Properties

Label 9702.2.a.ea.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.27133\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.79793 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.79793 q^{5} -1.00000 q^{8} +1.79793 q^{10} -1.00000 q^{11} -3.63899 q^{13} +1.00000 q^{16} +6.11582 q^{17} -1.84106 q^{19} -1.79793 q^{20} +1.00000 q^{22} +7.37109 q^{23} -1.76744 q^{25} +3.63899 q^{26} +10.4243 q^{29} -7.97958 q^{31} -1.00000 q^{32} -6.11582 q^{34} +10.1995 q^{37} +1.84106 q^{38} +1.79793 q^{40} +8.17680 q^{41} -2.28577 q^{43} -1.00000 q^{44} -7.37109 q^{46} -0.669485 q^{47} +1.76744 q^{50} -3.63899 q^{52} -11.2848 q^{53} +1.79793 q^{55} -10.4243 q^{58} -13.5706 q^{59} +0.274758 q^{61} +7.97958 q^{62} +1.00000 q^{64} +6.54266 q^{65} +3.88163 q^{67} +6.11582 q^{68} -10.8738 q^{71} -3.85892 q^{73} -10.1995 q^{74} -1.84106 q^{76} -12.5174 q^{79} -1.79793 q^{80} -8.17680 q^{82} +5.27314 q^{83} -10.9958 q^{85} +2.28577 q^{86} +1.00000 q^{88} +1.98214 q^{89} +7.37109 q^{92} +0.669485 q^{94} +3.31010 q^{95} -4.12066 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} + 4 q^{16} + 4 q^{17} - 12 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} + 8 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{32} - 4 q^{34} + 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 4 q^{47} - 4 q^{50} - 8 q^{52} + 8 q^{53} - 4 q^{55} - 8 q^{58} - 24 q^{61} + 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} + 4 q^{68} - 8 q^{71} - 4 q^{73} - 8 q^{74} - 12 q^{76} - 8 q^{79} + 4 q^{80} - 12 q^{82} + 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} + 24 q^{89} + 8 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.79793 −0.804060 −0.402030 0.915627i \(-0.631695\pi\)
−0.402030 + 0.915627i \(0.631695\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.79793 0.568556
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.63899 −1.00927 −0.504637 0.863331i \(-0.668374\pi\)
−0.504637 + 0.863331i \(0.668374\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.11582 1.48330 0.741652 0.670785i \(-0.234043\pi\)
0.741652 + 0.670785i \(0.234043\pi\)
\(18\) 0 0
\(19\) −1.84106 −0.422368 −0.211184 0.977446i \(-0.567732\pi\)
−0.211184 + 0.977446i \(0.567732\pi\)
\(20\) −1.79793 −0.402030
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.37109 1.53698 0.768489 0.639863i \(-0.221009\pi\)
0.768489 + 0.639863i \(0.221009\pi\)
\(24\) 0 0
\(25\) −1.76744 −0.353488
\(26\) 3.63899 0.713665
\(27\) 0 0
\(28\) 0 0
\(29\) 10.4243 1.93574 0.967871 0.251446i \(-0.0809062\pi\)
0.967871 + 0.251446i \(0.0809062\pi\)
\(30\) 0 0
\(31\) −7.97958 −1.43318 −0.716588 0.697497i \(-0.754297\pi\)
−0.716588 + 0.697497i \(0.754297\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.11582 −1.04885
\(35\) 0 0
\(36\) 0 0
\(37\) 10.1995 1.67679 0.838395 0.545063i \(-0.183494\pi\)
0.838395 + 0.545063i \(0.183494\pi\)
\(38\) 1.84106 0.298659
\(39\) 0 0
\(40\) 1.79793 0.284278
\(41\) 8.17680 1.27700 0.638501 0.769621i \(-0.279555\pi\)
0.638501 + 0.769621i \(0.279555\pi\)
\(42\) 0 0
\(43\) −2.28577 −0.348576 −0.174288 0.984695i \(-0.555762\pi\)
−0.174288 + 0.984695i \(0.555762\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −7.37109 −1.08681
\(47\) −0.669485 −0.0976545 −0.0488272 0.998807i \(-0.515548\pi\)
−0.0488272 + 0.998807i \(0.515548\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.76744 0.249954
\(51\) 0 0
\(52\) −3.63899 −0.504637
\(53\) −11.2848 −1.55009 −0.775046 0.631905i \(-0.782273\pi\)
−0.775046 + 0.631905i \(0.782273\pi\)
\(54\) 0 0
\(55\) 1.79793 0.242433
\(56\) 0 0
\(57\) 0 0
\(58\) −10.4243 −1.36878
\(59\) −13.5706 −1.76674 −0.883371 0.468674i \(-0.844732\pi\)
−0.883371 + 0.468674i \(0.844732\pi\)
\(60\) 0 0
\(61\) 0.274758 0.0351791 0.0175896 0.999845i \(-0.494401\pi\)
0.0175896 + 0.999845i \(0.494401\pi\)
\(62\) 7.97958 1.01341
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.54266 0.811517
\(66\) 0 0
\(67\) 3.88163 0.474217 0.237108 0.971483i \(-0.423800\pi\)
0.237108 + 0.971483i \(0.423800\pi\)
\(68\) 6.11582 0.741652
\(69\) 0 0
\(70\) 0 0
\(71\) −10.8738 −1.29049 −0.645244 0.763976i \(-0.723244\pi\)
−0.645244 + 0.763976i \(0.723244\pi\)
\(72\) 0 0
\(73\) −3.85892 −0.451653 −0.225826 0.974168i \(-0.572508\pi\)
−0.225826 + 0.974168i \(0.572508\pi\)
\(74\) −10.1995 −1.18567
\(75\) 0 0
