Properties

Label 3344.2.a.z.1.4
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(0\)
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: no (minimal twist has level 1672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.870457\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +O(q^{10})\) \(q+1.87046 q^{3} +2.83301 q^{5} +3.83301 q^{7} +0.498611 q^{9} +1.00000 q^{11} +6.12876 q^{13} +5.29902 q^{15} -3.70874 q^{17} -1.00000 q^{19} +7.16948 q^{21} -1.01807 q^{23} +3.02594 q^{25} -4.67874 q^{27} -3.56805 q^{29} +5.29902 q^{31} +1.87046 q^{33} +10.8590 q^{35} +1.68789 q^{37} +11.4636 q^{39} -6.71110 q^{41} +3.59858 q^{43} +1.41257 q^{45} +6.48461 q^{47} +7.69196 q^{49} -6.93704 q^{51} -6.34969 q^{53} +2.83301 q^{55} -1.87046 q^{57} -10.8133 q^{59} -0.743693 q^{61} +1.91118 q^{63} +17.3628 q^{65} -2.07574 q^{67} -1.90426 q^{69} +2.65687 q^{71} -11.0837 q^{73} +5.65990 q^{75} +3.83301 q^{77} +1.81581 q^{79} -10.2472 q^{81} -12.1416 q^{83} -10.5069 q^{85} -6.67389 q^{87} -1.51341 q^{89} +23.4916 q^{91} +9.91160 q^{93} -2.83301 q^{95} -13.3901 q^{97} +0.498611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 3 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 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
\(3\) 1.87046 1.07991 0.539955 0.841694i \(-0.318441\pi\)
0.539955 + 0.841694i \(0.318441\pi\)
\(4\) 0 0
\(5\) 2.83301 1.26696 0.633480 0.773759i \(-0.281626\pi\)
0.633480 + 0.773759i \(0.281626\pi\)
\(6\) 0 0
\(7\) 3.83301 1.44874 0.724371 0.689411i \(-0.242130\pi\)
0.724371 + 0.689411i \(0.242130\pi\)
\(8\) 0 0
\(9\) 0.498611 0.166204
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.12876 1.69981 0.849906 0.526934i \(-0.176659\pi\)
0.849906 + 0.526934i \(0.176659\pi\)
\(14\) 0 0
\(15\) 5.29902 1.36820
\(16\) 0 0
\(17\) −3.70874 −0.899502 −0.449751 0.893154i \(-0.648487\pi\)
−0.449751 + 0.893154i \(0.648487\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 7.16948 1.56451
\(22\) 0 0
\(23\) −1.01807 −0.212282 −0.106141 0.994351i \(-0.533850\pi\)
−0.106141 + 0.994351i \(0.533850\pi\)
\(24\) 0 0
\(25\) 3.02594 0.605188
\(26\) 0 0
\(27\) −4.67874 −0.900424
\(28\) 0 0
\(29\) −3.56805 −0.662571 −0.331285 0.943531i \(-0.607482\pi\)
−0.331285 + 0.943531i \(0.607482\pi\)
\(30\) 0 0
\(31\) 5.29902 0.951733 0.475866 0.879518i \(-0.342135\pi\)
0.475866 + 0.879518i \(0.342135\pi\)
\(32\) 0 0
\(33\) 1.87046 0.325605
\(34\) 0 0
\(35\) 10.8590 1.83550
\(36\) 0 0
\(37\) 1.68789 0.277488 0.138744 0.990328i \(-0.455693\pi\)
0.138744 + 0.990328i \(0.455693\pi\)
\(38\) 0 0
\(39\) 11.4636 1.83564
\(40\) 0 0
\(41\) −6.71110 −1.04810 −0.524049 0.851688i \(-0.675579\pi\)
−0.524049 + 0.851688i \(0.675579\pi\)
\(42\) 0 0
\(43\) 3.59858 0.548778 0.274389 0.961619i \(-0.411524\pi\)
0.274389 + 0.961619i \(0.411524\pi\)
\(44\) 0 0
\(45\) 1.41257 0.210573
\(46\) 0 0
\(47\) 6.48461 0.945877 0.472939 0.881095i \(-0.343193\pi\)
0.472939 + 0.881095i \(0.343193\pi\)
\(48\) 0 0
\(49\) 7.69196 1.09885
\(50\) 0 0
\(51\) −6.93704 −0.971380
\(52\) 0 0
\(53\) −6.34969 −0.872197 −0.436098 0.899899i \(-0.643640\pi\)
−0.436098 + 0.899899i \(0.643640\pi\)
\(54\) 0 0
\(55\) 2.83301 0.382003
\(56\) 0 0
\(57\) −1.87046 −0.247748
\(58\) 0 0
\(59\) −10.8133 −1.40777 −0.703884 0.710315i \(-0.748553\pi\)
−0.703884 + 0.710315i \(0.748553\pi\)
\(60\) 0 0
\(61\) −0.743693 −0.0952202 −0.0476101 0.998866i \(-0.515160\pi\)
−0.0476101 + 0.998866i \(0.515160\pi\)
\(62\) 0 0
\(63\) 1.91118 0.240786
\(64\) 0 0
\(65\) 17.3628 2.15359
\(66\) 0 0
\(67\) −2.07574 −0.253592 −0.126796 0.991929i \(-0.540469\pi\)
−0.126796 + 0.991929i \(0.540469\pi\)
\(68\) 0 0
\(69\) −1.90426 −0.229245
\(70\) 0 0
\(71\) 2.65687 0.315313 0.157656 0.987494i \(-0.449606\pi\)
0.157656 + 0.987494i \(0.449606\pi\)
\(72\) 0 0
\(73\) −11.0837 −1.29725 −0.648623 0.761110i \(-0.724655\pi\)
−0.648623 + 0.761110i \(0.724655\pi\)
\(74\) 0 0
\(75\) 5.65990 0.653548
\(76\) 0 0
\(77\) 3.83301 0.436812
\(78\) 0 0
\(79\) 1.81581 0.204295 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(80\) 0 0
\(81\) −10.2472 −1.13858
\(82\) 0 0
