211\) 4.04018 0.278137 0.139069 0.990283i \(-0.455589\pi\)
0.139069 + 0.990283i \(0.455589\pi\)
\(212\) 24.7716 1.70132
\(213\) 0 0
\(214\) 26.6552 1.82211
\(215\) 11.7389 0.800585
\(216\) 0 0
\(217\) 21.5182 1.46075
\(218\) −17.5977 −1.19187
\(219\) 0 0
\(220\) −19.1221 −1.28921
\(221\) −3.09317 −0.208069
\(222\) 0 0
\(223\) 21.4390 1.43566 0.717831 0.696218i \(-0.245135\pi\)
0.717831 + 0.696218i \(0.245135\pi\)
\(224\) −16.4529 −1.09930
\(225\) 0 0
\(226\) −19.0675 −1.26835
\(227\) −18.8647 −1.25209 −0.626047 0.779785i \(-0.715328\pi\)
−0.626047 + 0.779785i \(0.715328\pi\)
\(228\) 0 0
\(229\) 13.3048 0.879204 0.439602 0.898193i \(-0.355120\pi\)
0.439602 + 0.898193i \(0.355120\pi\)
\(230\) 35.1742 2.31932
\(231\) 0 0
\(232\) −22.7456 −1.49332
\(233\) −21.3160 −1.39646 −0.698229 0.715875i \(-0.746028\pi\)
−0.698229 + 0.715875i \(0.746028\pi\)
\(234\) 0 0
\(235\) 39.0436 2.54693
\(236\) −21.4390 −1.39556
\(237\) 0 0
\(238\) −17.2013 −1.11499
\(239\) −13.9123 −0.899909 −0.449954 0.893052i \(-0.648560\pi\)
−0.449954 + 0.893052i \(0.648560\pi\)
\(240\) 0 0
\(241\) 0.565726 0.0364416 0.0182208 0.999834i \(-0.494200\pi\)
0.0182208 + 0.999834i \(0.494200\pi\)
\(242\) −2.57641 −0.165618
\(243\) 0 0
\(244\) −0.899270 −0.0575699
\(245\) −9.65238 −0.616668
\(246\) 0 0
\(247\) 2.68783 0.171023
\(248\) −67.7548 −4.30243
\(249\) 0 0
\(250\) −74.3484 −4.70221
\(251\) −12.4985 −0.788898 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(252\) 0 0
\(253\) 3.31128 0.208178
\(254\) −21.6561 −1.35882
\(255\) 0 0
\(256\) −24.5754 −1.53596
\(257\) −14.7640 −0.920952 −0.460476 0.887672i \(-0.652321\pi\)
−0.460476 + 0.887672i \(0.652321\pi\)
\(258\) 0 0
\(259\) 4.44568 0.276241
\(260\) −19.1221 −1.18590
\(261\) 0 0
\(262\) −1.62344 −0.100297
\(263\) 15.8445 0.977014 0.488507 0.872560i \(-0.337542\pi\)
0.488507 + 0.872560i \(0.337542\pi\)
\(264\) 0 0
\(265\) 22.0214 1.35276
\(266\) 14.9472 0.916471
\(267\) 0 0
\(268\) −26.8378 −1.63938
\(269\) 2.34289 0.142848 0.0714242 0.997446i \(-0.477246\pi\)
0.0714242 + 0.997446i \(0.477246\pi\)
\(270\) 0 0
\(271\) 26.0038 1.57962 0.789810 0.613351i \(-0.210179\pi\)
0.789810 + 0.613351i \(0.210179\pi\)
\(272\) 25.4704 1.54437
\(273\) 0 0
\(274\) 12.4092 0.749665
\(275\) −11.9991 −0.723574
\(276\) 0 0
\(277\) 28.0019 1.68247 0.841235 0.540669i \(-0.181829\pi\)
0.841235 + 0.540669i \(0.181829\pi\)
\(278\) 27.2049 1.63164
\(279\) 0 0
\(280\) −60.4826 −3.61453
\(281\) 20.4632 1.22073 0.610367 0.792119i \(-0.291022\pi\)
0.610367 + 0.792119i \(0.291022\pi\)
\(282\) 0 0
\(283\) −9.17810 −0.545582 −0.272791 0.962073i \(-0.587947\pi\)
−0.272791 + 0.962073i \(0.587947\pi\)
\(284\) 17.5450 1.04110
\(285\) 0 0
\(286\) −2.57641 −0.152347
\(287\) −20.4922 −1.20961
\(288\) 0 0
\(289\) −7.43233 −0.437196
\(290\) −35.5508 −2.08761
\(291\) 0 0
\(292\) 34.6700 2.02891
\(293\) −18.7380 −1.09468 −0.547342 0.836909i \(-0.684360\pi\)
−0.547342 + 0.836909i \(0.684360\pi\)
\(294\) 0 0
\(295\) −19.0588 −1.10964
\(296\) −13.9982 −0.813630
\(297\) 0 0
\(298\) −7.21522 −0.417966
\(299\) 3.31128 0.191496
\(300\) 0 0
\(301\) 6.14546 0.354219
\(302\) 21.2451 1.22252
\(303\) 0 0
\(304\) −22.1327 −1.26940
\(305\) −0.799430 −0.0457752
\(306\) 0 0
\(307\) 3.49288 0.199349 0.0996746 0.995020i \(-0.468220\pi\)
0.0996746 + 0.995020i \(0.468220\pi\)
