Properties

Label 364.2.g.a.337.4
Level $364$
Weight $2$
Character 364.337
Analytic conductor $2.907$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [364,2,Mod(337,364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(364, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("364.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.41589892096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(-1.29363i\) of defining polynomial
Character \(\chi\) \(=\) 364.337
Dual form 364.2.g.a.337.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.29363 q^{3} +2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} +O(q^{10})\) \(q-1.29363 q^{3} +2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} -1.03291i q^{11} +(-1.79846 + 3.12499i) q^{13} -3.62016i q^{15} -6.95635 q^{17} +2.53774i q^{19} +1.29363i q^{21} -3.83136 q^{23} -2.83136 q^{25} +5.59691 q^{27} -5.10175 q^{29} +3.78880i q^{31} +1.33619i q^{33} +2.79846 q^{35} +6.20741i q^{37} +(2.32653 - 4.04257i) q^{39} -0.990338i q^{41} +0.560979 q^{43} -3.71224i q^{45} +7.19080i q^{47} -1.00000 q^{49} +8.99892 q^{51} +1.17830 q^{53} +2.89054 q^{55} -3.28288i q^{57} -11.9234i q^{59} +13.8469 q^{61} +1.32653i q^{63} +(-8.74515 - 5.03291i) q^{65} -13.7811i q^{67} +4.95635 q^{69} -3.80432i q^{71} -3.19080i q^{73} +3.66272 q^{75} -1.03291 q^{77} +7.33240 q^{79} -3.26072 q^{81} +16.6260i q^{83} -19.4670i q^{85} +6.59975 q^{87} +11.8964i q^{89} +(3.12499 + 1.79846i) q^{91} -4.90128i q^{93} -7.10175 q^{95} +8.60355i q^{97} +1.37018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 2 q^{13} + 4 q^{17} - 6 q^{23} + 2 q^{25} + 12 q^{27} - 2 q^{29} + 6 q^{35} - 6 q^{43} - 8 q^{49} - 8 q^{51} + 22 q^{53} - 20 q^{55} + 8 q^{61} - 6 q^{65} - 20 q^{69} - 20 q^{75} - 26 q^{79} - 24 q^{81} - 32 q^{87} - 10 q^{91} - 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/364\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(183\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −1.29363 −0.746875 −0.373438 0.927655i \(-0.621821\pi\)
−0.373438 + 0.927655i \(0.621821\pi\)
\(4\) 0 0
\(5\) 2.79846i 1.25151i 0.780020 + 0.625754i \(0.215209\pi\)
−0.780020 + 0.625754i \(0.784791\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.32653 −0.442177
\(10\) 0 0
\(11\) 1.03291i 0.311433i −0.987802 0.155716i \(-0.950231\pi\)
0.987802 0.155716i \(-0.0497686\pi\)
\(12\) 0 0
\(13\) −1.79846 + 3.12499i −0.498802 + 0.866716i
\(14\) 0 0
\(15\) 3.62016i 0.934721i
\(16\) 0 0
\(17\) −6.95635 −1.68716 −0.843581 0.537001i \(-0.819557\pi\)
−0.843581 + 0.537001i \(0.819557\pi\)
\(18\) 0 0
\(19\) 2.53774i 0.582197i 0.956693 + 0.291098i \(0.0940207\pi\)
−0.956693 + 0.291098i \(0.905979\pi\)
\(20\) 0 0
\(21\) 1.29363i 0.282292i
\(22\) 0 0
\(23\) −3.83136 −0.798894 −0.399447 0.916756i \(-0.630798\pi\)
−0.399447 + 0.916756i \(0.630798\pi\)
\(24\) 0 0
\(25\) −2.83136 −0.566272
\(26\) 0 0
\(27\) 5.59691 1.07713
\(28\) 0 0
\(29\) −5.10175 −0.947370 −0.473685 0.880694i \(-0.657077\pi\)
−0.473685 + 0.880694i \(0.657077\pi\)
\(30\) 0 0
\(31\) 3.78880i 0.680488i 0.940337 + 0.340244i \(0.110510\pi\)
−0.940337 + 0.340244i \(0.889490\pi\)
\(32\) 0 0
\(33\) 1.33619i 0.232601i
\(34\) 0 0
\(35\) 2.79846 0.473026
\(36\) 0 0
\(37\) 6.20741i 1.02049i 0.860029 + 0.510246i \(0.170446\pi\)
−0.860029 + 0.510246i \(0.829554\pi\)
\(38\) 0 0
\(39\) 2.32653 4.04257i 0.372543 0.647329i
\(40\) 0 0
\(41\) 0.990338i 0.154665i −0.997005 0.0773324i \(-0.975360\pi\)
0.997005 0.0773324i \(-0.0246403\pi\)
\(42\) 0 0
\(43\) 0.560979 0.0855485 0.0427743 0.999085i \(-0.486380\pi\)
0.0427743 + 0.999085i \(0.486380\pi\)
\(44\) 0 0
\(45\) 3.71224i 0.553388i
\(46\) 0 0
\(47\) 7.19080i 1.04889i 0.851446 + 0.524443i \(0.175726\pi\)
−0.851446 + 0.524443i \(0.824274\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.99892 1.26010
\(52\) 0 0
\(53\) 1.17830 0.161852 0.0809260 0.996720i \(-0.474212\pi\)
0.0809260 + 0.996720i \(0.474212\pi\)
\(54\) 0 0
\(55\) 2.89054 0.389760
\(56\) 0 0
\(57\) 3.28288i 0.434828i
\(58\) 0 0
\(59\) 11.9234i 1.55230i −0.630548 0.776150i \(-0.717170\pi\)
0.630548 0.776150i \(-0.282830\pi\)
\(60\) 0 0
\(61\) 13.8469 1.77291 0.886456 0.462812i \(-0.153160\pi\)
0.886456 + 0.462812i \(0.153160\pi\)
\(62\) 0 0
\(63\) 1.32653i 0.167127i
\(64\) 0 0
\(65\) −8.74515 5.03291i −1.08470 0.624255i
\(66\) 0 0
\(67\) 13.7811i 1.68363i −0.539769 0.841813i \(-0.681488\pi\)
0.539769 0.841813i \(-0.318512\pi\)
\(68\) 0 0
\(69\) 4.95635 0.596674