\(83\) −12.1416 −1.33272 −0.666358 0.745632i \(-0.732148\pi\)
−0.666358 + 0.745632i \(0.732148\pi\)
\(84\) 0 0
\(85\) −10.5069 −1.13963
\(86\) 0 0
\(87\) −6.67389 −0.715516
\(88\) 0 0
\(89\) −1.51341 −0.160421 −0.0802104 0.996778i \(-0.525559\pi\)
−0.0802104 + 0.996778i \(0.525559\pi\)
\(90\) 0 0
\(91\) 23.4916 2.46259
\(92\) 0 0
\(93\) 9.91160 1.02778
\(94\) 0 0
\(95\) −2.83301 −0.290661
\(96\) 0 0
\(97\) −13.3901 −1.35955 −0.679777 0.733419i \(-0.737923\pi\)
−0.679777 + 0.733419i \(0.737923\pi\)
\(98\) 0 0
\(99\) 0.498611 0.0501123
\(100\) 0 0
\(101\) −18.4417 −1.83502 −0.917509 0.397715i \(-0.869803\pi\)
−0.917509 + 0.397715i \(0.869803\pi\)
\(102\) 0 0
\(103\) 13.4662 1.32686 0.663431 0.748237i \(-0.269100\pi\)
0.663431 + 0.748237i \(0.269100\pi\)
\(104\) 0 0
\(105\) 20.3112 1.98217
\(106\) 0 0
\(107\) 16.3720 1.58274 0.791370 0.611338i \(-0.209368\pi\)
0.791370 + 0.611338i \(0.209368\pi\)
\(108\) 0 0
\(109\) 7.60741 0.728658 0.364329 0.931270i \(-0.381298\pi\)
0.364329 + 0.931270i \(0.381298\pi\)
\(110\) 0 0
\(111\) 3.15713 0.299662
\(112\) 0 0
\(113\) −10.0046 −0.941150 −0.470575 0.882360i \(-0.655953\pi\)
−0.470575 + 0.882360i \(0.655953\pi\)
\(114\) 0 0
\(115\) −2.88420 −0.268953
\(116\) 0 0
\(117\) 3.05586 0.282515
\(118\) 0 0
\(119\) −14.2156 −1.30315
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −12.5528 −1.13185
\(124\) 0 0
\(125\) −5.59253 −0.500211
\(126\) 0 0
\(127\) 20.6361 1.83116 0.915578 0.402140i \(-0.131734\pi\)
0.915578 + 0.402140i \(0.131734\pi\)
\(128\) 0 0
\(129\) 6.73099 0.592630
\(130\) 0 0
\(131\) −6.06512 −0.529912 −0.264956 0.964261i \(-0.585357\pi\)
−0.264956 + 0.964261i \(0.585357\pi\)
\(132\) 0 0
\(133\) −3.83301 −0.332364
\(134\) 0 0
\(135\) −13.2549 −1.14080
\(136\) 0 0
\(137\) 10.2035 0.871748 0.435874 0.900008i \(-0.356439\pi\)
0.435874 + 0.900008i \(0.356439\pi\)
\(138\) 0 0
\(139\) 21.7492 1.84474 0.922372 0.386304i \(-0.126248\pi\)
0.922372 + 0.386304i \(0.126248\pi\)
\(140\) 0 0
\(141\) 12.1292 1.02146
\(142\) 0 0
\(143\) 6.12876 0.512513
\(144\) 0 0
\(145\) −10.1083 −0.839451
\(146\) 0 0
\(147\) 14.3875 1.18666
\(148\) 0 0
\(149\) 11.6216 0.952075 0.476038 0.879425i \(-0.342073\pi\)
0.476038 + 0.879425i \(0.342073\pi\)
\(150\) 0 0
\(151\) −19.2317 −1.56506 −0.782528 0.622616i \(-0.786070\pi\)
−0.782528 + 0.622616i \(0.786070\pi\)
\(152\) 0 0
\(153\) −1.84922 −0.149500
\(154\) 0 0
\(155\) 15.0122 1.20581
\(156\) 0 0
\(157\) −5.03393 −0.401752 −0.200876 0.979617i \(-0.564379\pi\)
−0.200876 + 0.979617i \(0.564379\pi\)
\(158\) 0 0
\(159\) −11.8768 −0.941893
\(160\) 0 0
\(161\) −3.90227 −0.307542
\(162\) 0 0
\(163\) −13.3585 −1.04632 −0.523158 0.852236i \(-0.675246\pi\)
−0.523158 + 0.852236i \(0.675246\pi\)
\(164\) 0 0
\(165\) 5.29902 0.412528
\(166\) 0 0
\(167\) 6.88619 0.532869 0.266435 0.963853i \(-0.414154\pi\)
0.266435 + 0.963853i \(0.414154\pi\)
\(168\) 0 0
\(169\) 24.5617 1.88936
\(170\) 0 0
\(171\) −0.498611 −0.0381297
\(172\) 0 0
\(173\) −14.3759 −1.09298 −0.546490 0.837465i \(-0.684036\pi\)
−0.546490 + 0.837465i \(0.684036\pi\)
\(174\) 0 0
\(175\) 11.5985 0.876761
\(176\) 0 0
\(177\) −20.2258 −1.52026
\(178\) 0 0
\(179\) 2.40313 0.179619 0.0898094 0.995959i \(-0.471374\pi\)
0.0898094 + 0.995959i \(0.471374\pi\)
\(180\) 0 0
\(181\) −6.94819 −0.516455 −0.258227 0.966084i \(-0.583138\pi\)
−0.258227 + 0.966084i \(0.583138\pi\)
\(182\) 0 0
\(183\) −1.39105 −0.102829
\(184\) 0 0
\(185\) 4.78182 0.351566
\(186\) 0 0
\(187\) −3.70874 −0.271210
\(188\) 0 0
\(189\) −17.9337 −1.30448
\(190\) 0 0
\(191\) −15.1886 −1.09901 −0.549504 0.835491i \(-0.685183\pi\)
−0.549504 + 0.835491i \(0.685183\pi\)
\(192\) 0 0
\(193\) −10.9331 −0.786983 −0.393491 0.919328i \(-0.628733\pi\)
−0.393491 + 0.919328i \(0.628733\pi\)
\(194\) 0 0
\(195\) 32.4764 2.32569
\(196\) 0 0
\(197\) −12.3440 −0.879470 −0.439735 0.898128i \(-0.644928\pi\)
−0.439735 + 0.898128i \(0.644928\pi\)
\(198\) 0 0
\(199\) −12.9921 −0.920982 −0.460491 0.887664i \(-0.652327\pi\)
−0.460491 + 0.887664i \(0.652327\pi\)
\(200\) 0 0
\(201\) −3.88258 −0.273856
\(202\) 0 0
\(203\) −13.6764 −0.959894