\(308\) −10.0107 −0.570412
\(309\) 0 0
\(310\) −105.899 −6.01467
\(311\) 4.19951 0.238132 0.119066 0.992886i \(-0.462010\pi\)
0.119066 + 0.992886i \(0.462010\pi\)
\(312\) 0 0
\(313\) −12.4678 −0.704720 −0.352360 0.935864i \(-0.614621\pi\)
−0.352360 + 0.935864i \(0.614621\pi\)
\(314\) 34.2452 1.93257
\(315\) 0 0
\(316\) −79.7950 −4.48882
\(317\) 8.27494 0.464767 0.232383 0.972624i \(-0.425348\pi\)
0.232383 + 0.972624i \(0.425348\pi\)
\(318\) 0 0
\(319\) −3.34673 −0.187381
\(320\) 13.0700 0.730636
\(321\) 0 0
\(322\) 18.4142 1.02618
\(323\) −8.31391 −0.462599
\(324\) 0 0
\(325\) −11.9991 −0.665591
\(326\) 7.34495 0.406799
\(327\) 0 0
\(328\) 64.5242 3.56275
\(329\) 20.4399 1.12689
\(330\) 0 0
\(331\) −23.5852 −1.29636 −0.648179 0.761488i \(-0.724469\pi\)
−0.648179 + 0.761488i \(0.724469\pi\)
\(332\) 1.01158 0.0555176
\(333\) 0 0
\(334\) −3.35174 −0.183399
\(335\) −23.8582 −1.30351
\(336\) 0 0
\(337\) 0.664508 0.0361981 0.0180990 0.999836i \(-0.494239\pi\)
0.0180990 + 0.999836i \(0.494239\pi\)
\(338\) −2.57641 −0.140138
\(339\) 0 0
\(340\) 59.1478 3.20774
\(341\) −9.96928 −0.539867
\(342\) 0 0
\(343\) −20.1623 −1.08866
\(344\) −19.3504 −1.04330
\(345\) 0 0
\(346\) 21.9729 1.18127
\(347\) 1.33916 0.0718899 0.0359450 0.999354i \(-0.488556\pi\)
0.0359450 + 0.999354i \(0.488556\pi\)
\(348\) 0 0
\(349\) −1.48448 −0.0794626 −0.0397313 0.999210i \(-0.512650\pi\)
−0.0397313 + 0.999210i \(0.512650\pi\)
\(350\) −66.7277 −3.56675
\(351\) 0 0
\(352\) 7.62255 0.406283
\(353\) −6.04843 −0.321925 −0.160963 0.986960i \(-0.551460\pi\)
−0.160963 + 0.986960i \(0.551460\pi\)
\(354\) 0 0
\(355\) 15.5971 0.827807
\(356\) 7.09317 0.375937
\(357\) 0 0
\(358\) 49.7371 2.62869
\(359\) 16.7101 0.881925 0.440963 0.897525i \(-0.354637\pi\)
0.440963 + 0.897525i \(0.354637\pi\)
\(360\) 0 0
\(361\) −11.7756 −0.619766
\(362\) 67.8633 3.56682
\(363\) 0 0
\(364\) −10.0107 −0.524703
\(365\) 30.8208 1.61323
\(366\) 0 0
\(367\) 18.2943 0.954953 0.477476 0.878645i \(-0.341552\pi\)
0.477476 + 0.878645i \(0.341552\pi\)
\(368\) −27.2664 −1.42136
\(369\) 0 0
\(370\) −21.8789 −1.13743
\(371\) 11.5285 0.598530
\(372\) 0 0
\(373\) 16.5748 0.858211 0.429106 0.903254i \(-0.358829\pi\)
0.429106 + 0.903254i \(0.358829\pi\)
\(374\) 7.96928 0.412082
\(375\) 0 0
\(376\) −64.3596 −3.31910
\(377\) −3.34673 −0.172365
\(378\) 0 0
\(379\) −26.9952 −1.38665 −0.693326 0.720625i \(-0.743855\pi\)
−0.693326 + 0.720625i \(0.743855\pi\)
\(380\) −51.3970 −2.63661
\(381\) 0 0
\(382\) −4.79252 −0.245207
\(383\) −9.35039 −0.477783 −0.238891 0.971046i \(-0.576784\pi\)
−0.238891 + 0.971046i \(0.576784\pi\)
\(384\) 0 0
\(385\) −8.89927 −0.453549
\(386\) −42.3149 −2.15377
\(387\) 0 0
\(388\) 56.3328 2.85987
\(389\) −17.8258 −0.903802 −0.451901 0.892068i \(-0.649254\pi\)
−0.451901 + 0.892068i \(0.649254\pi\)
\(390\) 0 0
\(391\) −10.2423 −0.517977
\(392\) 15.9110 0.803628
\(393\) 0 0
\(394\) −45.3266 −2.28352
\(395\) −70.9359 −3.56917
\(396\) 0 0
\(397\) 36.0101 1.80730 0.903649 0.428275i \(-0.140878\pi\)
0.903649 + 0.428275i \(0.140878\pi\)
\(398\) −1.11887 −0.0560839
\(399\) 0 0
\(400\) 98.8057 4.94028
\(401\) −23.4604 −1.17156 −0.585778 0.810472i \(-0.699211\pi\)
−0.585778 + 0.810472i \(0.699211\pi\)
\(402\) 0 0
\(403\) −9.96928 −0.496605
\(404\) 61.2954 3.04956
\(405\) 0 0
\(406\) −18.6113 −0.923664
