Properties

Label 1143.2.a.e.1.2
Level $1143$
Weight $2$
Character 1143.1
Self dual yes
Analytic conductor $9.127$
Analytic rank $0$
Dimension $3$
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] = [1143,2,Mod(1,1143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1143.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.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: \(9.12690095103\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 127)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 1143.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.34730 q^{2} -0.184793 q^{4} +3.53209 q^{5} -2.53209 q^{7} -2.94356 q^{8} +O(q^{10})\) \(q+1.34730 q^{2} -0.184793 q^{4} +3.53209 q^{5} -2.53209 q^{7} -2.94356 q^{8} +4.75877 q^{10} +2.57398 q^{11} +3.94356 q^{13} -3.41147 q^{14} -3.59627 q^{16} +5.65270 q^{17} -0.532089 q^{19} -0.652704 q^{20} +3.46791 q^{22} -0.411474 q^{23} +7.47565 q^{25} +5.31315 q^{26} +0.467911 q^{28} -0.162504 q^{29} +9.29086 q^{31} +1.04189 q^{32} +7.61587 q^{34} -8.94356 q^{35} +5.43376 q^{37} -0.716881 q^{38} -10.3969 q^{40} -3.51754 q^{41} -12.1925 q^{43} -0.475652 q^{44} -0.554378 q^{46} +6.14796 q^{47} -0.588526 q^{49} +10.0719 q^{50} -0.728741 q^{52} +10.0719 q^{53} +9.09152 q^{55} +7.45336 q^{56} -0.218941 q^{58} -2.57398 q^{59} -12.9709 q^{61} +12.5175 q^{62} +8.59627 q^{64} +13.9290 q^{65} +0.532089 q^{67} -1.04458 q^{68} -12.0496 q^{70} -6.84255 q^{71} -10.4192 q^{73} +7.32089 q^{74} +0.0983261 q^{76} -6.51754 q^{77} +0.568926 q^{79} -12.7023 q^{80} -4.73917 q^{82} -15.4807 q^{83} +19.9659 q^{85} -16.4270 q^{86} -7.57667 q^{88} +13.0223 q^{89} -9.98545 q^{91} +0.0760373 q^{92} +8.28312 q^{94} -1.87939 q^{95} -15.2344 q^{97} -0.792919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{10} - 3 q^{13} + 3 q^{16} + 18 q^{17} + 3 q^{19} - 3 q^{20} + 15 q^{22} + 9 q^{23} + 3 q^{25} - 6 q^{26} + 6 q^{28} - 3 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} + 6 q^{38} - 3 q^{40} + 12 q^{41} - 9 q^{43} + 18 q^{44} + 9 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 21 q^{52} - 3 q^{53} - 3 q^{55} + 9 q^{56} - 18 q^{58} - 3 q^{61} + 15 q^{62} + 12 q^{64} + 9 q^{65} - 3 q^{67} + 9 q^{68} - 9 q^{70} - 3 q^{71} + 3 q^{73} - 24 q^{74} + 12 q^{76} + 3 q^{77} + 9 q^{79} - 12 q^{80} - 12 q^{83} + 39 q^{85} + 9 q^{86} - 6 q^{88} + 33 q^{89} - 12 q^{91} + 18 q^{92} + 33 q^{94} - 15 q^{97} - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34730 0.952682 0.476341 0.879261i \(-0.341963\pi\)
0.476341 + 0.879261i \(0.341963\pi\)
\(3\) 0 0
\(4\) −0.184793 −0.0923963
\(5\) 3.53209 1.57960 0.789799 0.613366i \(-0.210185\pi\)
0.789799 + 0.613366i \(0.210185\pi\)
\(6\) 0 0
\(7\) −2.53209 −0.957040 −0.478520 0.878077i \(-0.658827\pi\)
−0.478520 + 0.878077i \(0.658827\pi\)
\(8\) −2.94356 −1.04071
\(9\) 0 0
\(10\) 4.75877 1.50486
\(11\) 2.57398 0.776084 0.388042 0.921642i \(-0.373152\pi\)
0.388042 + 0.921642i \(0.373152\pi\)
\(12\) 0 0
\(13\) 3.94356 1.09375 0.546874 0.837215i \(-0.315818\pi\)
0.546874 + 0.837215i \(0.315818\pi\)
\(14\) −3.41147 −0.911755
\(15\) 0 0
\(16\) −3.59627 −0.899067
\(17\) 5.65270 1.37098 0.685491 0.728081i \(-0.259588\pi\)
0.685491 + 0.728081i \(0.259588\pi\)
\(18\) 0 0
\(19\) −0.532089 −0.122070 −0.0610348 0.998136i \(-0.519440\pi\)
−0.0610348 + 0.998136i \(0.519440\pi\)
\(20\) −0.652704 −0.145949
\(21\) 0 0
\(22\) 3.46791 0.739361
\(23\) −0.411474 −0.0857983 −0.0428991 0.999079i \(-0.513659\pi\)
−0.0428991 + 0.999079i \(0.513659\pi\)
\(24\) 0 0
\(25\) 7.47565 1.49513
\(26\) 5.31315 1.04199
\(27\) 0 0
\(28\) 0.467911 0.0884269
\(29\) −0.162504 −0.0301762 −0.0150881 0.999886i \(-0.504803\pi\)
−0.0150881 + 0.999886i \(0.504803\pi\)
\(30\) 0 0
\(31\) 9.29086 1.66869 0.834344 0.551244i \(-0.185847\pi\)
0.834344 + 0.551244i \(0.185847\pi\)
\(32\) 1.04189 0.184182
\(33\) 0 0
\(34\) 7.61587 1.30611
\(35\) −8.94356 −1.51174
\(36\) 0 0
\(37\) 5.43376 0.893305 0.446653 0.894707i \(-0.352616\pi\)
0.446653 + 0.894707i \(0.352616\pi\)
\(38\) −0.716881 −0.116294
\(39\) 0 0
\(40\) −10.3969 −1.64390
\(41\) −3.51754 −0.549348 −0.274674 0.961537i \(-0.588570\pi\)
−0.274674 + 0.961537i \(0.588570\pi\)
\(42\) 0 0
\(43\) −12.1925 −1.85934 −0.929672 0.368388i \(-0.879910\pi\)
−0.929672 + 0.368388i \(0.879910\pi\)
\(44\) −0.475652 −0.0717072
\(45\) 0 0
\(46\) −0.554378 −0.0817385
\(47\) 6.14796 0.896772 0.448386 0.893840i \(-0.351999\pi\)
0.448386 + 0.893840i \(0.351999\pi\)
\(48\) 0 0
\(49\) −0.588526 −0.0840751
\(50\) 10.0719 1.42438
\(51\) 0 0
\(52\) −0.728741 −0.101058
\(53\) 10.0719 1.38348 0.691742 0.722145i \(-0.256843\pi\)