\(76\) −1.84106 −0.211184
\(77\) 0 0
\(78\) 0 0
\(79\) −12.5174 −1.40832 −0.704159 0.710043i \(-0.748676\pi\)
−0.704159 + 0.710043i \(0.748676\pi\)
\(80\) −1.79793 −0.201015
\(81\) 0 0
\(82\) −8.17680 −0.902977
\(83\) 5.27314 0.578802 0.289401 0.957208i \(-0.406544\pi\)
0.289401 + 0.957208i \(0.406544\pi\)
\(84\) 0 0
\(85\) −10.9958 −1.19266
\(86\) 2.28577 0.246481
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 1.98214 0.210106 0.105053 0.994467i \(-0.466499\pi\)
0.105053 + 0.994467i \(0.466499\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.37109 0.768489
\(93\) 0 0
\(94\) 0.669485 0.0690522
\(95\) 3.31010 0.339609
\(96\) 0 0
\(97\) −4.12066 −0.418390 −0.209195 0.977874i \(-0.567084\pi\)
−0.209195 + 0.977874i \(0.567084\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.76744 −0.176744
\(101\) 6.90367 0.686941 0.343470 0.939163i \(-0.388397\pi\)
0.343470 + 0.939163i \(0.388397\pi\)
\(102\) 0 0
\(103\) −12.6154 −1.24303 −0.621514 0.783403i \(-0.713482\pi\)
−0.621514 + 0.783403i \(0.713482\pi\)
\(104\) 3.63899 0.356832
\(105\) 0 0
\(106\) 11.2848 1.09608
\(107\) −16.9417 −1.63782 −0.818908 0.573925i \(-0.805420\pi\)
−0.818908 + 0.573925i \(0.805420\pi\)
\(108\) 0 0
\(109\) 9.76326 0.935151 0.467576 0.883953i \(-0.345128\pi\)
0.467576 + 0.883953i \(0.345128\pi\)
\(110\) −1.79793 −0.171426
\(111\) 0 0
\(112\) 0 0
\(113\) 12.9348 1.21681 0.608404 0.793628i \(-0.291810\pi\)
0.608404 + 0.793628i \(0.291810\pi\)
\(114\) 0 0
\(115\) −13.2527 −1.23582
\(116\) 10.4243 0.967871
\(117\) 0 0
\(118\) 13.5706 1.24928
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.274758 −0.0248754
\(123\) 0 0
\(124\) −7.97958 −0.716588
\(125\) 12.1674 1.08829
\(126\) 0 0
\(127\) −7.43208 −0.659490 −0.329745 0.944070i \(-0.606963\pi\)
−0.329745 + 0.944070i \(0.606963\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.54266 −0.573829
\(131\) 1.64154 0.143422 0.0717112 0.997425i \(-0.477154\pi\)
0.0717112 + 0.997425i \(0.477154\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3.88163 −0.335322
\(135\) 0 0
\(136\) −6.11582 −0.524427
\(137\) −14.5307 −1.24144 −0.620721 0.784032i \(-0.713160\pi\)
−0.620721 + 0.784032i \(0.713160\pi\)
\(138\) 0 0
\(139\) 8.26535 0.701058 0.350529 0.936552i \(-0.386002\pi\)
0.350529 + 0.936552i \(0.386002\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.8738 0.912513
\(143\) 3.63899 0.304308
\(144\) 0 0
\(145\) −18.7422 −1.55645
\(146\) 3.85892 0.319367
\(147\) 0 0
\(148\) 10.1995 0.838395
\(149\) 7.55045 0.618557 0.309278 0.950972i \(-0.399913\pi\)
0.309278 + 0.950972i \(0.399913\pi\)
\(150\) 0 0
\(151\) 12.3990 1.00902 0.504509 0.863406i \(-0.331673\pi\)
0.504509 + 0.863406i \(0.331673\pi\)
\(152\) 1.84106 0.149330
\(153\) 0 0
\(154\) 0 0
\(155\) 14.3468 1.15236
\(156\) 0 0
\(157\) −4.54011 −0.362340 −0.181170 0.983452i \(-0.557988\pi\)
−0.181170 + 0.983452i \(0.557988\pi\)
\(158\) 12.5174 0.995831
\(159\) 0 0
\(160\) 1.79793 0.142139
\(161\) 0 0
\(162\) 0 0
\(163\) 15.8275 1.23971 0.619853 0.784718i \(-0.287192\pi\)
0.619853 + 0.784718i \(0.287192\pi\)
\(164\) 8.17680 0.638501
\(165\) 0 0
\(166\) −5.27314 −0.409275
\(167\) −22.0844 −1.70894 −0.854471 0.519499i \(-0.826118\pi\)
−0.854471 + 0.519499i \(0.826118\pi\)
\(168\) 0 0
\(169\) 0.242256 0.0186350
\(170\) 10.9958 0.843341
\(171\) 0 0
\(172\) −2.28577 −0.174288
\(173\) −10.2628 −0.780266 −0.390133 0.920758i \(-0.627571\pi\)
−0.390133 + 0.920758i \(0.627571\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −1.98214 −0.148567
\(179\) 12.3954 0.926477 0.463239 0.886234i \(-0.346687\pi\)
0.463239 + 0.886234i \(0.346687\pi\)
\(180\) 0 0
\(181\) 22.4506 1.66874 0.834370 0.551204i \(-0.185831\pi\)
0.834370 + 0.551204i \(0.185831\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.37109 −0.543404
\(185\) −18.3380 −1.34824
\(186\) 0 0
\(187\) −6.11582 −0.447233
\(188\) −0.669485 −0.0488272
\(189\) 0 0
\(190\) −3.31010 −0.240140
\(191\) 13.7142 0.992327 0.496164 0.868229i \(-0.334742\pi\)
0.496164 + 0.868229i \(0.334742\pi\)
\(192\) 0 0
\(193\) 3.77883 0.272006 0.136003 0.990708i \(-0.456574\pi\)
0.136003 + 0.990708i \(0.456574\pi\)
\(194\) 4.12066 0.295846
\(195\) 0 0
\(196\) 0 0
\(197\) 21.6316 1.54119 0.770594 0.637327i \(-0.219960\pi\)
0.770594 + 0.637327i \(0.219960\pi\)
\(198\) 0 0
\(199\) 17.8323 1.26410 0.632051 0.774927i \(-0.282213\pi\)