\(407\) −2.05966 −0.102094
\(408\) 0 0
\(409\) 14.8798 0.735758 0.367879 0.929874i \(-0.380084\pi\)
0.367879 + 0.929874i \(0.380084\pi\)
\(410\) 100.850 4.98062
\(411\) 0 0
\(412\) −21.4171 −1.05514
\(413\) −9.97753 −0.490962
\(414\) 0 0
\(415\) 0.899270 0.0441434
\(416\) 7.62255 0.373726
\(417\) 0 0
\(418\) −6.92497 −0.338712
\(419\) 27.9935 1.36757 0.683786 0.729683i \(-0.260332\pi\)
0.683786 + 0.729683i \(0.260332\pi\)
\(420\) 0 0
\(421\) −13.3185 −0.649103 −0.324551 0.945868i \(-0.605213\pi\)
−0.324551 + 0.945868i \(0.605213\pi\)
\(422\) −10.4092 −0.506710
\(423\) 0 0
\(424\) −36.3001 −1.76289
\(425\) 37.1152 1.80035
\(426\) 0 0
\(427\) −0.418513 −0.0202533
\(428\) −47.9831 −2.31935
\(429\) 0 0
\(430\) −30.2442 −1.45851
\(431\) −12.2181 −0.588526 −0.294263 0.955725i \(-0.595074\pi\)
−0.294263 + 0.955725i \(0.595074\pi\)
\(432\) 0 0
\(433\) −27.2663 −1.31034 −0.655168 0.755483i \(-0.727402\pi\)
−0.655168 + 0.755483i \(0.727402\pi\)
\(434\) −55.4397 −2.66119
\(435\) 0 0
\(436\) 31.6784 1.51712
\(437\) 8.90016 0.425752
\(438\) 0 0
\(439\) −34.0428 −1.62477 −0.812386 0.583120i \(-0.801832\pi\)
−0.812386 + 0.583120i \(0.801832\pi\)
\(440\) 28.0214 1.33587
\(441\) 0 0
\(442\) 7.96928 0.379060
\(443\) 39.2983 1.86712 0.933558 0.358425i \(-0.116686\pi\)
0.933558 + 0.358425i \(0.116686\pi\)
\(444\) 0 0
\(445\) 6.30566 0.298917
\(446\) −55.2358 −2.61549
\(447\) 0 0
\(448\) 6.84233 0.323270
\(449\) −14.8508 −0.700854 −0.350427 0.936590i \(-0.613964\pi\)
−0.350427 + 0.936590i \(0.613964\pi\)
\(450\) 0 0
\(451\) 9.49394 0.447052
\(452\) 34.3241 1.61447
\(453\) 0 0
\(454\) 48.6033 2.28107
\(455\) −8.89927 −0.417204
\(456\) 0 0
\(457\) 19.7213 0.922525 0.461263 0.887264i \(-0.347397\pi\)
0.461263 + 0.887264i \(0.347397\pi\)
\(458\) −34.2786 −1.60173
\(459\) 0 0
\(460\) −63.3186 −2.95224
\(461\) 29.8161 1.38867 0.694337 0.719650i \(-0.255698\pi\)
0.694337 + 0.719650i \(0.255698\pi\)
\(462\) 0 0
\(463\) −16.5769 −0.770395 −0.385198 0.922834i \(-0.625867\pi\)
−0.385198 + 0.922834i \(0.625867\pi\)
\(464\) 27.5583 1.27936
\(465\) 0 0
\(466\) 54.9189 2.54407
\(467\) 1.67187 0.0773651 0.0386825 0.999252i \(-0.487684\pi\)
0.0386825 + 0.999252i \(0.487684\pi\)
\(468\) 0 0
\(469\) −12.4901 −0.576739
\(470\) −100.593 −4.63999
\(471\) 0 0
\(472\) 31.4165 1.44606
\(473\) −2.84717 −0.130913
\(474\) 0 0
\(475\) −32.2516 −1.47981
\(476\) 30.9647 1.41927
\(477\) 0 0
\(478\) 35.8437 1.63945
\(479\) 20.0579 0.916468 0.458234 0.888832i \(-0.348482\pi\)
0.458234 + 0.888832i \(0.348482\pi\)
\(480\) 0 0
\(481\) −2.05966 −0.0939126
\(482\) −1.45754 −0.0663893
\(483\) 0 0
\(484\) 4.63791 0.210814
\(485\) 50.0786 2.27395
\(486\) 0 0
\(487\) 4.74733 0.215122 0.107561 0.994198i \(-0.465696\pi\)
0.107561 + 0.994198i \(0.465696\pi\)
\(488\) 1.31778 0.0596532
\(489\) 0 0
\(490\) 24.8685 1.12345
\(491\) −42.4522 −1.91584 −0.957919 0.287037i \(-0.907330\pi\)
−0.957919 + 0.287037i \(0.907330\pi\)
\(492\) 0 0
\(493\) 10.3520 0.466230
\(494\) −6.92497 −0.311569
\(495\) 0 0
\(496\) 82.0912 3.68600
\(497\) 8.16529 0.366263
\(498\) 0 0
\(499\) −20.6880 −0.926122 −0.463061 0.886326i \(-0.653249\pi\)
−0.463061 + 0.886326i \(0.653249\pi\)
\(500\) 133.838 5.98541
\(501\) 0 0
\(502\) 32.2013 1.43722
\(503\) −3.88526 −0.173235 −0.0866175 0.996242i \(-0.527606\pi\)
−0.0866175 + 0.996242i \(0.527606\pi\)