0.691742 + 0.722145i \(0.256843\pi\)
\(54\) 0 0
\(55\) 9.09152 1.22590
\(56\) 7.45336 0.995998
\(57\) 0 0
\(58\) −0.218941 −0.0287483
\(59\) −2.57398 −0.335103 −0.167552 0.985863i \(-0.553586\pi\)
−0.167552 + 0.985863i \(0.553586\pi\)
\(60\) 0 0
\(61\) −12.9709 −1.66075 −0.830377 0.557202i \(-0.811875\pi\)
−0.830377 + 0.557202i \(0.811875\pi\)
\(62\) 12.5175 1.58973
\(63\) 0 0
\(64\) 8.59627 1.07453
\(65\) 13.9290 1.72768
\(66\) 0 0
\(67\) 0.532089 0.0650050 0.0325025 0.999472i \(-0.489652\pi\)
0.0325025 + 0.999472i \(0.489652\pi\)
\(68\) −1.04458 −0.126674
\(69\) 0 0
\(70\) −12.0496 −1.44021
\(71\) −6.84255 −0.812061 −0.406031 0.913859i \(-0.633087\pi\)
−0.406031 + 0.913859i \(0.633087\pi\)
\(72\) 0 0
\(73\) −10.4192 −1.21948 −0.609738 0.792603i \(-0.708726\pi\)
−0.609738 + 0.792603i \(0.708726\pi\)
\(74\) 7.32089 0.851036
\(75\) 0 0
\(76\) 0.0983261 0.0112788
\(77\) −6.51754 −0.742743
\(78\) 0 0
\(79\) 0.568926 0.0640091 0.0320046 0.999488i \(-0.489811\pi\)
0.0320046 + 0.999488i \(0.489811\pi\)
\(80\) −12.7023 −1.42016
\(81\) 0 0
\(82\) −4.73917 −0.523354
\(83\) −15.4807 −1.69923 −0.849614 0.527405i \(-0.823165\pi\)
−0.849614 + 0.527405i \(0.823165\pi\)
\(84\) 0 0
\(85\) 19.9659 2.16560
\(86\) −16.4270 −1.77136
\(87\) 0 0
\(88\) −7.57667 −0.807675
\(89\) 13.0223 1.38036 0.690180 0.723638i \(-0.257531\pi\)
0.690180 + 0.723638i \(0.257531\pi\)
\(90\) 0 0
\(91\) −9.98545 −1.04676
\(92\) 0.0760373 0.00792744
\(93\) 0 0
\(94\) 8.28312 0.854338
\(95\) −1.87939 −0.192821
\(96\) 0 0
\(97\) −15.2344 −1.54682 −0.773411 0.633905i \(-0.781451\pi\)
−0.773411 + 0.633905i \(0.781451\pi\)
\(98\) −0.792919 −0.0800969
\(99\) 0 0
\(100\) −1.38144 −0.138144
\(101\) 8.43107 0.838923 0.419462 0.907773i \(-0.362219\pi\)
0.419462 + 0.907773i \(0.362219\pi\)
\(102\) 0 0
\(103\) −3.10607 −0.306050 −0.153025 0.988222i \(-0.548901\pi\)
−0.153025 + 0.988222i \(0.548901\pi\)
\(104\) −11.6081 −1.13827
\(105\) 0 0
\(106\) 13.5699 1.31802
\(107\) −7.50980 −0.726000 −0.363000 0.931789i \(-0.618247\pi\)
−0.363000 + 0.931789i \(0.618247\pi\)
\(108\) 0 0
\(109\) 11.4534 1.09703 0.548517 0.836140i \(-0.315193\pi\)
0.548517 + 0.836140i \(0.315193\pi\)
\(110\) 12.2490 1.16789
\(111\) 0 0
\(112\) 9.10607 0.860442
\(113\) 9.80066 0.921968 0.460984 0.887408i \(-0.347496\pi\)
0.460984 + 0.887408i \(0.347496\pi\)
\(114\) 0 0
\(115\) −1.45336 −0.135527
\(116\) 0.0300295 0.00278817
\(117\) 0 0
\(118\) −3.46791 −0.319247
\(119\) −14.3131 −1.31208
\(120\) 0 0
\(121\) −4.37464 −0.397694
\(122\) −17.4757 −1.58217
\(123\) 0 0
\(124\) −1.71688 −0.154181
\(125\) 8.74422 0.782107
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 9.49794 0.839507
\(129\) 0 0
\(130\) 18.7665 1.64593
\(131\) −6.76146 −0.590751 −0.295376 0.955381i \(-0.595445\pi\)
−0.295376 + 0.955381i \(0.595445\pi\)
\(132\) 0 0
\(133\) 1.34730 0.116825
\(134\) 0.716881 0.0619291
\(135\) 0 0
\(136\) −16.6391 −1.42679
\(137\) 11.0000 0.939793 0.469897 0.882721i \(-0.344291\pi\)
0.469897 + 0.882721i \(0.344291\pi\)
\(138\) 0 0
\(139\) 9.25671 0.785144 0.392572 0.919721i \(-0.371585\pi\)
0.392572 + 0.919721i \(0.371585\pi\)
\(140\) 1.65270 0.139679
\(141\) 0 0
\(142\) −9.21894 −0.773636
\(143\) 10.1506 0.848840
\(144\) 0 0
\(145\) −0.573978 −0.0476663
\(146\) −14.0378 −1.16177
\(147\) 0 0
\(148\) −1.00412 −0.0825381
\(149\) −18.9632 −1.55352 −0.776761 0.629795i \(-0.783139\pi\)
−0.776761 + 0.629795i \(0.783139\pi\)
\(150\) 0 0
\(151\) 17.0993 1.39152 0.695759 0.718275i \(-0.255068\pi\)
0.695759 + 0.718275i \(0.255068\pi\)
\(152\) 1.56624 0.127039
\(153\) 0 0
\(154\) −8.78106 −0.707598
\(155\) 32.8161 2.63586
\(156\) 0 0
\(157\) 3.41921 0.272883 0.136442 0.990648i \(-0.456433\pi\)
0.136442 + 0.990648i \(0.456433\pi\)
\(158\) 0.766511 0.0609804
\(159\) 0 0
\(160\) 3.68004 0.290933
\(161\) 1.04189 0.0821124
\(162\) 0 0
\(163\) 0.453363 0.0355101 0.0177551 0.999842i \(-0.494348\pi\)
0.0177551 + 0.999842i \(0.494348\pi\)
\(164\) 0.650015 0.0507577
\(165\) 0 0
\(166\) −20.8571 −1.61882
\(167\) 7.82295 0.605358 0.302679 0.953093i \(-0.402119\pi\)
0.302679 + 0.953093i \(0.402119\pi\)
\(168\) 0 0
\(169\) 2.55169 0.196284
\(170\) 26.8999 2.06313
\(171\) 0 0
\(172\) 2.25309 0.171796
\(173\) −19.8580 −1.50978 −0.754889 0.655853i \(-0.772309\pi\)
−0.754889 + 0.655853i \(0.772309\pi\)
\(174\) 0 0
\(175\) −18.9290 −1.43090
\(176\) −9.25671 −0.697751
\(177\) 0 0
\(178\) 17.5449 1.31504
\(179\) −11.8452 −0.885355 −0.442677 0.896681i \(-0.645971\pi\)
−0.442677 + 0.896681i \(0.645971\pi\)
\(180\) 0 0
\(181\) 4.47565 0.332673 0.166336 0.986069i \(-0.446806\pi\)
