Properties

Label 6348.2.a.t.1.2
Level $6348$
Weight $2$
Character 6348.1
Self dual yes
Analytic conductor $50.689$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6348,2,Mod(1,6348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6348.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6348.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: \(50.6890352031\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 276)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.92165\) of defining polynomial
Character \(\chi\) \(=\) 6348.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.68761 q^{5} +0.168413 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.68761 q^{5} +0.168413 q^{7} +1.00000 q^{9} +5.32400 q^{11} +3.84191 q^{13} -2.68761 q^{15} +6.65709 q^{17} +6.89290 q^{19} +0.168413 q^{21} +2.22326 q^{25} +1.00000 q^{27} -0.569888 q^{29} -6.95948 q^{31} +5.32400 q^{33} -0.452628 q^{35} -8.85711 q^{37} +3.84191 q^{39} +4.86755 q^{41} +6.75736 q^{43} -2.68761 q^{45} -2.26730 q^{47} -6.97164 q^{49} +6.65709 q^{51} -4.87576 q^{53} -14.3088 q^{55} +6.89290 q^{57} +2.28099 q^{59} +13.9609 q^{61} +0.168413 q^{63} -10.3255 q^{65} +1.61122 q^{67} -9.49666 q^{71} +3.83123 q^{73} +2.22326 q^{75} +0.896630 q^{77} -2.62588 q^{79} +1.00000 q^{81} +8.96954 q^{83} -17.8917 q^{85} -0.569888 q^{87} -6.28643 q^{89} +0.647026 q^{91} -6.95948 q^{93} -18.5254 q^{95} +13.2783 q^{97} +5.32400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.68761 −1.20194 −0.600968 0.799273i \(-0.705218\pi\)
−0.600968 + 0.799273i \(0.705218\pi\)
\(6\) 0 0
\(7\) 0.168413 0.0636541 0.0318270 0.999493i \(-0.489867\pi\)
0.0318270 + 0.999493i \(0.489867\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.32400 1.60525 0.802623 0.596487i \(-0.203437\pi\)
0.802623 + 0.596487i \(0.203437\pi\)
\(12\) 0 0
\(13\) 3.84191 1.06555 0.532776 0.846256i \(-0.321149\pi\)
0.532776 + 0.846256i \(0.321149\pi\)
\(14\) 0 0
\(15\) −2.68761 −0.693938
\(16\) 0 0
\(17\) 6.65709 1.61458 0.807291 0.590154i \(-0.200933\pi\)
0.807291 + 0.590154i \(0.200933\pi\)
\(18\) 0 0
\(19\) 6.89290 1.58134 0.790669 0.612243i \(-0.209733\pi\)
0.790669 + 0.612243i \(0.209733\pi\)
\(20\) 0 0
\(21\) 0.168413 0.0367507
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 2.22326 0.444651
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.569888 −0.105826 −0.0529128 0.998599i \(-0.516851\pi\)
−0.0529128 + 0.998599i \(0.516851\pi\)
\(30\) 0 0
\(31\) −6.95948 −1.24996 −0.624980 0.780641i \(-0.714893\pi\)
−0.624980 + 0.780641i \(0.714893\pi\)
\(32\) 0 0
\(33\) 5.32400 0.926789
\(34\) 0 0
\(35\) −0.452628 −0.0765082
\(36\) 0 0
\(37\) −8.85711 −1.45610 −0.728050 0.685524i \(-0.759573\pi\)
−0.728050 + 0.685524i \(0.759573\pi\)
\(38\) 0 0
\(39\) 3.84191 0.615197
\(40\) 0 0
\(41\) 4.86755 0.760183 0.380092 0.924949i \(-0.375892\pi\)
0.380092 + 0.924949i \(0.375892\pi\)
\(42\) 0 0
\(43\) 6.75736 1.03049 0.515244 0.857044i \(-0.327701\pi\)
0.515244 + 0.857044i \(0.327701\pi\)
\(44\) 0 0
\(45\) −2.68761 −0.400645
\(46\) 0 0
\(47\) −2.26730 −0.330720 −0.165360 0.986233i \(-0.552879\pi\)