\(504\) 0 0
\(505\) 54.4902 2.42478
\(506\) −8.53122 −0.379259
\(507\) 0 0
\(508\) 38.9840 1.72963
\(509\) −4.35776 −0.193154 −0.0965771 0.995326i \(-0.530789\pi\)
−0.0965771 + 0.995326i \(0.530789\pi\)
\(510\) 0 0
\(511\) 16.1351 0.713776
\(512\) 49.1608 2.17262
\(513\) 0 0
\(514\) 38.0381 1.67779
\(515\) −19.0393 −0.838971
\(516\) 0 0
\(517\) −9.46972 −0.416478
\(518\) −11.4539 −0.503256
\(519\) 0 0
\(520\) 28.0214 1.22882
\(521\) 21.4185 0.938360 0.469180 0.883103i \(-0.344550\pi\)
0.469180 + 0.883103i \(0.344550\pi\)
\(522\) 0 0
\(523\) 27.8835 1.21926 0.609630 0.792686i \(-0.291318\pi\)
0.609630 + 0.792686i \(0.291318\pi\)
\(524\) 2.92243 0.127667
\(525\) 0 0
\(526\) −40.8220 −1.77992
\(527\) 30.8366 1.34326
\(528\) 0 0
\(529\) −12.0354 −0.523280
\(530\) −56.7362 −2.46446
\(531\) 0 0
\(532\) −26.9071 −1.16657
\(533\) 9.49394 0.411228
\(534\) 0 0
\(535\) −42.6559 −1.84417
\(536\) 39.3279 1.69871
\(537\) 0 0
\(538\) −6.03625 −0.260241
\(539\) 2.34111 0.100839
\(540\) 0 0
\(541\) 16.1874 0.695950 0.347975 0.937504i \(-0.386869\pi\)
0.347975 + 0.937504i \(0.386869\pi\)
\(542\) −66.9967 −2.87775
\(543\) 0 0
\(544\) −23.5778 −1.01089
\(545\) 28.1614 1.20630
\(546\) 0 0
\(547\) −16.5264 −0.706617 −0.353309 0.935507i \(-0.614943\pi\)
−0.353309 + 0.935507i \(0.614943\pi\)
\(548\) −22.3383 −0.954244
\(549\) 0 0
\(550\) 30.9147 1.31821
\(551\) −8.99544 −0.383219
\(552\) 0 0
\(553\) −37.1359 −1.57918
\(554\) −72.1445 −3.06513
\(555\) 0 0
\(556\) −48.9727 −2.07691
\(557\) 27.0075 1.14434 0.572172 0.820133i \(-0.306101\pi\)
0.572172 + 0.820133i \(0.306101\pi\)
\(558\) 0 0
\(559\) −2.84717 −0.120422
\(560\) 73.2803 3.09666
\(561\) 0 0
\(562\) −52.7217 −2.22393
\(563\) −37.7276 −1.59003 −0.795015 0.606589i \(-0.792537\pi\)
−0.795015 + 0.606589i \(0.792537\pi\)
\(564\) 0 0
\(565\) 30.5134 1.28371
\(566\) 23.6466 0.993941
\(567\) 0 0
\(568\) −25.7103 −1.07878
\(569\) 7.39694 0.310096 0.155048 0.987907i \(-0.450447\pi\)
0.155048 + 0.987907i \(0.450447\pi\)
\(570\) 0 0
\(571\) 24.3578 1.01934 0.509670 0.860370i \(-0.329767\pi\)
0.509670 + 0.860370i \(0.329767\pi\)
\(572\) 4.63791 0.193921
\(573\) 0 0
\(574\) 52.7963 2.20368
\(575\) −39.7324 −1.65695
\(576\) 0 0
\(577\) 4.42797 0.184339 0.0921693 0.995743i \(-0.470620\pi\)
0.0921693 + 0.995743i \(0.470620\pi\)
\(578\) 19.1488 0.796483
\(579\) 0 0
\(580\) 63.9964 2.65731
\(581\) 0.470780 0.0195313
\(582\) 0 0
\(583\) −5.34111 −0.221206
\(584\) −50.8051 −2.10233
\(585\) 0 0
\(586\) 48.2768 1.99430
\(587\) −9.80799 −0.404819 −0.202410 0.979301i \(-0.564877\pi\)
−0.202410 + 0.979301i \(0.564877\pi\)
\(588\) 0 0
\(589\) −26.7958 −1.10410
\(590\) 49.1033 2.02155
\(591\) 0 0
\(592\) 16.9601 0.697057
\(593\) −23.4213 −0.961796 −0.480898 0.876777i \(-0.659689\pi\)
−0.480898 + 0.876777i \(0.659689\pi\)
\(594\) 0 0
\(595\) 27.5269 1.12849
\(596\) 12.9884 0.532026
\(597\) 0 0
\(598\) −8.53122 −0.348868
\(599\) 2.61327 0.106775 0.0533876 0.998574i \(-0.482998\pi\)
0.0533876 + 0.998574i \(0.482998\pi\)
\(600\) 0 0
\(601\) −27.4904 −1.12136 −0.560679 0.828034i \(-0.689460\pi\)
−0.560679 + 0.828034i \(0.689460\pi\)
\(602\) −15.8333 −0.645316
\(603\) 0 0
\(604\) −38.2442 −1.55614
\(605\) 4.12300 0.167624
\(606\) 0 0
\(607\) 35.0458 1.42246 0.711232 0.702958i \(-0.248138\pi\)