0.166336 + 0.986069i \(0.446806\pi\)
\(182\) −13.4534 −0.997230
\(183\) 0 0
\(184\) 1.21120 0.0892909
\(185\) 19.1925 1.41106
\(186\) 0 0
\(187\) 14.5499 1.06400
\(188\) −1.13610 −0.0828583
\(189\) 0 0
\(190\) −2.53209 −0.183697
\(191\) −5.74422 −0.415637 −0.207819 0.978167i \(-0.566636\pi\)
−0.207819 + 0.978167i \(0.566636\pi\)
\(192\) 0 0
\(193\) −8.91447 −0.641677 −0.320839 0.947134i \(-0.603965\pi\)
−0.320839 + 0.947134i \(0.603965\pi\)
\(194\) −20.5253 −1.47363
\(195\) 0 0
\(196\) 0.108755 0.00776823
\(197\) −17.6236 −1.25563 −0.627815 0.778363i \(-0.716051\pi\)
−0.627815 + 0.778363i \(0.716051\pi\)
\(198\) 0 0
\(199\) −14.1925 −1.00608 −0.503041 0.864263i \(-0.667786\pi\)
−0.503041 + 0.864263i \(0.667786\pi\)
\(200\) −22.0051 −1.55599
\(201\) 0 0
\(202\) 11.3592 0.799227
\(203\) 0.411474 0.0288798
\(204\) 0 0
\(205\) −12.4243 −0.867748
\(206\) −4.18479 −0.291568
\(207\) 0 0
\(208\) −14.1821 −0.983352
\(209\) −1.36959 −0.0947362
\(210\) 0 0
\(211\) −16.4192 −1.13035 −0.565173 0.824973i \(-0.691190\pi\)
−0.565173 + 0.824973i \(0.691190\pi\)
\(212\) −1.86122 −0.127829
\(213\) 0 0
\(214\) −10.1179 −0.691647
\(215\) −43.0651 −2.93702
\(216\) 0 0
\(217\) −23.5253 −1.59700
\(218\) 15.4311 1.04512
\(219\) 0 0
\(220\) −1.68004 −0.113269
\(221\) 22.2918 1.49951
\(222\) 0 0
\(223\) −23.0283 −1.54209 −0.771044 0.636782i \(-0.780265\pi\)
−0.771044 + 0.636782i \(0.780265\pi\)
\(224\) −2.63816 −0.176269
\(225\) 0 0
\(226\) 13.2044 0.878343
\(227\) 7.12061 0.472612 0.236306 0.971679i \(-0.424063\pi\)
0.236306 + 0.971679i \(0.424063\pi\)
\(228\) 0 0
\(229\) −9.34730 −0.617687 −0.308843 0.951113i \(-0.599942\pi\)
−0.308843 + 0.951113i \(0.599942\pi\)
\(230\) −1.95811 −0.129114
\(231\) 0 0
\(232\) 0.478340 0.0314046
\(233\) 13.6800 0.896210 0.448105 0.893981i \(-0.352099\pi\)
0.448105 + 0.893981i \(0.352099\pi\)
\(234\) 0 0
\(235\) 21.7151 1.41654
\(236\) 0.475652 0.0309623
\(237\) 0 0
\(238\) −19.2841 −1.25000
\(239\) 16.4415 1.06351 0.531756 0.846897i \(-0.321532\pi\)
0.531756 + 0.846897i \(0.321532\pi\)
\(240\) 0 0
\(241\) 8.52259 0.548989 0.274494 0.961589i \(-0.411490\pi\)
0.274494 + 0.961589i \(0.411490\pi\)
\(242\) −5.89393 −0.378876
\(243\) 0 0
\(244\) 2.39693 0.153447
\(245\) −2.07873 −0.132805
\(246\) 0 0
\(247\) −2.09833 −0.133513
\(248\) −27.3482 −1.73661
\(249\) 0 0
\(250\) 11.7811 0.745100
\(251\) −16.2713 −1.02703 −0.513516 0.858080i \(-0.671657\pi\)
−0.513516 + 0.858080i \(0.671657\pi\)
\(252\) 0 0
\(253\) −1.05913 −0.0665866
\(254\) −1.34730 −0.0845369
\(255\) 0 0
\(256\) −4.39599 −0.274750
\(257\) −5.78611 −0.360928 −0.180464 0.983582i \(-0.557760\pi\)
−0.180464 + 0.983582i \(0.557760\pi\)
\(258\) 0 0
\(259\) −13.7588 −0.854928
\(260\) −2.57398 −0.159631
\(261\) 0 0
\(262\) −9.10969 −0.562798
\(263\) −28.4270 −1.75288 −0.876441 0.481510i \(-0.840089\pi\)
−0.876441 + 0.481510i \(0.840089\pi\)
\(264\) 0 0
\(265\) 35.5749 2.18535
\(266\) 1.81521 0.111298
\(267\) 0 0
\(268\) −0.0983261 −0.00600622
\(269\) −6.84255 −0.417198 −0.208599 0.978001i \(-0.566890\pi\)
−0.208599 + 0.978001i \(0.566890\pi\)
\(270\) 0 0
\(271\) −26.0523 −1.58257 −0.791283 0.611450i \(-0.790586\pi\)
−0.791283 + 0.611450i \(0.790586\pi\)
\(272\) −20.3286 −1.23260
\(273\) 0 0
\(274\) 14.8203 0.895325
\(275\) 19.2422 1.16035
\(276\) 0 0
\(277\) −5.98814 −0.359792 −0.179896 0.983686i \(-0.557576\pi\)
−0.179896 + 0.983686i \(0.557576\pi\)
\(278\) 12.4715 0.747993
\(279\) 0 0
\(280\) 26.3259 1.57328
\(281\) −8.22937 −0.490923 −0.245462 0.969406i \(-0.578940\pi\)
−0.245462 + 0.969406i \(0.578940\pi\)
\(282\) 0 0
\(283\) 1.92808 0.114613 0.0573063 0.998357i \(-0.481749\pi\)
0.0573063 + 0.998357i \(0.481749\pi\)
\(284\) 1.26445 0.0750314
\(285\) 0 0
\(286\) 13.6759 0.808674
\(287\) 8.90673 0.525747
\(288\) 0 0
\(289\) 14.9531 0.879592
\(290\) −0.773318 −0.0454108
\(291\) 0 0
\(292\) 1.92539 0.112675
\(293\) 31.5577 1.84362 0.921810 0.387643i \(-0.126711\pi\)
0.921810 + 0.387643i \(0.126711\pi\)
\(294\) 0 0
\(295\) −9.09152 −0.529329
\(296\) −15.9946 −0.929669
\(297\) 0 0
\(298\) −25.5490 −1.48001
\(299\) −1.62267 −0.0938417
\(300\) 0 0
\(301\) 30.8726 1.77947
\(302\) 23.0378 1.32567
\(303\) 0 0
\(304\) 1.91353 0.109749
\(305\) −45.8144 −2.62332
\(306\) 0 0
\(307\) 0.830689 0.0474099 0.0237050 0.999719i \(-0.492454\pi\)
0.0237050 + 0.999719i \(0.492454\pi\)
\(308\) 1.20439 0.0686267
\(309\) 0 0
\(310\) 44.2131 2.51113
\(311\) 0.615867 0.0349226 0.0174613 0.999848i \(-0.494442\pi\)
0.0174613 + 0.999848i \(0.494442\pi\)
\(312\) 0 0
\(313\) 2.85298 0.161260 0.0806299 0.996744i \(-0.474307\pi\)
0.0806299 + 0.996744i \(0.474307\pi\)