0.632051 + 0.774927i \(0.282213\pi\)
\(200\) 1.76744 0.124977
\(201\) 0 0
\(202\) −6.90367 −0.485741
\(203\) 0 0
\(204\) 0 0
\(205\) −14.7013 −1.02679
\(206\) 12.6154 0.878953
\(207\) 0 0
\(208\) −3.63899 −0.252319
\(209\) 1.84106 0.127349
\(210\) 0 0
\(211\) −8.27020 −0.569344 −0.284672 0.958625i \(-0.591885\pi\)
−0.284672 + 0.958625i \(0.591885\pi\)
\(212\) −11.2848 −0.775046
\(213\) 0 0
\(214\) 16.9417 1.15811
\(215\) 4.10965 0.280276
\(216\) 0 0
\(217\) 0 0
\(218\) −9.76326 −0.661252
\(219\) 0 0
\(220\) 1.79793 0.121217
\(221\) −22.2554 −1.49706
\(222\) 0 0
\(223\) 9.10574 0.609765 0.304883 0.952390i \(-0.401383\pi\)
0.304883 + 0.952390i \(0.401383\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.9348 −0.860413
\(227\) −0.620456 −0.0411811 −0.0205905 0.999788i \(-0.506555\pi\)
−0.0205905 + 0.999788i \(0.506555\pi\)
\(228\) 0 0
\(229\) −18.0011 −1.18954 −0.594772 0.803895i \(-0.702758\pi\)
−0.594772 + 0.803895i \(0.702758\pi\)
\(230\) 13.2527 0.873858
\(231\) 0 0
\(232\) −10.4243 −0.684388
\(233\) 19.8697 1.30171 0.650853 0.759204i \(-0.274412\pi\)
0.650853 + 0.759204i \(0.274412\pi\)
\(234\) 0 0
\(235\) 1.20369 0.0785201
\(236\) −13.5706 −0.883371
\(237\) 0 0
\(238\) 0 0
\(239\) 17.4201 1.12681 0.563407 0.826180i \(-0.309490\pi\)
0.563407 + 0.826180i \(0.309490\pi\)
\(240\) 0 0
\(241\) −18.4149 −1.18621 −0.593104 0.805126i \(-0.702098\pi\)
−0.593104 + 0.805126i \(0.702098\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 0.274758 0.0175896
\(245\) 0 0
\(246\) 0 0
\(247\) 6.69959 0.426285
\(248\) 7.97958 0.506704
\(249\) 0 0
\(250\) −12.1674 −0.769534
\(251\) 21.8022 1.37614 0.688072 0.725642i \(-0.258457\pi\)
0.688072 + 0.725642i \(0.258457\pi\)
\(252\) 0 0
\(253\) −7.37109 −0.463416
\(254\) 7.43208 0.466330
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −4.00229 −0.249656 −0.124828 0.992178i \(-0.539838\pi\)
−0.124828 + 0.992178i \(0.539838\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.54266 0.405759
\(261\) 0 0
\(262\) −1.64154 −0.101415
\(263\) 1.13946 0.0702619 0.0351309 0.999383i \(-0.488815\pi\)
0.0351309 + 0.999383i \(0.488815\pi\)
\(264\) 0 0
\(265\) 20.2894 1.24637
\(266\) 0 0
\(267\) 0 0
\(268\) 3.88163 0.237108
\(269\) 3.85892 0.235283 0.117641 0.993056i \(-0.462467\pi\)
0.117641 + 0.993056i \(0.462467\pi\)
\(270\) 0 0
\(271\) 22.0202 1.33763 0.668815 0.743429i \(-0.266802\pi\)
0.668815 + 0.743429i \(0.266802\pi\)
\(272\) 6.11582 0.370826
\(273\) 0 0
\(274\) 14.5307 0.877832
\(275\) 1.76744 0.106581
\(276\) 0 0
\(277\) −13.1307 −0.788950 −0.394475 0.918907i \(-0.629073\pi\)
−0.394475 + 0.918907i \(0.629073\pi\)
\(278\) −8.26535 −0.495723
\(279\) 0 0
\(280\) 0 0
\(281\) 16.1064 0.960828 0.480414 0.877042i \(-0.340486\pi\)
0.480414 + 0.877042i \(0.340486\pi\)
\(282\) 0 0
\(283\) 10.6085 0.630610 0.315305 0.948990i \(-0.397893\pi\)
0.315305 + 0.948990i \(0.397893\pi\)
\(284\) −10.8738 −0.645244
\(285\) 0 0
\(286\) −3.63899 −0.215178
\(287\) 0 0
\(288\) 0 0
\(289\) 20.4032 1.20019
\(290\) 18.7422 1.10058
\(291\) 0 0
\(292\) −3.85892 −0.225826
\(293\) −3.69313 −0.215755 −0.107877 0.994164i \(-0.534405\pi\)
−0.107877 + 0.994164i \(0.534405\pi\)
\(294\) 0 0
\(295\) 24.3990 1.42057
\(296\) −10.1995 −0.592835
\(297\) 0 0
\(298\) −7.55045 −0.437386
\(299\) −26.8233 −1.55123
\(300\) 0 0
\(301\) 0 0
\(302\) −12.3990 −0.713484
\(303\) 0 0
\(304\) −1.84106 −0.105592
\(305\) −0.493996 −0.0282861
\(306\) 0 0
\(307\) −19.1223 −1.09137 −0.545683 0.837992i \(-0.683730\pi\)
−0.545683 + 0.837992i \(0.683730\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −14.3468 −0.814841
\(311\) −13.1190 −0.743913 −0.371956 0.928250i \(-0.621313\pi\)
−0.371956 + 0.928250i \(0.621313\pi\)
\(312\) 0 0
\(313\) 12.6248 0.713594 0.356797 0.934182i \(-0.383869\pi\)
0.356797 + 0.934182i \(0.383869\pi\)
\(314\) 4.54011 0.256213
\(315\) 0 0
\(316\) −12.5174 −0.704159
\(317\) 29.4555 1.65438 0.827192 0.561919i \(-0.189937\pi\)
0.827192 + 0.561919i \(0.189937\pi\)
\(318\) 0 0
\(319\) −10.4243 −0.583648
\(320\) −1.79793 −0.100507
\(321\) 0 0
\(322\) 0 0
\(323\) −11.2596 −0.626499
\(324\) 0 0
\(325\) 6.43169 0.356766
\(326\) −15.8275 −0.876604
\(327\) 0 0
\(328\) −8.17680 −0.451489
\(329\) 0 0
\(330\) 0 0
\(331\) 2.28901 0.125815 0.0629077 0.998019i \(-0.479963\pi\)
