Properties

Label 1672.2.a.i.1.4
Level $1672$
Weight $2$
Character 1672.1
Self dual yes
Analytic conductor $13.351$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1672,2,Mod(1,1672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1672.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.3509872180\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.106392688.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 12x^{3} + 25x^{2} - 10x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.972165\) of defining polynomial
Character \(\chi\) \(=\) 1672.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0278351 q^{3} -0.122408 q^{5} -0.877592 q^{7} -2.99923 q^{9} +O(q^{10})\) \(q-0.0278351 q^{3} -0.122408 q^{5} -0.877592 q^{7} -2.99923 q^{9} -1.00000 q^{11} +1.68436 q^{13} +0.00340723 q^{15} +6.34040 q^{17} +1.00000 q^{19} +0.0244279 q^{21} -6.12835 q^{23} -4.98502 q^{25} +0.166989 q^{27} +3.64572 q^{29} +0.00340723 q^{31} +0.0278351 q^{33} +0.107424 q^{35} -8.21050 q^{37} -0.0468844 q^{39} -11.1723 q^{41} +0.0339670 q^{43} +0.367128 q^{45} -6.10979 q^{47} -6.22983 q^{49} -0.176486 q^{51} +7.24999 q^{53} +0.122408 q^{55} -0.0278351 q^{57} -14.2031 q^{59} -4.05412 q^{61} +2.63210 q^{63} -0.206179 q^{65} -8.58181 q^{67} +0.170583 q^{69} +0.0136263 q^{71} +3.73280 q^{73} +0.138759 q^{75} +0.877592 q^{77} -0.356156 q^{79} +8.99303 q^{81} -15.5146 q^{83} -0.776113 q^{85} -0.101479 q^{87} +5.31740 q^{89} -1.47818 q^{91} -9.48408e-5 q^{93} -0.122408 q^{95} +16.7120 q^{97} +2.99923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} - 6 q^{11} - 3 q^{13} - q^{15} - q^{17} + 6 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - 16 q^{27} - 7 q^{29} - q^{31} + 4 q^{33} - 32 q^{35} + 5 q^{37} - 13 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 7 q^{49} - 35 q^{51} - 11 q^{53} + 3 q^{55} - 4 q^{57} - 18 q^{59} + 16 q^{61} - 6 q^{63} + 10 q^{65} - 18 q^{67} - 2 q^{69} - 7 q^{71} - 21 q^{73} - 23 q^{75} + 3 q^{77} - 22 q^{79} - 10 q^{81} - 16 q^{83} - 3 q^{87} - 13 q^{89} - 7 q^{91} - 9 q^{93} - 3 q^{95} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.0278351 −0.0160706 −0.00803531 0.999968i \(-0.502558\pi\)
−0.00803531 + 0.999968i \(0.502558\pi\)
\(4\) 0 0
\(5\) −0.122408 −0.0547424 −0.0273712 0.999625i \(-0.508714\pi\)
−0.0273712 + 0.999625i \(0.508714\pi\)
\(6\) 0 0
\(7\) −0.877592 −0.331699 −0.165849 0.986151i \(-0.553037\pi\)
−0.165849 + 0.986151i \(0.553037\pi\)
\(8\) 0 0
\(9\) −2.99923 −0.999742
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.68436 0.467157 0.233579 0.972338i \(-0.424956\pi\)
0.233579 + 0.972338i \(0.424956\pi\)
\(14\) 0 0
\(15\) 0.00340723 0.000879744 0
\(16\) 0 0
\(17\) 6.34040 1.53777 0.768886 0.639386i \(-0.220811\pi\)
0.768886 + 0.639386i \(0.220811\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.0244279 0.00533060
\(22\) 0 0
\(23\) −6.12835 −1.27785 −0.638925 0.769269i \(-0.720621\pi\)
−0.638925 + 0.769269i \(0.720621\pi\)
\(24\) 0 0
\(25\) −4.98502 −0.997003
\(26\) 0 0
\(27\) 0.166989 0.0321371
\(28\) 0 0
\(29\) 3.64572 0.676994 0.338497 0.940968i \(-0.390082\pi\)
0.338497 + 0.940968i \(0.390082\pi\)
\(30\) 0 0
\(31\) 0.00340723 0.000611957 0 0.000305979 1.00000i \(-0.499903\pi\)
0.000305979 1.00000i \(0.499903\pi\)
\(32\) 0 0
\(33\) 0.0278351 0.00484547
\(34\) 0 0
\(35\) 0.107424 0.0181580
\(36\) 0 0
\(37\) −8.21050 −1.34980 −0.674899 0.737911i \(-0.735813\pi\)
−0.674899 + 0.737911i \(0.735813\pi\)
\(38\) 0 0
\(39\) −0.0468844 −0.00750751
\(40\) 0 0
\(41\) −11.1723 −1.74483 −0.872414 0.488768i \(-0.837446\pi\)
−0.872414 + 0.488768i \(0.837446\pi\)
\(42\) 0 0
\(43\) 0.0339670 0.00517992 0.00258996 0.999997i \(-0.499176\pi\)
0.00258996 + 0.999997i \(0.499176\pi\)
\(44\) 0 0
\(45\) 0.367128 0.0547282
\(46\) 0 0
\(47\) −6.10979 −0.891205 −0.445602 0.895231i \(-0.647010\pi\)
−0.445602 + 0.895231i \(0.647010\pi\)
\(48\) 0 0
\(49\) −6.22983 −0.889976
\(50\) 0 0
\(51\) −0.176486 −0.0247130
\(52\) 0 0
\(53\) 7.24999 0.995862 0.497931 0.867217i \(-0.334093\pi\)
0.497931 + 0.867217i \(0.334093\pi\)
\(54\) 0 0
\(55\) 0.122408 0.0165054
\(56\) 0 0
\(57\) −0.0278351 −0.00368685
\(58\) 0 0
\(59\) −14.2031 −1.84909 −0.924543 0.381077i \(-0.875553\pi\)
−0.924543 + 0.381077i \(0.875553\pi\)
\(60\) 0 0
\(61\) −4.05412 −0.519077 −0.259538 0.965733i \(-0.583570\pi\)