\(314\) 4.60670 0.259971
\(315\) 0 0
\(316\) −0.105133 −0.00591420
\(317\) 22.7493 1.27773 0.638863 0.769320i \(-0.279405\pi\)
0.638863 + 0.769320i \(0.279405\pi\)
\(318\) 0 0
\(319\) −0.418281 −0.0234193
\(320\) 30.3628 1.69733
\(321\) 0 0
\(322\) 1.40373 0.0782270
\(323\) −3.00774 −0.167355
\(324\) 0 0
\(325\) 29.4807 1.63530
\(326\) 0.610815 0.0338299
\(327\) 0 0
\(328\) 10.3541 0.571710
\(329\) −15.5672 −0.858246
\(330\) 0 0
\(331\) 30.4388 1.67307 0.836534 0.547915i \(-0.184578\pi\)
0.836534 + 0.547915i \(0.184578\pi\)
\(332\) 2.86072 0.157002
\(333\) 0 0
\(334\) 10.5398 0.576714
\(335\) 1.87939 0.102682
\(336\) 0 0
\(337\) 12.9855 0.707363 0.353681 0.935366i \(-0.384930\pi\)
0.353681 + 0.935366i \(0.384930\pi\)
\(338\) 3.43788 0.186996
\(339\) 0 0
\(340\) −3.68954 −0.200093
\(341\) 23.9145 1.29504
\(342\) 0 0
\(343\) 19.2148 1.03750
\(344\) 35.8895 1.93503
\(345\) 0 0
\(346\) −26.7547 −1.43834
\(347\) 29.6382 1.59106 0.795530 0.605914i \(-0.207193\pi\)
0.795530 + 0.605914i \(0.207193\pi\)
\(348\) 0 0
\(349\) −23.6486 −1.26588 −0.632940 0.774201i \(-0.718152\pi\)
−0.632940 + 0.774201i \(0.718152\pi\)
\(350\) −25.5030 −1.36319
\(351\) 0 0
\(352\) 2.68180 0.142940
\(353\) 17.9118 0.953348 0.476674 0.879080i \(-0.341842\pi\)
0.476674 + 0.879080i \(0.341842\pi\)
\(354\) 0 0
\(355\) −24.1685 −1.28273
\(356\) −2.40642 −0.127540
\(357\) 0 0
\(358\) −15.9590 −0.843462
\(359\) 6.10607 0.322266 0.161133 0.986933i \(-0.448485\pi\)
0.161133 + 0.986933i \(0.448485\pi\)
\(360\) 0 0
\(361\) −18.7169 −0.985099
\(362\) 6.03003 0.316931
\(363\) 0 0
\(364\) 1.84524 0.0967167
\(365\) −36.8016 −1.92628
\(366\) 0 0
\(367\) 7.84793 0.409658 0.204829 0.978798i \(-0.434336\pi\)
0.204829 + 0.978798i \(0.434336\pi\)
\(368\) 1.47977 0.0771384
\(369\) 0 0
\(370\) 25.8580 1.34429
\(371\) −25.5030 −1.32405
\(372\) 0 0
\(373\) 6.66456 0.345078 0.172539 0.985003i \(-0.444803\pi\)
0.172539 + 0.985003i \(0.444803\pi\)
\(374\) 19.6031 1.01365
\(375\) 0 0
\(376\) −18.0969 −0.933276
\(377\) −0.640844 −0.0330051
\(378\) 0 0
\(379\) 2.69728 0.138550 0.0692750 0.997598i \(-0.477931\pi\)
0.0692750 + 0.997598i \(0.477931\pi\)
\(380\) 0.347296 0.0178159
\(381\) 0 0
\(382\) −7.73917 −0.395970
\(383\) 28.2686 1.44446 0.722228 0.691655i \(-0.243118\pi\)
0.722228 + 0.691655i \(0.243118\pi\)
\(384\) 0 0
\(385\) −23.0205 −1.17324
\(386\) −12.0104 −0.611315
\(387\) 0 0
\(388\) 2.81521 0.142921
\(389\) 28.0523 1.42231 0.711154 0.703036i \(-0.248173\pi\)
0.711154 + 0.703036i \(0.248173\pi\)
\(390\) 0 0
\(391\) −2.32594 −0.117628
\(392\) 1.73236 0.0874975
\(393\) 0 0
\(394\) −23.7442 −1.19622
\(395\) 2.00950 0.101109
\(396\) 0 0
\(397\) 4.51249 0.226475 0.113238 0.993568i \(-0.463878\pi\)
0.113238 + 0.993568i \(0.463878\pi\)
\(398\) −19.1215 −0.958477
\(399\) 0 0
\(400\) −26.8844 −1.34422
\(401\) −8.57398 −0.428164 −0.214082 0.976816i \(-0.568676\pi\)
−0.214082 + 0.976816i \(0.568676\pi\)
\(402\) 0 0
\(403\) 36.6391 1.82512
\(404\) −1.55800 −0.0775134
\(405\) 0 0
\(406\) 0.554378 0.0275133
\(407\) 13.9864 0.693279
\(408\) 0 0
\(409\) −1.10338 −0.0545586 −0.0272793 0.999628i \(-0.508684\pi\)
−0.0272793 + 0.999628i \(0.508684\pi\)
\(410\) −16.7392 −0.826689
\(411\) 0 0
\(412\) 0.573978 0.0282779
\(413\) 6.51754 0.320707
\(414\) 0 0
\(415\) −54.6792 −2.68410
\(416\) 4.10876 0.201448
\(417\) 0 0
\(418\) −1.84524 −0.0902535
\(419\) 31.7939 1.55323 0.776616 0.629975i \(-0.216935\pi\)
0.776616 + 0.629975i \(0.216935\pi\)
\(420\) 0 0
\(421\) 3.45067 0.168176 0.0840878 0.996458i \(-0.473202\pi\)
0.0840878 + 0.996458i \(0.473202\pi\)
\(422\) −22.1215 −1.07686
\(423\) 0 0
\(424\) −29.6473 −1.43980
\(425\) 42.2576 2.04980
\(426\) 0 0
\(427\) 32.8435 1.58941
\(428\) 1.38775 0.0670797
\(429\) 0 0
\(430\) −58.0215 −2.79804
\(431\) −19.3773 −0.933373 −0.466686 0.884423i \(-0.654552\pi\)
−0.466686 + 0.884423i \(0.654552\pi\)
\(432\) 0 0
\(433\) −36.1857 −1.73898 −0.869488 0.493955i \(-0.835551\pi\)
−0.869488 + 0.493955i \(0.835551\pi\)
\(434\) −31.6955 −1.52143
\(435\) 0 0
\(436\) −2.11650 −0.101362
\(437\) 0.218941 0.0104734
\(438\) 0 0
\(439\) −27.5827 −1.31645 −0.658224 0.752822i \(-0.728692\pi\)
−0.658224 + 0.752822i \(0.728692\pi\)
\(440\) −26.7615 −1.27580
\(441\) 0 0
\(442\) 30.0337 1.42856
\(443\) −5.07697 −0.241214 −0.120607 0.992700i \(-0.538484\pi\)
−0.120607 + 0.992700i \(0.538484\pi\)
\(444\) 0 0
\(445\) 45.9959 2.18041
\(446\) −31.0259 −1.46912
\(447\) 0 0
\(448\) −21.7665 −1.02837
\(449\) −1.12836 −0.0532504 −0.0266252 0.999645i \(-0.508476\pi\)
−0.0266252 + 0.999645i \(0.508476\pi\)
\(450\) 0 0