\(204\) 0 0
\(205\) −19.0126 −1.32790
\(206\) 0 0
\(207\) −0.507620 −0.0352820
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 9.42821 0.649065 0.324532 0.945875i \(-0.394793\pi\)
0.324532 + 0.945875i \(0.394793\pi\)
\(212\) 0 0
\(213\) 4.96957 0.340509
\(214\) 0 0
\(215\) 10.1948 0.695280
\(216\) 0 0
\(217\) 20.3112 1.37881
\(218\) 0 0
\(219\) −20.7316 −1.40091
\(220\) 0 0
\(221\) −22.7300 −1.52898
\(222\) 0 0
\(223\) −10.9595 −0.733900 −0.366950 0.930241i \(-0.619598\pi\)
−0.366950 + 0.930241i \(0.619598\pi\)
\(224\) 0 0
\(225\) 1.50877 0.100584
\(226\) 0 0
\(227\) 19.1738 1.27261 0.636306 0.771437i \(-0.280462\pi\)
0.636306 + 0.771437i \(0.280462\pi\)
\(228\) 0 0
\(229\) 4.44137 0.293494 0.146747 0.989174i \(-0.453120\pi\)
0.146747 + 0.989174i \(0.453120\pi\)
\(230\) 0 0
\(231\) 7.16948 0.471717
\(232\) 0 0
\(233\) 24.9949 1.63747 0.818736 0.574170i \(-0.194675\pi\)
0.818736 + 0.574170i \(0.194675\pi\)
\(234\) 0 0
\(235\) 18.3710 1.19839
\(236\) 0 0
\(237\) 3.39640 0.220620
\(238\) 0 0
\(239\) 8.50862 0.550377 0.275188 0.961390i \(-0.411260\pi\)
0.275188 + 0.961390i \(0.411260\pi\)
\(240\) 0 0
\(241\) 20.4857 1.31960 0.659800 0.751442i \(-0.270641\pi\)
0.659800 + 0.751442i \(0.270641\pi\)
\(242\) 0 0
\(243\) −5.13076 −0.329138
\(244\) 0 0
\(245\) 21.7914 1.39220
\(246\) 0 0
\(247\) −6.12876 −0.389964
\(248\) 0 0
\(249\) −22.7104 −1.43921
\(250\) 0 0
\(251\) −14.0342 −0.885828 −0.442914 0.896564i \(-0.646055\pi\)
−0.442914 + 0.896564i \(0.646055\pi\)
\(252\) 0 0
\(253\) −1.01807 −0.0640055
\(254\) 0 0
\(255\) −19.6527 −1.23070
\(256\) 0 0
\(257\) 16.4673 1.02720 0.513601 0.858029i \(-0.328311\pi\)
0.513601 + 0.858029i \(0.328311\pi\)
\(258\) 0 0
\(259\) 6.46971 0.402008
\(260\) 0 0
\(261\) −1.77907 −0.110122
\(262\) 0 0
\(263\) 27.0368 1.66716 0.833580 0.552399i \(-0.186288\pi\)
0.833580 + 0.552399i \(0.186288\pi\)
\(264\) 0 0
\(265\) −17.9887 −1.10504
\(266\) 0 0
\(267\) −2.83076 −0.173240
\(268\) 0 0
\(269\) 2.69523 0.164331 0.0821657 0.996619i \(-0.473816\pi\)
0.0821657 + 0.996619i \(0.473816\pi\)
\(270\) 0 0
\(271\) −0.753780 −0.0457889 −0.0228944 0.999738i \(-0.507288\pi\)
−0.0228944 + 0.999738i \(0.507288\pi\)
\(272\) 0 0
\(273\) 43.9400 2.65937
\(274\) 0 0
\(275\) 3.02594 0.182471
\(276\) 0 0
\(277\) 12.0493 0.723972 0.361986 0.932184i \(-0.382099\pi\)
0.361986 + 0.932184i \(0.382099\pi\)
\(278\) 0 0
\(279\) 2.64215 0.158181
\(280\) 0 0
\(281\) 30.3906 1.81295 0.906475 0.422258i \(-0.138763\pi\)
0.906475 + 0.422258i \(0.138763\pi\)
\(282\) 0 0
\(283\) −9.49645 −0.564505 −0.282253 0.959340i \(-0.591082\pi\)
−0.282253 + 0.959340i \(0.591082\pi\)
\(284\) 0 0
\(285\) −5.29902 −0.313887
\(286\) 0 0
\(287\) −25.7237 −1.51842
\(288\) 0 0
\(289\) −3.24523 −0.190896
\(290\) 0 0
\(291\) −25.0455 −1.46819
\(292\) 0 0
\(293\) −17.9766 −1.05020 −0.525101 0.851040i \(-0.675972\pi\)
−0.525101 + 0.851040i \(0.675972\pi\)
\(294\) 0 0
\(295\) −30.6341 −1.78359
\(296\) 0 0
\(297\) −4.67874 −0.271488
\(298\) 0 0
\(299\) −6.23950 −0.360840
\(300\) 0 0
\(301\) 13.7934 0.795037
\(302\) 0 0
\(303\) −34.4944 −1.98165
\(304\) 0 0
\(305\) −2.10689 −0.120640
\(306\) 0 0
\(307\) −11.4216 −0.651867 −0.325933 0.945393i \(-0.605679\pi\)
−0.325933 + 0.945393i \(0.605679\pi\)
\(308\) 0 0
\(309\) 25.1879 1.43289
\(310\) 0 0
\(311\) −25.6594 −1.45501 −0.727507 0.686101i \(-0.759321\pi\)
−0.727507 + 0.686101i \(0.759321\pi\)
\(312\) 0 0
\(313\) −11.6229 −0.656962 −0.328481 0.944511i \(-0.606537\pi\)
−0.328481 + 0.944511i \(0.606537\pi\)
\(314\) 0 0
\(315\) 5.41439 0.305066
\(316\) 0 0
\(317\) 30.1542 1.69363 0.846813 0.531890i \(-0.178518\pi\)
0.846813 + 0.531890i \(0.178518\pi\)
\(318\) 0 0
\(319\) −3.56805 −0.199773
\(320\) 0 0
\(321\) 30.6231 1.70921
\(322\) 0 0
\(323\) 3.70874 0.206360
\(324\) 0 0
\(325\) 18.5453 1.02871
\(326\) 0 0
\(327\) 14.2293 0.786884
\(328\) 0 0
\(329\) 24.8556 1.37033
\(330\) 0 0
\(331\) 25.4171 1.39705 0.698526 0.715585i \(-0.253840\pi\)
0.698526 + 0.715585i \(0.253840\pi\)
\(332\) 0 0
\(333\) 0.841602 0.0461195
\(334\) 0 0
\(335\) −5.88058 −0.321291