0.0629077 + 0.998019i \(0.479963\pi\)
\(332\) 5.27314 0.289401
\(333\) 0 0
\(334\) 22.0844 1.20840
\(335\) −6.97891 −0.381299
\(336\) 0 0
\(337\) 5.35093 0.291484 0.145742 0.989323i \(-0.453443\pi\)
0.145742 + 0.989323i \(0.453443\pi\)
\(338\) −0.242256 −0.0131770
\(339\) 0 0
\(340\) −10.9958 −0.596332
\(341\) 7.97958 0.432119
\(342\) 0 0
\(343\) 0 0
\(344\) 2.28577 0.123240
\(345\) 0 0
\(346\) 10.2628 0.551731
\(347\) −20.8197 −1.11766 −0.558830 0.829282i \(-0.688750\pi\)
−0.558830 + 0.829282i \(0.688750\pi\)
\(348\) 0 0
\(349\) 23.1843 1.24103 0.620514 0.784195i \(-0.286924\pi\)
0.620514 + 0.784195i \(0.286924\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 12.1243 0.645310 0.322655 0.946517i \(-0.395425\pi\)
0.322655 + 0.946517i \(0.395425\pi\)
\(354\) 0 0
\(355\) 19.5504 1.03763
\(356\) 1.98214 0.105053
\(357\) 0 0
\(358\) −12.3954 −0.655118
\(359\) −12.3312 −0.650815 −0.325408 0.945574i \(-0.605501\pi\)
−0.325408 + 0.945574i \(0.605501\pi\)
\(360\) 0 0
\(361\) −15.6105 −0.821605
\(362\) −22.4506 −1.17998
\(363\) 0 0
\(364\) 0 0
\(365\) 6.93808 0.363156
\(366\) 0 0
\(367\) 18.0860 0.944081 0.472041 0.881577i \(-0.343518\pi\)
0.472041 + 0.881577i \(0.343518\pi\)
\(368\) 7.37109 0.384245
\(369\) 0 0
\(370\) 18.3380 0.953349
\(371\) 0 0
\(372\) 0 0
\(373\) 6.95365 0.360046 0.180023 0.983662i \(-0.442383\pi\)
0.180023 + 0.983662i \(0.442383\pi\)
\(374\) 6.11582 0.316241
\(375\) 0 0
\(376\) 0.669485 0.0345261
\(377\) −37.9339 −1.95370
\(378\) 0 0
\(379\) 5.14307 0.264182 0.132091 0.991238i \(-0.457831\pi\)
0.132091 + 0.991238i \(0.457831\pi\)
\(380\) 3.31010 0.169804
\(381\) 0 0
\(382\) −13.7142 −0.701681
\(383\) −18.8810 −0.964772 −0.482386 0.875959i \(-0.660230\pi\)
−0.482386 + 0.875959i \(0.660230\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.77883 −0.192337
\(387\) 0 0
\(388\) −4.12066 −0.209195
\(389\) 30.3990 1.54129 0.770646 0.637263i \(-0.219934\pi\)
0.770646 + 0.637263i \(0.219934\pi\)
\(390\) 0 0
\(391\) 45.0802 2.27980
\(392\) 0 0
\(393\) 0 0
\(394\) −21.6316 −1.08978
\(395\) 22.5054 1.13237
\(396\) 0 0
\(397\) 2.08009 0.104397 0.0521983 0.998637i \(-0.483377\pi\)
0.0521983 + 0.998637i \(0.483377\pi\)
\(398\) −17.8323 −0.893855
\(399\) 0 0
\(400\) −1.76744 −0.0883719
\(401\) 21.8444 1.09086 0.545429 0.838157i \(-0.316367\pi\)
0.545429 + 0.838157i \(0.316367\pi\)
\(402\) 0 0
\(403\) 29.0376 1.44647
\(404\) 6.90367 0.343470
\(405\) 0 0
\(406\) 0 0
\(407\) −10.1995 −0.505571
\(408\) 0 0
\(409\) −33.7571 −1.66918 −0.834591 0.550871i \(-0.814296\pi\)
−0.834591 + 0.550871i \(0.814296\pi\)
\(410\) 14.7013 0.726048
\(411\) 0 0
\(412\) −12.6154 −0.621514
\(413\) 0 0
\(414\) 0 0
\(415\) −9.48074 −0.465391
\(416\) 3.63899 0.178416
\(417\) 0 0
\(418\) −1.84106 −0.0900491
\(419\) 22.7371 1.11078 0.555389 0.831590i \(-0.312569\pi\)
0.555389 + 0.831590i \(0.312569\pi\)
\(420\) 0 0
\(421\) −21.0192 −1.02441 −0.512207 0.858862i \(-0.671172\pi\)
−0.512207 + 0.858862i \(0.671172\pi\)
\(422\) 8.27020 0.402587
\(423\) 0 0
\(424\) 11.2848 0.548040
\(425\) −10.8093 −0.524329
\(426\) 0 0
\(427\) 0 0
\(428\) −16.9417 −0.818908
\(429\) 0 0
\(430\) −4.10965 −0.198185
\(431\) −7.03305 −0.338770 −0.169385 0.985550i \(-0.554178\pi\)
−0.169385 + 0.985550i \(0.554178\pi\)
\(432\) 0 0
\(433\) −5.77293 −0.277429 −0.138715 0.990332i \(-0.544297\pi\)
−0.138715 + 0.990332i \(0.544297\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.76326 0.467576
\(437\) −13.5706 −0.649170
\(438\) 0 0
\(439\) −31.2275 −1.49041 −0.745203 0.666838i \(-0.767647\pi\)
−0.745203 + 0.666838i \(0.767647\pi\)
\(440\) −1.79793 −0.0857131
\(441\) 0 0
\(442\) 22.2554 1.05858
\(443\) 35.8953 1.70544 0.852720 0.522369i \(-0.174952\pi\)
0.852720 + 0.522369i \(0.174952\pi\)
\(444\) 0 0
\(445\) −3.56375 −0.168938
\(446\) −9.10574 −0.431169
\(447\) 0 0
\(448\) 0 0
\(449\) 24.5495 1.15856 0.579282 0.815127i \(-0.303333\pi\)
0.579282 + 0.815127i \(0.303333\pi\)
\(450\) 0 0
\(451\) −8.17680 −0.385031
\(452\) 12.9348 0.608404
\(453\) 0 0
\(454\) 0.620456 0.0291194
\(455\) 0 0
\(456\) 0 0
\(457\) 26.2771 1.22919 0.614594 0.788843i \(-0.289320\pi\)
0.614594 + 0.788843i \(0.289320\pi\)
\(458\) 18.0011 0.841134
\(459\) 0 0
\(460\) −13.2527 −0.617911
\(461\) 20.8330 0.970289 0.485144 0.874434i \(-0.338767\pi\)
0.485144 + 0.874434i \(0.338767\pi\)
\(462\) 0 0