\(451\) −9.05407 −0.426340
\(452\) −1.81109 −0.0851864
\(453\) 0 0
\(454\) 9.59358 0.450249
\(455\) −35.2695 −1.65346
\(456\) 0 0
\(457\) 35.3678 1.65444 0.827219 0.561880i \(-0.189922\pi\)
0.827219 + 0.561880i \(0.189922\pi\)
\(458\) −12.5936 −0.588459
\(459\) 0 0
\(460\) 0.268571 0.0125222
\(461\) 19.1685 0.892766 0.446383 0.894842i \(-0.352712\pi\)
0.446383 + 0.894842i \(0.352712\pi\)
\(462\) 0 0
\(463\) 12.3601 0.574422 0.287211 0.957867i \(-0.407272\pi\)
0.287211 + 0.957867i \(0.407272\pi\)
\(464\) 0.584407 0.0271304
\(465\) 0 0
\(466\) 18.4311 0.853803
\(467\) 19.1566 0.886463 0.443232 0.896407i \(-0.353832\pi\)
0.443232 + 0.896407i \(0.353832\pi\)
\(468\) 0 0
\(469\) −1.34730 −0.0622124
\(470\) 29.2567 1.34951
\(471\) 0 0
\(472\) 7.57667 0.348744
\(473\) −31.3833 −1.44301
\(474\) 0 0
\(475\) −3.97771 −0.182510
\(476\) 2.64496 0.121232
\(477\) 0 0
\(478\) 22.1516 1.01319
\(479\) −24.0104 −1.09706 −0.548532 0.836129i \(-0.684813\pi\)
−0.548532 + 0.836129i \(0.684813\pi\)
\(480\) 0 0
\(481\) 21.4284 0.977050
\(482\) 11.4825 0.523012
\(483\) 0 0
\(484\) 0.808400 0.0367455
\(485\) −53.8093 −2.44336
\(486\) 0 0
\(487\) −0.811089 −0.0367539 −0.0183770 0.999831i \(-0.505850\pi\)
−0.0183770 + 0.999831i \(0.505850\pi\)
\(488\) 38.1807 1.72836
\(489\) 0 0
\(490\) −2.80066 −0.126521
\(491\) 5.59358 0.252435 0.126217 0.992003i \(-0.459716\pi\)
0.126217 + 0.992003i \(0.459716\pi\)
\(492\) 0 0
\(493\) −0.918586 −0.0413710
\(494\) −2.82707 −0.127196
\(495\) 0 0
\(496\) −33.4124 −1.50026
\(497\) 17.3259 0.777175
\(498\) 0 0
\(499\) 29.3928 1.31580 0.657901 0.753104i \(-0.271444\pi\)
0.657901 + 0.753104i \(0.271444\pi\)
\(500\) −1.61587 −0.0722638
\(501\) 0 0
\(502\) −21.9222 −0.978436
\(503\) −19.6774 −0.877370 −0.438685 0.898641i \(-0.644556\pi\)
−0.438685 + 0.898641i \(0.644556\pi\)
\(504\) 0 0
\(505\) 29.7793 1.32516
\(506\) −1.42696 −0.0634359
\(507\) 0 0
\(508\) 0.184793 0.00819884
\(509\) −11.5149 −0.510387 −0.255193 0.966890i \(-0.582139\pi\)
−0.255193 + 0.966890i \(0.582139\pi\)
\(510\) 0 0
\(511\) 26.3824 1.16709
\(512\) −24.9186 −1.10126
\(513\) 0 0
\(514\) −7.79561 −0.343849
\(515\) −10.9709 −0.483436
\(516\) 0 0
\(517\) 15.8247 0.695970
\(518\) −18.5371 −0.814475
\(519\) 0 0
\(520\) −41.0009 −1.79801
\(521\) 4.21213 0.184537 0.0922685 0.995734i \(-0.470588\pi\)
0.0922685 + 0.995734i \(0.470588\pi\)
\(522\) 0 0
\(523\) 40.4124 1.76711 0.883556 0.468326i \(-0.155143\pi\)
0.883556 + 0.468326i \(0.155143\pi\)
\(524\) 1.24947 0.0545832
\(525\) 0 0
\(526\) −38.2995 −1.66994
\(527\) 52.5185 2.28774
\(528\) 0 0
\(529\) −22.8307 −0.992639
\(530\) 47.9299 2.08194
\(531\) 0 0
\(532\) −0.248970 −0.0107942
\(533\) −13.8716 −0.600848
\(534\) 0 0
\(535\) −26.5253 −1.14679
\(536\) −1.56624 −0.0676511
\(537\) 0 0
\(538\) −9.21894 −0.397457
\(539\) −1.51485 −0.0652493
\(540\) 0 0
\(541\) 15.2635 0.656230 0.328115 0.944638i \(-0.393587\pi\)
0.328115 + 0.944638i \(0.393587\pi\)
\(542\) −35.1002 −1.50768
\(543\) 0 0
\(544\) 5.88949 0.252510
\(545\) 40.4543 1.73287
\(546\) 0 0
\(547\) −5.14971 −0.220186 −0.110093 0.993921i \(-0.535115\pi\)
−0.110093 + 0.993921i \(0.535115\pi\)
\(548\) −2.03272 −0.0868334
\(549\) 0 0
\(550\) 25.9249 1.10544
\(551\) 0.0864665 0.00368360
\(552\) 0 0
\(553\) −1.44057 −0.0612593
\(554\) −8.06780 −0.342768
\(555\) 0 0
\(556\) −1.71057 −0.0725444
\(557\) −5.26176 −0.222948 −0.111474 0.993767i \(-0.535557\pi\)
−0.111474 + 0.993767i \(0.535557\pi\)
\(558\) 0 0
\(559\) −48.0820 −2.03365
\(560\) 32.1634 1.35915
\(561\) 0 0
\(562\) −11.0874 −0.467694
\(563\) −4.39961 −0.185422 −0.0927108 0.995693i \(-0.529553\pi\)
−0.0927108 + 0.995693i \(0.529553\pi\)
\(564\) 0 0
\(565\) 34.6168 1.45634
\(566\) 2.59770 0.109189
\(567\) 0 0
\(568\) 20.1415 0.845117
\(569\) −15.1625 −0.635645 −0.317823 0.948150i \(-0.602952\pi\)
−0.317823 + 0.948150i \(0.602952\pi\)
\(570\) 0 0
\(571\) −13.9240 −0.582700 −0.291350 0.956617i \(-0.594104\pi\)
−0.291350 + 0.956617i \(0.594104\pi\)
\(572\) −1.87576 −0.0784296
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −3.07604 −0.128280
\(576\) 0 0
\(577\) −38.6117 −1.60743 −0.803714 0.595016i \(-0.797146\pi\)
−0.803714 + 0.595016i \(0.797146\pi\)
\(578\) 20.1462 0.837971
\(579\) 0 0
\(580\) 0.106067 0.00440419
\(581\) 39.1985 1.62623
\(582\) 0 0
\(583\) 25.9249 1.07370
\(584\) 30.6696 1.26912
\(585\) 0 0
\(586\) 42.5175 1.75638
\(587\) 20.4260 0.843072 0.421536 0.906812i \(-0.361491\pi\)
0.421536 + 0.906812i \(0.361491\pi\)
\(588\) 0 0
\(589\) −4.94356 −0.203696
\(590\) −12.2490 −0.504282
\(591\) 0 0
\(592\) −19.5413 −0.803141