−0.259538 + 0.965733i \(0.583570\pi\)
\(62\) 0 0
\(63\) 2.63210 0.331613
\(64\) 0 0
\(65\) −0.206179 −0.0255733
\(66\) 0 0
\(67\) −8.58181 −1.04843 −0.524217 0.851585i \(-0.675642\pi\)
−0.524217 + 0.851585i \(0.675642\pi\)
\(68\) 0 0
\(69\) 0.170583 0.0205358
\(70\) 0 0
\(71\) 0.0136263 0.00161714 0.000808572 1.00000i \(-0.499743\pi\)
0.000808572 1.00000i \(0.499743\pi\)
\(72\) 0 0
\(73\) 3.73280 0.436891 0.218446 0.975849i \(-0.429901\pi\)
0.218446 + 0.975849i \(0.429901\pi\)
\(74\) 0 0
\(75\) 0.138759 0.0160225
\(76\) 0 0
\(77\) 0.877592 0.100011
\(78\) 0 0
\(79\) −0.356156 −0.0400707 −0.0200353 0.999799i \(-0.506378\pi\)
−0.0200353 + 0.999799i \(0.506378\pi\)
\(80\) 0 0
\(81\) 8.99303 0.999225
\(82\) 0 0
\(83\) −15.5146 −1.70294 −0.851472 0.524400i \(-0.824290\pi\)
−0.851472 + 0.524400i \(0.824290\pi\)
\(84\) 0 0
\(85\) −0.776113 −0.0841813
\(86\) 0 0
\(87\) −0.101479 −0.0108797
\(88\) 0 0
\(89\) 5.31740 0.563644 0.281822 0.959467i \(-0.409061\pi\)
0.281822 + 0.959467i \(0.409061\pi\)
\(90\) 0 0
\(91\) −1.47818 −0.154956
\(92\) 0 0
\(93\) −9.48408e−5 0 −9.83453e−6 0
\(94\) 0 0
\(95\) −0.122408 −0.0125588
\(96\) 0 0
\(97\) 16.7120 1.69685 0.848424 0.529317i \(-0.177552\pi\)
0.848424 + 0.529317i \(0.177552\pi\)
\(98\) 0 0
\(99\) 2.99923 0.301433
\(100\) 0 0
\(101\) −11.0126 −1.09579 −0.547896 0.836547i \(-0.684571\pi\)
−0.547896 + 0.836547i \(0.684571\pi\)
\(102\) 0 0
\(103\) −6.43605 −0.634162 −0.317081 0.948398i \(-0.602703\pi\)
−0.317081 + 0.948398i \(0.602703\pi\)
\(104\) 0 0
\(105\) −0.00299016 −0.000291810 0
\(106\) 0 0
\(107\) 6.58366 0.636467 0.318233 0.948012i \(-0.396910\pi\)
0.318233 + 0.948012i \(0.396910\pi\)
\(108\) 0 0
\(109\) 14.1528 1.35559 0.677794 0.735252i \(-0.262936\pi\)
0.677794 + 0.735252i \(0.262936\pi\)
\(110\) 0 0
\(111\) 0.228540 0.0216921
\(112\) 0 0
\(113\) 4.82796 0.454176 0.227088 0.973874i \(-0.427079\pi\)
0.227088 + 0.973874i \(0.427079\pi\)
\(114\) 0 0
\(115\) 0.750157 0.0699525
\(116\) 0 0
\(117\) −5.05178 −0.467037
\(118\) 0 0
\(119\) −5.56428 −0.510077
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.310984 0.0280405
\(124\) 0 0
\(125\) 1.22224 0.109321
\(126\) 0 0
\(127\) 1.05382 0.0935114 0.0467557 0.998906i \(-0.485112\pi\)
0.0467557 + 0.998906i \(0.485112\pi\)
\(128\) 0 0
\(129\) −0.000945476 0 −8.32445e−5 0
\(130\) 0 0
\(131\) 9.20418 0.804173 0.402086 0.915602i \(-0.368285\pi\)
0.402086 + 0.915602i \(0.368285\pi\)
\(132\) 0 0
\(133\) −0.877592 −0.0760969
\(134\) 0 0
\(135\) −0.0204408 −0.00175926
\(136\) 0 0
\(137\) −6.50230 −0.555529 −0.277764 0.960649i \(-0.589593\pi\)
−0.277764 + 0.960649i \(0.589593\pi\)
\(138\) 0 0
\(139\) −11.8310 −1.00349 −0.501747 0.865015i \(-0.667309\pi\)
−0.501747 + 0.865015i \(0.667309\pi\)
\(140\) 0 0
\(141\) 0.170067 0.0143222
\(142\) 0 0
\(143\) −1.68436 −0.140853
\(144\) 0 0
\(145\) −0.446264 −0.0370602
\(146\) 0 0
\(147\) 0.173408 0.0143025
\(148\) 0 0
\(149\) −18.2102 −1.49184 −0.745921 0.666035i \(-0.767990\pi\)
−0.745921 + 0.666035i \(0.767990\pi\)
\(150\) 0 0
\(151\) −14.9002 −1.21257 −0.606283 0.795249i \(-0.707340\pi\)
−0.606283 + 0.795249i \(0.707340\pi\)
\(152\) 0 0
\(153\) −19.0163 −1.53738
\(154\) 0 0
\(155\) −0.000417071 0 −3.35000e−5 0
\(156\) 0 0
\(157\) 19.8091 1.58094 0.790471 0.612500i \(-0.209836\pi\)
0.790471 + 0.612500i \(0.209836\pi\)
\(158\) 0 0
\(159\) −0.201804 −0.0160041
\(160\) 0 0
\(161\) 5.37820 0.423861
\(162\) 0 0
\(163\) −17.2833 −1.35373 −0.676865 0.736107i \(-0.736662\pi\)
−0.676865 + 0.736107i \(0.736662\pi\)
\(164\) 0 0
\(165\) −0.00340723 −0.000265253 0
\(166\) 0 0
\(167\) −11.9578 −0.925320 −0.462660 0.886536i \(-0.653105\pi\)
−0.462660 + 0.886536i \(0.653105\pi\)
\(168\) 0 0
\(169\) −10.1629 −0.781764
\(170\) 0 0
\(171\) −2.99923 −0.229356
\(172\) 0 0
\(173\) −12.0077 −0.912930 −0.456465 0.889741i \(-0.650885\pi\)
−0.456465 + 0.889741i \(0.650885\pi\)
\(174\) 0 0
\(175\) 4.37481 0.330705
\(176\) 0 0
\(177\) 0.395345 0.0297160
\(178\) 0 0
\(179\) 12.4981 0.934153 0.467076 0.884217i \(-0.345307\pi\)
0.467076 + 0.884217i \(0.345307\pi\)
\(180\) 0 0
\(181\) 4.84332 0.360001 0.180001 0.983666i \(-0.442390\pi\)
0.180001 + 0.983666i \(0.442390\pi\)
\(182\) 0 0
\(183\) 0.112847 0.00834189
\(184\) 0 0
\(185\) 1.00503 0.0738911