−0.165360 + 0.986233i \(0.552879\pi\)
\(48\) 0 0
\(49\) −6.97164 −0.995948
\(50\) 0 0
\(51\) 6.65709 0.932179
\(52\) 0 0
\(53\) −4.87576 −0.669737 −0.334868 0.942265i \(-0.608692\pi\)
−0.334868 + 0.942265i \(0.608692\pi\)
\(54\) 0 0
\(55\) −14.3088 −1.92940
\(56\) 0 0
\(57\) 6.89290 0.912986
\(58\) 0 0
\(59\) 2.28099 0.296959 0.148480 0.988915i \(-0.452562\pi\)
0.148480 + 0.988915i \(0.452562\pi\)
\(60\) 0 0
\(61\) 13.9609 1.78751 0.893757 0.448551i \(-0.148060\pi\)
0.893757 + 0.448551i \(0.148060\pi\)
\(62\) 0 0
\(63\) 0.168413 0.0212180
\(64\) 0 0
\(65\) −10.3255 −1.28073
\(66\) 0 0
\(67\) 1.61122 0.196842 0.0984211 0.995145i \(-0.468621\pi\)
0.0984211 + 0.995145i \(0.468621\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.49666 −1.12705 −0.563523 0.826100i \(-0.690554\pi\)
−0.563523 + 0.826100i \(0.690554\pi\)
\(72\) 0 0
\(73\) 3.83123 0.448411 0.224206 0.974542i \(-0.428021\pi\)
0.224206 + 0.974542i \(0.428021\pi\)
\(74\) 0 0
\(75\) 2.22326 0.256719
\(76\) 0 0
\(77\) 0.896630 0.102180
\(78\) 0 0
\(79\) −2.62588 −0.295435 −0.147717 0.989030i \(-0.547193\pi\)
−0.147717 + 0.989030i \(0.547193\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.96954 0.984535 0.492268 0.870444i \(-0.336168\pi\)
0.492268 + 0.870444i \(0.336168\pi\)
\(84\) 0 0
\(85\) −17.8917 −1.94062
\(86\) 0 0
\(87\) −0.569888 −0.0610984
\(88\) 0 0
\(89\) −6.28643 −0.666361 −0.333180 0.942863i \(-0.608122\pi\)
−0.333180 + 0.942863i \(0.608122\pi\)
\(90\) 0 0
\(91\) 0.647026 0.0678268
\(92\) 0 0
\(93\) −6.95948 −0.721665
\(94\) 0 0
\(95\) −18.5254 −1.90067
\(96\) 0 0
\(97\) 13.2783 1.34821 0.674105 0.738635i \(-0.264529\pi\)
0.674105 + 0.738635i \(0.264529\pi\)
\(98\) 0 0
\(99\) 5.32400 0.535082
\(100\) 0 0
\(101\) −7.26663 −0.723056 −0.361528 0.932361i \(-0.617745\pi\)
−0.361528 + 0.932361i \(0.617745\pi\)
\(102\) 0 0
\(103\) 4.28252 0.421969 0.210984 0.977489i \(-0.432333\pi\)
0.210984 + 0.977489i \(0.432333\pi\)
\(104\) 0 0
\(105\) −0.452628 −0.0441720
\(106\) 0 0
\(107\) −9.71161 −0.938857 −0.469428 0.882971i \(-0.655540\pi\)
−0.469428 + 0.882971i \(0.655540\pi\)
\(108\) 0 0
\(109\) 8.85005 0.847681 0.423841 0.905737i \(-0.360682\pi\)
0.423841 + 0.905737i \(0.360682\pi\)
\(110\) 0 0
\(111\) −8.85711 −0.840679
\(112\) 0 0
\(113\) −8.54541 −0.803884 −0.401942 0.915665i \(-0.631665\pi\)
−0.401942 + 0.915665i \(0.631665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.84191 0.355184
\(118\) 0 0
\(119\) 1.12114 0.102775
\(120\) 0 0
\(121\) 17.3450 1.57681
\(122\) 0 0
\(123\) 4.86755 0.438892
\(124\) 0 0
\(125\) 7.46281 0.667494
\(126\) 0 0
\(127\) −15.1659 −1.34575 −0.672877 0.739755i \(-0.734942\pi\)
−0.672877 + 0.739755i \(0.734942\pi\)
\(128\) 0 0
\(129\) 6.75736 0.594952
\(130\) 0 0
\(131\) −1.75962 −0.153739 −0.0768696 0.997041i \(-0.524493\pi\)
−0.0768696 + 0.997041i \(0.524493\pi\)
\(132\) 0 0
\(133\) 1.16085 0.100659
\(134\) 0 0
\(135\) −2.68761 −0.231313
\(136\) 0 0
\(137\) −14.6865 −1.25475 −0.627375 0.778717i \(-0.715871\pi\)
−0.627375 + 0.778717i \(0.715871\pi\)
\(138\) 0 0
\(139\) 22.3528 1.89594 0.947972 0.318355i \(-0.103130\pi\)