\(336\) 0 0
\(337\) −9.43868 −0.514158 −0.257079 0.966390i \(-0.582760\pi\)
−0.257079 + 0.966390i \(0.582760\pi\)
\(338\) 0 0
\(339\) −18.7131 −1.01636
\(340\) 0 0
\(341\) 5.29902 0.286958
\(342\) 0 0
\(343\) 2.65229 0.143210
\(344\) 0 0
\(345\) −5.39477 −0.290445
\(346\) 0 0
\(347\) 35.1560 1.88727 0.943636 0.330985i \(-0.107381\pi\)
0.943636 + 0.330985i \(0.107381\pi\)
\(348\) 0 0
\(349\) 21.5526 1.15368 0.576841 0.816856i \(-0.304285\pi\)
0.576841 + 0.816856i \(0.304285\pi\)
\(350\) 0 0
\(351\) −28.6749 −1.53055
\(352\) 0 0
\(353\) 8.48984 0.451869 0.225934 0.974143i \(-0.427457\pi\)
0.225934 + 0.974143i \(0.427457\pi\)
\(354\) 0 0
\(355\) 7.52695 0.399489
\(356\) 0 0
\(357\) −26.5898 −1.40728
\(358\) 0 0
\(359\) 2.61783 0.138164 0.0690820 0.997611i \(-0.477993\pi\)
0.0690820 + 0.997611i \(0.477993\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.87046 0.0981735
\(364\) 0 0
\(365\) −31.4002 −1.64356
\(366\) 0 0
\(367\) 4.37102 0.228166 0.114083 0.993471i \(-0.463607\pi\)
0.114083 + 0.993471i \(0.463607\pi\)
\(368\) 0 0
\(369\) −3.34622 −0.174197
\(370\) 0 0
\(371\) −24.3384 −1.26359
\(372\) 0 0
\(373\) −2.66161 −0.137813 −0.0689066 0.997623i \(-0.521951\pi\)
−0.0689066 + 0.997623i \(0.521951\pi\)
\(374\) 0 0
\(375\) −10.4606 −0.540182
\(376\) 0 0
\(377\) −21.8677 −1.12625
\(378\) 0 0
\(379\) −17.7952 −0.914078 −0.457039 0.889447i \(-0.651090\pi\)
−0.457039 + 0.889447i \(0.651090\pi\)
\(380\) 0 0
\(381\) 38.5989 1.97748
\(382\) 0 0
\(383\) 32.3374 1.65236 0.826181 0.563405i \(-0.190509\pi\)
0.826181 + 0.563405i \(0.190509\pi\)
\(384\) 0 0
\(385\) 10.8590 0.553423
\(386\) 0 0
\(387\) 1.79429 0.0912088
\(388\) 0 0
\(389\) 13.1909 0.668804 0.334402 0.942431i \(-0.391466\pi\)
0.334402 + 0.942431i \(0.391466\pi\)
\(390\) 0 0
\(391\) 3.77576 0.190948
\(392\) 0 0
\(393\) −11.3445 −0.572256
\(394\) 0 0
\(395\) 5.14421 0.258833
\(396\) 0 0
\(397\) 30.8885 1.55025 0.775124 0.631809i \(-0.217687\pi\)
0.775124 + 0.631809i \(0.217687\pi\)
\(398\) 0 0
\(399\) −7.16948 −0.358923
\(400\) 0 0
\(401\) −26.8786 −1.34225 −0.671126 0.741343i \(-0.734189\pi\)
−0.671126 + 0.741343i \(0.734189\pi\)
\(402\) 0 0
\(403\) 32.4764 1.61777
\(404\) 0 0
\(405\) −29.0305 −1.44254
\(406\) 0 0
\(407\) 1.68789 0.0836658
\(408\) 0 0
\(409\) 28.5926 1.41381 0.706906 0.707307i \(-0.250090\pi\)
0.706906 + 0.707307i \(0.250090\pi\)
\(410\) 0 0
\(411\) 19.0853 0.941408
\(412\) 0 0
\(413\) −41.4474 −2.03949
\(414\) 0 0
\(415\) −34.3973 −1.68850
\(416\) 0 0
\(417\) 40.6810 1.99216
\(418\) 0 0
\(419\) −17.3779 −0.848965 −0.424482 0.905436i \(-0.639544\pi\)
−0.424482 + 0.905436i \(0.639544\pi\)
\(420\) 0 0
\(421\) 8.93888 0.435654 0.217827 0.975987i \(-0.430103\pi\)
0.217827 + 0.975987i \(0.430103\pi\)
\(422\) 0 0
\(423\) 3.23329 0.157208
\(424\) 0 0
\(425\) −11.2224 −0.544368
\(426\) 0 0
\(427\) −2.85058 −0.137949
\(428\) 0 0
\(429\) 11.4636 0.553467
\(430\) 0 0
\(431\) 13.0145 0.626887 0.313444 0.949607i \(-0.398517\pi\)
0.313444 + 0.949607i \(0.398517\pi\)
\(432\) 0 0
\(433\) −5.00494 −0.240522 −0.120261 0.992742i \(-0.538373\pi\)
−0.120261 + 0.992742i \(0.538373\pi\)
\(434\) 0 0
\(435\) −18.9072 −0.906531
\(436\) 0 0
\(437\) 1.01807 0.0487009
\(438\) 0 0
\(439\) −24.6582 −1.17687 −0.588437 0.808543i \(-0.700256\pi\)
−0.588437 + 0.808543i \(0.700256\pi\)
\(440\) 0 0
\(441\) 3.83529 0.182633
\(442\) 0 0
\(443\) −9.81103 −0.466136 −0.233068 0.972460i \(-0.574877\pi\)
−0.233068 + 0.972460i \(0.574877\pi\)
\(444\) 0 0
\(445\) −4.28749 −0.203247
\(446\) 0 0
\(447\) 21.7376 1.02815
\(448\) 0 0
\(449\) −19.6064 −0.925285 −0.462642 0.886545i \(-0.653099\pi\)
−0.462642 + 0.886545i \(0.653099\pi\)
\(450\) 0 0
\(451\) −6.71110 −0.316013
\(452\) 0 0
\(453\) −35.9721 −1.69012
\(454\) 0 0
\(455\) 66.5519 3.12000
\(456\) 0 0
\(457\) 25.8767 1.21046 0.605230 0.796051i \(-0.293081\pi\)
0.605230 + 0.796051i \(0.293081\pi\)
\(458\) 0 0
\(459\) 17.3522 0.809934
\(460\) 0 0
\(461\) −34.2353 −1.59450 −0.797249 0.603650i \(-0.793712\pi\)
−0.797249 + 0.603650i \(0.793712\pi\)
\(462\) 0 0
\(463\) 21.5544 1.00172 0.500859 0.865529i \(-0.333018\pi\)