\(70\) 0 0
\(71\) 3.80432i 0.451490i −0.974186 0.225745i \(-0.927518\pi\)
0.974186 0.225745i \(-0.0724816\pi\)
\(72\) 0 0
\(73\) 3.19080i 0.373455i −0.982412 0.186727i \(-0.940212\pi\)
0.982412 0.186727i \(-0.0597881\pi\)
\(74\) 0 0
\(75\) 3.66272 0.422935
\(76\) 0 0
\(77\) −1.03291 −0.117710
\(78\) 0 0
\(79\) 7.33240 0.824959 0.412480 0.910967i \(-0.364663\pi\)
0.412480 + 0.910967i \(0.364663\pi\)
\(80\) 0 0
\(81\) −3.26072 −0.362302
\(82\) 0 0
\(83\) 16.6260i 1.82494i 0.409140 + 0.912472i \(0.365829\pi\)
−0.409140 + 0.912472i \(0.634171\pi\)
\(84\) 0 0
\(85\) 19.4670i 2.11150i
\(86\) 0 0
\(87\) 6.59975 0.707568
\(88\) 0 0
\(89\) 11.8964i 1.26102i 0.776182 + 0.630508i \(0.217154\pi\)
−0.776182 + 0.630508i \(0.782846\pi\)
\(90\) 0 0
\(91\) 3.12499 + 1.79846i 0.327588 + 0.188530i
\(92\) 0 0
\(93\) 4.90128i 0.508240i
\(94\) 0 0
\(95\) −7.10175 −0.728624
\(96\) 0 0
\(97\) 8.60355i 0.873558i 0.899569 + 0.436779i \(0.143881\pi\)
−0.899569 + 0.436779i \(0.856119\pi\)
\(98\) 0 0
\(99\) 1.37018i 0.137708i
\(100\) 0 0
\(101\) 8.36910 0.832756 0.416378 0.909192i \(-0.363299\pi\)
0.416378 + 0.909192i \(0.363299\pi\)
\(102\) 0 0
\(103\) 0.521442 0.0513792 0.0256896 0.999670i \(-0.491822\pi\)
0.0256896 + 0.999670i \(0.491822\pi\)
\(104\) 0 0
\(105\) −3.62016 −0.353291
\(106\) 0 0
\(107\) 9.43523 0.912138 0.456069 0.889944i \(-0.349257\pi\)
0.456069 + 0.889944i \(0.349257\pi\)
\(108\) 0 0
\(109\) 13.6860i 1.31088i 0.755248 + 0.655439i \(0.227516\pi\)
−0.755248 + 0.655439i \(0.772484\pi\)
\(110\) 0 0
\(111\) 8.03007i 0.762180i
\(112\) 0 0
\(113\) −12.2655 −1.15384 −0.576921 0.816800i \(-0.695746\pi\)
−0.576921 + 0.816800i \(0.695746\pi\)
\(114\) 0 0
\(115\) 10.7219i 0.999823i
\(116\) 0 0
\(117\) 2.38571 4.14539i 0.220559 0.383242i
\(118\) 0 0
\(119\) 6.95635i 0.637688i
\(120\) 0 0
\(121\) 9.93311 0.903010
\(122\) 0 0
\(123\) 1.28113i 0.115515i
\(124\) 0 0
\(125\) 6.06884i 0.542814i
\(126\) 0 0
\(127\) −10.9331 −0.970156 −0.485078 0.874471i \(-0.661209\pi\)
−0.485078 + 0.874471i \(0.661209\pi\)
\(128\) 0 0
\(129\) −0.725697 −0.0638941
\(130\) 0 0
\(131\) −6.58725 −0.575531 −0.287765 0.957701i \(-0.592912\pi\)
−0.287765 + 0.957701i \(0.592912\pi\)
\(132\) 0 0
\(133\) 2.53774 0.220050
\(134\) 0 0
\(135\) 15.6627i 1.34803i
\(136\) 0 0
\(137\) 12.8333i 1.09642i 0.836340 + 0.548212i \(0.184691\pi\)
−0.836340 + 0.548212i \(0.815309\pi\)
\(138\) 0 0
\(139\) −18.5998 −1.57761 −0.788805 0.614643i \(-0.789300\pi\)
−0.788805 + 0.614643i \(0.789300\pi\)
\(140\) 0 0
\(141\) 9.30221i 0.783387i
\(142\) 0 0
\(143\) 3.22782 + 1.85764i 0.269924 + 0.155343i
\(144\) 0 0
\(145\) 14.2770i 1.18564i
\(146\) 0 0
\(147\) 1.29363 0.106696
\(148\) 0 0
\(149\) 9.97676i 0.817328i 0.912685 + 0.408664i \(0.134005\pi\)
−0.912685 + 0.408664i \(0.865995\pi\)
\(150\) 0 0
\(151\) 14.0543i 1.14372i −0.820350 0.571861i \(-0.806222\pi\)
0.820350 0.571861i \(-0.193778\pi\)
\(152\) 0 0
\(153\) 9.22782 0.746025
\(154\) 0 0
\(155\) −10.6028 −0.851636
\(156\) 0 0
\(157\) −7.76858 −0.620000 −0.310000 0.950737i \(-0.600329\pi\)
−0.310000 + 0.950737i \(0.600329\pi\)
\(158\) 0 0
\(159\) −1.52428 −0.120883
\(160\) 0 0
\(161\) 3.83136i 0.301954i
\(162\) 0 0
\(163\) 7.39342i 0.579098i −0.957163 0.289549i \(-0.906495\pi\)
0.957163 0.289549i \(-0.0935053\pi\)
\(164\) 0 0
\(165\) −3.73928 −0.291102
\(166\) 0 0
\(167\) 15.7122i 1.21585i −0.793995 0.607925i \(-0.792002\pi\)
0.793995 0.607925i \(-0.207998\pi\)
\(168\) 0 0
\(169\) −6.53110 11.2403i −0.502393 0.864640i
\(170\) 0 0
\(171\) 3.36639i 0.257434i
\(172\) 0 0
\(173\) −0.415585 −0.0315963 −0.0157982 0.999875i \(-0.505029\pi\)
−0.0157982 + 0.999875i \(0.505029\pi\)
\(174\) 0 0
\(175\) 2.83136i 0.214031i
\(176\) 0 0
\(177\) 15.4245i 1.15938i
\(178\) 0 0
\(179\) 3.90792 0.292091 0.146046 0.989278i \(-0.453345\pi\)
0.146046 + 0.989278i \(0.453345\pi\)
\(180\) 0 0
\(181\) 4.79151 0.356150 0.178075 0.984017i \(-0.443013\pi\)
0.178075 + 0.984017i \(0.443013\pi\)
\(182\) 0 0
\(183\) −17.9127 −1.32414
\(184\) 0 0
\(185\) −17.3712 −1.27715
\(186\) 0 0
\(187\) 7.18525i 0.525437i
\(188\) 0 0
\(189\) 5.59691i 0.407116i
\(190\) 0 0
\(191\) −8.62905 −0.624376 −0.312188 0.950020i \(-0.601062\pi\)
−0.312188 + 0.950020i \(0.601062\pi\)
\(192\) 0 0
\(193\) 24.4535i 1.76020i 0.474789 + 0.880100i \(0.342524\pi\)
−0.474789 + 0.880100i \(0.657476\pi\)
\(194\) 0 0
\(195\) 11.3129 + 6.51070i 0.810137 + 0.466241i
\(196\) 0 0
\(197\) 7.14128i 0.508795i −0.967100 0.254398i \(-0.918123\pi\)