\(186\) 0 0
\(187\) −6.34040 −0.463656
\(188\) 0 0
\(189\) −0.146548 −0.0106598
\(190\) 0 0
\(191\) −9.43667 −0.682813 −0.341407 0.939916i \(-0.610903\pi\)
−0.341407 + 0.939916i \(0.610903\pi\)
\(192\) 0 0
\(193\) 16.7679 1.20698 0.603489 0.797371i \(-0.293777\pi\)
0.603489 + 0.797371i \(0.293777\pi\)
\(194\) 0 0
\(195\) 0.00573901 0.000410979 0
\(196\) 0 0
\(197\) 26.4397 1.88375 0.941875 0.335963i \(-0.109062\pi\)
0.941875 + 0.335963i \(0.109062\pi\)
\(198\) 0 0
\(199\) 19.2973 1.36795 0.683974 0.729506i \(-0.260250\pi\)
0.683974 + 0.729506i \(0.260250\pi\)
\(200\) 0 0
\(201\) 0.238876 0.0168490
\(202\) 0 0
\(203\) −3.19946 −0.224558
\(204\) 0 0
\(205\) 1.36758 0.0955160
\(206\) 0 0
\(207\) 18.3803 1.27752
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 9.97964 0.687026 0.343513 0.939148i \(-0.388383\pi\)
0.343513 + 0.939148i \(0.388383\pi\)
\(212\) 0 0
\(213\) −0.000379290 0 −2.59885e−5 0
\(214\) 0 0
\(215\) −0.00415782 −0.000283561 0
\(216\) 0 0
\(217\) −0.00299016 −0.000202985 0
\(218\) 0 0
\(219\) −0.103903 −0.00702112
\(220\) 0 0
\(221\) 10.6795 0.718382
\(222\) 0 0
\(223\) 16.2692 1.08947 0.544734 0.838609i \(-0.316631\pi\)
0.544734 + 0.838609i \(0.316631\pi\)
\(224\) 0 0
\(225\) 14.9512 0.996746
\(226\) 0 0
\(227\) −25.1015 −1.66605 −0.833023 0.553238i \(-0.813392\pi\)
−0.833023 + 0.553238i \(0.813392\pi\)
\(228\) 0 0
\(229\) −3.06007 −0.202215 −0.101107 0.994876i \(-0.532239\pi\)
−0.101107 + 0.994876i \(0.532239\pi\)
\(230\) 0 0
\(231\) −0.0244279 −0.00160724
\(232\) 0 0
\(233\) −22.1783 −1.45295 −0.726473 0.687195i \(-0.758842\pi\)
−0.726473 + 0.687195i \(0.758842\pi\)
\(234\) 0 0
\(235\) 0.747885 0.0487866
\(236\) 0 0
\(237\) 0.00991364 0.000643960 0
\(238\) 0 0
\(239\) 21.9023 1.41674 0.708370 0.705842i \(-0.249431\pi\)
0.708370 + 0.705842i \(0.249431\pi\)
\(240\) 0 0
\(241\) −2.89608 −0.186553 −0.0932765 0.995640i \(-0.529734\pi\)
−0.0932765 + 0.995640i \(0.529734\pi\)
\(242\) 0 0
\(243\) −0.751290 −0.0481953
\(244\) 0 0
\(245\) 0.762579 0.0487194
\(246\) 0 0
\(247\) 1.68436 0.107173
\(248\) 0 0
\(249\) 0.431850 0.0273674
\(250\) 0 0
\(251\) −7.46432 −0.471143 −0.235572 0.971857i \(-0.575696\pi\)
−0.235572 + 0.971857i \(0.575696\pi\)
\(252\) 0 0
\(253\) 6.12835 0.385286
\(254\) 0 0
\(255\) 0.0216032 0.00135285
\(256\) 0 0
\(257\) 5.24751 0.327331 0.163665 0.986516i \(-0.447668\pi\)
0.163665 + 0.986516i \(0.447668\pi\)
\(258\) 0 0
\(259\) 7.20547 0.447726
\(260\) 0 0
\(261\) −10.9343 −0.676819
\(262\) 0 0
\(263\) 18.6240 1.14841 0.574204 0.818712i \(-0.305312\pi\)
0.574204 + 0.818712i \(0.305312\pi\)
\(264\) 0 0
\(265\) −0.887454 −0.0545158
\(266\) 0 0
\(267\) −0.148011 −0.00905810
\(268\) 0 0
\(269\) −15.0400 −0.917005 −0.458503 0.888693i \(-0.651614\pi\)
−0.458503 + 0.888693i \(0.651614\pi\)
\(270\) 0 0
\(271\) −28.2895 −1.71847 −0.859234 0.511583i \(-0.829059\pi\)
−0.859234 + 0.511583i \(0.829059\pi\)
\(272\) 0 0
\(273\) 0.0411454 0.00249023
\(274\) 0 0
\(275\) 4.98502 0.300608
\(276\) 0 0
\(277\) 22.3581 1.34337 0.671685 0.740837i \(-0.265571\pi\)
0.671685 + 0.740837i \(0.265571\pi\)
\(278\) 0 0
\(279\) −0.0102191 −0.000611799 0
\(280\) 0 0
\(281\) 12.3487 0.736662 0.368331 0.929695i \(-0.379929\pi\)
0.368331 + 0.929695i \(0.379929\pi\)
\(282\) 0 0
\(283\) −12.3996 −0.737082 −0.368541 0.929611i \(-0.620143\pi\)
−0.368541 + 0.929611i \(0.620143\pi\)
\(284\) 0 0
\(285\) 0.00340723 0.000201827 0
\(286\) 0 0
\(287\) 9.80477 0.578757
\(288\) 0 0
\(289\) 23.2006 1.36474
\(290\) 0 0
\(291\) −0.465181 −0.0272694
\(292\) 0 0
\(293\) −7.24377 −0.423186 −0.211593 0.977358i \(-0.567865\pi\)
−0.211593 + 0.977358i \(0.567865\pi\)
\(294\) 0 0
\(295\) 1.73857 0.101223
\(296\) 0 0
\(297\) −0.166989 −0.00968970
\(298\) 0 0
\(299\) −10.3224 −0.596957
\(300\) 0 0
\(301\) −0.0298092 −0.00171817
\(302\) 0 0
\(303\) 0.306536 0.0176100
\(304\) 0 0
\(305\) 0.496255 0.0284155
\(306\) 0 0
\(307\) −18.1408 −1.03535 −0.517674 0.855578i \(-0.673202\pi\)
−0.517674 + 0.855578i \(0.673202\pi\)
\(308\) 0 0
\(309\) 0.179148 0.0101914
\(310\) 0 0
\(311\) 9.59406 0.544029 0.272015 0.962293i \(-0.412310\pi\)
0.272015 + 0.962293i \(0.412310\pi\)
\(312\) 0 0
\(313\) 29.2280 1.65207 0.826033 0.563622i \(-0.190592\pi\)
0.826033 + 0.563622i \(0.190592\pi\)
\(314\) 0 0
\(315\) −0.322189 −0.0181533