0.711232 + 0.702958i \(0.248138\pi\)
\(608\) 20.4882 0.830904
\(609\) 0 0
\(610\) 2.05966 0.0833934
\(611\) −9.46972 −0.383104
\(612\) 0 0
\(613\) −15.8368 −0.639641 −0.319821 0.947478i \(-0.603623\pi\)
−0.319821 + 0.947478i \(0.603623\pi\)
\(614\) −8.99911 −0.363175
\(615\) 0 0
\(616\) 14.6696 0.591054
\(617\) −19.0168 −0.765588 −0.382794 0.923834i \(-0.625038\pi\)
−0.382794 + 0.923834i \(0.625038\pi\)
\(618\) 0 0
\(619\) −26.7417 −1.07484 −0.537419 0.843315i \(-0.680601\pi\)
−0.537419 + 0.843315i \(0.680601\pi\)
\(620\) 190.634 7.65603
\(621\) 0 0
\(622\) −10.8197 −0.433830
\(623\) 3.30110 0.132256
\(624\) 0 0
\(625\) 58.9831 2.35932
\(626\) 32.1222 1.28386
\(627\) 0 0
\(628\) −61.6462 −2.45995
\(629\) 6.37088 0.254024
\(630\) 0 0
\(631\) −39.8528 −1.58651 −0.793257 0.608888i \(-0.791616\pi\)
−0.793257 + 0.608888i \(0.791616\pi\)
\(632\) 116.931 4.65126
\(633\) 0 0
\(634\) −21.3197 −0.846712
\(635\) 34.6559 1.37528
\(636\) 0 0
\(637\) 2.34111 0.0927581
\(638\) 8.62255 0.341370
\(639\) 0 0
\(640\) 29.1818 1.15351
\(641\) −9.85468 −0.389236 −0.194618 0.980879i \(-0.562347\pi\)
−0.194618 + 0.980879i \(0.562347\pi\)
\(642\) 0 0
\(643\) 10.3078 0.406499 0.203249 0.979127i \(-0.434850\pi\)
0.203249 + 0.979127i \(0.434850\pi\)
\(644\) −33.1482 −1.30622
\(645\) 0 0
\(646\) 21.4201 0.842762
\(647\) 24.4455 0.961052 0.480526 0.876980i \(-0.340446\pi\)
0.480526 + 0.876980i \(0.340446\pi\)
\(648\) 0 0
\(649\) 4.62255 0.181451
\(650\) 30.9147 1.21257
\(651\) 0 0
\(652\) −13.2219 −0.517811
\(653\) −26.5723 −1.03986 −0.519928 0.854210i \(-0.674041\pi\)
−0.519928 + 0.854210i \(0.674041\pi\)
\(654\) 0 0
\(655\) 2.59797 0.101511
\(656\) −78.1770 −3.05230
\(657\) 0 0
\(658\) −52.6616 −2.05296
\(659\) −22.5480 −0.878345 −0.439172 0.898403i \(-0.644728\pi\)
−0.439172 + 0.898403i \(0.644728\pi\)
\(660\) 0 0
\(661\) −44.8824 −1.74572 −0.872861 0.487969i \(-0.837738\pi\)
−0.872861 + 0.487969i \(0.837738\pi\)
\(662\) 60.7651 2.36170
\(663\) 0 0
\(664\) −1.48236 −0.0575267
\(665\) −23.9198 −0.927568
\(666\) 0 0
\(667\) −11.0819 −0.429094
\(668\) 6.03361 0.233448
\(669\) 0 0
\(670\) 61.4686 2.37474
\(671\) 0.193895 0.00748525
\(672\) 0 0
\(673\) −0.766928 −0.0295629 −0.0147815 0.999891i \(-0.504705\pi\)
−0.0147815 + 0.999891i \(0.504705\pi\)
\(674\) −1.71205 −0.0659457
\(675\) 0 0
\(676\) 4.63791 0.178381
\(677\) −22.6545 −0.870682 −0.435341 0.900266i \(-0.643372\pi\)
−0.435341 + 0.900266i \(0.643372\pi\)
\(678\) 0 0
\(679\) 26.2168 1.00611
\(680\) −86.6748 −3.32383
\(681\) 0 0
\(682\) 25.6850 0.983529
\(683\) −6.69712 −0.256258 −0.128129 0.991757i \(-0.540897\pi\)
−0.128129 + 0.991757i \(0.540897\pi\)
\(684\) 0 0
\(685\) −19.8582 −0.758743
\(686\) 51.9464 1.98332
\(687\) 0 0
\(688\) 23.4448 0.893824
\(689\) −5.34111 −0.203480
\(690\) 0 0
\(691\) −22.1563 −0.842865 −0.421433 0.906860i \(-0.638473\pi\)
−0.421433 + 0.906860i \(0.638473\pi\)
\(692\) −39.5544 −1.50363
\(693\) 0 0
\(694\) −3.45023 −0.130969
\(695\) −43.5356 −1.65140
\(696\) 0 0
\(697\) −29.3663 −1.11233
\(698\) 3.82465 0.144765
\(699\) 0 0
\(700\) 120.119 4.54009
\(701\) −14.7084 −0.555527 −0.277763 0.960650i \(-0.589593\pi\)
−0.277763 + 0.960650i \(0.589593\pi\)
\(702\) 0 0
\(703\) −5.53603 −0.208796
\(704\) −3.17003 −0.119475
\(705\) 0 0
\(706\) 15.5833 0.586484
\(707\) 28.5264 1.07285
\(708\) 0 0