\(593\) −8.53209 −0.350371 −0.175185 0.984535i \(-0.556053\pi\)
−0.175185 + 0.984535i \(0.556053\pi\)
\(594\) 0 0
\(595\) −50.5553 −2.07257
\(596\) 3.50425 0.143540
\(597\) 0 0
\(598\) −2.18622 −0.0894013
\(599\) 19.2294 0.785691 0.392845 0.919605i \(-0.371491\pi\)
0.392845 + 0.919605i \(0.371491\pi\)
\(600\) 0 0
\(601\) 6.25166 0.255010 0.127505 0.991838i \(-0.459303\pi\)
0.127505 + 0.991838i \(0.459303\pi\)
\(602\) 41.5945 1.69527
\(603\) 0 0
\(604\) −3.15982 −0.128571
\(605\) −15.4516 −0.628197
\(606\) 0 0
\(607\) 33.9718 1.37887 0.689437 0.724345i \(-0.257858\pi\)
0.689437 + 0.724345i \(0.257858\pi\)
\(608\) −0.554378 −0.0224830
\(609\) 0 0
\(610\) −61.7256 −2.49919
\(611\) 24.2449 0.980842
\(612\) 0 0
\(613\) −17.2422 −0.696404 −0.348202 0.937419i \(-0.613208\pi\)
−0.348202 + 0.937419i \(0.613208\pi\)
\(614\) 1.11918 0.0451666
\(615\) 0 0
\(616\) 19.1848 0.772977
\(617\) 11.5193 0.463749 0.231875 0.972746i \(-0.425514\pi\)
0.231875 + 0.972746i \(0.425514\pi\)
\(618\) 0 0
\(619\) −5.55531 −0.223287 −0.111643 0.993748i \(-0.535611\pi\)
−0.111643 + 0.993748i \(0.535611\pi\)
\(620\) −6.06418 −0.243543
\(621\) 0 0
\(622\) 0.829755 0.0332702
\(623\) −32.9736 −1.32106
\(624\) 0 0
\(625\) −6.49289 −0.259716
\(626\) 3.84381 0.153629
\(627\) 0 0
\(628\) −0.631845 −0.0252134
\(629\) 30.7155 1.22471
\(630\) 0 0
\(631\) 1.01960 0.0405896 0.0202948 0.999794i \(-0.493540\pi\)
0.0202948 + 0.999794i \(0.493540\pi\)
\(632\) −1.67467 −0.0666147
\(633\) 0 0
\(634\) 30.6500 1.21727
\(635\) −3.53209 −0.140167
\(636\) 0 0
\(637\) −2.32089 −0.0919570
\(638\) −0.563549 −0.0223111
\(639\) 0 0
\(640\) 33.5476 1.32608
\(641\) −12.9709 −0.512320 −0.256160 0.966634i \(-0.582457\pi\)
−0.256160 + 0.966634i \(0.582457\pi\)
\(642\) 0 0
\(643\) −11.8479 −0.467237 −0.233618 0.972328i \(-0.575057\pi\)
−0.233618 + 0.972328i \(0.575057\pi\)
\(644\) −0.192533 −0.00758688
\(645\) 0 0
\(646\) −4.05232 −0.159436
\(647\) 36.1293 1.42039 0.710194 0.704006i \(-0.248607\pi\)
0.710194 + 0.704006i \(0.248607\pi\)
\(648\) 0 0
\(649\) −6.62536 −0.260068
\(650\) 39.7192 1.55792
\(651\) 0 0
\(652\) −0.0837781 −0.00328100
\(653\) −39.9887 −1.56488 −0.782440 0.622726i \(-0.786025\pi\)
−0.782440 + 0.622726i \(0.786025\pi\)
\(654\) 0 0
\(655\) −23.8821 −0.933150
\(656\) 12.6500 0.493900
\(657\) 0 0
\(658\) −20.9736 −0.817636
\(659\) 12.8598 0.500946 0.250473 0.968124i \(-0.419414\pi\)
0.250473 + 0.968124i \(0.419414\pi\)
\(660\) 0 0
\(661\) 3.58946 0.139614 0.0698069 0.997561i \(-0.477762\pi\)
0.0698069 + 0.997561i \(0.477762\pi\)
\(662\) 41.0101 1.59390
\(663\) 0 0
\(664\) 45.5684 1.76840
\(665\) 4.75877 0.184537
\(666\) 0 0
\(667\) 0.0668661 0.00258907
\(668\) −1.44562 −0.0559328
\(669\) 0 0
\(670\) 2.53209 0.0978231
\(671\) −33.3868 −1.28888
\(672\) 0 0
\(673\) 34.2449 1.32004 0.660021 0.751247i \(-0.270547\pi\)
0.660021 + 0.751247i \(0.270547\pi\)
\(674\) 17.4953 0.673892
\(675\) 0 0
\(676\) −0.471533 −0.0181359
\(677\) 8.76382 0.336821 0.168411 0.985717i \(-0.446137\pi\)
0.168411 + 0.985717i \(0.446137\pi\)
\(678\) 0 0
\(679\) 38.5749 1.48037
\(680\) −58.7707 −2.25376
\(681\) 0 0
\(682\) 32.2199 1.23376
\(683\) −19.4439 −0.743999 −0.371999 0.928233i \(-0.621328\pi\)
−0.371999 + 0.928233i \(0.621328\pi\)
\(684\) 0 0
\(685\) 38.8530 1.48450
\(686\) 25.8881 0.988411
\(687\) 0 0
\(688\) 43.8476 1.67167
\(689\) 39.7192 1.51318
\(690\) 0 0
\(691\) 2.58441 0.0983155 0.0491577 0.998791i \(-0.484346\pi\)
0.0491577 + 0.998791i \(0.484346\pi\)
\(692\) 3.66962 0.139498
\(693\) 0 0
\(694\) 39.9314 1.51577
\(695\) 32.6955 1.24021
\(696\) 0 0
\(697\) −19.8836 −0.753146
\(698\) −31.8617 −1.20598
\(699\) 0 0
\(700\) 3.49794 0.132210
\(701\) −25.5458 −0.964852 −0.482426 0.875937i \(-0.660244\pi\)
−0.482426 + 0.875937i \(0.660244\pi\)
\(702\) 0 0
\(703\) −2.89124 −0.109045
\(704\) 22.1266 0.833928
\(705\) 0 0
\(706\) 24.1325 0.908238
\(707\) −21.3482 −0.802883
\(708\) 0 0
\(709\) 24.4192 0.917083 0.458541 0.888673i \(-0.348372\pi\)
0.458541 + 0.888673i \(0.348372\pi\)
\(710\) −32.5621 −1.22203
\(711\) 0 0
\(712\) −38.3319 −1.43655
\(713\) −3.82295 −0.143171
\(714\) 0 0
\(715\) 35.8530 1.34083
\(716\) 2.18891 0.0818035
\(717\) 0 0
\(718\) 8.22668 0.307017
\(719\) −28.9127 −1.07826 −0.539131 0.842222i \(-0.681247\pi\)
−0.539131 + 0.842222i \(0.681247\pi\)
\(720\) 0 0
\(721\) 7.86484 0.292902
\(722\) −25.2172 −0.938486
\(723\) 0 0
\(724\) −0.827067 −0.0307377
\(725\) −1.21482 −0.0451174
\(726\) 0 0
\(727\) −51.7752 −1.92024 −0.960118 0.279596i \(-0.909799\pi\)
−0.960118 + 0.279596i \(0.909799\pi\)
\(728\) 29.3928 1.08937
\(729\) 0 0
\(730\) −49.5827 −1.83514