0.967100 0.254398i \(-0.0818772\pi\)
\(198\) 0 0
\(199\) −1.00284 −0.0710892 −0.0355446 0.999368i \(-0.511317\pi\)
−0.0355446 + 0.999368i \(0.511317\pi\)
\(200\) 0 0
\(201\) 17.8276i 1.25746i
\(202\) 0 0
\(203\) 5.10175i 0.358072i
\(204\) 0 0
\(205\) 2.77142 0.193564
\(206\) 0 0
\(207\) 5.08242 0.353253
\(208\) 0 0
\(209\) 2.62124 0.181315
\(210\) 0 0
\(211\) 26.2655 1.80819 0.904096 0.427329i \(-0.140546\pi\)
0.904096 + 0.427329i \(0.140546\pi\)
\(212\) 0 0
\(213\) 4.92137i 0.337207i
\(214\) 0 0
\(215\) 1.56988i 0.107065i
\(216\) 0 0
\(217\) 3.78880 0.257200
\(218\) 0 0
\(219\) 4.12770i 0.278924i
\(220\) 0 0
\(221\) 12.5107 21.7385i 0.841561 1.46229i
\(222\) 0 0
\(223\) 17.7540i 1.18890i 0.804133 + 0.594449i \(0.202630\pi\)
−0.804133 + 0.594449i \(0.797370\pi\)
\(224\) 0 0
\(225\) 3.75589 0.250393
\(226\) 0 0
\(227\) 17.8276i 1.18326i 0.806211 + 0.591629i \(0.201515\pi\)
−0.806211 + 0.591629i \(0.798485\pi\)
\(228\) 0 0
\(229\) 4.21423i 0.278484i −0.990258 0.139242i \(-0.955533\pi\)
0.990258 0.139242i \(-0.0444667\pi\)
\(230\) 0 0
\(231\) 1.33619 0.0879150
\(232\) 0 0
\(233\) −28.2880 −1.85321 −0.926604 0.376039i \(-0.877286\pi\)
−0.926604 + 0.376039i \(0.877286\pi\)
\(234\) 0 0
\(235\) −20.1231 −1.31269
\(236\) 0 0
\(237\) −9.48538 −0.616142
\(238\) 0 0
\(239\) 8.94385i 0.578530i 0.957249 + 0.289265i \(0.0934108\pi\)
−0.957249 + 0.289265i \(0.906589\pi\)
\(240\) 0 0
\(241\) 17.0019i 1.09519i −0.836743 0.547596i \(-0.815543\pi\)
0.836743 0.547596i \(-0.184457\pi\)
\(242\) 0 0
\(243\) −12.5726 −0.806532
\(244\) 0 0
\(245\) 2.79846i 0.178787i
\(246\) 0 0
\(247\) −7.93040 4.56401i −0.504599 0.290401i
\(248\) 0 0
\(249\) 21.5079i 1.36301i
\(250\) 0 0
\(251\) −6.70637 −0.423303 −0.211651 0.977345i \(-0.567884\pi\)
−0.211651 + 0.977345i \(0.567884\pi\)
\(252\) 0 0
\(253\) 3.95743i 0.248802i
\(254\) 0 0
\(255\) 25.1831i 1.57703i
\(256\) 0 0
\(257\) −14.6656 −0.914813 −0.457406 0.889258i \(-0.651221\pi\)
−0.457406 + 0.889258i \(0.651221\pi\)
\(258\) 0 0
\(259\) 6.20741 0.385710
\(260\) 0 0
\(261\) 6.76762 0.418905
\(262\) 0 0
\(263\) −6.02627 −0.371596 −0.185798 0.982588i \(-0.559487\pi\)
−0.185798 + 0.982588i \(0.559487\pi\)
\(264\) 0 0
\(265\) 3.29742i 0.202559i
\(266\) 0 0
\(267\) 15.3895i 0.941822i
\(268\) 0 0
\(269\) −0.825495 −0.0503313 −0.0251657 0.999683i \(-0.508011\pi\)
−0.0251657 + 0.999683i \(0.508011\pi\)
\(270\) 0 0
\(271\) 9.82757i 0.596982i −0.954412 0.298491i \(-0.903517\pi\)
0.954412 0.298491i \(-0.0964833\pi\)
\(272\) 0 0
\(273\) −4.04257 2.32653i −0.244667 0.140808i
\(274\) 0 0
\(275\) 2.92453i 0.176356i
\(276\) 0 0
\(277\) 27.8517 1.67344 0.836722 0.547627i \(-0.184469\pi\)
0.836722 + 0.547627i \(0.184469\pi\)
\(278\) 0 0
\(279\) 5.02596i 0.300896i
\(280\) 0 0
\(281\) 4.04257i 0.241159i 0.992704 + 0.120580i \(0.0384753\pi\)
−0.992704 + 0.120580i \(0.961525\pi\)
\(282\) 0 0
\(283\) −29.7596 −1.76902 −0.884512 0.466517i \(-0.845509\pi\)
−0.884512 + 0.466517i \(0.845509\pi\)
\(284\) 0 0
\(285\) 9.18700 0.544191
\(286\) 0 0
\(287\) −0.990338 −0.0584578
\(288\) 0 0
\(289\) 31.3908 1.84652
\(290\) 0 0
\(291\) 11.1298i 0.652439i
\(292\) 0 0
\(293\) 29.8091i 1.74147i −0.491755 0.870733i \(-0.663645\pi\)
0.491755 0.870733i \(-0.336355\pi\)
\(294\) 0 0
\(295\) 33.3673 1.94272
\(296\) 0 0
\(297\) 5.78108i 0.335452i
\(298\) 0 0
\(299\) 6.89054 11.9730i 0.398490 0.692414i
\(300\) 0 0
\(301\) 0.560979i 0.0323343i
\(302\) 0 0
\(303\) −10.8265 −0.621965
\(304\) 0 0
\(305\) 38.7499i 2.21881i
\(306\) 0 0
\(307\) 18.5928i 1.06115i 0.847639 + 0.530574i \(0.178023\pi\)
−0.847639 + 0.530574i \(0.821977\pi\)
\(308\) 0 0
\(309\) −0.674552 −0.0383739
\(310\) 0 0
\(311\) 15.9119 0.902283 0.451142 0.892452i \(-0.351017\pi\)
0.451142 + 0.892452i \(0.351017\pi\)
\(312\) 0 0
\(313\) −1.79575 −0.101502 −0.0507508 0.998711i \(-0.516161\pi\)
−0.0507508 + 0.998711i \(0.516161\pi\)
\(314\) 0 0
\(315\) −3.71224 −0.209161
\(316\) 0 0
\(317\) 16.0311i 0.900394i −0.892929 0.450197i \(-0.851354\pi\)
0.892929 0.450197i \(-0.148646\pi\)
\(318\) 0 0
\(319\) 5.26962i 0.295042i
\(320\) 0 0
\(321\) −12.2057 −0.681253
\(322\) 0 0
\(323\) 17.6534i 0.982260i
\(324\) 0 0
\(325\) 5.09208 8.84797i 0.282458 0.490797i
\(326\) 0 0
\(327\) 17.7045i 0.979063i
\(328\) 0 0
\(329\) 7.19080 0.396442
\(330\) 0 0
\(331\) 6.81791i 0.374746i 0.982289 + 0.187373i \(0.0599973\pi\)
−0.982289 + 0.187373i \(0.940003\pi\)
\(332\) 0 0
\(333\) 8.23432i 0.451238i