\(463\) 17.8697 0.830474 0.415237 0.909713i \(-0.363699\pi\)
0.415237 + 0.909713i \(0.363699\pi\)
\(464\) 10.4243 0.483936
\(465\) 0 0
\(466\) −19.8697 −0.920445
\(467\) −26.8642 −1.24312 −0.621562 0.783365i \(-0.713502\pi\)
−0.621562 + 0.783365i \(0.713502\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.20369 −0.0555221
\(471\) 0 0
\(472\) 13.5706 0.624638
\(473\) 2.28577 0.105100
\(474\) 0 0
\(475\) 3.25396 0.149302
\(476\) 0 0
\(477\) 0 0
\(478\) −17.4201 −0.796778
\(479\) −15.5050 −0.708443 −0.354221 0.935162i \(-0.615254\pi\)
−0.354221 + 0.935162i \(0.615254\pi\)
\(480\) 0 0
\(481\) −37.1159 −1.69234
\(482\) 18.4149 0.838775
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 7.40867 0.336411
\(486\) 0 0
\(487\) 10.8858 0.493283 0.246641 0.969107i \(-0.420673\pi\)
0.246641 + 0.969107i \(0.420673\pi\)
\(488\) −0.274758 −0.0124377
\(489\) 0 0
\(490\) 0 0
\(491\) −7.06975 −0.319053 −0.159527 0.987194i \(-0.550997\pi\)
−0.159527 + 0.987194i \(0.550997\pi\)
\(492\) 0 0
\(493\) 63.7531 2.87129
\(494\) −6.69959 −0.301429
\(495\) 0 0
\(496\) −7.97958 −0.358294
\(497\) 0 0
\(498\) 0 0
\(499\) 18.8678 0.844637 0.422319 0.906447i \(-0.361216\pi\)
0.422319 + 0.906447i \(0.361216\pi\)
\(500\) 12.1674 0.544143
\(501\) 0 0
\(502\) −21.8022 −0.973081
\(503\) 28.9752 1.29194 0.645969 0.763364i \(-0.276453\pi\)
0.645969 + 0.763364i \(0.276453\pi\)
\(504\) 0 0
\(505\) −12.4123 −0.552342
\(506\) 7.37109 0.327685
\(507\) 0 0
\(508\) −7.43208 −0.329745
\(509\) −16.3539 −0.724874 −0.362437 0.932008i \(-0.618055\pi\)
−0.362437 + 0.932008i \(0.618055\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 4.00229 0.176534
\(515\) 22.6816 0.999469
\(516\) 0 0
\(517\) 0.669485 0.0294439
\(518\) 0 0
\(519\) 0 0
\(520\) −6.54266 −0.286915
\(521\) −21.6949 −0.950470 −0.475235 0.879859i \(-0.657637\pi\)
−0.475235 + 0.879859i \(0.657637\pi\)
\(522\) 0 0
\(523\) 37.8171 1.65363 0.826814 0.562475i \(-0.190151\pi\)
0.826814 + 0.562475i \(0.190151\pi\)
\(524\) 1.64154 0.0717112
\(525\) 0 0
\(526\) −1.13946 −0.0496826
\(527\) −48.8017 −2.12583
\(528\) 0 0
\(529\) 31.3329 1.36230
\(530\) −20.2894 −0.881314
\(531\) 0 0
\(532\) 0 0
\(533\) −29.7553 −1.28885
\(534\) 0 0
\(535\) 30.4600 1.31690
\(536\) −3.88163 −0.167661
\(537\) 0 0
\(538\) −3.85892 −0.166370
\(539\) 0 0
\(540\) 0 0
\(541\) 1.68212 0.0723198 0.0361599 0.999346i \(-0.488487\pi\)
0.0361599 + 0.999346i \(0.488487\pi\)
\(542\) −22.0202 −0.945847
\(543\) 0 0
\(544\) −6.11582 −0.262213
\(545\) −17.5537 −0.751917
\(546\) 0 0
\(547\) 13.6665 0.584339 0.292170 0.956366i \(-0.405623\pi\)
0.292170 + 0.956366i \(0.405623\pi\)
\(548\) −14.5307 −0.620721
\(549\) 0 0
\(550\) −1.76744 −0.0753638
\(551\) −19.1917 −0.817595
\(552\) 0 0
\(553\) 0 0
\(554\) 13.1307 0.557872
\(555\) 0 0
\(556\) 8.26535 0.350529
\(557\) 8.99583 0.381165 0.190583 0.981671i \(-0.438962\pi\)
0.190583 + 0.981671i \(0.438962\pi\)
\(558\) 0 0
\(559\) 8.31788 0.351809
\(560\) 0 0
\(561\) 0 0
\(562\) −16.1064 −0.679408
\(563\) 21.7461 0.916489 0.458244 0.888826i \(-0.348478\pi\)
0.458244 + 0.888826i \(0.348478\pi\)
\(564\) 0 0
\(565\) −23.2560 −0.978386
\(566\) −10.6085 −0.445908
\(567\) 0 0
\(568\) 10.8738 0.456256
\(569\) −32.4549 −1.36058 −0.680290 0.732943i \(-0.738146\pi\)
−0.680290 + 0.732943i \(0.738146\pi\)
\(570\) 0 0
\(571\) −20.2031 −0.845474 −0.422737 0.906252i \(-0.638931\pi\)
−0.422737 + 0.906252i \(0.638931\pi\)
\(572\) 3.63899 0.152154
\(573\) 0 0
\(574\) 0 0
\(575\) −13.0279 −0.543303
\(576\) 0 0
\(577\) −24.4458 −1.01769 −0.508845 0.860858i \(-0.669927\pi\)
−0.508845 + 0.860858i \(0.669927\pi\)
\(578\) −20.4032 −0.848661
\(579\) 0 0
\(580\) −18.7422 −0.778226
\(581\) 0 0
\(582\) 0 0
\(583\) 11.2848 0.467370
\(584\) 3.85892 0.159683
\(585\) 0 0
\(586\) 3.69313 0.152562
\(587\) 16.8894 0.697101 0.348550 0.937290i \(-0.386674\pi\)
0.348550 + 0.937290i \(0.386674\pi\)
\(588\) 0 0
\(589\) 14.6909 0.605327
\(590\) −24.3990 −1.00449
\(591\) 0 0
\(592\) 10.1995 0.419197
\(593\) −22.2781 −0.914852 −0.457426 0.889248i \(-0.651229\pi\)
−0.457426 + 0.889248i \(0.651229\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.55045 0.309278
\(597\) 0 0
\(598\) 26.8233 1.09689
\(599\) 6.25179 0.255441 0.127721 0.991810i \(-0.459234\pi\)
0.127721 + 0.991810i \(0.459234\pi\)
\(600\) 0 0