\(709\) 28.8284 1.08267 0.541336 0.840806i \(-0.317919\pi\)
0.541336 + 0.840806i \(0.317919\pi\)
\(710\) −40.1846 −1.50810
\(711\) 0 0
\(712\) −10.3943 −0.389542
\(713\) −33.0110 −1.23627
\(714\) 0 0
\(715\) 4.12300 0.154191
\(716\) −89.5339 −3.34604
\(717\) 0 0
\(718\) −43.0522 −1.60669
\(719\) 8.10420 0.302236 0.151118 0.988516i \(-0.451713\pi\)
0.151118 + 0.988516i \(0.451713\pi\)
\(720\) 0 0
\(721\) −9.96733 −0.371203
\(722\) 30.3387 1.12909
\(723\) 0 0
\(724\) −122.164 −4.54018
\(725\) 40.1577 1.49142
\(726\) 0 0
\(727\) −19.1116 −0.708810 −0.354405 0.935092i \(-0.615317\pi\)
−0.354405 + 0.935092i \(0.615317\pi\)
\(728\) 14.6696 0.543691
\(729\) 0 0
\(730\) −79.4072 −2.93899
\(731\) 8.80677 0.325730
\(732\) 0 0
\(733\) −27.4827 −1.01510 −0.507548 0.861623i \(-0.669448\pi\)
−0.507548 + 0.861623i \(0.669448\pi\)
\(734\) −47.1336 −1.73973
\(735\) 0 0
\(736\) 25.2404 0.930373
\(737\) 5.78662 0.213153
\(738\) 0 0
\(739\) −3.99458 −0.146943 −0.0734715 0.997297i \(-0.523408\pi\)
−0.0734715 + 0.997297i \(0.523408\pi\)
\(740\) 39.3851 1.44783
\(741\) 0 0
\(742\) −29.7022 −1.09040
\(743\) −27.9386 −1.02497 −0.512483 0.858697i \(-0.671274\pi\)
−0.512483 + 0.858697i \(0.671274\pi\)
\(744\) 0 0
\(745\) 11.5464 0.423028
\(746\) −42.7036 −1.56349
\(747\) 0 0
\(748\) −14.3458 −0.524536
\(749\) −22.3309 −0.815955
\(750\) 0 0
\(751\) −16.3149 −0.595341 −0.297670 0.954669i \(-0.596210\pi\)
−0.297670 + 0.954669i \(0.596210\pi\)
\(752\) 77.9776 2.84355
\(753\) 0 0
\(754\) 8.62255 0.314015
\(755\) −33.9982 −1.23732
\(756\) 0 0
\(757\) 50.8067 1.84660 0.923301 0.384077i \(-0.125480\pi\)
0.923301 + 0.384077i \(0.125480\pi\)
\(758\) 69.5509 2.52620
\(759\) 0 0
\(760\) 75.3168 2.73203
\(761\) 7.06895 0.256249 0.128125 0.991758i \(-0.459104\pi\)
0.128125 + 0.991758i \(0.459104\pi\)
\(762\) 0 0
\(763\) 14.7429 0.533728
\(764\) 8.62722 0.312122
\(765\) 0 0
\(766\) 24.0905 0.870425
\(767\) 4.62255 0.166911
\(768\) 0 0
\(769\) −19.0465 −0.686834 −0.343417 0.939183i \(-0.611584\pi\)
−0.343417 + 0.939183i \(0.611584\pi\)
\(770\) 22.9282 0.826275
\(771\) 0 0
\(772\) 76.1728 2.74152
\(773\) 34.6902 1.24772 0.623861 0.781536i \(-0.285563\pi\)
0.623861 + 0.781536i \(0.285563\pi\)
\(774\) 0 0
\(775\) 119.622 4.29697
\(776\) −82.5496 −2.96336
\(777\) 0 0
\(778\) 45.9266 1.64655
\(779\) 25.5181 0.914282
\(780\) 0 0
\(781\) −3.78295 −0.135364
\(782\) 26.3885 0.943650
\(783\) 0 0
\(784\) −19.2777 −0.688488
\(785\) −54.8020 −1.95597
\(786\) 0 0
\(787\) −29.2246 −1.04174 −0.520872 0.853635i \(-0.674393\pi\)
−0.520872 + 0.853635i \(0.674393\pi\)
\(788\) 81.5943 2.90668
\(789\) 0 0
\(790\) 182.760 6.50232
\(791\) 15.9742 0.567976
\(792\) 0 0
\(793\) 0.193895 0.00688543
\(794\) −92.7771 −3.29254
\(795\) 0 0
\(796\) 2.01413 0.0713888
\(797\) 3.30655 0.117124 0.0585620 0.998284i \(-0.481348\pi\)
0.0585620 + 0.998284i \(0.481348\pi\)
\(798\) 0 0
\(799\) 29.2914 1.03626
\(800\) −91.4638 −3.23374
\(801\) 0 0
\(802\) 60.4437 2.13434
\(803\) −7.47534 −0.263799
\(804\) 0 0
\(805\) −29.4679 −1.03861
\(806\) 25.6850 0.904715
\(807\) 0 0
\(808\) −89.8218 −3.15992
\(809\) 9.10142 0.319989 0.159994 0.987118i \(-0.448852\pi\)
0.159994 + 0.987118i \(0.448852\pi\)
\(810\) 0 0
\(811\) 25.0849 0.880849 0.440425 0.897790i \(-0.354828\pi\)
0.440425 + 0.897790i \(0.354828\pi\)
\(812\) 33.5030 1.17573