\(334\) 0 0
\(335\) 38.5658 2.10707
\(336\) 0 0
\(337\) 0.418615 0.0228034 0.0114017 0.999935i \(-0.496371\pi\)
0.0114017 + 0.999935i \(0.496371\pi\)
\(338\) 0 0
\(339\) 15.8670 0.861776
\(340\) 0 0
\(341\) 3.91347 0.211926
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 13.8701i 0.746743i
\(346\) 0 0
\(347\) −10.0658 −0.540361 −0.270180 0.962810i \(-0.587083\pi\)
−0.270180 + 0.962810i \(0.587083\pi\)
\(348\) 0 0
\(349\) 8.27233i 0.442808i −0.975182 0.221404i \(-0.928936\pi\)
0.975182 0.221404i \(-0.0710639\pi\)
\(350\) 0 0
\(351\) −10.0658 + 17.4903i −0.537273 + 0.933563i
\(352\) 0 0
\(353\) 16.3908i 0.872395i 0.899851 + 0.436197i \(0.143675\pi\)
−0.899851 + 0.436197i \(0.856325\pi\)
\(354\) 0 0
\(355\) 10.6462 0.565044
\(356\) 0 0
\(357\) 8.99892i 0.476273i
\(358\) 0 0
\(359\) 32.1512i 1.69687i −0.529297 0.848437i \(-0.677544\pi\)
0.529297 0.848437i \(-0.322456\pi\)
\(360\) 0 0
\(361\) 12.5599 0.661047
\(362\) 0 0
\(363\) −12.8497 −0.674436
\(364\) 0 0
\(365\) 8.92931 0.467382
\(366\) 0 0
\(367\) −7.79575 −0.406935 −0.203467 0.979082i \(-0.565221\pi\)
−0.203467 + 0.979082i \(0.565221\pi\)
\(368\) 0 0
\(369\) 1.31371i 0.0683892i
\(370\) 0 0
\(371\) 1.17830i 0.0611743i
\(372\) 0 0
\(373\) 31.7371 1.64329 0.821643 0.570003i \(-0.193058\pi\)
0.821643 + 0.570003i \(0.193058\pi\)
\(374\) 0 0
\(375\) 7.85081i 0.405414i
\(376\) 0 0
\(377\) 9.17527 15.9429i 0.472550 0.821101i
\(378\) 0 0
\(379\) 30.3236i 1.55762i 0.627261 + 0.778809i \(0.284176\pi\)
−0.627261 + 0.778809i \(0.715824\pi\)
\(380\) 0 0
\(381\) 14.1434 0.724586
\(382\) 0 0
\(383\) 14.0193i 0.716354i 0.933654 + 0.358177i \(0.116602\pi\)
−0.933654 + 0.358177i \(0.883398\pi\)
\(384\) 0 0
\(385\) 2.89054i 0.147316i
\(386\) 0 0
\(387\) −0.744156 −0.0378276
\(388\) 0 0
\(389\) 19.8887 1.00840 0.504198 0.863588i \(-0.331788\pi\)
0.504198 + 0.863588i \(0.331788\pi\)
\(390\) 0 0
\(391\) 26.6523 1.34786
\(392\) 0 0
\(393\) 8.52144 0.429850
\(394\) 0 0
\(395\) 20.5194i 1.03244i
\(396\) 0 0
\(397\) 14.7219i 0.738871i 0.929256 + 0.369436i \(0.120449\pi\)
−0.929256 + 0.369436i \(0.879551\pi\)
\(398\) 0 0
\(399\) −3.28288 −0.164350
\(400\) 0 0
\(401\) 19.6412i 0.980836i −0.871487 0.490418i \(-0.836844\pi\)
0.871487 0.490418i \(-0.163156\pi\)
\(402\) 0 0
\(403\) −11.8399 6.81399i −0.589789 0.339429i
\(404\) 0 0
\(405\) 9.12499i 0.453424i
\(406\) 0 0
\(407\) 6.41166 0.317814
\(408\) 0 0
\(409\) 24.1132i 1.19232i −0.802866 0.596160i \(-0.796693\pi\)
0.802866 0.596160i \(-0.203307\pi\)
\(410\) 0 0
\(411\) 16.6015i 0.818892i
\(412\) 0 0
\(413\) −11.9234 −0.586714
\(414\) 0 0
\(415\) −46.5272 −2.28393
\(416\) 0 0
\(417\) 24.0611 1.17828
\(418\) 0 0
\(419\) −19.7818 −0.966406 −0.483203 0.875508i \(-0.660527\pi\)
−0.483203 + 0.875508i \(0.660527\pi\)
\(420\) 0 0
\(421\) 19.1237i 0.932033i 0.884776 + 0.466016i \(0.154311\pi\)
−0.884776 + 0.466016i \(0.845689\pi\)
\(422\) 0 0
\(423\) 9.53882i 0.463793i
\(424\) 0 0
\(425\) 19.6959 0.955394
\(426\) 0 0
\(427\) 13.8469i 0.670098i
\(428\) 0 0
\(429\) −4.17559 2.40309i −0.201599 0.116022i
\(430\) 0 0
\(431\) 8.87588i 0.427536i −0.976884 0.213768i \(-0.931426\pi\)
0.976884 0.213768i \(-0.0685736\pi\)
\(432\) 0 0
\(433\) −36.9341 −1.77494 −0.887470 0.460866i \(-0.847539\pi\)
−0.887470 + 0.460866i \(0.847539\pi\)
\(434\) 0 0
\(435\) 18.4691i 0.885527i
\(436\) 0 0
\(437\) 9.72299i 0.465113i
\(438\) 0 0
\(439\) −20.6591 −0.986006 −0.493003 0.870028i \(-0.664101\pi\)
−0.493003 + 0.870028i \(0.664101\pi\)
\(440\) 0 0
\(441\) 1.32653 0.0631682
\(442\) 0 0
\(443\) 2.85492 0.135641 0.0678207 0.997698i \(-0.478395\pi\)
0.0678207 + 0.997698i \(0.478395\pi\)
\(444\) 0 0
\(445\) −33.2916 −1.57817
\(446\) 0 0
\(447\) 12.9062i 0.610442i
\(448\) 0 0
\(449\) 30.0443i 1.41788i 0.705269 + 0.708940i \(0.250826\pi\)
−0.705269 + 0.708940i \(0.749174\pi\)
\(450\) 0 0
\(451\) −1.02293 −0.0481677
\(452\) 0 0
\(453\) 18.1810i 0.854218i
\(454\) 0 0
\(455\) −5.03291 + 8.74515i −0.235946 + 0.409979i
\(456\) 0 0
\(457\) 28.6491i 1.34015i −0.742293 0.670075i \(-0.766262\pi\)
0.742293 0.670075i \(-0.233738\pi\)
\(458\) 0 0
\(459\) −38.9341 −1.81729
\(460\) 0 0
\(461\) 0.922363i 0.0429587i −0.999769 0.0214794i \(-0.993162\pi\)
0.999769 0.0214794i \(-0.00683762\pi\)
\(462\) 0 0
\(463\) 26.0929i 1.21264i 0.795220 + 0.606321i \(0.207355\pi\)
−0.795220 + 0.606321i \(0.792645\pi\)
\(464\) 0 0
\(465\) 13.7160 0.636066
\(466\) 0 0
\(467\) 29.5553 1.36766 0.683829 0.729642i \(-0.260313\pi\)