0.500859 + 0.865529i \(0.333018\pi\)
\(464\) 0 0
\(465\) 28.0796 1.30216
\(466\) 0 0
\(467\) −6.66020 −0.308198 −0.154099 0.988055i \(-0.549247\pi\)
−0.154099 + 0.988055i \(0.549247\pi\)
\(468\) 0 0
\(469\) −7.95632 −0.367389
\(470\) 0 0
\(471\) −9.41576 −0.433855
\(472\) 0 0
\(473\) 3.59858 0.165463
\(474\) 0 0
\(475\) −3.02594 −0.138840
\(476\) 0 0
\(477\) −3.16602 −0.144962
\(478\) 0 0
\(479\) 8.06349 0.368430 0.184215 0.982886i \(-0.441026\pi\)
0.184215 + 0.982886i \(0.441026\pi\)
\(480\) 0 0
\(481\) 10.3447 0.471677
\(482\) 0 0
\(483\) −7.29903 −0.332117
\(484\) 0 0
\(485\) −37.9341 −1.72250
\(486\) 0 0
\(487\) −23.8768 −1.08196 −0.540981 0.841035i \(-0.681947\pi\)
−0.540981 + 0.841035i \(0.681947\pi\)
\(488\) 0 0
\(489\) −24.9865 −1.12993
\(490\) 0 0
\(491\) −2.42952 −0.109643 −0.0548214 0.998496i \(-0.517459\pi\)
−0.0548214 + 0.998496i \(0.517459\pi\)
\(492\) 0 0
\(493\) 13.2330 0.595984
\(494\) 0 0
\(495\) 1.41257 0.0634902
\(496\) 0 0
\(497\) 10.1838 0.456807
\(498\) 0 0
\(499\) −30.9479 −1.38542 −0.692709 0.721217i \(-0.743583\pi\)
−0.692709 + 0.721217i \(0.743583\pi\)
\(500\) 0 0
\(501\) 12.8803 0.575450
\(502\) 0 0
\(503\) 10.1030 0.450472 0.225236 0.974304i \(-0.427685\pi\)
0.225236 + 0.974304i \(0.427685\pi\)
\(504\) 0 0
\(505\) −52.2455 −2.32490
\(506\) 0 0
\(507\) 45.9416 2.04034
\(508\) 0 0
\(509\) 30.0363 1.33134 0.665668 0.746248i \(-0.268147\pi\)
0.665668 + 0.746248i \(0.268147\pi\)
\(510\) 0 0
\(511\) −42.4839 −1.87937
\(512\) 0 0
\(513\) 4.67874 0.206572
\(514\) 0 0
\(515\) 38.1498 1.68108
\(516\) 0 0
\(517\) 6.48461 0.285193
\(518\) 0 0
\(519\) −26.8895 −1.18032
\(520\) 0 0
\(521\) −26.3804 −1.15574 −0.577872 0.816127i \(-0.696117\pi\)
−0.577872 + 0.816127i \(0.696117\pi\)
\(522\) 0 0
\(523\) −22.2738 −0.973965 −0.486982 0.873412i \(-0.661902\pi\)
−0.486982 + 0.873412i \(0.661902\pi\)
\(524\) 0 0
\(525\) 21.6944 0.946823
\(526\) 0 0
\(527\) −19.6527 −0.856085
\(528\) 0 0
\(529\) −21.9635 −0.954936
\(530\) 0 0
\(531\) −5.39161 −0.233976
\(532\) 0 0
\(533\) −41.1307 −1.78157
\(534\) 0 0
\(535\) 46.3820 2.00527
\(536\) 0 0
\(537\) 4.49496 0.193972
\(538\) 0 0
\(539\) 7.69196 0.331316
\(540\) 0 0
\(541\) 25.9664 1.11638 0.558192 0.829712i \(-0.311495\pi\)
0.558192 + 0.829712i \(0.311495\pi\)
\(542\) 0 0
\(543\) −12.9963 −0.557724
\(544\) 0 0
\(545\) 21.5519 0.923180
\(546\) 0 0
\(547\) 9.05138 0.387009 0.193505 0.981099i \(-0.438015\pi\)
0.193505 + 0.981099i \(0.438015\pi\)
\(548\) 0 0
\(549\) −0.370813 −0.0158259
\(550\) 0 0
\(551\) 3.56805 0.152004
\(552\) 0 0
\(553\) 6.96002 0.295970
\(554\) 0 0
\(555\) 8.94419 0.379660
\(556\) 0 0
\(557\) 9.46486 0.401039 0.200519 0.979690i \(-0.435737\pi\)
0.200519 + 0.979690i \(0.435737\pi\)
\(558\) 0 0
\(559\) 22.0548 0.932819
\(560\) 0 0
\(561\) −6.93704 −0.292882
\(562\) 0 0
\(563\) 14.9480 0.629982 0.314991 0.949095i \(-0.397999\pi\)
0.314991 + 0.949095i \(0.397999\pi\)
\(564\) 0 0
\(565\) −28.3430 −1.19240
\(566\) 0 0
\(567\) −39.2777 −1.64951
\(568\) 0 0
\(569\) −37.8266 −1.58577 −0.792887 0.609369i \(-0.791423\pi\)
−0.792887 + 0.609369i \(0.791423\pi\)
\(570\) 0 0
\(571\) −21.6033 −0.904071 −0.452035 0.892000i \(-0.649302\pi\)
−0.452035 + 0.892000i \(0.649302\pi\)
\(572\) 0 0
\(573\) −28.4096 −1.18683
\(574\) 0 0
\(575\) −3.08062 −0.128471
\(576\) 0 0
\(577\) −27.5079 −1.14517 −0.572584 0.819846i \(-0.694059\pi\)
−0.572584 + 0.819846i \(0.694059\pi\)
\(578\) 0 0
\(579\) −20.4499 −0.849870
\(580\) 0 0
\(581\) −46.5389 −1.93076
\(582\) 0 0
\(583\) −6.34969 −0.262977
\(584\) 0 0
\(585\) 8.65729 0.357935
\(586\) 0 0
\(587\) 16.4534 0.679106 0.339553 0.940587i \(-0.389724\pi\)
0.339553 + 0.940587i \(0.389724\pi\)
\(588\) 0 0
\(589\) −5.29902 −0.218342
\(590\) 0 0
\(591\) −23.0888 −0.949748
\(592\) 0 0
\(593\) −0.937222 −0.0384871 −0.0192435 0.999815i \(-0.506126\pi\)
−0.0192435 + 0.999815i \(0.506126\pi\)
\(594\) 0 0
\(595\) −40.2730 −1.65103
\(596\) 0 0
\(597\) −24.3011 −0.994577
\(598\) 0 0
\(599\) −9.50619 −0.388412 −0.194206 0.980961i \(-0.562213\pi\)
−0.194206 + 0.980961i \(0.562213\pi\)