\(813\) 0 0
\(814\) 5.30655 0.185994
\(815\) −11.7540 −0.411725
\(816\) 0 0
\(817\) −7.65272 −0.267735
\(818\) −38.3365 −1.34040
\(819\) 0 0
\(820\) −181.544 −6.33980
\(821\) −6.29231 −0.219603 −0.109802 0.993954i \(-0.535022\pi\)
−0.109802 + 0.993954i \(0.535022\pi\)
\(822\) 0 0
\(823\) −32.7466 −1.14147 −0.570737 0.821133i \(-0.693343\pi\)
−0.570737 + 0.821133i \(0.693343\pi\)
\(824\) 31.3844 1.09333
\(825\) 0 0
\(826\) 25.7063 0.894435
\(827\) 6.16948 0.214534 0.107267 0.994230i \(-0.465790\pi\)
0.107267 + 0.994230i \(0.465790\pi\)
\(828\) 0 0
\(829\) 16.0410 0.557127 0.278564 0.960418i \(-0.410142\pi\)
0.278564 + 0.960418i \(0.410142\pi\)
\(830\) −2.31689 −0.0804205
\(831\) 0 0
\(832\) −3.17003 −0.109901
\(833\) −7.24144 −0.250901
\(834\) 0 0
\(835\) 5.36374 0.185620
\(836\) 12.4659 0.431144
\(837\) 0 0
\(838\) −72.1229 −2.49144
\(839\) −27.1837 −0.938484 −0.469242 0.883070i \(-0.655473\pi\)
−0.469242 + 0.883070i \(0.655473\pi\)
\(840\) 0 0
\(841\) −17.7994 −0.613773
\(842\) 34.3139 1.18254
\(843\) 0 0
\(844\) 18.7380 0.644988
\(845\) 4.12300 0.141835
\(846\) 0 0
\(847\) 2.15845 0.0741651
\(848\) 43.9809 1.51031
\(849\) 0 0
\(850\) −95.6242 −3.27988
\(851\) −6.82012 −0.233791
\(852\) 0 0
\(853\) 24.2748 0.831152 0.415576 0.909558i \(-0.363580\pi\)
0.415576 + 0.909558i \(0.363580\pi\)
\(854\) 1.07826 0.0368974
\(855\) 0 0
\(856\) 70.3140 2.40328
\(857\) −16.7036 −0.570584 −0.285292 0.958441i \(-0.592091\pi\)
−0.285292 + 0.958441i \(0.592091\pi\)
\(858\) 0 0
\(859\) −28.9414 −0.987468 −0.493734 0.869613i \(-0.664368\pi\)
−0.493734 + 0.869613i \(0.664368\pi\)
\(860\) 54.4439 1.85652
\(861\) 0 0
\(862\) 31.4789 1.07218
\(863\) 33.6601 1.14580 0.572901 0.819625i \(-0.305818\pi\)
0.572901 + 0.819625i \(0.305818\pi\)
\(864\) 0 0
\(865\) −35.1630 −1.19558
\(866\) 70.2493 2.38717
\(867\) 0 0
\(868\) 99.7993 3.38741
\(869\) 17.2049 0.583637
\(870\) 0 0
\(871\) 5.78662 0.196072
\(872\) −46.4213 −1.57202
\(873\) 0 0
\(874\) −22.9305 −0.775636
\(875\) 62.2870 2.10568
\(876\) 0 0
\(877\) −56.6214 −1.91197 −0.955985 0.293417i \(-0.905208\pi\)
−0.955985 + 0.293417i \(0.905208\pi\)
\(878\) 87.7083 2.96001
\(879\) 0 0
\(880\) −33.9505 −1.14447
\(881\) −35.8236 −1.20693 −0.603465 0.797390i \(-0.706214\pi\)
−0.603465 + 0.797390i \(0.706214\pi\)
\(882\) 0 0
\(883\) −29.2843 −0.985493 −0.492747 0.870173i \(-0.664007\pi\)
−0.492747 + 0.870173i \(0.664007\pi\)
\(884\) −14.3458 −0.482503
\(885\) 0 0
\(886\) −101.249 −3.40151
\(887\) 36.8342 1.23677 0.618385 0.785875i \(-0.287787\pi\)
0.618385 + 0.785875i \(0.287787\pi\)
\(888\) 0 0
\(889\) 18.1428 0.608491
\(890\) −16.2460 −0.544567
\(891\) 0 0
\(892\) 99.4322 3.32924
\(893\) −25.4530 −0.851754
\(894\) 0 0
\(895\) −79.5935 −2.66052
\(896\) 15.2771 0.510371
\(897\) 0 0
\(898\) 38.2619 1.27682
\(899\) 33.3644 1.11277
\(900\) 0 0
\(901\) 16.5209 0.550392
\(902\) −24.4603 −0.814440
\(903\) 0 0
\(904\) −50.2983 −1.67290
\(905\) −108.601 −3.61001
\(906\) 0 0
\(907\) 0.491159 0.0163087 0.00815433 0.999967i \(-0.497404\pi\)
0.00815433 + 0.999967i \(0.497404\pi\)
\(908\) −87.4929 −2.90355
\(909\) 0 0
\(910\) 22.9282 0.760063
\(911\) 44.1275 1.46201 0.731005 0.682373i \(-0.239052\pi\)
0.731005 + 0.682373i \(0.239052\pi\)
\(912\) 0 0
\(913\) −0.218111 −0.00721841
\(914\) −50.8103 −1.68066
\(915\) 0 0