0.683829 + 0.729642i \(0.260313\pi\)
\(468\) 0 0
\(469\) −13.7811 −0.636351
\(470\) 0 0
\(471\) 10.0496 0.463063
\(472\) 0 0
\(473\) 0.579438i 0.0266426i
\(474\) 0 0
\(475\) 7.18525i 0.329682i
\(476\) 0 0
\(477\) −1.56305 −0.0715672
\(478\) 0 0
\(479\) 15.8954i 0.726280i −0.931735 0.363140i \(-0.881705\pi\)
0.931735 0.363140i \(-0.118295\pi\)
\(480\) 0 0
\(481\) −19.3981 11.1638i −0.884476 0.509024i
\(482\) 0 0
\(483\) 4.95635i 0.225522i
\(484\) 0 0
\(485\) −24.0767 −1.09326
\(486\) 0 0
\(487\) 0.0851336i 0.00385777i 0.999998 + 0.00192889i \(0.000613984\pi\)
−0.999998 + 0.00192889i \(0.999386\pi\)
\(488\) 0 0
\(489\) 9.56433i 0.432514i
\(490\) 0 0
\(491\) 2.56324 0.115678 0.0578388 0.998326i \(-0.481579\pi\)
0.0578388 + 0.998326i \(0.481579\pi\)
\(492\) 0 0
\(493\) 35.4895 1.59837
\(494\) 0 0
\(495\) −3.83439 −0.172343
\(496\) 0 0
\(497\) −3.80432 −0.170647
\(498\) 0 0
\(499\) 28.6941i 1.28452i 0.766485 + 0.642262i \(0.222004\pi\)
−0.766485 + 0.642262i \(0.777996\pi\)
\(500\) 0 0
\(501\) 20.3258i 0.908088i
\(502\) 0 0
\(503\) −32.7056 −1.45827 −0.729136 0.684369i \(-0.760078\pi\)
−0.729136 + 0.684369i \(0.760078\pi\)
\(504\) 0 0
\(505\) 23.4206i 1.04220i
\(506\) 0 0
\(507\) 8.44881 + 14.5408i 0.375225 + 0.645778i
\(508\) 0 0
\(509\) 26.0938i 1.15659i 0.815828 + 0.578295i \(0.196282\pi\)
−0.815828 + 0.578295i \(0.803718\pi\)
\(510\) 0 0
\(511\) −3.19080 −0.141153
\(512\) 0 0
\(513\) 14.2035i 0.627099i
\(514\) 0 0
\(515\) 1.45923i 0.0643015i
\(516\) 0 0
\(517\) 7.42741 0.326657
\(518\) 0 0
\(519\) 0.537611 0.0235985
\(520\) 0 0
\(521\) −1.75570 −0.0769185 −0.0384592 0.999260i \(-0.512245\pi\)
−0.0384592 + 0.999260i \(0.512245\pi\)
\(522\) 0 0
\(523\) −32.3779 −1.41579 −0.707893 0.706319i \(-0.750354\pi\)
−0.707893 + 0.706319i \(0.750354\pi\)
\(524\) 0 0
\(525\) 3.66272i 0.159854i
\(526\) 0 0
\(527\) 26.3562i 1.14809i
\(528\) 0 0
\(529\) −8.32066 −0.361768
\(530\) 0 0
\(531\) 15.8168i 0.686392i
\(532\) 0 0
\(533\) 3.09480 + 1.78108i 0.134050 + 0.0771472i
\(534\) 0 0
\(535\) 26.4041i 1.14155i
\(536\) 0 0
\(537\) −5.05538 −0.218156
\(538\) 0 0
\(539\) 1.03291i 0.0444904i
\(540\) 0 0
\(541\) 8.52536i 0.366534i −0.983063 0.183267i \(-0.941333\pi\)
0.983063 0.183267i \(-0.0586673\pi\)
\(542\) 0 0
\(543\) −6.19842 −0.266000
\(544\) 0 0
\(545\) −38.2996 −1.64057
\(546\) 0 0
\(547\) 1.14355 0.0488945 0.0244473 0.999701i \(-0.492217\pi\)
0.0244473 + 0.999701i \(0.492217\pi\)
\(548\) 0 0
\(549\) −18.3683 −0.783941
\(550\) 0 0
\(551\) 12.9469i 0.551556i
\(552\) 0 0
\(553\) 7.33240i 0.311805i
\(554\) 0 0
\(555\) 22.4718 0.953875
\(556\) 0 0
\(557\) 42.8275i 1.81466i 0.420421 + 0.907329i \(0.361882\pi\)
−0.420421 + 0.907329i \(0.638118\pi\)
\(558\) 0 0
\(559\) −1.00890 + 1.75305i −0.0426718 + 0.0741463i
\(560\) 0 0
\(561\) 9.29503i 0.392436i
\(562\) 0 0
\(563\) 13.5754 0.572136 0.286068 0.958209i \(-0.407652\pi\)
0.286068 + 0.958209i \(0.407652\pi\)
\(564\) 0 0
\(565\) 34.3245i 1.44404i
\(566\) 0 0
\(567\) 3.26072i 0.136937i
\(568\) 0 0
\(569\) −8.87109 −0.371896 −0.185948 0.982560i \(-0.559536\pi\)
−0.185948 + 0.982560i \(0.559536\pi\)
\(570\) 0 0
\(571\) 6.28093 0.262849 0.131424 0.991326i \(-0.458045\pi\)
0.131424 + 0.991326i \(0.458045\pi\)
\(572\) 0 0
\(573\) 11.1628 0.466331
\(574\) 0 0
\(575\) 10.8480 0.452392
\(576\) 0 0
\(577\) 25.1005i 1.04495i −0.852656 0.522473i \(-0.825010\pi\)
0.852656 0.522473i \(-0.174990\pi\)
\(578\) 0 0
\(579\) 31.6336i 1.31465i
\(580\) 0 0
\(581\) 16.6260 0.689764
\(582\) 0 0
\(583\) 1.21707i 0.0504060i
\(584\) 0 0
\(585\) 11.6007 + 6.67631i 0.479630 + 0.276031i
\(586\) 0 0
\(587\) 28.2963i 1.16791i 0.811784 + 0.583957i \(0.198497\pi\)
−0.811784 + 0.583957i \(0.801503\pi\)
\(588\) 0 0
\(589\) −9.61496 −0.396178
\(590\) 0 0
\(591\) 9.23815i 0.380007i
\(592\) 0 0
\(593\) 12.8643i 0.528272i −0.964485 0.264136i \(-0.914913\pi\)
0.964485 0.264136i \(-0.0850868\pi\)
\(594\) 0 0
\(595\) −19.4670 −0.798071
\(596\) 0 0
\(597\) 1.29730 0.0530948
\(598\) 0 0
\(599\) 25.5685 1.04470 0.522350 0.852731i \(-0.325056\pi\)
0.522350 + 0.852731i \(0.325056\pi\)
\(600\) 0 0
\(601\) 6.63763 0.270755 0.135377 0.990794i \(-0.456775\pi\)
0.135377 + 0.990794i \(0.456775\pi\)
\(602\) 0 0
\(603\) 18.2810i 0.744461i
\(604\) 0 0
\(605\) 27.7974i 1.13012i
\(606\) 0 0
\(607\) 22.3226 0.906047 0.453023 0.891499i \(-0.350345\pi\)
0.453023 + 0.891499i \(0.350345\pi\)
\(608\) 0 0
\(609\) 6.59975i 0.267435i
\(610\) 0 0