\(316\) 0 0
\(317\) 12.8728 0.723008 0.361504 0.932371i \(-0.382263\pi\)
0.361504 + 0.932371i \(0.382263\pi\)
\(318\) 0 0
\(319\) −3.64572 −0.204121
\(320\) 0 0
\(321\) −0.183257 −0.0102284
\(322\) 0 0
\(323\) 6.34040 0.352789
\(324\) 0 0
\(325\) −8.39656 −0.465758
\(326\) 0 0
\(327\) −0.393944 −0.0217851
\(328\) 0 0
\(329\) 5.36191 0.295611
\(330\) 0 0
\(331\) 8.48289 0.466262 0.233131 0.972445i \(-0.425103\pi\)
0.233131 + 0.972445i \(0.425103\pi\)
\(332\) 0 0
\(333\) 24.6251 1.34945
\(334\) 0 0
\(335\) 1.05048 0.0573938
\(336\) 0 0
\(337\) −13.7212 −0.747442 −0.373721 0.927541i \(-0.621918\pi\)
−0.373721 + 0.927541i \(0.621918\pi\)
\(338\) 0 0
\(339\) −0.134387 −0.00729890
\(340\) 0 0
\(341\) −0.00340723 −0.000184512 0
\(342\) 0 0
\(343\) 11.6104 0.626903
\(344\) 0 0
\(345\) −0.0208807 −0.00112418
\(346\) 0 0
\(347\) −13.8279 −0.742318 −0.371159 0.928569i \(-0.621040\pi\)
−0.371159 + 0.928569i \(0.621040\pi\)
\(348\) 0 0
\(349\) 30.1644 1.61466 0.807331 0.590099i \(-0.200911\pi\)
0.807331 + 0.590099i \(0.200911\pi\)
\(350\) 0 0
\(351\) 0.281270 0.0150131
\(352\) 0 0
\(353\) −32.9396 −1.75320 −0.876598 0.481223i \(-0.840193\pi\)
−0.876598 + 0.481223i \(0.840193\pi\)
\(354\) 0 0
\(355\) −0.00166796 −8.85263e−5 0
\(356\) 0 0
\(357\) 0.154883 0.00819726
\(358\) 0 0
\(359\) 35.8135 1.89017 0.945083 0.326830i \(-0.105980\pi\)
0.945083 + 0.326830i \(0.105980\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.0278351 −0.00146097
\(364\) 0 0
\(365\) −0.456923 −0.0239165
\(366\) 0 0
\(367\) 29.6271 1.54652 0.773260 0.634089i \(-0.218625\pi\)
0.773260 + 0.634089i \(0.218625\pi\)
\(368\) 0 0
\(369\) 33.5084 1.74438
\(370\) 0 0
\(371\) −6.36253 −0.330326
\(372\) 0 0
\(373\) 3.64878 0.188927 0.0944633 0.995528i \(-0.469886\pi\)
0.0944633 + 0.995528i \(0.469886\pi\)
\(374\) 0 0
\(375\) −0.0340213 −0.00175685
\(376\) 0 0
\(377\) 6.14071 0.316263
\(378\) 0 0
\(379\) 2.07155 0.106408 0.0532041 0.998584i \(-0.483057\pi\)
0.0532041 + 0.998584i \(0.483057\pi\)
\(380\) 0 0
\(381\) −0.0293332 −0.00150279
\(382\) 0 0
\(383\) 14.4545 0.738592 0.369296 0.929312i \(-0.379599\pi\)
0.369296 + 0.929312i \(0.379599\pi\)
\(384\) 0 0
\(385\) −0.107424 −0.00547483
\(386\) 0 0
\(387\) −0.101875 −0.00517858
\(388\) 0 0
\(389\) 6.37041 0.322992 0.161496 0.986873i \(-0.448368\pi\)
0.161496 + 0.986873i \(0.448368\pi\)
\(390\) 0 0
\(391\) −38.8562 −1.96504
\(392\) 0 0
\(393\) −0.256199 −0.0129236
\(394\) 0 0
\(395\) 0.0435962 0.00219356
\(396\) 0 0
\(397\) −13.9040 −0.697824 −0.348912 0.937155i \(-0.613449\pi\)
−0.348912 + 0.937155i \(0.613449\pi\)
\(398\) 0 0
\(399\) 0.0244279 0.00122292
\(400\) 0 0
\(401\) −25.6324 −1.28002 −0.640011 0.768366i \(-0.721070\pi\)
−0.640011 + 0.768366i \(0.721070\pi\)
\(402\) 0 0
\(403\) 0.00573901 0.000285880 0
\(404\) 0 0
\(405\) −1.10082 −0.0547000
\(406\) 0 0
\(407\) 8.21050 0.406979
\(408\) 0 0
\(409\) 2.60766 0.128941 0.0644703 0.997920i \(-0.479464\pi\)
0.0644703 + 0.997920i \(0.479464\pi\)
\(410\) 0 0
\(411\) 0.180992 0.00892769
\(412\) 0 0
\(413\) 12.4645 0.613340
\(414\) 0 0
\(415\) 1.89910 0.0932232
\(416\) 0 0
\(417\) 0.329318 0.0161268
\(418\) 0 0
\(419\) −4.20940 −0.205643 −0.102821 0.994700i \(-0.532787\pi\)
−0.102821 + 0.994700i \(0.532787\pi\)
\(420\) 0 0
\(421\) −34.8223 −1.69714 −0.848568 0.529086i \(-0.822535\pi\)
−0.848568 + 0.529086i \(0.822535\pi\)
\(422\) 0 0
\(423\) 18.3246 0.890974
\(424\) 0 0
\(425\) −31.6070 −1.53316
\(426\) 0 0
\(427\) 3.55787 0.172177
\(428\) 0 0
\(429\) 0.0468844 0.00226360
\(430\) 0 0
\(431\) −21.1564 −1.01907 −0.509535 0.860450i \(-0.670182\pi\)
−0.509535 + 0.860450i \(0.670182\pi\)
\(432\) 0 0
\(433\) 4.66913 0.224384 0.112192 0.993687i \(-0.464213\pi\)
0.112192 + 0.993687i \(0.464213\pi\)
\(434\) 0 0
\(435\) 0.0124218 0.000595581 0
\(436\) 0 0
\(437\) −6.12835 −0.293159
\(438\) 0 0
\(439\) 36.2453 1.72989 0.864947 0.501864i \(-0.167352\pi\)
0.864947 + 0.501864i \(0.167352\pi\)
\(440\) 0 0
\(441\) 18.6847 0.889746
\(442\) 0 0
\(443\) −5.79016 −0.275099 −0.137549 0.990495i \(-0.543923\pi\)
−0.137549 + 0.990495i \(0.543923\pi\)
\(444\) 0 0
\(445\) −0.650891 −0.0308552
\(446\) 0 0
\(447\) 0.506884 0.0239748
\(448\) 0 0
\(449\) −19.4020 −0.915635 −0.457818 0.889046i \(-0.651369\pi\)
−0.457818 + 0.889046i \(0.651369\pi\)