\(916\) 61.7064 2.03884
\(917\) 1.36007 0.0449136
\(918\) 0 0
\(919\) −43.8564 −1.44669 −0.723345 0.690487i \(-0.757396\pi\)
−0.723345 + 0.690487i \(0.757396\pi\)
\(920\) 92.7865 3.05908
\(921\) 0 0
\(922\) −76.8186 −2.52989
\(923\) −3.78295 −0.124517
\(924\) 0 0
\(925\) 24.7141 0.812596
\(926\) 42.7090 1.40351
\(927\) 0 0
\(928\) −25.5106 −0.837426
\(929\) 21.9917 0.721526 0.360763 0.932658i \(-0.382516\pi\)
0.360763 + 0.932658i \(0.382516\pi\)
\(930\) 0 0
\(931\) 6.29251 0.206229
\(932\) −98.8618 −3.23833
\(933\) 0 0
\(934\) −4.30744 −0.140944
\(935\) −12.7531 −0.417071
\(936\) 0 0
\(937\) −15.9198 −0.520076 −0.260038 0.965598i \(-0.583735\pi\)
−0.260038 + 0.965598i \(0.583735\pi\)
\(938\) 32.1797 1.05070
\(939\) 0 0
\(940\) 181.081 5.90621
\(941\) −7.79435 −0.254088 −0.127044 0.991897i \(-0.540549\pi\)
−0.127044 + 0.991897i \(0.540549\pi\)
\(942\) 0 0
\(943\) 31.4371 1.02373
\(944\) −38.0640 −1.23888
\(945\) 0 0
\(946\) 7.33549 0.238497
\(947\) −37.4253 −1.21616 −0.608079 0.793876i \(-0.708060\pi\)
−0.608079 + 0.793876i \(0.708060\pi\)
\(948\) 0 0
\(949\) −7.47534 −0.242660
\(950\) 83.0935 2.69591
\(951\) 0 0
\(952\) −45.3754 −1.47063
\(953\) −40.9457 −1.32636 −0.663181 0.748459i \(-0.730794\pi\)
−0.663181 + 0.748459i \(0.730794\pi\)
\(954\) 0 0
\(955\) 7.66940 0.248176
\(956\) −64.5238 −2.08685
\(957\) 0 0
\(958\) −51.6774 −1.66962
\(959\) −10.3960 −0.335706
\(960\) 0 0
\(961\) 68.3865 2.20602
\(962\) 5.30655 0.171090
\(963\) 0 0
\(964\) 2.62379 0.0845065
\(965\) 67.7158 2.17985
\(966\) 0 0
\(967\) −37.0738 −1.19221 −0.596107 0.802905i \(-0.703287\pi\)
−0.596107 + 0.802905i \(0.703287\pi\)
\(968\) −6.79636 −0.218443
\(969\) 0 0
\(970\) −129.023 −4.14268
\(971\) 9.13718 0.293226 0.146613 0.989194i \(-0.453163\pi\)
0.146613 + 0.989194i \(0.453163\pi\)
\(972\) 0 0
\(973\) −22.7915 −0.730662
\(974\) −12.2311 −0.391909
\(975\) 0 0
\(976\) −1.59662 −0.0511064
\(977\) −24.8647 −0.795491 −0.397745 0.917496i \(-0.630207\pi\)
−0.397745 + 0.917496i \(0.630207\pi\)
\(978\) 0 0
\(979\) −1.52939 −0.0488794
\(980\) −44.7669 −1.43003
\(981\) 0 0
\(982\) 109.374 3.49028
\(983\) 22.0037 0.701808 0.350904 0.936411i \(-0.385874\pi\)
0.350904 + 0.936411i \(0.385874\pi\)
\(984\) 0 0
\(985\) 72.5354 2.31117
\(986\) −26.6710 −0.849377
\(987\) 0 0
\(988\) 12.4659 0.396594
\(989\) −9.42777 −0.299786
\(990\) 0 0
\(991\) 44.3616 1.40919 0.704596 0.709608i \(-0.251128\pi\)
0.704596 + 0.709608i \(0.251128\pi\)
\(992\) −75.9913 −2.41273
\(993\) 0 0
\(994\) −21.0372 −0.667259
\(995\) 1.79051 0.0567630
\(996\) 0 0
\(997\) −0.262678 −0.00831910 −0.00415955 0.999991i \(-0.501324\pi\)
−0.00415955 + 0.999991i \(0.501324\pi\)
\(998\) 53.3009 1.68721
\(999\) 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 1287.2.a.k.1.1 4
3.2 odd 2 143.2.a.b.1.4 4
12.11 even 2 2288.2.a.x.1.3 4
15.14 odd 2 3575.2.a.k.1.1 4
21.20 even 2 7007.2.a.n.1.4 4
24.5 odd 2 9152.2.a.ch.1.3 4
24.11 even 2 9152.2.a.cg.1.2 4
33.32 even 2 1573.2.a.f.1.1 4
39.38 odd 2 1859.2.a.i.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
143.2.a.b.1.4 4 3.2 odd 2
1287.2.a.k.1.1 4 1.1 even 1 trivial
1573.2.a.f.1.1 4 33.32 even 2
1859.2.a.i.1.1 4 39.38 odd 2
2288.2.a.x.1.3 4 12.11 even 2
3575.2.a.k.1.1 4 15.14 odd 2
7007.2.a.n.1.4 4 21.20 even 2
9152.2.a.cg.1.2 4 24.11 even 2
9152.2.a.ch.1.3 4 24.5 odd 2