\(601\) −31.3043 −1.27693 −0.638465 0.769651i \(-0.720430\pi\)
−0.638465 + 0.769651i \(0.720430\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 12.3990 0.504509
\(605\) −1.79793 −0.0730964
\(606\) 0 0
\(607\) 25.0348 1.01613 0.508066 0.861318i \(-0.330361\pi\)
0.508066 + 0.861318i \(0.330361\pi\)
\(608\) 1.84106 0.0746648
\(609\) 0 0
\(610\) 0.493996 0.0200013
\(611\) 2.43625 0.0985602
\(612\) 0 0
\(613\) 1.17157 0.0473194 0.0236597 0.999720i \(-0.492468\pi\)
0.0236597 + 0.999720i \(0.492468\pi\)
\(614\) 19.1223 0.771713
\(615\) 0 0
\(616\) 0 0
\(617\) −5.06651 −0.203970 −0.101985 0.994786i \(-0.532519\pi\)
−0.101985 + 0.994786i \(0.532519\pi\)
\(618\) 0 0
\(619\) 24.6812 0.992021 0.496010 0.868317i \(-0.334798\pi\)
0.496010 + 0.868317i \(0.334798\pi\)
\(620\) 14.3468 0.576180
\(621\) 0 0
\(622\) 13.1190 0.526026
\(623\) 0 0
\(624\) 0 0
\(625\) −13.0390 −0.521559
\(626\) −12.6248 −0.504587
\(627\) 0 0
\(628\) −4.54011 −0.181170
\(629\) 62.3784 2.48719
\(630\) 0 0
\(631\) −25.8408 −1.02871 −0.514353 0.857579i \(-0.671968\pi\)
−0.514353 + 0.857579i \(0.671968\pi\)
\(632\) 12.5174 0.497915
\(633\) 0 0
\(634\) −29.4555 −1.16983
\(635\) 13.3624 0.530270
\(636\) 0 0
\(637\) 0 0
\(638\) 10.4243 0.412702
\(639\) 0 0
\(640\) 1.79793 0.0710695
\(641\) 3.19684 0.126267 0.0631337 0.998005i \(-0.479891\pi\)
0.0631337 + 0.998005i \(0.479891\pi\)
\(642\) 0 0
\(643\) −20.7623 −0.818787 −0.409393 0.912358i \(-0.634260\pi\)
−0.409393 + 0.912358i \(0.634260\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 11.2596 0.443002
\(647\) 28.4056 1.11674 0.558370 0.829592i \(-0.311427\pi\)
0.558370 + 0.829592i \(0.311427\pi\)
\(648\) 0 0
\(649\) 13.5706 0.532693
\(650\) −6.43169 −0.252272
\(651\) 0 0
\(652\) 15.8275 0.619853
\(653\) −16.7825 −0.656750 −0.328375 0.944548i \(-0.606501\pi\)
−0.328375 + 0.944548i \(0.606501\pi\)
\(654\) 0 0
\(655\) −2.95138 −0.115320
\(656\) 8.17680 0.319251
\(657\) 0 0
\(658\) 0 0
\(659\) 15.3426 0.597662 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(660\) 0 0
\(661\) 6.68735 0.260108 0.130054 0.991507i \(-0.458485\pi\)
0.130054 + 0.991507i \(0.458485\pi\)
\(662\) −2.28901 −0.0889649
\(663\) 0 0
\(664\) −5.27314 −0.204637
\(665\) 0 0
\(666\) 0 0
\(667\) 76.8384 2.97519
\(668\) −22.0844 −0.854471
\(669\) 0 0
\(670\) 6.97891 0.269619
\(671\) −0.274758 −0.0106069
\(672\) 0 0
\(673\) 12.4921 0.481536 0.240768 0.970583i \(-0.422601\pi\)
0.240768 + 0.970583i \(0.422601\pi\)
\(674\) −5.35093 −0.206110
\(675\) 0 0
\(676\) 0.242256 0.00931752
\(677\) 16.8830 0.648866 0.324433 0.945909i \(-0.394826\pi\)
0.324433 + 0.945909i \(0.394826\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 10.9958 0.421671
\(681\) 0 0
\(682\) −7.97958 −0.305554
\(683\) 6.76970 0.259036 0.129518 0.991577i \(-0.458657\pi\)
0.129518 + 0.991577i \(0.458657\pi\)
\(684\) 0 0
\(685\) 26.1252 0.998193
\(686\) 0 0
\(687\) 0 0
\(688\) −2.28577 −0.0871440
\(689\) 41.0654 1.56447
\(690\) 0 0
\(691\) 42.8876 1.63152 0.815760 0.578391i \(-0.196319\pi\)
0.815760 + 0.578391i \(0.196319\pi\)
\(692\) −10.2628 −0.390133
\(693\) 0 0
\(694\) 20.8197 0.790305
\(695\) −14.8605 −0.563693
\(696\) 0 0
\(697\) 50.0078 1.89418
\(698\) −23.1843 −0.877540
\(699\) 0 0
\(700\) 0 0
\(701\) 33.8990 1.28035 0.640173 0.768231i \(-0.278863\pi\)
0.640173 + 0.768231i \(0.278863\pi\)
\(702\) 0 0
\(703\) −18.7779 −0.708222
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −12.1243 −0.456303
\(707\) 0 0
\(708\) 0 0
\(709\) −28.6407 −1.07562 −0.537812 0.843065i \(-0.680749\pi\)
−0.537812 + 0.843065i \(0.680749\pi\)
\(710\) −19.5504 −0.733715
\(711\) 0 0
\(712\) −1.98214 −0.0742837
\(713\) −58.8182 −2.20276
\(714\) 0 0
\(715\) −6.54266 −0.244682
\(716\) 12.3954 0.463239
\(717\) 0 0
\(718\) 12.3312 0.460196
\(719\) 2.52225 0.0940639 0.0470319 0.998893i \(-0.485024\pi\)
0.0470319 + 0.998893i \(0.485024\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.6105 0.580963
\(723\) 0 0
\(724\) 22.4506 0.834370
\(725\) −18.4243 −0.684261
\(726\) 0 0
\(727\) 30.0452 1.11431 0.557157 0.830407i \(-0.311892\pi\)
0.557157 + 0.830407i \(0.311892\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6.93808 −0.256790
\(731\) −13.9793 −0.517044
\(732\) 0 0
\(733\) 33.5866 1.24055 0.620275 0.784385i \(-0.287021\pi\)
0.620275 + 0.784385i \(0.287021\pi\)
\(734\) −18.0860 −0.667566
\(735\) 0 0