\(600\) 0 0
\(601\) −33.0715 −1.34901 −0.674507 0.738268i \(-0.735644\pi\)
−0.674507 + 0.738268i \(0.735644\pi\)
\(602\) 0 0
\(603\) −1.03498 −0.0421478
\(604\) 0 0
\(605\) 2.83301 0.115178
\(606\) 0 0
\(607\) 16.8867 0.685411 0.342705 0.939443i \(-0.388657\pi\)
0.342705 + 0.939443i \(0.388657\pi\)
\(608\) 0 0
\(609\) −25.5811 −1.03660
\(610\) 0 0
\(611\) 39.7426 1.60781
\(612\) 0 0
\(613\) −12.5772 −0.507987 −0.253993 0.967206i \(-0.581744\pi\)
−0.253993 + 0.967206i \(0.581744\pi\)
\(614\) 0 0
\(615\) −35.5623 −1.43401
\(616\) 0 0
\(617\) −21.2054 −0.853696 −0.426848 0.904323i \(-0.640376\pi\)
−0.426848 + 0.904323i \(0.640376\pi\)
\(618\) 0 0
\(619\) −2.83787 −0.114064 −0.0570319 0.998372i \(-0.518164\pi\)
−0.0570319 + 0.998372i \(0.518164\pi\)
\(620\) 0 0
\(621\) 4.76328 0.191144
\(622\) 0 0
\(623\) −5.80090 −0.232408
\(624\) 0 0
\(625\) −30.9734 −1.23894
\(626\) 0 0
\(627\) −1.87046 −0.0746989
\(628\) 0 0
\(629\) −6.25996 −0.249601
\(630\) 0 0
\(631\) −21.1498 −0.841962 −0.420981 0.907070i \(-0.638314\pi\)
−0.420981 + 0.907070i \(0.638314\pi\)
\(632\) 0 0
\(633\) 17.6351 0.700931
\(634\) 0 0
\(635\) 58.4622 2.32000
\(636\) 0 0
\(637\) 47.1422 1.86784
\(638\) 0 0
\(639\) 1.32475 0.0524061
\(640\) 0 0
\(641\) 35.8882 1.41750 0.708749 0.705461i \(-0.249260\pi\)
0.708749 + 0.705461i \(0.249260\pi\)
\(642\) 0 0
\(643\) 23.9922 0.946162 0.473081 0.881019i \(-0.343142\pi\)
0.473081 + 0.881019i \(0.343142\pi\)
\(644\) 0 0
\(645\) 19.0689 0.750839
\(646\) 0 0
\(647\) −3.96962 −0.156062 −0.0780310 0.996951i \(-0.524863\pi\)
−0.0780310 + 0.996951i \(0.524863\pi\)
\(648\) 0 0
\(649\) −10.8133 −0.424458
\(650\) 0 0
\(651\) 37.9912 1.48899
\(652\) 0 0
\(653\) −9.43621 −0.369268 −0.184634 0.982807i \(-0.559110\pi\)
−0.184634 + 0.982807i \(0.559110\pi\)
\(654\) 0 0
\(655\) −17.1825 −0.671377
\(656\) 0 0
\(657\) −5.52644 −0.215607
\(658\) 0 0
\(659\) 31.8841 1.24203 0.621014 0.783799i \(-0.286721\pi\)
0.621014 + 0.783799i \(0.286721\pi\)
\(660\) 0 0
\(661\) −27.9815 −1.08835 −0.544177 0.838970i \(-0.683158\pi\)
−0.544177 + 0.838970i \(0.683158\pi\)
\(662\) 0 0
\(663\) −42.5155 −1.65116
\(664\) 0 0
\(665\) −10.8590 −0.421092
\(666\) 0 0
\(667\) 3.63253 0.140652
\(668\) 0 0
\(669\) −20.4992 −0.792545
\(670\) 0 0
\(671\) −0.743693 −0.0287100
\(672\) 0 0
\(673\) 33.3736 1.28646 0.643228 0.765675i \(-0.277595\pi\)
0.643228 + 0.765675i \(0.277595\pi\)
\(674\) 0 0
\(675\) −14.1576 −0.544926
\(676\) 0 0
\(677\) 18.6319 0.716081 0.358040 0.933706i \(-0.383445\pi\)
0.358040 + 0.933706i \(0.383445\pi\)
\(678\) 0 0
\(679\) −51.3242 −1.96964
\(680\) 0 0
\(681\) 35.8638 1.37431
\(682\) 0 0
\(683\) 38.0254 1.45500 0.727501 0.686106i \(-0.240681\pi\)
0.727501 + 0.686106i \(0.240681\pi\)
\(684\) 0 0
\(685\) 28.9067 1.10447
\(686\) 0 0
\(687\) 8.30739 0.316947
\(688\) 0 0
\(689\) −38.9157 −1.48257
\(690\) 0 0
\(691\) −16.7840 −0.638495 −0.319247 0.947671i \(-0.603430\pi\)
−0.319247 + 0.947671i \(0.603430\pi\)
\(692\) 0 0
\(693\) 1.91118 0.0725997
\(694\) 0 0
\(695\) 61.6157 2.33722
\(696\) 0 0
\(697\) 24.8897 0.942766
\(698\) 0 0
\(699\) 46.7520 1.76832
\(700\) 0 0
\(701\) −31.4816 −1.18904 −0.594522 0.804079i \(-0.702659\pi\)
−0.594522 + 0.804079i \(0.702659\pi\)
\(702\) 0 0
\(703\) −1.68789 −0.0636601
\(704\) 0 0
\(705\) 34.3621 1.29415
\(706\) 0 0
\(707\) −70.6872 −2.65847
\(708\) 0 0
\(709\) 38.9461 1.46265 0.731326 0.682028i \(-0.238902\pi\)
0.731326 + 0.682028i \(0.238902\pi\)
\(710\) 0 0
\(711\) 0.905382 0.0339545
\(712\) 0 0
\(713\) −5.39477 −0.202036
\(714\) 0 0
\(715\) 17.3628 0.649333
\(716\) 0 0
\(717\) 15.9150 0.594357
\(718\) 0 0
\(719\) −28.0860 −1.04743 −0.523715 0.851894i \(-0.675454\pi\)
−0.523715 + 0.851894i \(0.675454\pi\)
\(720\) 0 0
\(721\) 51.6160 1.92228
\(722\) 0 0
\(723\) 38.3176 1.42505
\(724\) 0 0
\(725\) −10.7967 −0.400980
\(726\) 0 0
\(727\) −44.4663 −1.64916 −0.824582 0.565742i \(-0.808590\pi\)
−0.824582 + 0.565742i \(0.808590\pi\)
\(728\) 0 0
\(729\) 21.1448 0.783140
\(730\) 0 0
\(731\) −13.3462 −0.493627
\(732\) 0 0
\(733\) 27.3248 1.00926 0.504631 0.863335i \(-0.331628\pi\)