\(450\) 0 0
\(451\) 11.1723 0.526085
\(452\) 0 0
\(453\) 0.414750 0.0194867
\(454\) 0 0
\(455\) 0.180941 0.00848263
\(456\) 0 0
\(457\) 24.6795 1.15446 0.577228 0.816583i \(-0.304134\pi\)
0.577228 + 0.816583i \(0.304134\pi\)
\(458\) 0 0
\(459\) 1.05878 0.0494195
\(460\) 0 0
\(461\) 3.63779 0.169429 0.0847145 0.996405i \(-0.473002\pi\)
0.0847145 + 0.996405i \(0.473002\pi\)
\(462\) 0 0
\(463\) 13.1814 0.612592 0.306296 0.951936i \(-0.400910\pi\)
0.306296 + 0.951936i \(0.400910\pi\)
\(464\) 0 0
\(465\) 1.16092e−5 0 5.38365e−7 0
\(466\) 0 0
\(467\) −31.6892 −1.46640 −0.733200 0.680013i \(-0.761974\pi\)
−0.733200 + 0.680013i \(0.761974\pi\)
\(468\) 0 0
\(469\) 7.53133 0.347764
\(470\) 0 0
\(471\) −0.551390 −0.0254067
\(472\) 0 0
\(473\) −0.0339670 −0.00156180
\(474\) 0 0
\(475\) −4.98502 −0.228728
\(476\) 0 0
\(477\) −21.7443 −0.995605
\(478\) 0 0
\(479\) 41.2270 1.88371 0.941854 0.336022i \(-0.109082\pi\)
0.941854 + 0.336022i \(0.109082\pi\)
\(480\) 0 0
\(481\) −13.8294 −0.630568
\(482\) 0 0
\(483\) −0.149703 −0.00681171
\(484\) 0 0
\(485\) −2.04568 −0.0928895
\(486\) 0 0
\(487\) 3.21468 0.145671 0.0728354 0.997344i \(-0.476795\pi\)
0.0728354 + 0.997344i \(0.476795\pi\)
\(488\) 0 0
\(489\) 0.481082 0.0217553
\(490\) 0 0
\(491\) −5.92336 −0.267317 −0.133659 0.991027i \(-0.542673\pi\)
−0.133659 + 0.991027i \(0.542673\pi\)
\(492\) 0 0
\(493\) 23.1153 1.04106
\(494\) 0 0
\(495\) −0.367128 −0.0165012
\(496\) 0 0
\(497\) −0.0119583 −0.000536404 0
\(498\) 0 0
\(499\) −13.8155 −0.618468 −0.309234 0.950986i \(-0.600073\pi\)
−0.309234 + 0.950986i \(0.600073\pi\)
\(500\) 0 0
\(501\) 0.332846 0.0148705
\(502\) 0 0
\(503\) 22.8789 1.02012 0.510061 0.860138i \(-0.329623\pi\)
0.510061 + 0.860138i \(0.329623\pi\)
\(504\) 0 0
\(505\) 1.34802 0.0599862
\(506\) 0 0
\(507\) 0.282887 0.0125634
\(508\) 0 0
\(509\) −37.4761 −1.66110 −0.830549 0.556946i \(-0.811973\pi\)
−0.830549 + 0.556946i \(0.811973\pi\)
\(510\) 0 0
\(511\) −3.27588 −0.144916
\(512\) 0 0
\(513\) 0.166989 0.00737275
\(514\) 0 0
\(515\) 0.787821 0.0347156
\(516\) 0 0
\(517\) 6.10979 0.268708
\(518\) 0 0
\(519\) 0.334237 0.0146714
\(520\) 0 0
\(521\) 25.4757 1.11611 0.558056 0.829803i \(-0.311547\pi\)
0.558056 + 0.829803i \(0.311547\pi\)
\(522\) 0 0
\(523\) 32.3594 1.41498 0.707488 0.706725i \(-0.249828\pi\)
0.707488 + 0.706725i \(0.249828\pi\)
\(524\) 0 0
\(525\) −0.121773 −0.00531463
\(526\) 0 0
\(527\) 0.0216032 0.000941051 0
\(528\) 0 0
\(529\) 14.5567 0.632900
\(530\) 0 0
\(531\) 42.5983 1.84861
\(532\) 0 0
\(533\) −18.8183 −0.815109
\(534\) 0 0
\(535\) −0.805891 −0.0348417
\(536\) 0 0
\(537\) −0.347887 −0.0150124
\(538\) 0 0
\(539\) 6.22983 0.268338
\(540\) 0 0
\(541\) 18.2211 0.783388 0.391694 0.920096i \(-0.371889\pi\)
0.391694 + 0.920096i \(0.371889\pi\)
\(542\) 0 0
\(543\) −0.134815 −0.00578545
\(544\) 0 0
\(545\) −1.73241 −0.0742081
\(546\) 0 0
\(547\) −30.5039 −1.30425 −0.652127 0.758110i \(-0.726123\pi\)
−0.652127 + 0.758110i \(0.726123\pi\)
\(548\) 0 0
\(549\) 12.1592 0.518943
\(550\) 0 0
\(551\) 3.64572 0.155313
\(552\) 0 0
\(553\) 0.312560 0.0132914
\(554\) 0 0
\(555\) −0.0279751 −0.00118748
\(556\) 0 0
\(557\) −22.4094 −0.949519 −0.474759 0.880116i \(-0.657465\pi\)
−0.474759 + 0.880116i \(0.657465\pi\)
\(558\) 0 0
\(559\) 0.0572127 0.00241984
\(560\) 0 0
\(561\) 0.176486 0.00745124
\(562\) 0 0
\(563\) −41.7915 −1.76130 −0.880652 0.473764i \(-0.842895\pi\)
−0.880652 + 0.473764i \(0.842895\pi\)
\(564\) 0 0
\(565\) −0.590979 −0.0248627
\(566\) 0 0
\(567\) −7.89221 −0.331442
\(568\) 0 0
\(569\) −11.4500 −0.480009 −0.240004 0.970772i \(-0.577149\pi\)
−0.240004 + 0.970772i \(0.577149\pi\)
\(570\) 0 0
\(571\) 1.39669 0.0584497 0.0292248 0.999573i \(-0.490696\pi\)
0.0292248 + 0.999573i \(0.490696\pi\)
\(572\) 0 0
\(573\) 0.262671 0.0109732
\(574\) 0 0
\(575\) 30.5499 1.27402
\(576\) 0 0
\(577\) −8.85413 −0.368602 −0.184301 0.982870i \(-0.559002\pi\)
−0.184301 + 0.982870i \(0.559002\pi\)
\(578\) 0 0
\(579\) −0.466736 −0.0193969
\(580\) 0 0
\(581\) 13.6155 0.564864
\(582\) 0 0
\(583\) −7.24999 −0.300264
\(584\) 0 0
\(585\) 0.618376 0.0255667
\(586\) 0 0
\(587\) −10.1705 −0.419783 −0.209891 0.977725i \(-0.567311\pi\)
−0.209891 + 0.977725i \(0.567311\pi\)
\(588\) 0 0
\(589\) 0.00340723 0.000140393 0
\(590\) 0 0
\(591\) −0.735952 −0.0302730