\(731\) −68.9208 −2.54913
\(732\) 0 0
\(733\) −36.7023 −1.35563 −0.677816 0.735232i \(-0.737073\pi\)
−0.677816 + 0.735232i \(0.737073\pi\)
\(734\) 10.5735 0.390274
\(735\) 0 0
\(736\) −0.428710 −0.0158025
\(737\) 1.36959 0.0504493
\(738\) 0 0
\(739\) −47.8699 −1.76092 −0.880461 0.474118i \(-0.842767\pi\)
−0.880461 + 0.474118i \(0.842767\pi\)
\(740\) −3.54664 −0.130377
\(741\) 0 0
\(742\) −34.3601 −1.26140
\(743\) −24.6631 −0.904803 −0.452401 0.891814i \(-0.649433\pi\)
−0.452401 + 0.891814i \(0.649433\pi\)
\(744\) 0 0
\(745\) −66.9796 −2.45394
\(746\) 8.97914 0.328750
\(747\) 0 0
\(748\) −2.68872 −0.0983093
\(749\) 19.0155 0.694811
\(750\) 0 0
\(751\) 9.86753 0.360071 0.180036 0.983660i \(-0.442379\pi\)
0.180036 + 0.983660i \(0.442379\pi\)
\(752\) −22.1097 −0.806257
\(753\) 0 0
\(754\) −0.863407 −0.0314434
\(755\) 60.3961 2.19804
\(756\) 0 0
\(757\) 20.8939 0.759403 0.379701 0.925109i \(-0.376027\pi\)
0.379701 + 0.925109i \(0.376027\pi\)
\(758\) 3.63404 0.131994
\(759\) 0 0
\(760\) 5.53209 0.200670
\(761\) 38.2959 1.38823 0.694113 0.719866i \(-0.255797\pi\)
0.694113 + 0.719866i \(0.255797\pi\)
\(762\) 0 0
\(763\) −29.0009 −1.04990
\(764\) 1.06149 0.0384033
\(765\) 0 0
\(766\) 38.0861 1.37611
\(767\) −10.1506 −0.366519
\(768\) 0 0
\(769\) 9.25402 0.333709 0.166854 0.985982i \(-0.446639\pi\)
0.166854 + 0.985982i \(0.446639\pi\)
\(770\) −31.0155 −1.11772
\(771\) 0 0
\(772\) 1.64733 0.0592886
\(773\) 11.6509 0.419056 0.209528 0.977803i \(-0.432807\pi\)
0.209528 + 0.977803i \(0.432807\pi\)
\(774\) 0 0
\(775\) 69.4552 2.49491
\(776\) 44.8435 1.60979
\(777\) 0 0
\(778\) 37.7948 1.35501
\(779\) 1.87164 0.0670586
\(780\) 0 0
\(781\) −17.6126 −0.630227
\(782\) −3.13373 −0.112062
\(783\) 0 0
\(784\) 2.11650 0.0755891
\(785\) 12.0770 0.431046
\(786\) 0 0
\(787\) 47.5672 1.69559 0.847793 0.530327i \(-0.177931\pi\)
0.847793 + 0.530327i \(0.177931\pi\)
\(788\) 3.25671 0.116016
\(789\) 0 0
\(790\) 2.70739 0.0963245
\(791\) −24.8161 −0.882360
\(792\) 0 0
\(793\) −51.1516 −1.81645
\(794\) 6.07966 0.215759
\(795\) 0 0
\(796\) 2.62267 0.0929582
\(797\) −26.1046 −0.924674 −0.462337 0.886704i \(-0.652989\pi\)
−0.462337 + 0.886704i \(0.652989\pi\)
\(798\) 0 0
\(799\) 34.7526 1.22946
\(800\) 7.78880 0.275376
\(801\) 0 0
\(802\) −11.5517 −0.407904
\(803\) −26.8188 −0.946416
\(804\) 0 0
\(805\) 3.68004 0.129705
\(806\) 49.3637 1.73876
\(807\) 0 0
\(808\) −24.8174 −0.873073
\(809\) −15.1361 −0.532157 −0.266078 0.963951i \(-0.585728\pi\)
−0.266078 + 0.963951i \(0.585728\pi\)
\(810\) 0 0
\(811\) −0.696957 −0.0244735 −0.0122367 0.999925i \(-0.503895\pi\)
−0.0122367 + 0.999925i \(0.503895\pi\)
\(812\) −0.0760373 −0.00266839
\(813\) 0 0
\(814\) 18.8438 0.660475
\(815\) 1.60132 0.0560918
\(816\) 0 0
\(817\) 6.48751 0.226969
\(818\) −1.48658 −0.0519770
\(819\) 0 0
\(820\) 2.29591 0.0801767
\(821\) −23.1881 −0.809270 −0.404635 0.914478i \(-0.632601\pi\)
−0.404635 + 0.914478i \(0.632601\pi\)
\(822\) 0 0
\(823\) −25.7229 −0.896643 −0.448321 0.893872i \(-0.647978\pi\)
−0.448321 + 0.893872i \(0.647978\pi\)
\(824\) 9.14290 0.318508
\(825\) 0 0
\(826\) 8.78106 0.305532
\(827\) 29.2481 1.01706 0.508529 0.861045i \(-0.330189\pi\)
0.508529 + 0.861045i \(0.330189\pi\)
\(828\) 0 0
\(829\) 4.38238 0.152206 0.0761031 0.997100i \(-0.475752\pi\)
0.0761031 + 0.997100i \(0.475752\pi\)
\(830\) −73.6691 −2.55709
\(831\) 0 0
\(832\) 33.8999 1.17527
\(833\) −3.32676 −0.115265
\(834\) 0 0
\(835\) 27.6313 0.956222
\(836\) 0.253089 0.00875327
\(837\) 0 0
\(838\) 42.8357 1.47974
\(839\) −25.8753 −0.893313 −0.446657 0.894705i \(-0.647385\pi\)
−0.446657 + 0.894705i \(0.647385\pi\)
\(840\) 0 0
\(841\) −28.9736 −0.999089
\(842\) 4.64908 0.160218
\(843\) 0 0
\(844\) 3.03415 0.104440
\(845\) 9.01279 0.310050
\(846\) 0 0
\(847\) 11.0770 0.380609
\(848\) −36.2213 −1.24384
\(849\) 0 0
\(850\) 56.9336 1.95281
\(851\) −2.23585 −0.0766440
\(852\) 0 0
\(853\) −35.5262 −1.21639 −0.608197 0.793786i \(-0.708107\pi\)
−0.608197 + 0.793786i \(0.708107\pi\)
\(854\) 44.2499 1.51420
\(855\) 0 0
\(856\) 22.1056 0.755553
\(857\) 35.9632 1.22848 0.614239 0.789120i \(-0.289463\pi\)
0.614239 + 0.789120i \(0.289463\pi\)
\(858\) 0 0
\(859\) −34.0806 −1.16281 −0.581407 0.813613i \(-0.697498\pi\)
−0.581407 + 0.813613i \(0.697498\pi\)
\(860\) 7.95811 0.271369
\(861\) 0 0
\(862\) −26.1070 −0.889208
\(863\) 13.6391 0.464280 0.232140 0.972682i \(-0.425427\pi\)
0.232140 + 0.972682i \(0.425427\pi\)
\(864\) 0 0
\(865\) −70.1403 −2.38484
\(866\) −48.7529 −1.65669
\(867\) 0 0
\(868\) 4.34730 0.147557
\(869\) 1.46440 0.0496764
\(870\) 0 0
\(871\) 2.09833 0.0710991