0.504631 + 0.863335i \(0.331628\pi\)
\(734\) 0 0
\(735\) 40.7599 1.50345
\(736\) 0 0
\(737\) −2.07574 −0.0764608
\(738\) 0 0
\(739\) 32.4957 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(740\) 0 0
\(741\) −11.4636 −0.421125
\(742\) 0 0
\(743\) −3.71370 −0.136242 −0.0681212 0.997677i \(-0.521700\pi\)
−0.0681212 + 0.997677i \(0.521700\pi\)
\(744\) 0 0
\(745\) 32.9240 1.20624
\(746\) 0 0
\(747\) −6.05394 −0.221502
\(748\) 0 0
\(749\) 62.7540 2.29298
\(750\) 0 0
\(751\) −10.8240 −0.394974 −0.197487 0.980305i \(-0.563278\pi\)
−0.197487 + 0.980305i \(0.563278\pi\)
\(752\) 0 0
\(753\) −26.2503 −0.956614
\(754\) 0 0
\(755\) −54.4837 −1.98286
\(756\) 0 0
\(757\) 16.0604 0.583724 0.291862 0.956461i \(-0.405725\pi\)
0.291862 + 0.956461i \(0.405725\pi\)
\(758\) 0 0
\(759\) −1.90426 −0.0691201
\(760\) 0 0
\(761\) −24.4280 −0.885515 −0.442757 0.896641i \(-0.646000\pi\)
−0.442757 + 0.896641i \(0.646000\pi\)
\(762\) 0 0
\(763\) 29.1593 1.05564
\(764\) 0 0
\(765\) −5.23885 −0.189411
\(766\) 0 0
\(767\) −66.2719 −2.39294
\(768\) 0 0
\(769\) −29.2657 −1.05535 −0.527674 0.849447i \(-0.676936\pi\)
−0.527674 + 0.849447i \(0.676936\pi\)
\(770\) 0 0
\(771\) 30.8014 1.10928
\(772\) 0 0
\(773\) 17.0485 0.613194 0.306597 0.951839i \(-0.400810\pi\)
0.306597 + 0.951839i \(0.400810\pi\)
\(774\) 0 0
\(775\) 16.0345 0.575978
\(776\) 0 0
\(777\) 12.1013 0.434132
\(778\) 0 0
\(779\) 6.71110 0.240450
\(780\) 0 0
\(781\) 2.65687 0.0950704
\(782\) 0 0
\(783\) 16.6940 0.596595
\(784\) 0 0
\(785\) −14.2612 −0.509003
\(786\) 0 0
\(787\) −37.5808 −1.33961 −0.669806 0.742536i \(-0.733623\pi\)
−0.669806 + 0.742536i \(0.733623\pi\)
\(788\) 0 0
\(789\) 50.5712 1.80038
\(790\) 0 0
\(791\) −38.3476 −1.36348
\(792\) 0 0
\(793\) −4.55792 −0.161856
\(794\) 0 0
\(795\) −33.6472 −1.19334
\(796\) 0 0
\(797\) 17.4740 0.618959 0.309480 0.950906i \(-0.399845\pi\)
0.309480 + 0.950906i \(0.399845\pi\)
\(798\) 0 0
\(799\) −24.0497 −0.850819
\(800\) 0 0
\(801\) −0.754600 −0.0266625
\(802\) 0 0
\(803\) −11.0837 −0.391135
\(804\) 0 0
\(805\) −11.0552 −0.389643
\(806\) 0 0
\(807\) 5.04132 0.177463
\(808\) 0 0
\(809\) −36.0380 −1.26703 −0.633514 0.773731i \(-0.718388\pi\)
−0.633514 + 0.773731i \(0.718388\pi\)
\(810\) 0 0
\(811\) 51.5131 1.80887 0.904435 0.426611i \(-0.140293\pi\)
0.904435 + 0.426611i \(0.140293\pi\)
\(812\) 0 0
\(813\) −1.40991 −0.0494478
\(814\) 0 0
\(815\) −37.8447 −1.32564
\(816\) 0 0
\(817\) −3.59858 −0.125898
\(818\) 0 0
\(819\) 11.7132 0.409291
\(820\) 0 0
\(821\) 39.3130 1.37203 0.686017 0.727586i \(-0.259358\pi\)
0.686017 + 0.727586i \(0.259358\pi\)
\(822\) 0 0
\(823\) 24.3573 0.849040 0.424520 0.905418i \(-0.360443\pi\)
0.424520 + 0.905418i \(0.360443\pi\)
\(824\) 0 0
\(825\) 5.65990 0.197052
\(826\) 0 0
\(827\) 0.290693 0.0101084 0.00505420 0.999987i \(-0.498391\pi\)
0.00505420 + 0.999987i \(0.498391\pi\)
\(828\) 0 0
\(829\) −44.2644 −1.53737 −0.768684 0.639629i \(-0.779088\pi\)
−0.768684 + 0.639629i \(0.779088\pi\)
\(830\) 0 0
\(831\) 22.5377 0.781824
\(832\) 0 0
\(833\) −28.5275 −0.988419
\(834\) 0 0
\(835\) 19.5086 0.675124
\(836\) 0 0
\(837\) −24.7928 −0.856963
\(838\) 0 0
\(839\) −14.9485 −0.516080 −0.258040 0.966134i \(-0.583077\pi\)
−0.258040 + 0.966134i \(0.583077\pi\)
\(840\) 0 0
\(841\) −16.2690 −0.561000
\(842\) 0 0
\(843\) 56.8443 1.95782
\(844\) 0 0
\(845\) 69.5835 2.39374
\(846\) 0 0
\(847\) 3.83301 0.131704
\(848\) 0 0
\(849\) −17.7627 −0.609614
\(850\) 0 0
\(851\) −1.71839 −0.0589058
\(852\) 0 0
\(853\) 5.52071 0.189026 0.0945128 0.995524i \(-0.469871\pi\)
0.0945128 + 0.995524i \(0.469871\pi\)
\(854\) 0 0
\(855\) −1.41257 −0.0483088
\(856\) 0 0
\(857\) −30.5165 −1.04242 −0.521212 0.853427i \(-0.674520\pi\)
−0.521212 + 0.853427i \(0.674520\pi\)
\(858\) 0 0
\(859\) −55.2698 −1.88578 −0.942891 0.333101i \(-0.891905\pi\)
−0.942891 + 0.333101i \(0.891905\pi\)
\(860\) 0 0
\(861\) −48.1151 −1.63976
\(862\) 0 0
\(863\) 21.7474 0.740291 0.370145 0.928974i \(-0.379308\pi\)
0.370145 + 0.928974i \(0.379308\pi\)
\(864\) 0 0
\(865\) −40.7271 −1.38476
\(866\) 0 0
\(867\) −6.07007 −0.206150