\(736\) −7.37109 −0.271702
\(737\) −3.88163 −0.142982
\(738\) 0 0
\(739\) −29.7380 −1.09393 −0.546965 0.837155i \(-0.684217\pi\)
−0.546965 + 0.837155i \(0.684217\pi\)
\(740\) −18.3380 −0.674120
\(741\) 0 0
\(742\) 0 0
\(743\) −1.59262 −0.0584276 −0.0292138 0.999573i \(-0.509300\pi\)
−0.0292138 + 0.999573i \(0.509300\pi\)
\(744\) 0 0
\(745\) −13.5752 −0.497357
\(746\) −6.95365 −0.254591
\(747\) 0 0
\(748\) −6.11582 −0.223616
\(749\) 0 0
\(750\) 0 0
\(751\) 0.305924 0.0111633 0.00558166 0.999984i \(-0.498223\pi\)
0.00558166 + 0.999984i \(0.498223\pi\)
\(752\) −0.669485 −0.0244136
\(753\) 0 0
\(754\) 37.9339 1.38147
\(755\) −22.2926 −0.811312
\(756\) 0 0
\(757\) 40.4921 1.47171 0.735856 0.677138i \(-0.236780\pi\)
0.735856 + 0.677138i \(0.236780\pi\)
\(758\) −5.14307 −0.186805
\(759\) 0 0
\(760\) −3.31010 −0.120070
\(761\) 48.6887 1.76496 0.882482 0.470346i \(-0.155871\pi\)
0.882482 + 0.470346i \(0.155871\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 13.7142 0.496164
\(765\) 0 0
\(766\) 18.8810 0.682197
\(767\) 49.3833 1.78313
\(768\) 0 0
\(769\) −14.7255 −0.531017 −0.265508 0.964109i \(-0.585540\pi\)
−0.265508 + 0.964109i \(0.585540\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.77883 0.136003
\(773\) 28.9612 1.04166 0.520830 0.853660i \(-0.325622\pi\)
0.520830 + 0.853660i \(0.325622\pi\)
\(774\) 0 0
\(775\) 14.1034 0.506610
\(776\) 4.12066 0.147923
\(777\) 0 0
\(778\) −30.3990 −1.08986
\(779\) −15.0540 −0.539365
\(780\) 0 0
\(781\) 10.8738 0.389097
\(782\) −45.0802 −1.61207
\(783\) 0 0
\(784\) 0 0
\(785\) 8.16281 0.291343
\(786\) 0 0
\(787\) 2.14203 0.0763551 0.0381775 0.999271i \(-0.487845\pi\)
0.0381775 + 0.999271i \(0.487845\pi\)
\(788\) 21.6316 0.770594
\(789\) 0 0
\(790\) −22.5054 −0.800708
\(791\) 0 0
\(792\) 0 0
\(793\) −0.999840 −0.0355054
\(794\) −2.08009 −0.0738196
\(795\) 0 0
\(796\) 17.8323 0.632051
\(797\) 7.38735 0.261673 0.130837 0.991404i \(-0.458234\pi\)
0.130837 + 0.991404i \(0.458234\pi\)
\(798\) 0 0
\(799\) −4.09445 −0.144851
\(800\) 1.76744 0.0624884
\(801\) 0 0
\(802\) −21.8444 −0.771353
\(803\) 3.85892 0.136178
\(804\) 0 0
\(805\) 0 0
\(806\) −29.0376 −1.02281
\(807\) 0 0
\(808\) −6.90367 −0.242870
\(809\) −8.49214 −0.298568 −0.149284 0.988794i \(-0.547697\pi\)
−0.149284 + 0.988794i \(0.547697\pi\)
\(810\) 0 0
\(811\) 5.97995 0.209984 0.104992 0.994473i \(-0.466518\pi\)
0.104992 + 0.994473i \(0.466518\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 10.1995 0.357493
\(815\) −28.4568 −0.996797
\(816\) 0 0
\(817\) 4.20823 0.147227
\(818\) 33.7571 1.18029
\(819\) 0 0
\(820\) −14.7013 −0.513393
\(821\) −20.6430 −0.720445 −0.360223 0.932866i \(-0.617299\pi\)
−0.360223 + 0.932866i \(0.617299\pi\)
\(822\) 0 0
\(823\) −32.8269 −1.14427 −0.572137 0.820158i \(-0.693886\pi\)
−0.572137 + 0.820158i \(0.693886\pi\)
\(824\) 12.6154 0.439477
\(825\) 0 0
\(826\) 0 0
\(827\) −7.22287 −0.251164 −0.125582 0.992083i \(-0.540080\pi\)
−0.125582 + 0.992083i \(0.540080\pi\)
\(828\) 0 0
\(829\) −52.2625 −1.81515 −0.907577 0.419887i \(-0.862070\pi\)
−0.907577 + 0.419887i \(0.862070\pi\)
\(830\) 9.48074 0.329081
\(831\) 0 0
\(832\) −3.63899 −0.126159
\(833\) 0 0
\(834\) 0 0
\(835\) 39.7062 1.37409
\(836\) 1.84106 0.0636743
\(837\) 0 0
\(838\) −22.7371 −0.785439
\(839\) −20.5644 −0.709963 −0.354981 0.934873i \(-0.615513\pi\)
−0.354981 + 0.934873i \(0.615513\pi\)
\(840\) 0 0
\(841\) 79.6659 2.74710
\(842\) 21.0192 0.724370
\(843\) 0 0
\(844\) −8.27020 −0.284672
\(845\) −0.435559 −0.0149837
\(846\) 0 0
\(847\) 0 0
\(848\) −11.2848 −0.387523
\(849\) 0 0
\(850\) 10.8093 0.370757
\(851\) 75.1815 2.57719
\(852\) 0 0
\(853\) −24.8004 −0.849148 −0.424574 0.905393i \(-0.639576\pi\)
−0.424574 + 0.905393i \(0.639576\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.9417 0.579055
\(857\) −30.1658 −1.03044 −0.515222 0.857056i \(-0.672291\pi\)
−0.515222 + 0.857056i \(0.672291\pi\)
\(858\) 0 0
\(859\) −26.3550 −0.899219 −0.449610 0.893225i \(-0.648437\pi\)
−0.449610 + 0.893225i \(0.648437\pi\)
\(860\) 4.10965 0.140138
\(861\) 0 0
\(862\) 7.03305 0.239547
\(863\) 16.8489 0.573545 0.286772 0.957999i \(-0.407418\pi\)
0.286772 + 0.957999i \(0.407418\pi\)
\(864\) 0 0
\(865\) 18.4518 0.627381
\(866\) 5.77293 0.196172
\(867\) 0 0
\(868\) 0 0
\(869\) 12.5174 0.424624
\(870\) 0 0
\(871\) −14.1252 −0.478615