0.947972 + 0.318355i \(0.103130\pi\)
\(140\) 0 0
\(141\) −2.26730 −0.190941
\(142\) 0 0
\(143\) 20.4543 1.71047
\(144\) 0 0
\(145\) 1.53164 0.127196
\(146\) 0 0
\(147\) −6.97164 −0.575011
\(148\) 0 0
\(149\) 6.86314 0.562250 0.281125 0.959671i \(-0.409292\pi\)
0.281125 + 0.959671i \(0.409292\pi\)
\(150\) 0 0
\(151\) −1.51838 −0.123564 −0.0617820 0.998090i \(-0.519678\pi\)
−0.0617820 + 0.998090i \(0.519678\pi\)
\(152\) 0 0
\(153\) 6.65709 0.538194
\(154\) 0 0
\(155\) 18.7044 1.50237
\(156\) 0 0
\(157\) 6.59043 0.525974 0.262987 0.964799i \(-0.415292\pi\)
0.262987 + 0.964799i \(0.415292\pi\)
\(158\) 0 0
\(159\) −4.87576 −0.386673
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.45066 0.191950 0.0959751 0.995384i \(-0.469403\pi\)
0.0959751 + 0.995384i \(0.469403\pi\)
\(164\) 0 0
\(165\) −14.3088 −1.11394
\(166\) 0 0
\(167\) 10.3057 0.797479 0.398740 0.917064i \(-0.369448\pi\)
0.398740 + 0.917064i \(0.369448\pi\)
\(168\) 0 0
\(169\) 1.76024 0.135403
\(170\) 0 0
\(171\) 6.89290 0.527113
\(172\) 0 0
\(173\) −3.01452 −0.229190 −0.114595 0.993412i \(-0.536557\pi\)
−0.114595 + 0.993412i \(0.536557\pi\)
\(174\) 0 0
\(175\) 0.374425 0.0283039
\(176\) 0 0
\(177\) 2.28099 0.171449
\(178\) 0 0
\(179\) −13.5122 −1.00995 −0.504973 0.863135i \(-0.668498\pi\)
−0.504973 + 0.863135i \(0.668498\pi\)
\(180\) 0 0
\(181\) −13.8996 −1.03315 −0.516575 0.856242i \(-0.672793\pi\)
−0.516575 + 0.856242i \(0.672793\pi\)
\(182\) 0 0
\(183\) 13.9609 1.03202
\(184\) 0 0
\(185\) 23.8045 1.75014
\(186\) 0 0
\(187\) 35.4423 2.59180
\(188\) 0 0
\(189\) 0.168413 0.0122502
\(190\) 0 0
\(191\) −12.7981 −0.926035 −0.463018 0.886349i \(-0.653233\pi\)
−0.463018 + 0.886349i \(0.653233\pi\)
\(192\) 0 0
\(193\) 16.1381 1.16164 0.580822 0.814031i \(-0.302731\pi\)
0.580822 + 0.814031i \(0.302731\pi\)
\(194\) 0 0
\(195\) −10.3255 −0.739428
\(196\) 0 0
\(197\) 13.1747 0.938661 0.469330 0.883023i \(-0.344495\pi\)
0.469330 + 0.883023i \(0.344495\pi\)
\(198\) 0 0
\(199\) 9.25611 0.656149 0.328074 0.944652i \(-0.393600\pi\)
0.328074 + 0.944652i \(0.393600\pi\)
\(200\) 0 0
\(201\) 1.61122 0.113647
\(202\) 0 0
\(203\) −0.0959765 −0.00673623
\(204\) 0 0
\(205\) −13.0821 −0.913692
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 36.6978 2.53844
\(210\) 0 0
\(211\) −14.7860 −1.01791 −0.508955 0.860793i \(-0.669968\pi\)
−0.508955 + 0.860793i \(0.669968\pi\)
\(212\) 0 0
\(213\) −9.49666 −0.650700
\(214\) 0 0
\(215\) −18.1612 −1.23858
\(216\) 0 0
\(217\) −1.17207 −0.0795650
\(218\) 0 0
\(219\) 3.83123 0.258890
\(220\) 0 0
\(221\) 25.5759 1.72042
\(222\) 0 0
\(223\) 14.9643 1.00209 0.501043 0.865422i \(-0.332950\pi\)
0.501043 + 0.865422i \(0.332950\pi\)
\(224\) 0 0
\(225\) 2.22326 0.148217
\(226\) 0 0
\(227\) −19.3027 −1.28117 −0.640583 0.767889i \(-0.721307\pi\)
−0.640583 + 0.767889i \(0.721307\pi\)
\(228\) 0 0
\(229\) 26.6442 1.76070 0.880349 0.474326i \(-0.157308\pi\)
0.880349 + 0.474326i \(0.157308\pi\)
\(230\) 0 0
\(231\) 0.896630 0.0589939
\(232\) 0 0
\(233\) −14.4481 −0.946527 −0.473263 0.880921i \(-0.656924\pi\)
−0.473263 + 0.880921i \(0.656924\pi\)