\(592\) 0 0
\(593\) −5.01632 −0.205996 −0.102998 0.994682i \(-0.532843\pi\)
−0.102998 + 0.994682i \(0.532843\pi\)
\(594\) 0 0
\(595\) 0.681111 0.0279228
\(596\) 0 0
\(597\) −0.537143 −0.0219838
\(598\) 0 0
\(599\) 27.9408 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(600\) 0 0
\(601\) −28.9161 −1.17951 −0.589756 0.807581i \(-0.700776\pi\)
−0.589756 + 0.807581i \(0.700776\pi\)
\(602\) 0 0
\(603\) 25.7388 1.04816
\(604\) 0 0
\(605\) −0.122408 −0.00497658
\(606\) 0 0
\(607\) −3.30116 −0.133990 −0.0669950 0.997753i \(-0.521341\pi\)
−0.0669950 + 0.997753i \(0.521341\pi\)
\(608\) 0 0
\(609\) 0.0890574 0.00360879
\(610\) 0 0
\(611\) −10.2911 −0.416333
\(612\) 0 0
\(613\) −34.7837 −1.40490 −0.702451 0.711732i \(-0.747911\pi\)
−0.702451 + 0.711732i \(0.747911\pi\)
\(614\) 0 0
\(615\) −0.0380668 −0.00153500
\(616\) 0 0
\(617\) −0.452766 −0.0182277 −0.00911383 0.999958i \(-0.502901\pi\)
−0.00911383 + 0.999958i \(0.502901\pi\)
\(618\) 0 0
\(619\) 30.7171 1.23462 0.617312 0.786718i \(-0.288222\pi\)
0.617312 + 0.786718i \(0.288222\pi\)
\(620\) 0 0
\(621\) −1.02337 −0.0410664
\(622\) 0 0
\(623\) −4.66651 −0.186960
\(624\) 0 0
\(625\) 24.7755 0.991019
\(626\) 0 0
\(627\) 0.0278351 0.00111163
\(628\) 0 0
\(629\) −52.0578 −2.07568
\(630\) 0 0
\(631\) 15.0741 0.600092 0.300046 0.953925i \(-0.402998\pi\)
0.300046 + 0.953925i \(0.402998\pi\)
\(632\) 0 0
\(633\) −0.277784 −0.0110409
\(634\) 0 0
\(635\) −0.128996 −0.00511904
\(636\) 0 0
\(637\) −10.4933 −0.415759
\(638\) 0 0
\(639\) −0.0408683 −0.00161673
\(640\) 0 0
\(641\) 35.2312 1.39155 0.695774 0.718261i \(-0.255061\pi\)
0.695774 + 0.718261i \(0.255061\pi\)
\(642\) 0 0
\(643\) 26.1788 1.03239 0.516195 0.856471i \(-0.327348\pi\)
0.516195 + 0.856471i \(0.327348\pi\)
\(644\) 0 0
\(645\) 0.000115733 0 4.55700e−6 0
\(646\) 0 0
\(647\) −24.2822 −0.954633 −0.477316 0.878732i \(-0.658390\pi\)
−0.477316 + 0.878732i \(0.658390\pi\)
\(648\) 0 0
\(649\) 14.2031 0.557520
\(650\) 0 0
\(651\) 8.32315e−5 0 3.26210e−6 0
\(652\) 0 0
\(653\) −50.2439 −1.96620 −0.983098 0.183078i \(-0.941394\pi\)
−0.983098 + 0.183078i \(0.941394\pi\)
\(654\) 0 0
\(655\) −1.12666 −0.0440223
\(656\) 0 0
\(657\) −11.1955 −0.436779
\(658\) 0 0
\(659\) 9.25601 0.360563 0.180281 0.983615i \(-0.442299\pi\)
0.180281 + 0.983615i \(0.442299\pi\)
\(660\) 0 0
\(661\) 22.8800 0.889927 0.444964 0.895549i \(-0.353217\pi\)
0.444964 + 0.895549i \(0.353217\pi\)
\(662\) 0 0
\(663\) −0.297266 −0.0115448
\(664\) 0 0
\(665\) 0.107424 0.00416572
\(666\) 0 0
\(667\) −22.3423 −0.865096
\(668\) 0 0
\(669\) −0.452856 −0.0175084
\(670\) 0 0
\(671\) 4.05412 0.156508
\(672\) 0 0
\(673\) −42.5307 −1.63944 −0.819718 0.572767i \(-0.805870\pi\)
−0.819718 + 0.572767i \(0.805870\pi\)
\(674\) 0 0
\(675\) −0.832444 −0.0320408
\(676\) 0 0
\(677\) 29.8688 1.14795 0.573976 0.818872i \(-0.305400\pi\)
0.573976 + 0.818872i \(0.305400\pi\)
\(678\) 0 0
\(679\) −14.6663 −0.562842
\(680\) 0 0
\(681\) 0.698704 0.0267744
\(682\) 0 0
\(683\) −45.1162 −1.72632 −0.863162 0.504928i \(-0.831519\pi\)
−0.863162 + 0.504928i \(0.831519\pi\)
\(684\) 0 0
\(685\) 0.795931 0.0304109
\(686\) 0 0
\(687\) 0.0851773 0.00324972
\(688\) 0 0
\(689\) 12.2116 0.465224
\(690\) 0 0
\(691\) −14.2491 −0.542062 −0.271031 0.962571i \(-0.587365\pi\)
−0.271031 + 0.962571i \(0.587365\pi\)
\(692\) 0 0
\(693\) −2.63210 −0.0999851
\(694\) 0 0
\(695\) 1.44821 0.0549336
\(696\) 0 0
\(697\) −70.8371 −2.68315
\(698\) 0 0
\(699\) 0.617335 0.0233497
\(700\) 0 0
\(701\) 4.86294 0.183671 0.0918354 0.995774i \(-0.470727\pi\)
0.0918354 + 0.995774i \(0.470727\pi\)
\(702\) 0 0
\(703\) −8.21050 −0.309665
\(704\) 0 0
\(705\) −0.0208175 −0.000784032 0
\(706\) 0 0
\(707\) 9.66454 0.363473
\(708\) 0 0
\(709\) −25.0524 −0.940863 −0.470432 0.882436i \(-0.655902\pi\)
−0.470432 + 0.882436i \(0.655902\pi\)
\(710\) 0 0
\(711\) 1.06819 0.0400603
\(712\) 0 0
\(713\) −0.0208807 −0.000781989 0
\(714\) 0 0
\(715\) 0.206179 0.00771064
\(716\) 0 0
\(717\) −0.609652 −0.0227679
\(718\) 0 0
\(719\) −45.9734 −1.71452 −0.857260 0.514884i \(-0.827835\pi\)
−0.857260 + 0.514884i \(0.827835\pi\)
\(720\) 0 0
\(721\) 5.64822 0.210351
\(722\) 0 0
\(723\) 0.0806128 0.00299802
\(724\) 0 0
\(725\) −18.1740 −0.674965
\(726\) 0 0
\(727\) 26.4967 0.982709 0.491354 0.870960i \(-0.336502\pi\)