\(868\) 0 0
\(869\) 1.81581 0.0615972
\(870\) 0 0
\(871\) −12.7217 −0.431058
\(872\) 0 0
\(873\) −6.67642 −0.225963
\(874\) 0 0
\(875\) −21.4362 −0.724676
\(876\) 0 0
\(877\) −38.7805 −1.30952 −0.654762 0.755835i \(-0.727231\pi\)
−0.654762 + 0.755835i \(0.727231\pi\)
\(878\) 0 0
\(879\) −33.6244 −1.13412
\(880\) 0 0
\(881\) −51.5855 −1.73796 −0.868979 0.494849i \(-0.835223\pi\)
−0.868979 + 0.494849i \(0.835223\pi\)
\(882\) 0 0
\(883\) 7.82960 0.263487 0.131743 0.991284i \(-0.457942\pi\)
0.131743 + 0.991284i \(0.457942\pi\)
\(884\) 0 0
\(885\) −57.2998 −1.92611
\(886\) 0 0
\(887\) −26.0841 −0.875820 −0.437910 0.899019i \(-0.644281\pi\)
−0.437910 + 0.899019i \(0.644281\pi\)
\(888\) 0 0
\(889\) 79.0983 2.65287
\(890\) 0 0
\(891\) −10.2472 −0.343295
\(892\) 0 0
\(893\) −6.48461 −0.216999
\(894\) 0 0
\(895\) 6.80810 0.227570
\(896\) 0 0
\(897\) −11.6707 −0.389674
\(898\) 0 0
\(899\) −18.9072 −0.630590
\(900\) 0 0
\(901\) 23.5494 0.784543
\(902\) 0 0
\(903\) 25.7999 0.858568
\(904\) 0 0
\(905\) −19.6843 −0.654328
\(906\) 0 0
\(907\) 52.5096 1.74355 0.871777 0.489903i \(-0.162968\pi\)
0.871777 + 0.489903i \(0.162968\pi\)
\(908\) 0 0
\(909\) −9.19523 −0.304987
\(910\) 0 0
\(911\) 10.2679 0.340190 0.170095 0.985428i \(-0.445592\pi\)
0.170095 + 0.985428i \(0.445592\pi\)
\(912\) 0 0
\(913\) −12.1416 −0.401829
\(914\) 0 0
\(915\) −3.94085 −0.130280
\(916\) 0 0
\(917\) −23.2476 −0.767705
\(918\) 0 0
\(919\) 18.6639 0.615665 0.307833 0.951441i \(-0.400396\pi\)
0.307833 + 0.951441i \(0.400396\pi\)
\(920\) 0 0
\(921\) −21.3637 −0.703957
\(922\) 0 0
\(923\) 16.2833 0.535973
\(924\) 0 0
\(925\) 5.10747 0.167933
\(926\) 0 0
\(927\) 6.71438 0.220529
\(928\) 0 0
\(929\) −24.0742 −0.789849 −0.394925 0.918713i \(-0.629229\pi\)
−0.394925 + 0.918713i \(0.629229\pi\)
\(930\) 0 0
\(931\) −7.69196 −0.252094
\(932\) 0 0
\(933\) −47.9949 −1.57128
\(934\) 0 0
\(935\) −10.5069 −0.343612
\(936\) 0 0
\(937\) 9.95255 0.325136 0.162568 0.986697i \(-0.448022\pi\)
0.162568 + 0.986697i \(0.448022\pi\)
\(938\) 0 0
\(939\) −21.7400 −0.709460
\(940\) 0 0
\(941\) −28.7364 −0.936779 −0.468390 0.883522i \(-0.655166\pi\)
−0.468390 + 0.883522i \(0.655166\pi\)
\(942\) 0 0
\(943\) 6.83236 0.222492
\(944\) 0 0
\(945\) −50.8062 −1.65273
\(946\) 0 0
\(947\) −1.35752 −0.0441135 −0.0220567 0.999757i \(-0.507021\pi\)
−0.0220567 + 0.999757i \(0.507021\pi\)
\(948\) 0 0
\(949\) −67.9292 −2.20508
\(950\) 0 0
\(951\) 56.4021 1.82896
\(952\) 0 0
\(953\) −57.1898 −1.85256 −0.926280 0.376835i \(-0.877012\pi\)
−0.926280 + 0.376835i \(0.877012\pi\)
\(954\) 0 0
\(955\) −43.0294 −1.39240
\(956\) 0 0
\(957\) −6.67389 −0.215736
\(958\) 0 0
\(959\) 39.1103 1.26294
\(960\) 0 0
\(961\) −2.92035 −0.0942050
\(962\) 0 0
\(963\) 8.16324 0.263057
\(964\) 0 0
\(965\) −30.9736 −0.997076
\(966\) 0 0
\(967\) −31.9125 −1.02624 −0.513118 0.858318i \(-0.671510\pi\)
−0.513118 + 0.858318i \(0.671510\pi\)
\(968\) 0 0
\(969\) 6.93704 0.222850
\(970\) 0 0
\(971\) −22.4554 −0.720627 −0.360314 0.932831i \(-0.617330\pi\)
−0.360314 + 0.932831i \(0.617330\pi\)
\(972\) 0 0
\(973\) 83.3649 2.67256
\(974\) 0 0
\(975\) 34.6881 1.11091
\(976\) 0 0
\(977\) 39.6048 1.26707 0.633535 0.773714i \(-0.281603\pi\)
0.633535 + 0.773714i \(0.281603\pi\)
\(978\) 0 0
\(979\) −1.51341 −0.0483687
\(980\) 0 0
\(981\) 3.79313 0.121105
\(982\) 0 0
\(983\) 50.2521 1.60279 0.801396 0.598134i \(-0.204091\pi\)
0.801396 + 0.598134i \(0.204091\pi\)
\(984\) 0 0
\(985\) −34.9705 −1.11425
\(986\) 0 0
\(987\) 46.4913 1.47983
\(988\) 0 0
\(989\) −3.66360 −0.116496
\(990\) 0 0
\(991\) −35.6965 −1.13394 −0.566968 0.823740i \(-0.691884\pi\)
−0.566968 + 0.823740i \(0.691884\pi\)
\(992\) 0 0
\(993\) 47.5416 1.50869
\(994\) 0 0
\(995\) −36.8066 −1.16685
\(996\) 0 0
\(997\) 42.6983 1.35227 0.676134 0.736778i \(-0.263654\pi\)
0.676134 + 0.736778i \(0.263654\pi\)
\(998\) 0 0
\(999\) −7.89722 −0.249857
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 3344.2.a.z.1.4 6
4.3 odd 2 1672.2.a.i.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.i.1.3 6 4.3 odd 2
3344.2.a.z.1.4 6 1.1 even 1 trivial