\(234\) 0 0
\(235\) 6.09362 0.397504
\(236\) 0 0
\(237\) −2.62588 −0.170569
\(238\) 0 0
\(239\) −24.1116 −1.55965 −0.779823 0.626000i \(-0.784691\pi\)
−0.779823 + 0.626000i \(0.784691\pi\)
\(240\) 0 0
\(241\) 11.2125 0.722259 0.361129 0.932516i \(-0.382391\pi\)
0.361129 + 0.932516i \(0.382391\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 18.7371 1.19707
\(246\) 0 0
\(247\) 26.4819 1.68500
\(248\) 0 0
\(249\) 8.96954 0.568422
\(250\) 0 0
\(251\) 28.6735 1.80986 0.904929 0.425563i \(-0.139924\pi\)
0.904929 + 0.425563i \(0.139924\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −17.8917 −1.12042
\(256\) 0 0
\(257\) 18.6188 1.16141 0.580705 0.814114i \(-0.302777\pi\)
0.580705 + 0.814114i \(0.302777\pi\)
\(258\) 0 0
\(259\) −1.49165 −0.0926867
\(260\) 0 0
\(261\) −0.569888 −0.0352752
\(262\) 0 0
\(263\) 14.3106 0.882427 0.441214 0.897402i \(-0.354548\pi\)
0.441214 + 0.897402i \(0.354548\pi\)
\(264\) 0 0
\(265\) 13.1041 0.804981
\(266\) 0 0
\(267\) −6.28643 −0.384724
\(268\) 0 0
\(269\) −7.01605 −0.427776 −0.213888 0.976858i \(-0.568613\pi\)
−0.213888 + 0.976858i \(0.568613\pi\)
\(270\) 0 0
\(271\) 10.6915 0.649462 0.324731 0.945806i \(-0.394726\pi\)
0.324731 + 0.945806i \(0.394726\pi\)
\(272\) 0 0
\(273\) 0.647026 0.0391598
\(274\) 0 0
\(275\) 11.8366 0.713774
\(276\) 0 0
\(277\) 22.1540 1.33111 0.665553 0.746351i \(-0.268196\pi\)
0.665553 + 0.746351i \(0.268196\pi\)
\(278\) 0 0
\(279\) −6.95948 −0.416653
\(280\) 0 0
\(281\) −7.06273 −0.421327 −0.210664 0.977559i \(-0.567562\pi\)
−0.210664 + 0.977559i \(0.567562\pi\)
\(282\) 0 0
\(283\) −26.9335 −1.60103 −0.800515 0.599313i \(-0.795441\pi\)
−0.800515 + 0.599313i \(0.795441\pi\)
\(284\) 0 0
\(285\) −18.5254 −1.09735
\(286\) 0 0
\(287\) 0.819758 0.0483888
\(288\) 0 0
\(289\) 27.3168 1.60687
\(290\) 0 0
\(291\) 13.2783 0.778390
\(292\) 0 0
\(293\) −28.5308 −1.66679 −0.833393 0.552681i \(-0.813605\pi\)
−0.833393 + 0.552681i \(0.813605\pi\)
\(294\) 0 0
\(295\) −6.13041 −0.356926
\(296\) 0 0
\(297\) 5.32400 0.308930
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.13803 0.0655948
\(302\) 0 0
\(303\) −7.26663 −0.417457
\(304\) 0 0
\(305\) −37.5216 −2.14848
\(306\) 0 0
\(307\) −11.6303 −0.663774 −0.331887 0.943319i \(-0.607685\pi\)
−0.331887 + 0.943319i \(0.607685\pi\)
\(308\) 0 0
\(309\) 4.28252 0.243624
\(310\) 0 0
\(311\) 26.2524 1.48864 0.744319 0.667825i \(-0.232774\pi\)
0.744319 + 0.667825i \(0.232774\pi\)
\(312\) 0 0
\(313\) 25.6165 1.44793 0.723964 0.689837i \(-0.242318\pi\)
0.723964 + 0.689837i \(0.242318\pi\)
\(314\) 0 0
\(315\) −0.452628 −0.0255027
\(316\) 0 0
\(317\) 20.1963 1.13434 0.567169 0.823602i \(-0.308039\pi\)
0.567169 + 0.823602i \(0.308039\pi\)
\(318\) 0 0
\(319\) −3.03408 −0.169876
\(320\) 0 0
\(321\) −9.71161 −0.542049
\(322\) 0 0
\(323\) 45.8866 2.55320
\(324\) 0 0
\(325\) 8.54154 0.473799
\(326\) 0 0
\(327\) 8.85005 0.489409
\(328\) 0 0
\(329\) −0.381843 −0.0210517
\(330\) 0 0
\(331\) −21.5420 −1.18406 −0.592028 0.805917i \(-0.701673\pi\)
−0.592028 + 0.805917i \(0.701673\pi\)
\(332\) 0 0
\(333\) −8.85711 −0.485366