0.491354 + 0.870960i \(0.336502\pi\)
\(728\) 0 0
\(729\) −26.9582 −0.998451
\(730\) 0 0
\(731\) 0.215364 0.00796554
\(732\) 0 0
\(733\) −0.180886 −0.00668116 −0.00334058 0.999994i \(-0.501063\pi\)
−0.00334058 + 0.999994i \(0.501063\pi\)
\(734\) 0 0
\(735\) −0.0212265 −0.000782951 0
\(736\) 0 0
\(737\) 8.58181 0.316115
\(738\) 0 0
\(739\) 45.3719 1.66903 0.834516 0.550984i \(-0.185747\pi\)
0.834516 + 0.550984i \(0.185747\pi\)
\(740\) 0 0
\(741\) −0.0468844 −0.00172234
\(742\) 0 0
\(743\) −48.6104 −1.78334 −0.891672 0.452683i \(-0.850467\pi\)
−0.891672 + 0.452683i \(0.850467\pi\)
\(744\) 0 0
\(745\) 2.22907 0.0816669
\(746\) 0 0
\(747\) 46.5317 1.70250
\(748\) 0 0
\(749\) −5.77777 −0.211115
\(750\) 0 0
\(751\) 43.6401 1.59245 0.796225 0.605000i \(-0.206827\pi\)
0.796225 + 0.605000i \(0.206827\pi\)
\(752\) 0 0
\(753\) 0.207770 0.00757157
\(754\) 0 0
\(755\) 1.82390 0.0663787
\(756\) 0 0
\(757\) 40.0661 1.45623 0.728114 0.685456i \(-0.240397\pi\)
0.728114 + 0.685456i \(0.240397\pi\)
\(758\) 0 0
\(759\) −0.170583 −0.00619179
\(760\) 0 0
\(761\) 24.0137 0.870494 0.435247 0.900311i \(-0.356661\pi\)
0.435247 + 0.900311i \(0.356661\pi\)
\(762\) 0 0
\(763\) −12.4204 −0.449647
\(764\) 0 0
\(765\) 2.32774 0.0841596
\(766\) 0 0
\(767\) −23.9231 −0.863814
\(768\) 0 0
\(769\) 30.8574 1.11275 0.556374 0.830932i \(-0.312192\pi\)
0.556374 + 0.830932i \(0.312192\pi\)
\(770\) 0 0
\(771\) −0.146065 −0.00526041
\(772\) 0 0
\(773\) −18.0873 −0.650554 −0.325277 0.945619i \(-0.605458\pi\)
−0.325277 + 0.945619i \(0.605458\pi\)
\(774\) 0 0
\(775\) −0.0169851 −0.000610123 0
\(776\) 0 0
\(777\) −0.200565 −0.00719523
\(778\) 0 0
\(779\) −11.1723 −0.400291
\(780\) 0 0
\(781\) −0.0136263 −0.000487587 0
\(782\) 0 0
\(783\) 0.608797 0.0217566
\(784\) 0 0
\(785\) −2.42479 −0.0865445
\(786\) 0 0
\(787\) −44.3431 −1.58066 −0.790330 0.612681i \(-0.790091\pi\)
−0.790330 + 0.612681i \(0.790091\pi\)
\(788\) 0 0
\(789\) −0.518403 −0.0184556
\(790\) 0 0
\(791\) −4.23698 −0.150650
\(792\) 0 0
\(793\) −6.82860 −0.242491
\(794\) 0 0
\(795\) 0.0247024 0.000876103 0
\(796\) 0 0
\(797\) −11.4047 −0.403976 −0.201988 0.979388i \(-0.564740\pi\)
−0.201988 + 0.979388i \(0.564740\pi\)
\(798\) 0 0
\(799\) −38.7385 −1.37047
\(800\) 0 0
\(801\) −15.9481 −0.563498
\(802\) 0 0
\(803\) −3.73280 −0.131728
\(804\) 0 0
\(805\) −0.658332 −0.0232032
\(806\) 0 0
\(807\) 0.418641 0.0147368
\(808\) 0 0
\(809\) −11.4641 −0.403057 −0.201528 0.979483i \(-0.564591\pi\)
−0.201528 + 0.979483i \(0.564591\pi\)
\(810\) 0 0
\(811\) −33.7845 −1.18633 −0.593167 0.805080i \(-0.702122\pi\)
−0.593167 + 0.805080i \(0.702122\pi\)
\(812\) 0 0
\(813\) 0.787443 0.0276168
\(814\) 0 0
\(815\) 2.11560 0.0741064
\(816\) 0 0
\(817\) 0.0339670 0.00118836
\(818\) 0 0
\(819\) 4.43340 0.154916
\(820\) 0 0
\(821\) 14.9537 0.521889 0.260944 0.965354i \(-0.415966\pi\)
0.260944 + 0.965354i \(0.415966\pi\)
\(822\) 0 0
\(823\) −18.9606 −0.660925 −0.330462 0.943819i \(-0.607205\pi\)
−0.330462 + 0.943819i \(0.607205\pi\)
\(824\) 0 0
\(825\) −0.138759 −0.00483095
\(826\) 0 0
\(827\) 28.1451 0.978699 0.489350 0.872088i \(-0.337234\pi\)
0.489350 + 0.872088i \(0.337234\pi\)
\(828\) 0 0
\(829\) 31.4452 1.09214 0.546069 0.837740i \(-0.316124\pi\)
0.546069 + 0.837740i \(0.316124\pi\)
\(830\) 0 0
\(831\) −0.622342 −0.0215888
\(832\) 0 0
\(833\) −39.4996 −1.36858
\(834\) 0 0
\(835\) 1.46372 0.0506542
\(836\) 0 0
\(837\) 0.000568971 0 1.96665e−5 0
\(838\) 0 0
\(839\) 49.6194 1.71305 0.856526 0.516104i \(-0.172618\pi\)
0.856526 + 0.516104i \(0.172618\pi\)
\(840\) 0 0
\(841\) −15.7087 −0.541679
\(842\) 0 0
\(843\) −0.343728 −0.0118386
\(844\) 0 0
\(845\) 1.24402 0.0427956
\(846\) 0 0
\(847\) −0.877592 −0.0301544
\(848\) 0 0
\(849\) 0.345146 0.0118454
\(850\) 0 0
\(851\) 50.3168 1.72484
\(852\) 0 0
\(853\) −27.9643 −0.957478 −0.478739 0.877957i \(-0.658906\pi\)
−0.478739 + 0.877957i \(0.658906\pi\)
\(854\) 0 0
\(855\) 0.367128 0.0125555
\(856\) 0 0
\(857\) −1.38737 −0.0473916 −0.0236958 0.999719i \(-0.507543\pi\)
−0.0236958 + 0.999719i \(0.507543\pi\)
\(858\) 0 0
\(859\) 18.3432 0.625864 0.312932 0.949776i \(-0.398689\pi\)
0.312932 + 0.949776i \(0.398689\pi\)
\(860\) 0 0
\(861\) −0.272917 −0.00930098
\(862\) 0 0
\(863\) −45.2423 −1.54006 −0.770032 0.638005i \(-0.779760\pi\)