\(872\) −9.76326 −0.330626
\(873\) 0 0
\(874\) 13.5706 0.459032
\(875\) 0 0
\(876\) 0 0
\(877\) 39.3054 1.32725 0.663624 0.748066i \(-0.269018\pi\)
0.663624 + 0.748066i \(0.269018\pi\)
\(878\) 31.2275 1.05388
\(879\) 0 0
\(880\) 1.79793 0.0606083
\(881\) 12.1055 0.407843 0.203922 0.978987i \(-0.434631\pi\)
0.203922 + 0.978987i \(0.434631\pi\)
\(882\) 0 0
\(883\) 21.0614 0.708773 0.354386 0.935099i \(-0.384690\pi\)
0.354386 + 0.935099i \(0.384690\pi\)
\(884\) −22.2554 −0.748530
\(885\) 0 0
\(886\) −35.8953 −1.20593
\(887\) −45.1762 −1.51687 −0.758434 0.651750i \(-0.774035\pi\)
−0.758434 + 0.651750i \(0.774035\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 3.56375 0.119457
\(891\) 0 0
\(892\) 9.10574 0.304883
\(893\) 1.23256 0.0412461
\(894\) 0 0
\(895\) −22.2861 −0.744943
\(896\) 0 0
\(897\) 0 0
\(898\) −24.5495 −0.819228
\(899\) −83.1815 −2.77426
\(900\) 0 0
\(901\) −69.0160 −2.29926
\(902\) 8.17680 0.272258
\(903\) 0 0
\(904\) −12.9348 −0.430206
\(905\) −40.3647 −1.34177
\(906\) 0 0
\(907\) −33.5283 −1.11329 −0.556644 0.830751i \(-0.687911\pi\)
−0.556644 + 0.830751i \(0.687911\pi\)
\(908\) −0.620456 −0.0205905
\(909\) 0 0
\(910\) 0 0
\(911\) 27.3400 0.905813 0.452906 0.891558i \(-0.350387\pi\)
0.452906 + 0.891558i \(0.350387\pi\)
\(912\) 0 0
\(913\) −5.27314 −0.174515
\(914\) −26.2771 −0.869168
\(915\) 0 0
\(916\) −18.0011 −0.594772
\(917\) 0 0
\(918\) 0 0
\(919\) −12.6118 −0.416026 −0.208013 0.978126i \(-0.566700\pi\)
−0.208013 + 0.978126i \(0.566700\pi\)
\(920\) 13.2527 0.436929
\(921\) 0 0
\(922\) −20.8330 −0.686098
\(923\) 39.5698 1.30246
\(924\) 0 0
\(925\) −18.0270 −0.592725
\(926\) −17.8697 −0.587234
\(927\) 0 0
\(928\) −10.4243 −0.342194
\(929\) −53.3407 −1.75005 −0.875027 0.484075i \(-0.839156\pi\)
−0.875027 + 0.484075i \(0.839156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 19.8697 0.650853
\(933\) 0 0
\(934\) 26.8642 0.879022
\(935\) 10.9958 0.359602
\(936\) 0 0
\(937\) 20.5927 0.672736 0.336368 0.941731i \(-0.390801\pi\)
0.336368 + 0.941731i \(0.390801\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.20369 0.0392600
\(941\) −27.9554 −0.911319 −0.455660 0.890154i \(-0.650597\pi\)
−0.455660 + 0.890154i \(0.650597\pi\)
\(942\) 0 0
\(943\) 60.2719 1.96272
\(944\) −13.5706 −0.441686
\(945\) 0 0
\(946\) −2.28577 −0.0743167
\(947\) 44.7266 1.45342 0.726710 0.686945i \(-0.241049\pi\)
0.726710 + 0.686945i \(0.241049\pi\)
\(948\) 0 0
\(949\) 14.0426 0.455841
\(950\) −3.25396 −0.105572
\(951\) 0 0
\(952\) 0 0
\(953\) 51.6527 1.67320 0.836598 0.547817i \(-0.184541\pi\)
0.836598 + 0.547817i \(0.184541\pi\)
\(954\) 0 0
\(955\) −24.6573 −0.797890
\(956\) 17.4201 0.563407
\(957\) 0 0
\(958\) 15.5050 0.500945
\(959\) 0 0
\(960\) 0 0
\(961\) 32.6738 1.05399
\(962\) 37.1159 1.19667
\(963\) 0 0
\(964\) −18.4149 −0.593104
\(965\) −6.79409 −0.218709
\(966\) 0 0
\(967\) −50.7320 −1.63143 −0.815715 0.578454i \(-0.803656\pi\)
−0.815715 + 0.578454i \(0.803656\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −7.40867 −0.237878
\(971\) −22.7299 −0.729436 −0.364718 0.931118i \(-0.618835\pi\)
−0.364718 + 0.931118i \(0.618835\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −10.8858 −0.348804
\(975\) 0 0
\(976\) 0.274758 0.00879478
\(977\) 27.4811 0.879198 0.439599 0.898194i \(-0.355120\pi\)
0.439599 + 0.898194i \(0.355120\pi\)
\(978\) 0 0
\(979\) −1.98214 −0.0633494
\(980\) 0 0
\(981\) 0 0
\(982\) 7.06975 0.225605
\(983\) 34.2044 1.09095 0.545475 0.838127i \(-0.316349\pi\)
0.545475 + 0.838127i \(0.316349\pi\)
\(984\) 0 0
\(985\) −38.8921 −1.23921
\(986\) −63.7531 −2.03031
\(987\) 0 0
\(988\) 6.69959 0.213143
\(989\) −16.8486 −0.535754
\(990\) 0 0
\(991\) −33.3890 −1.06064 −0.530318 0.847799i \(-0.677927\pi\)
−0.530318 + 0.847799i \(0.677927\pi\)
\(992\) 7.97958 0.253352
\(993\) 0 0
\(994\) 0 0
\(995\) −32.0614 −1.01641
\(996\) 0 0
\(997\) −7.74406 −0.245257 −0.122628 0.992453i \(-0.539132\pi\)
−0.122628 + 0.992453i \(0.539132\pi\)
\(998\) −18.8678 −0.597249
\(999\) 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 9702.2.a.ea.1.1 4
3.2 odd 2 3234.2.a.bl.1.4 4
7.6 odd 2 9702.2.a.dz.1.4 4
21.20 even 2 3234.2.a.bm.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bl.1.4 4 3.2 odd 2
3234.2.a.bm.1.1 yes 4 21.20 even 2
9702.2.a.dz.1.4 4 7.6 odd 2
9702.2.a.ea.1.1 4 1.1 even 1 trivial