\(334\) 0 0
\(335\) −4.33034 −0.236592
\(336\) 0 0
\(337\) −1.95917 −0.106723 −0.0533615 0.998575i \(-0.516994\pi\)
−0.0533615 + 0.998575i \(0.516994\pi\)
\(338\) 0 0
\(339\) −8.54541 −0.464123
\(340\) 0 0
\(341\) −37.0523 −2.00649
\(342\) 0 0
\(343\) −2.35300 −0.127050
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.75901 0.523891 0.261946 0.965083i \(-0.415636\pi\)
0.261946 + 0.965083i \(0.415636\pi\)
\(348\) 0 0
\(349\) −16.3641 −0.875951 −0.437975 0.898987i \(-0.644304\pi\)
−0.437975 + 0.898987i \(0.644304\pi\)
\(350\) 0 0
\(351\) 3.84191 0.205066
\(352\) 0 0
\(353\) −18.8249 −1.00195 −0.500975 0.865462i \(-0.667025\pi\)
−0.500975 + 0.865462i \(0.667025\pi\)
\(354\) 0 0
\(355\) 25.5233 1.35464
\(356\) 0 0
\(357\) 1.12114 0.0593370
\(358\) 0 0
\(359\) 13.1714 0.695158 0.347579 0.937651i \(-0.387004\pi\)
0.347579 + 0.937651i \(0.387004\pi\)
\(360\) 0 0
\(361\) 28.5120 1.50063
\(362\) 0 0
\(363\) 17.3450 0.910374
\(364\) 0 0
\(365\) −10.2968 −0.538962
\(366\) 0 0
\(367\) −12.2223 −0.638000 −0.319000 0.947755i \(-0.603347\pi\)
−0.319000 + 0.947755i \(0.603347\pi\)
\(368\) 0 0
\(369\) 4.86755 0.253394
\(370\) 0 0
\(371\) −0.821140 −0.0426315
\(372\) 0 0
\(373\) 25.7142 1.33143 0.665715 0.746206i \(-0.268127\pi\)
0.665715 + 0.746206i \(0.268127\pi\)
\(374\) 0 0
\(375\) 7.46281 0.385378
\(376\) 0 0
\(377\) −2.18946 −0.112763
\(378\) 0 0
\(379\) −19.3168 −0.992239 −0.496120 0.868254i \(-0.665242\pi\)
−0.496120 + 0.868254i \(0.665242\pi\)
\(380\) 0 0
\(381\) −15.1659 −0.776971
\(382\) 0 0
\(383\) −26.7155 −1.36510 −0.682548 0.730840i \(-0.739128\pi\)
−0.682548 + 0.730840i \(0.739128\pi\)
\(384\) 0 0
\(385\) −2.40979 −0.122814
\(386\) 0 0
\(387\) 6.75736 0.343496
\(388\) 0 0
\(389\) 5.66024 0.286986 0.143493 0.989651i \(-0.454167\pi\)
0.143493 + 0.989651i \(0.454167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.75962 −0.0887613
\(394\) 0 0
\(395\) 7.05735 0.355094
\(396\) 0 0
\(397\) 7.82688 0.392820 0.196410 0.980522i \(-0.437072\pi\)
0.196410 + 0.980522i \(0.437072\pi\)
\(398\) 0 0
\(399\) 1.16085 0.0581153
\(400\) 0 0
\(401\) 8.86169 0.442532 0.221266 0.975214i \(-0.428981\pi\)
0.221266 + 0.975214i \(0.428981\pi\)
\(402\) 0 0
\(403\) −26.7377 −1.33190
\(404\) 0 0
\(405\) −2.68761 −0.133548
\(406\) 0 0
\(407\) −47.1552 −2.33740
\(408\) 0 0
\(409\) 13.5450 0.669755 0.334878 0.942262i \(-0.391305\pi\)
0.334878 + 0.942262i \(0.391305\pi\)
\(410\) 0 0
\(411\) −14.6865 −0.724430
\(412\) 0 0
\(413\) 0.384147 0.0189027
\(414\) 0 0
\(415\) −24.1066 −1.18335
\(416\) 0 0
\(417\) 22.3528 1.09462
\(418\) 0 0
\(419\) −9.44850 −0.461590 −0.230795 0.973002i \(-0.574133\pi\)
−0.230795 + 0.973002i \(0.574133\pi\)
\(420\) 0 0
\(421\) 31.1359 1.51747 0.758735 0.651399i \(-0.225818\pi\)
0.758735 + 0.651399i \(0.225818\pi\)
\(422\) 0 0
\(423\) −2.26730 −0.110240
\(424\) 0 0
\(425\) 14.8004 0.717925
\(426\) 0 0
\(427\) 2.35120 0.113783
\(428\) 0 0
\(429\) 20.4543 0.987543
\(430\) 0 0
\(431\) 21.5752 1.03924 0.519621 0.854397i \(-0.326073\pi\)
0.519621 + 0.854397i \(0.326073\pi\)
\(432\) 0 0
\(433\) 0.528096 0.0253787 0.0126893 0.999919i \(-0.495961\pi\)