\(611\) −22.4712 12.9323i −0.909086 0.523187i
\(612\) 0 0
\(613\) 4.23671i 0.171119i 0.996333 + 0.0855596i \(0.0272678\pi\)
−0.996333 + 0.0855596i \(0.972732\pi\)
\(614\) 0 0
\(615\) −3.58518 −0.144568
\(616\) 0 0
\(617\) 14.0193i 0.564397i 0.959356 + 0.282198i \(0.0910636\pi\)
−0.959356 + 0.282198i \(0.908936\pi\)
\(618\) 0 0
\(619\) 23.8469i 0.958487i 0.877682 + 0.479244i \(0.159089\pi\)
−0.877682 + 0.479244i \(0.840911\pi\)
\(620\) 0 0
\(621\) −21.4438 −0.860510
\(622\) 0 0
\(623\) 11.8964 0.476619
\(624\) 0 0
\(625\) −31.1402 −1.24561
\(626\) 0 0
\(627\) −3.39090 −0.135420
\(628\) 0 0
\(629\) 43.1809i 1.72174i
\(630\) 0 0
\(631\) 7.18624i 0.286080i 0.989717 + 0.143040i \(0.0456877\pi\)
−0.989717 + 0.143040i \(0.954312\pi\)
\(632\) 0 0
\(633\) −33.9777 −1.35049
\(634\) 0 0
\(635\) 30.5958i 1.21416i
\(636\) 0 0
\(637\) 1.79846 3.12499i 0.0712575 0.123817i
\(638\) 0 0
\(639\) 5.04655i 0.199639i
\(640\) 0 0
\(641\) −8.05886 −0.318306 −0.159153 0.987254i \(-0.550876\pi\)
−0.159153 + 0.987254i \(0.550876\pi\)
\(642\) 0 0
\(643\) 28.4148i 1.12057i −0.828300 0.560286i \(-0.810691\pi\)
0.828300 0.560286i \(-0.189309\pi\)
\(644\) 0 0
\(645\) 2.03083i 0.0799640i
\(646\) 0 0
\(647\) 37.6791 1.48132 0.740659 0.671881i \(-0.234513\pi\)
0.740659 + 0.671881i \(0.234513\pi\)
\(648\) 0 0
\(649\) −12.3158 −0.483437
\(650\) 0 0
\(651\) −4.90128 −0.192097
\(652\) 0 0
\(653\) 37.0486 1.44982 0.724911 0.688842i \(-0.241881\pi\)
0.724911 + 0.688842i \(0.241881\pi\)
\(654\) 0 0
\(655\) 18.4341i 0.720282i
\(656\) 0 0
\(657\) 4.23269i 0.165133i
\(658\) 0 0
\(659\) 44.4553 1.73173 0.865867 0.500274i \(-0.166767\pi\)
0.865867 + 0.500274i \(0.166767\pi\)
\(660\) 0 0
\(661\) 40.3413i 1.56909i 0.620069 + 0.784547i \(0.287105\pi\)
−0.620069 + 0.784547i \(0.712895\pi\)
\(662\) 0 0
\(663\) −16.1842 + 28.1215i −0.628541 + 1.09215i
\(664\) 0 0
\(665\) 7.10175i 0.275394i
\(666\) 0 0
\(667\) 19.5466 0.756849
\(668\) 0 0
\(669\) 22.9671i 0.887959i
\(670\) 0 0
\(671\) 14.3025i 0.552143i
\(672\) 0 0
\(673\) −16.8024 −0.647684 −0.323842 0.946111i \(-0.604975\pi\)
−0.323842 + 0.946111i \(0.604975\pi\)
\(674\) 0 0
\(675\) −15.8469 −0.609947
\(676\) 0 0
\(677\) −46.5650 −1.78964 −0.894819 0.446429i \(-0.852696\pi\)
−0.894819 + 0.446429i \(0.852696\pi\)
\(678\) 0 0
\(679\) 8.60355 0.330174
\(680\) 0 0
\(681\) 23.0622i 0.883746i
\(682\) 0 0
\(683\) 5.79498i 0.221739i −0.993835 0.110869i \(-0.964636\pi\)
0.993835 0.110869i \(-0.0353635\pi\)
\(684\) 0 0
\(685\) −35.9135 −1.37218
\(686\) 0 0
\(687\) 5.45164i 0.207993i
\(688\) 0 0
\(689\) −2.11912 + 3.68217i −0.0807321 + 0.140280i
\(690\) 0 0
\(691\) 27.1213i 1.03174i −0.856666 0.515871i \(-0.827468\pi\)
0.856666 0.515871i \(-0.172532\pi\)
\(692\) 0 0
\(693\) 1.37018 0.0520489
\(694\) 0 0
\(695\) 52.0506i 1.97439i
\(696\) 0 0
\(697\) 6.88914i 0.260945i
\(698\) 0 0
\(699\) 36.5941 1.38412
\(700\) 0 0
\(701\) 11.0283 0.416535 0.208267 0.978072i \(-0.433218\pi\)
0.208267 + 0.978072i \(0.433218\pi\)
\(702\) 0 0
\(703\) −15.7528 −0.594127
\(704\) 0 0
\(705\) 26.0318 0.980415
\(706\) 0 0
\(707\) 8.36910i 0.314752i
\(708\) 0 0
\(709\) 49.7496i 1.86839i 0.356768 + 0.934193i \(0.383879\pi\)
−0.356768 + 0.934193i \(0.616121\pi\)
\(710\) 0 0
\(711\) −9.72666 −0.364778
\(712\) 0 0
\(713\) 14.5162i 0.543638i
\(714\) 0 0
\(715\) −5.19851 + 9.03291i −0.194413 + 0.337811i
\(716\) 0 0
\(717\) 11.5700i 0.432090i
\(718\) 0 0
\(719\) 28.5650 1.06529 0.532647 0.846337i \(-0.321197\pi\)
0.532647 + 0.846337i \(0.321197\pi\)
\(720\) 0 0
\(721\) 0.521442i 0.0194195i
\(722\) 0 0
\(723\) 21.9942i 0.817972i
\(724\) 0 0
\(725\) 14.4449 0.536470
\(726\) 0 0
\(727\) 36.5250 1.35464 0.677318 0.735691i \(-0.263142\pi\)
0.677318 + 0.735691i \(0.263142\pi\)
\(728\) 0 0
\(729\) 26.0464 0.964681
\(730\) 0 0
\(731\) −3.90237 −0.144334
\(732\) 0 0
\(733\) 18.7444i 0.692340i 0.938172 + 0.346170i \(0.112518\pi\)
−0.938172 + 0.346170i \(0.887482\pi\)
\(734\) 0 0
\(735\) 3.62016i 0.133532i
\(736\) 0 0
\(737\) −14.2345 −0.524336
\(738\) 0 0
\(739\) 11.1838i 0.411405i −0.978615 0.205702i \(-0.934052\pi\)
0.978615 0.205702i \(-0.0659478\pi\)
\(740\) 0 0
\(741\) 10.2590 + 5.90412i 0.376873 + 0.216893i
\(742\) 0 0
\(743\) 32.7986i 1.20326i 0.798774 + 0.601632i \(0.205482\pi\)
−0.798774 + 0.601632i \(0.794518\pi\)
\(744\) 0 0
\(745\) −27.9195 −1.02289
\(746\) 0 0
\(747\) 22.0549i 0.806948i
\(748\) 0 0
\(749\) 9.43523i 0.344756i
\(750\) 0 0
\(751\) −28.7451 −1.04893 −0.524463 0.851433i \(-0.675734\pi\)