−0.770032 + 0.638005i \(0.779760\pi\)
\(864\) 0 0
\(865\) 1.46984 0.0499760
\(866\) 0 0
\(867\) −0.645793 −0.0219323
\(868\) 0 0
\(869\) 0.356156 0.0120818
\(870\) 0 0
\(871\) −14.4549 −0.489784
\(872\) 0 0
\(873\) −50.1231 −1.69641
\(874\) 0 0
\(875\) −1.07263 −0.0362615
\(876\) 0 0
\(877\) −42.8202 −1.44593 −0.722967 0.690882i \(-0.757222\pi\)
−0.722967 + 0.690882i \(0.757222\pi\)
\(878\) 0 0
\(879\) 0.201631 0.00680086
\(880\) 0 0
\(881\) −32.4911 −1.09465 −0.547326 0.836920i \(-0.684354\pi\)
−0.547326 + 0.836920i \(0.684354\pi\)
\(882\) 0 0
\(883\) 13.0319 0.438558 0.219279 0.975662i \(-0.429630\pi\)
0.219279 + 0.975662i \(0.429630\pi\)
\(884\) 0 0
\(885\) −0.0483933 −0.00162672
\(886\) 0 0
\(887\) −39.6956 −1.33285 −0.666424 0.745573i \(-0.732176\pi\)
−0.666424 + 0.745573i \(0.732176\pi\)
\(888\) 0 0
\(889\) −0.924824 −0.0310176
\(890\) 0 0
\(891\) −8.99303 −0.301278
\(892\) 0 0
\(893\) −6.10979 −0.204456
\(894\) 0 0
\(895\) −1.52986 −0.0511377
\(896\) 0 0
\(897\) 0.287324 0.00959347
\(898\) 0 0
\(899\) 0.0124218 0.000414291 0
\(900\) 0 0
\(901\) 45.9678 1.53141
\(902\) 0 0
\(903\) 0.000829742 0 2.76121e−5 0
\(904\) 0 0
\(905\) −0.592860 −0.0197073
\(906\) 0 0
\(907\) −0.181821 −0.00603727 −0.00301863 0.999995i \(-0.500961\pi\)
−0.00301863 + 0.999995i \(0.500961\pi\)
\(908\) 0 0
\(909\) 33.0292 1.09551
\(910\) 0 0
\(911\) −31.9193 −1.05753 −0.528767 0.848767i \(-0.677345\pi\)
−0.528767 + 0.848767i \(0.677345\pi\)
\(912\) 0 0
\(913\) 15.5146 0.513457
\(914\) 0 0
\(915\) −0.0138133 −0.000456655 0
\(916\) 0 0
\(917\) −8.07751 −0.266743
\(918\) 0 0
\(919\) 28.8673 0.952246 0.476123 0.879379i \(-0.342042\pi\)
0.476123 + 0.879379i \(0.342042\pi\)
\(920\) 0 0
\(921\) 0.504950 0.0166387
\(922\) 0 0
\(923\) 0.0229516 0.000755461 0
\(924\) 0 0
\(925\) 40.9295 1.34575
\(926\) 0 0
\(927\) 19.3031 0.633999
\(928\) 0 0
\(929\) 24.5258 0.804666 0.402333 0.915493i \(-0.368199\pi\)
0.402333 + 0.915493i \(0.368199\pi\)
\(930\) 0 0
\(931\) −6.22983 −0.204174
\(932\) 0 0
\(933\) −0.267052 −0.00874289
\(934\) 0 0
\(935\) 0.776113 0.0253816
\(936\) 0 0
\(937\) −10.6373 −0.347506 −0.173753 0.984789i \(-0.555589\pi\)
−0.173753 + 0.984789i \(0.555589\pi\)
\(938\) 0 0
\(939\) −0.813566 −0.0265497
\(940\) 0 0
\(941\) 1.41861 0.0462453 0.0231226 0.999733i \(-0.492639\pi\)
0.0231226 + 0.999733i \(0.492639\pi\)
\(942\) 0 0
\(943\) 68.4681 2.22963
\(944\) 0 0
\(945\) 0.0179387 0.000583544 0
\(946\) 0 0
\(947\) −5.25867 −0.170884 −0.0854418 0.996343i \(-0.527230\pi\)
−0.0854418 + 0.996343i \(0.527230\pi\)
\(948\) 0 0
\(949\) 6.28738 0.204097
\(950\) 0 0
\(951\) −0.358316 −0.0116192
\(952\) 0 0
\(953\) −51.5273 −1.66913 −0.834567 0.550906i \(-0.814282\pi\)
−0.834567 + 0.550906i \(0.814282\pi\)
\(954\) 0 0
\(955\) 1.15512 0.0373788
\(956\) 0 0
\(957\) 0.101479 0.00328036
\(958\) 0 0
\(959\) 5.70636 0.184268
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −19.7459 −0.636302
\(964\) 0 0
\(965\) −2.05252 −0.0660728
\(966\) 0 0
\(967\) 47.8627 1.53916 0.769581 0.638550i \(-0.220465\pi\)
0.769581 + 0.638550i \(0.220465\pi\)
\(968\) 0 0
\(969\) −0.176486 −0.00566954
\(970\) 0 0
\(971\) 7.48025 0.240053 0.120026 0.992771i \(-0.461702\pi\)
0.120026 + 0.992771i \(0.461702\pi\)
\(972\) 0 0
\(973\) 10.3828 0.332857
\(974\) 0 0
\(975\) 0.233719 0.00748501
\(976\) 0 0
\(977\) −13.0292 −0.416841 −0.208420 0.978039i \(-0.566832\pi\)
−0.208420 + 0.978039i \(0.566832\pi\)
\(978\) 0 0
\(979\) −5.31740 −0.169945
\(980\) 0 0
\(981\) −42.4473 −1.35524
\(982\) 0 0
\(983\) 48.1827 1.53679 0.768394 0.639977i \(-0.221056\pi\)
0.768394 + 0.639977i \(0.221056\pi\)
\(984\) 0 0
\(985\) −3.23642 −0.103121
\(986\) 0 0
\(987\) −0.149249 −0.00475066
\(988\) 0 0
\(989\) −0.208162 −0.00661916
\(990\) 0 0
\(991\) −54.1756 −1.72095 −0.860473 0.509496i \(-0.829832\pi\)
−0.860473 + 0.509496i \(0.829832\pi\)
\(992\) 0 0
\(993\) −0.236122 −0.00749312
\(994\) 0 0
\(995\) −2.36214 −0.0748848
\(996\) 0 0
\(997\) −17.3227 −0.548617 −0.274308 0.961642i \(-0.588449\pi\)
−0.274308 + 0.961642i \(0.588449\pi\)
\(998\) 0 0
\(999\) −1.37106 −0.0433786
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1672.2.a.i.1.4 6
4.3 odd 2 3344.2.a.z.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.i.1.4 6 1.1 even 1 trivial
3344.2.a.z.1.3 6 4.3 odd 2