0.0126893 + 0.999919i \(0.495961\pi\)
\(434\) 0 0
\(435\) 1.53164 0.0734364
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 21.6220 1.03196 0.515980 0.856601i \(-0.327428\pi\)
0.515980 + 0.856601i \(0.327428\pi\)
\(440\) 0 0
\(441\) −6.97164 −0.331983
\(442\) 0 0
\(443\) 22.2377 1.05654 0.528272 0.849075i \(-0.322840\pi\)
0.528272 + 0.849075i \(0.322840\pi\)
\(444\) 0 0
\(445\) 16.8955 0.800923
\(446\) 0 0
\(447\) 6.86314 0.324615
\(448\) 0 0
\(449\) −41.7152 −1.96866 −0.984332 0.176325i \(-0.943579\pi\)
−0.984332 + 0.176325i \(0.943579\pi\)
\(450\) 0 0
\(451\) 25.9148 1.22028
\(452\) 0 0
\(453\) −1.51838 −0.0713397
\(454\) 0 0
\(455\) −1.73896 −0.0815235
\(456\) 0 0
\(457\) −19.6407 −0.918751 −0.459376 0.888242i \(-0.651927\pi\)
−0.459376 + 0.888242i \(0.651927\pi\)
\(458\) 0 0
\(459\) 6.65709 0.310726
\(460\) 0 0
\(461\) −14.8017 −0.689383 −0.344692 0.938716i \(-0.612017\pi\)
−0.344692 + 0.938716i \(0.612017\pi\)
\(462\) 0 0
\(463\) 15.2602 0.709200 0.354600 0.935018i \(-0.384617\pi\)
0.354600 + 0.935018i \(0.384617\pi\)
\(464\) 0 0
\(465\) 18.7044 0.867395
\(466\) 0 0
\(467\) −15.4487 −0.714881 −0.357440 0.933936i \(-0.616350\pi\)
−0.357440 + 0.933936i \(0.616350\pi\)
\(468\) 0 0
\(469\) 0.271351 0.0125298
\(470\) 0 0
\(471\) 6.59043 0.303671
\(472\) 0 0
\(473\) 35.9762 1.65419
\(474\) 0 0
\(475\) 15.3247 0.703144
\(476\) 0 0
\(477\) −4.87576 −0.223246
\(478\) 0 0
\(479\) −18.9013 −0.863622 −0.431811 0.901964i \(-0.642125\pi\)
−0.431811 + 0.901964i \(0.642125\pi\)
\(480\) 0 0
\(481\) −34.0282 −1.55155
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −35.6870 −1.62046
\(486\) 0 0
\(487\) 10.3167 0.467494 0.233747 0.972297i \(-0.424901\pi\)
0.233747 + 0.972297i \(0.424901\pi\)
\(488\) 0 0
\(489\) 2.45066 0.110822
\(490\) 0 0
\(491\) 6.96153 0.314170 0.157085 0.987585i \(-0.449790\pi\)
0.157085 + 0.987585i \(0.449790\pi\)
\(492\) 0 0
\(493\) −3.79380 −0.170864
\(494\) 0 0
\(495\) −14.3088 −0.643135
\(496\) 0 0
\(497\) −1.59936 −0.0717411
\(498\) 0 0
\(499\) −1.02074 −0.0456945 −0.0228472 0.999739i \(-0.507273\pi\)
−0.0228472 + 0.999739i \(0.507273\pi\)
\(500\) 0 0
\(501\) 10.3057 0.460425
\(502\) 0 0
\(503\) 14.4196 0.642937 0.321468 0.946920i \(-0.395824\pi\)
0.321468 + 0.946920i \(0.395824\pi\)
\(504\) 0 0
\(505\) 19.5299 0.869068
\(506\) 0 0
\(507\) 1.76024 0.0781750
\(508\) 0 0
\(509\) −25.0238 −1.10916 −0.554580 0.832131i \(-0.687121\pi\)
−0.554580 + 0.832131i \(0.687121\pi\)
\(510\) 0 0
\(511\) 0.645228 0.0285432
\(512\) 0 0
\(513\) 6.89290 0.304329
\(514\) 0 0
\(515\) −11.5097 −0.507180
\(516\) 0 0
\(517\) −12.0711 −0.530887
\(518\) 0 0
\(519\) −3.01452 −0.132323
\(520\) 0 0
\(521\) −34.4792 −1.51056 −0.755280 0.655402i \(-0.772499\pi\)
−0.755280 + 0.655402i \(0.772499\pi\)
\(522\) 0 0
\(523\) 12.5663 0.549485 0.274742 0.961518i \(-0.411407\pi\)
0.274742 + 0.961518i \(0.411407\pi\)
\(524\) 0 0
\(525\) 0.374425 0.0163412
\(526\) 0 0
\(527\) −46.3299 −2.01816
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 2.28099 0.0989864
\(532\) 0 0
\(533\) 18.7007 0.810016