−0.524463 + 0.851433i \(0.675734\pi\)
\(752\) 0 0
\(753\) 8.67554 0.316154
\(754\) 0 0
\(755\) 39.3304 1.43138
\(756\) 0 0
\(757\) 12.4952 0.454145 0.227072 0.973878i \(-0.427085\pi\)
0.227072 + 0.973878i \(0.427085\pi\)
\(758\) 0 0
\(759\) 5.11944i 0.185824i
\(760\) 0 0
\(761\) 29.6628i 1.07528i 0.843175 + 0.537639i \(0.180684\pi\)
−0.843175 + 0.537639i \(0.819316\pi\)
\(762\) 0 0
\(763\) 13.6860 0.495465
\(764\) 0 0
\(765\) 25.8236i 0.933656i
\(766\) 0 0
\(767\) 37.2606 + 21.4438i 1.34540 + 0.774291i
\(768\) 0 0
\(769\) 40.8488i 1.47305i −0.676412 0.736523i \(-0.736466\pi\)
0.676412 0.736523i \(-0.263534\pi\)
\(770\) 0 0
\(771\) 18.9718 0.683251
\(772\) 0 0
\(773\) 19.4045i 0.697932i 0.937135 + 0.348966i \(0.113467\pi\)
−0.937135 + 0.348966i \(0.886533\pi\)
\(774\) 0 0
\(775\) 10.7275i 0.385341i
\(776\) 0 0
\(777\) −8.03007 −0.288077
\(778\) 0 0
\(779\) 2.51322 0.0900453
\(780\) 0 0
\(781\) −3.92951 −0.140609
\(782\) 0 0
\(783\) −28.5540 −1.02044
\(784\) 0 0
\(785\) 21.7400i 0.775935i
\(786\) 0 0
\(787\) 10.0913i 0.359717i −0.983693 0.179858i \(-0.942436\pi\)
0.983693 0.179858i \(-0.0575639\pi\)
\(788\) 0 0
\(789\) 7.79575 0.277536
\(790\) 0 0
\(791\) 12.2655i 0.436111i
\(792\) 0 0
\(793\) −24.9030 + 43.2714i −0.884333 + 1.53661i
\(794\) 0 0
\(795\) 4.26563i 0.151286i
\(796\) 0 0
\(797\) −11.2735 −0.399329 −0.199665 0.979864i \(-0.563985\pi\)
−0.199665 + 0.979864i \(0.563985\pi\)
\(798\) 0 0
\(799\) 50.0217i 1.76964i
\(800\) 0 0
\(801\) 15.7810i 0.557593i
\(802\) 0 0
\(803\) −3.29579 −0.116306
\(804\) 0 0
\(805\) −10.7219 −0.377897
\(806\) 0 0
\(807\) 1.06788 0.0375912
\(808\) 0 0
\(809\) 17.4236 0.612581 0.306290 0.951938i \(-0.400912\pi\)
0.306290 + 0.951938i \(0.400912\pi\)
\(810\) 0 0
\(811\) 35.9555i 1.26257i 0.775552 + 0.631284i \(0.217472\pi\)
−0.775552 + 0.631284i \(0.782528\pi\)
\(812\) 0 0
\(813\) 12.7132i 0.445871i
\(814\) 0 0
\(815\) 20.6902 0.724745
\(816\) 0 0
\(817\) 1.42362i 0.0498061i
\(818\) 0 0
\(819\) −4.14539 2.38571i −0.144852 0.0833634i
\(820\) 0 0
\(821\) 15.1183i 0.527631i −0.964573 0.263816i \(-0.915019\pi\)
0.964573 0.263816i \(-0.0849810\pi\)
\(822\) 0 0
\(823\) 43.2335 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(824\) 0 0
\(825\) 3.78325i 0.131716i
\(826\) 0 0
\(827\) 50.4513i 1.75436i −0.480158 0.877182i \(-0.659421\pi\)
0.480158 0.877182i \(-0.340579\pi\)
\(828\) 0 0
\(829\) 1.79727 0.0624219 0.0312110 0.999513i \(-0.490064\pi\)
0.0312110 + 0.999513i \(0.490064\pi\)
\(830\) 0 0
\(831\) −36.0297 −1.24985
\(832\) 0 0
\(833\) 6.95635 0.241023
\(834\) 0 0
\(835\) 43.9700 1.52165
\(836\) 0 0
\(837\) 21.2056i 0.732971i
\(838\) 0 0
\(839\) 4.72104i 0.162988i 0.996674 + 0.0814942i \(0.0259692\pi\)
−0.996674 + 0.0814942i \(0.974031\pi\)
\(840\) 0 0
\(841\) −2.97220 −0.102490
\(842\) 0 0
\(843\) 5.22957i 0.180116i
\(844\) 0 0
\(845\) 31.4555 18.2770i 1.08210 0.628748i
\(846\) 0 0
\(847\) 9.93311i 0.341306i
\(848\) 0 0
\(849\) 38.4978 1.32124
\(850\) 0 0
\(851\) 23.7828i 0.815265i
\(852\) 0 0
\(853\) 48.8428i 1.67234i −0.548467 0.836172i \(-0.684788\pi\)
0.548467 0.836172i \(-0.315212\pi\)
\(854\) 0 0
\(855\) 9.42069 0.322181
\(856\) 0 0
\(857\) 14.9964 0.512267 0.256134 0.966641i \(-0.417551\pi\)
0.256134 + 0.966641i \(0.417551\pi\)
\(858\) 0 0
\(859\) −33.0464 −1.12753 −0.563764 0.825936i \(-0.690647\pi\)
−0.563764 + 0.825936i \(0.690647\pi\)
\(860\) 0 0
\(861\) 1.28113 0.0436607
\(862\) 0 0
\(863\) 47.1122i 1.60372i −0.597513 0.801859i \(-0.703844\pi\)
0.597513 0.801859i \(-0.296156\pi\)
\(864\) 0 0
\(865\) 1.16300i 0.0395430i
\(866\) 0 0
\(867\) −40.6080 −1.37912
\(868\) 0 0
\(869\) 7.57367i 0.256919i
\(870\) 0 0
\(871\) 43.0657 + 24.7847i 1.45923 + 0.839797i
\(872\) 0 0
\(873\) 11.4129i 0.386267i
\(874\) 0 0
\(875\) 6.06884 0.205164
\(876\) 0 0
\(877\) 6.74601i 0.227797i 0.993492 + 0.113898i \(0.0363338\pi\)
−0.993492 + 0.113898i \(0.963666\pi\)
\(878\) 0 0
\(879\) 38.5618i 1.30066i
\(880\) 0 0
\(881\) 22.5679 0.760333 0.380166 0.924918i \(-0.375867\pi\)
0.380166 + 0.924918i \(0.375867\pi\)
\(882\) 0 0
\(883\) −19.2628 −0.648245 −0.324122 0.946015i \(-0.605069\pi\)
−0.324122 + 0.946015i \(0.605069\pi\)
\(884\) 0 0
\(885\) −43.1647 −1.45097
\(886\) 0 0
\(887\) −0.852658 −0.0286295 −0.0143147 0.999898i \(-0.504557\pi\)
−0.0143147 + 0.999898i \(0.504557\pi\)
\(888\) 0 0
\(889\) 10.9331i 0.366685i
\(890\) 0 0
\(891\) 3.36802i 0.112833i
\(892\) 0 0
\(893\) −18.2483 −0.610658