\(872\) −33.7137 −1.14169
\(873\) 0 0
\(874\) 0.294978 0.00997778
\(875\) −22.1411 −0.748507
\(876\) 0 0
\(877\) −34.7888 −1.17473 −0.587367 0.809321i \(-0.699836\pi\)
−0.587367 + 0.809321i \(0.699836\pi\)
\(878\) −37.1620 −1.25416
\(879\) 0 0
\(880\) −32.6955 −1.10217
\(881\) −36.1735 −1.21872 −0.609359 0.792895i \(-0.708573\pi\)
−0.609359 + 0.792895i \(0.708573\pi\)
\(882\) 0 0
\(883\) 54.6154 1.83795 0.918977 0.394312i \(-0.129017\pi\)
0.918977 + 0.394312i \(0.129017\pi\)
\(884\) −4.11936 −0.138549
\(885\) 0 0
\(886\) −6.84018 −0.229800
\(887\) 39.3732 1.32202 0.661011 0.750376i \(-0.270127\pi\)
0.661011 + 0.750376i \(0.270127\pi\)
\(888\) 0 0
\(889\) 2.53209 0.0849235
\(890\) 61.9701 2.07724
\(891\) 0 0
\(892\) 4.25545 0.142483
\(893\) −3.27126 −0.109469
\(894\) 0 0
\(895\) −41.8384 −1.39850
\(896\) −24.0496 −0.803442
\(897\) 0 0
\(898\) −1.52023 −0.0507307
\(899\) −1.50980 −0.0503547
\(900\) 0 0
\(901\) 56.9336 1.89673
\(902\) −12.1985 −0.406166
\(903\) 0 0
\(904\) −28.8489 −0.959499
\(905\) 15.8084 0.525489
\(906\) 0 0
\(907\) −39.4356 −1.30944 −0.654719 0.755872i \(-0.727213\pi\)
−0.654719 + 0.755872i \(0.727213\pi\)
\(908\) −1.31584 −0.0436676
\(909\) 0 0
\(910\) −47.5185 −1.57522
\(911\) 42.8566 1.41990 0.709951 0.704251i \(-0.248717\pi\)
0.709951 + 0.704251i \(0.248717\pi\)
\(912\) 0 0
\(913\) −39.8470 −1.31874
\(914\) 47.6509 1.57615
\(915\) 0 0
\(916\) 1.72731 0.0570719
\(917\) 17.1206 0.565373
\(918\) 0 0
\(919\) 34.3583 1.13338 0.566688 0.823932i \(-0.308224\pi\)
0.566688 + 0.823932i \(0.308224\pi\)
\(920\) 4.27807 0.141044
\(921\) 0 0
\(922\) 25.8256 0.850522
\(923\) −26.9840 −0.888190
\(924\) 0 0
\(925\) 40.6209 1.33561
\(926\) 16.6527 0.547242
\(927\) 0 0
\(928\) −0.169311 −0.00555790
\(929\) −5.15476 −0.169122 −0.0845611 0.996418i \(-0.526949\pi\)
−0.0845611 + 0.996418i \(0.526949\pi\)
\(930\) 0 0
\(931\) 0.313148 0.0102630
\(932\) −2.52797 −0.0828064
\(933\) 0 0
\(934\) 25.8097 0.844518
\(935\) 51.3917 1.68069
\(936\) 0 0
\(937\) −13.8862 −0.453642 −0.226821 0.973936i \(-0.572833\pi\)
−0.226821 + 0.973936i \(0.572833\pi\)
\(938\) −1.81521 −0.0592686
\(939\) 0 0
\(940\) −4.01279 −0.130883
\(941\) 2.16519 0.0705832 0.0352916 0.999377i \(-0.488764\pi\)
0.0352916 + 0.999377i \(0.488764\pi\)
\(942\) 0 0
\(943\) 1.44738 0.0471331
\(944\) 9.25671 0.301280
\(945\) 0 0
\(946\) −42.2826 −1.37473
\(947\) 8.10019 0.263221 0.131610 0.991302i \(-0.457985\pi\)
0.131610 + 0.991302i \(0.457985\pi\)
\(948\) 0 0
\(949\) −41.0888 −1.33380
\(950\) −5.35916 −0.173874
\(951\) 0 0
\(952\) 42.1317 1.36549
\(953\) 5.63579 0.182561 0.0912806 0.995825i \(-0.470904\pi\)
0.0912806 + 0.995825i \(0.470904\pi\)
\(954\) 0 0
\(955\) −20.2891 −0.656540
\(956\) −3.03827 −0.0982646
\(957\) 0 0
\(958\) −32.3492 −1.04515
\(959\) −27.8530 −0.899420
\(960\) 0 0
\(961\) 55.3201 1.78452
\(962\) 28.8704 0.930819
\(963\) 0 0
\(964\) −1.57491 −0.0507245
\(965\) −31.4867 −1.01359
\(966\) 0 0
\(967\) 27.2540 0.876430 0.438215 0.898870i \(-0.355611\pi\)
0.438215 + 0.898870i \(0.355611\pi\)
\(968\) 12.8770 0.413883
\(969\) 0 0
\(970\) −72.4971 −2.32774
\(971\) −20.8179 −0.668078 −0.334039 0.942559i \(-0.608412\pi\)
−0.334039 + 0.942559i \(0.608412\pi\)
\(972\) 0 0
\(973\) −23.4388 −0.751414
\(974\) −1.09278 −0.0350148
\(975\) 0 0
\(976\) 46.6468 1.49313
\(977\) −33.1448 −1.06040 −0.530198 0.847874i \(-0.677882\pi\)
−0.530198 + 0.847874i \(0.677882\pi\)
\(978\) 0 0
\(979\) 33.5191 1.07127
\(980\) 0.384133 0.0122707
\(981\) 0 0
\(982\) 7.53621 0.240490
\(983\) −19.8749 −0.633912 −0.316956 0.948440i \(-0.602661\pi\)
−0.316956 + 0.948440i \(0.602661\pi\)
\(984\) 0 0
\(985\) −62.2481 −1.98339
\(986\) −1.23761 −0.0394135
\(987\) 0 0
\(988\) 0.387755 0.0123361
\(989\) 5.01691 0.159529
\(990\) 0 0
\(991\) 26.2490 0.833826 0.416913 0.908946i \(-0.363112\pi\)
0.416913 + 0.908946i \(0.363112\pi\)
\(992\) 9.68004 0.307342
\(993\) 0 0
\(994\) 23.3432 0.740401
\(995\) −50.1293 −1.58921
\(996\) 0 0
\(997\) 7.38001 0.233727 0.116864 0.993148i \(-0.462716\pi\)
0.116864 + 0.993148i \(0.462716\pi\)
\(998\) 39.6008 1.25354
\(999\) 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 1143.2.a.e.1.2 3
3.2 odd 2 127.2.a.a.1.2 3
12.11 even 2 2032.2.a.k.1.1 3
15.14 odd 2 3175.2.a.h.1.2 3
21.20 even 2 6223.2.a.e.1.2 3
24.5 odd 2 8128.2.a.bd.1.1 3
24.11 even 2 8128.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
127.2.a.a.1.2 3 3.2 odd 2
1143.2.a.e.1.2 3 1.1 even 1 trivial
2032.2.a.k.1.1 3 12.11 even 2
3175.2.a.h.1.2 3 15.14 odd 2
6223.2.a.e.1.2 3 21.20 even 2
8128.2.a.w.1.3 3 24.11 even 2
8128.2.a.bd.1.1 3 24.5 odd 2