\(534\) 0 0
\(535\) 26.1010 1.12845
\(536\) 0 0
\(537\) −13.5122 −0.583093
\(538\) 0 0
\(539\) −37.1170 −1.59874
\(540\) 0 0
\(541\) −13.1406 −0.564960 −0.282480 0.959273i \(-0.591157\pi\)
−0.282480 + 0.959273i \(0.591157\pi\)
\(542\) 0 0
\(543\) −13.8996 −0.596489
\(544\) 0 0
\(545\) −23.7855 −1.01886
\(546\) 0 0
\(547\) −35.0838 −1.50008 −0.750038 0.661395i \(-0.769965\pi\)
−0.750038 + 0.661395i \(0.769965\pi\)
\(548\) 0 0
\(549\) 13.9609 0.595838
\(550\) 0 0
\(551\) −3.92818 −0.167346
\(552\) 0 0
\(553\) −0.442233 −0.0188056
\(554\) 0 0
\(555\) 23.8045 1.01044
\(556\) 0 0
\(557\) 0.643547 0.0272680 0.0136340 0.999907i \(-0.495660\pi\)
0.0136340 + 0.999907i \(0.495660\pi\)
\(558\) 0 0
\(559\) 25.9611 1.09804
\(560\) 0 0
\(561\) 35.4423 1.49638
\(562\) 0 0
\(563\) 11.6418 0.490644 0.245322 0.969442i \(-0.421106\pi\)
0.245322 + 0.969442i \(0.421106\pi\)
\(564\) 0 0
\(565\) 22.9667 0.966218
\(566\) 0 0
\(567\) 0.168413 0.00707268
\(568\) 0 0
\(569\) 35.5814 1.49165 0.745825 0.666141i \(-0.232055\pi\)
0.745825 + 0.666141i \(0.232055\pi\)
\(570\) 0 0
\(571\) −4.13581 −0.173078 −0.0865392 0.996248i \(-0.527581\pi\)
−0.0865392 + 0.996248i \(0.527581\pi\)
\(572\) 0 0
\(573\) −12.7981 −0.534647
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.33356 −0.138778 −0.0693890 0.997590i \(-0.522105\pi\)
−0.0693890 + 0.997590i \(0.522105\pi\)
\(578\) 0 0
\(579\) 16.1381 0.670676
\(580\) 0 0
\(581\) 1.51059 0.0626697
\(582\) 0 0
\(583\) −25.9585 −1.07509
\(584\) 0 0
\(585\) −10.3255 −0.426909
\(586\) 0 0
\(587\) 13.8017 0.569656 0.284828 0.958579i \(-0.408063\pi\)
0.284828 + 0.958579i \(0.408063\pi\)
\(588\) 0 0
\(589\) −47.9710 −1.97661
\(590\) 0 0
\(591\) 13.1747 0.541936
\(592\) 0 0
\(593\) −36.1689 −1.48528 −0.742639 0.669692i \(-0.766426\pi\)
−0.742639 + 0.669692i \(0.766426\pi\)
\(594\) 0 0
\(595\) −3.01319 −0.123529
\(596\) 0 0
\(597\) 9.25611 0.378828
\(598\) 0 0
\(599\) 18.8219 0.769041 0.384520 0.923117i \(-0.374367\pi\)
0.384520 + 0.923117i \(0.374367\pi\)
\(600\) 0 0
\(601\) −44.4453 −1.81296 −0.906481 0.422247i \(-0.861241\pi\)
−0.906481 + 0.422247i \(0.861241\pi\)
\(602\) 0 0
\(603\) 1.61122 0.0656140
\(604\) 0 0
\(605\) −46.6165 −1.89523
\(606\) 0 0
\(607\) −24.5329 −0.995760 −0.497880 0.867246i \(-0.665888\pi\)
−0.497880 + 0.867246i \(0.665888\pi\)
\(608\) 0 0
\(609\) −0.0959765 −0.00388917
\(610\) 0 0
\(611\) −8.71075 −0.352399
\(612\) 0 0
\(613\) 2.27696 0.0919657 0.0459829 0.998942i \(-0.485358\pi\)
0.0459829 + 0.998942i \(0.485358\pi\)
\(614\) 0 0
\(615\) −13.0821 −0.527520
\(616\) 0 0
\(617\) 12.3066 0.495445 0.247722 0.968831i \(-0.420318\pi\)
0.247722 + 0.968831i \(0.420318\pi\)
\(618\) 0 0
\(619\) −8.92236 −0.358620 −0.179310 0.983793i \(-0.557387\pi\)
−0.179310 + 0.983793i \(0.557387\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.05872 −0.0424166
\(624\) 0 0
\(625\) −31.1734 −1.24694
\(626\) 0 0
\(627\) 36.6978 1.46557
\(628\) 0 0
\(629\) −58.9625 −2.35099
\(630\) 0 0
\(631\) 33.0833 1.31702 0.658512 0.752570i \(-0.271186\pi\)
0.658512 + 0.752570i \(0.271186\pi\)
\(632\) 0 0