\(894\) 0 0
\(895\) 10.9361i 0.365555i
\(896\) 0 0
\(897\) −8.91378 + 15.4885i −0.297623 + 0.517147i
\(898\) 0 0
\(899\) 19.3295i 0.644674i
\(900\) 0 0
\(901\) −8.19667 −0.273071
\(902\) 0 0
\(903\) 0.725697i 0.0241497i
\(904\) 0 0
\(905\) 13.4088i 0.445725i
\(906\) 0 0
\(907\) 46.7594 1.55262 0.776310 0.630351i \(-0.217089\pi\)
0.776310 + 0.630351i \(0.217089\pi\)
\(908\) 0 0
\(909\) −11.1019 −0.368226
\(910\) 0 0
\(911\) −33.7415 −1.11791 −0.558954 0.829199i \(-0.688797\pi\)
−0.558954 + 0.829199i \(0.688797\pi\)
\(912\) 0 0
\(913\) 17.1731 0.568347
\(914\) 0 0
\(915\) 50.1279i 1.65718i
\(916\) 0 0
\(917\) 6.58725i 0.217530i
\(918\) 0 0
\(919\) −23.4340 −0.773018 −0.386509 0.922286i \(-0.626319\pi\)
−0.386509 + 0.922286i \(0.626319\pi\)
\(920\) 0 0
\(921\) 24.0521i 0.792545i
\(922\) 0 0
\(923\) 11.8885 + 6.84191i 0.391314 + 0.225204i
\(924\) 0 0
\(925\) 17.5754i 0.577876i
\(926\) 0 0
\(927\) −0.691710 −0.0227187
\(928\) 0 0
\(929\) 10.9826i 0.360328i −0.983637 0.180164i \(-0.942337\pi\)
0.983637 0.180164i \(-0.0576629\pi\)
\(930\) 0 0
\(931\) 2.53774i 0.0831709i
\(932\) 0 0
\(933\) −20.5841 −0.673893
\(934\) 0 0
\(935\) −20.1076 −0.657589
\(936\) 0 0
\(937\) 47.5028 1.55185 0.775924 0.630826i \(-0.217284\pi\)
0.775924 + 0.630826i \(0.217284\pi\)
\(938\) 0 0
\(939\) 2.32302 0.0758090
\(940\) 0 0
\(941\) 8.08915i 0.263699i 0.991270 + 0.131849i \(0.0420915\pi\)
−0.991270 + 0.131849i \(0.957908\pi\)
\(942\) 0 0
\(943\) 3.79434i 0.123561i
\(944\) 0 0
\(945\) 15.6627 0.509508
\(946\) 0 0
\(947\) 18.7265i 0.608528i −0.952588 0.304264i \(-0.901589\pi\)
0.952588 0.304264i \(-0.0984105\pi\)
\(948\) 0 0
\(949\) 9.97121 + 5.73851i 0.323679 + 0.186280i
\(950\) 0 0
\(951\) 20.7382i 0.672482i
\(952\) 0 0
\(953\) 47.2451 1.53042 0.765209 0.643781i \(-0.222635\pi\)
0.765209 + 0.643781i \(0.222635\pi\)
\(954\) 0 0
\(955\) 24.1480i 0.781412i
\(956\) 0 0
\(957\) 6.81692i 0.220360i
\(958\) 0 0
\(959\) 12.8333 0.414409
\(960\) 0 0
\(961\) 16.6450 0.536936
\(962\) 0 0
\(963\) −12.5161 −0.403326
\(964\) 0 0
\(965\) −68.4320 −2.20290
\(966\) 0 0
\(967\) 39.7889i 1.27952i −0.768573 0.639762i \(-0.779033\pi\)
0.768573 0.639762i \(-0.220967\pi\)
\(968\) 0 0
\(969\) 22.8369i 0.733626i
\(970\) 0 0
\(971\) 10.0133 0.321341 0.160670 0.987008i \(-0.448634\pi\)
0.160670 + 0.987008i \(0.448634\pi\)
\(972\) 0 0
\(973\) 18.5998i 0.596281i
\(974\) 0 0
\(975\) −6.58725 + 11.4460i −0.210961 + 0.366564i
\(976\) 0 0
\(977\) 58.9087i 1.88466i 0.334691 + 0.942328i \(0.391368\pi\)
−0.334691 + 0.942328i \(0.608632\pi\)
\(978\) 0 0
\(979\) 12.2879 0.392722
\(980\) 0 0
\(981\) 18.1549i 0.579640i
\(982\) 0 0
\(983\) 5.15605i 0.164452i 0.996614 + 0.0822262i \(0.0262030\pi\)
−0.996614 + 0.0822262i \(0.973797\pi\)
\(984\) 0 0
\(985\) 19.9846 0.636761
\(986\) 0 0
\(987\) −9.30221 −0.296092
\(988\) 0 0
\(989\) −2.14931 −0.0683442
\(990\) 0 0
\(991\) −62.6905 −1.99143 −0.995715 0.0924728i \(-0.970523\pi\)
−0.995715 + 0.0924728i \(0.970523\pi\)
\(992\) 0 0
\(993\) 8.81982i 0.279889i
\(994\) 0 0
\(995\) 2.80640i 0.0889687i
\(996\) 0 0
\(997\) −14.1377 −0.447745 −0.223872 0.974618i \(-0.571870\pi\)
−0.223872 + 0.974618i \(0.571870\pi\)
\(998\) 0 0
\(999\) 34.7423i 1.09920i
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 364.2.g.a.337.4 yes 8
3.2 odd 2 3276.2.e.f.2521.1 8
4.3 odd 2 1456.2.k.d.337.6 8
7.2 even 3 2548.2.y.f.753.6 16
7.3 odd 6 2548.2.y.e.961.4 16
7.4 even 3 2548.2.y.f.961.5 16
7.5 odd 6 2548.2.y.e.753.3 16
7.6 odd 2 2548.2.g.g.2157.5 8
13.5 odd 4 4732.2.a.p.1.2 4
13.8 odd 4 4732.2.a.o.1.2 4
13.12 even 2 inner 364.2.g.a.337.3 8
39.38 odd 2 3276.2.e.f.2521.8 8
52.51 odd 2 1456.2.k.d.337.5 8
91.12 odd 6 2548.2.y.e.753.4 16
91.25 even 6 2548.2.y.f.961.6 16
91.38 odd 6 2548.2.y.e.961.3 16
91.51 even 6 2548.2.y.f.753.5 16
91.90 odd 2 2548.2.g.g.2157.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
364.2.g.a.337.3 8 13.12 even 2 inner
364.2.g.a.337.4 yes 8 1.1 even 1 trivial
1456.2.k.d.337.5 8 52.51 odd 2
1456.2.k.d.337.6 8 4.3 odd 2
2548.2.g.g.2157.5 8 7.6 odd 2
2548.2.g.g.2157.6 8 91.90 odd 2
2548.2.y.e.753.3 16 7.5 odd 6
2548.2.y.e.753.4 16 91.12 odd 6
2548.2.y.e.961.3 16 91.38 odd 6
2548.2.y.e.961.4 16 7.3 odd 6
2548.2.y.f.753.5 16 91.51 even 6
2548.2.y.f.753.6 16 7.2 even 3
2548.2.y.f.961.5 16 7.4 even 3
2548.2.y.f.961.6 16 91.25 even 6
3276.2.e.f.2521.1 8 3.2 odd 2
3276.2.e.f.2521.8 8 39.38 odd 2
4732.2.a.o.1.2 4 13.8 odd 4
4732.2.a.p.1.2 4 13.5 odd 4