\(633\) −14.7860 −0.587691
\(634\) 0 0
\(635\) 40.7600 1.61751
\(636\) 0 0
\(637\) −26.7844 −1.06124
\(638\) 0 0
\(639\) −9.49666 −0.375682
\(640\) 0 0
\(641\) −45.5559 −1.79935 −0.899675 0.436560i \(-0.856197\pi\)
−0.899675 + 0.436560i \(0.856197\pi\)
\(642\) 0 0
\(643\) 6.72806 0.265329 0.132664 0.991161i \(-0.457647\pi\)
0.132664 + 0.991161i \(0.457647\pi\)
\(644\) 0 0
\(645\) −18.1612 −0.715095
\(646\) 0 0
\(647\) −4.78380 −0.188070 −0.0940352 0.995569i \(-0.529977\pi\)
−0.0940352 + 0.995569i \(0.529977\pi\)
\(648\) 0 0
\(649\) 12.1440 0.476692
\(650\) 0 0
\(651\) −1.17207 −0.0459369
\(652\) 0 0
\(653\) 31.3121 1.22534 0.612668 0.790340i \(-0.290096\pi\)
0.612668 + 0.790340i \(0.290096\pi\)
\(654\) 0 0
\(655\) 4.72919 0.184785
\(656\) 0 0
\(657\) 3.83123 0.149470
\(658\) 0 0
\(659\) 7.80466 0.304026 0.152013 0.988378i \(-0.451424\pi\)
0.152013 + 0.988378i \(0.451424\pi\)
\(660\) 0 0
\(661\) −46.7053 −1.81663 −0.908313 0.418291i \(-0.862629\pi\)
−0.908313 + 0.418291i \(0.862629\pi\)
\(662\) 0 0
\(663\) 25.5759 0.993286
\(664\) 0 0
\(665\) −3.11992 −0.120985
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.9643 0.578555
\(670\) 0 0
\(671\) 74.3280 2.86940
\(672\) 0 0
\(673\) −43.8070 −1.68864 −0.844318 0.535842i \(-0.819994\pi\)
−0.844318 + 0.535842i \(0.819994\pi\)
\(674\) 0 0
\(675\) 2.22326 0.0855731
\(676\) 0 0
\(677\) −26.0495 −1.00116 −0.500582 0.865689i \(-0.666881\pi\)
−0.500582 + 0.865689i \(0.666881\pi\)
\(678\) 0 0
\(679\) 2.23624 0.0858191
\(680\) 0 0
\(681\) −19.3027 −0.739682
\(682\) 0 0
\(683\) 12.0695 0.461826 0.230913 0.972974i \(-0.425829\pi\)
0.230913 + 0.972974i \(0.425829\pi\)
\(684\) 0 0
\(685\) 39.4715 1.50813
\(686\) 0 0
\(687\) 26.6442 1.01654
\(688\) 0 0
\(689\) −18.7322 −0.713640
\(690\) 0 0
\(691\) −41.2368 −1.56872 −0.784360 0.620305i \(-0.787009\pi\)
−0.784360 + 0.620305i \(0.787009\pi\)
\(692\) 0 0
\(693\) 0.896630 0.0340602
\(694\) 0 0
\(695\) −60.0758 −2.27880
\(696\) 0 0
\(697\) 32.4037 1.22738
\(698\) 0 0
\(699\) −14.4481 −0.546477
\(700\) 0 0
\(701\) 15.9467 0.602299 0.301149 0.953577i \(-0.402630\pi\)
0.301149 + 0.953577i \(0.402630\pi\)
\(702\) 0 0
\(703\) −61.0511 −2.30259
\(704\) 0 0
\(705\) 6.09362 0.229499
\(706\) 0 0
\(707\) −1.22379 −0.0460255
\(708\) 0 0
\(709\) 5.43912 0.204270 0.102135 0.994771i \(-0.467433\pi\)
0.102135 + 0.994771i \(0.467433\pi\)
\(710\) 0 0
\(711\) −2.62588 −0.0984783
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −54.9732 −2.05588
\(716\) 0 0
\(717\) −24.1116 −0.900463
\(718\) 0 0
\(719\) −5.60236 −0.208933 −0.104466 0.994528i \(-0.533313\pi\)
−0.104466 + 0.994528i \(0.533313\pi\)
\(720\) 0 0
\(721\) 0.721231 0.0268600
\(722\) 0 0
\(723\) 11.2125 0.416996
\(724\) 0 0
\(725\) −1.26701 −0.0470555
\(726\) 0 0
\(727\) −39.2481 −1.45563 −0.727815 0.685773i \(-0.759464\pi\)
−0.727815 + 0.685773i \(0.759464\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 44.9844 1.66381
\(732\) 0 0
\(733\) 30.2116 1.11589 0.557946 0.829877i \(-0.311590\pi\)
0.557946 + 0.829877i \(0.311590\pi\)
\(734\) 0 0
\(735\) 18.7371 0.691127
\(736\) 0 0
\(737\) 8.57814 0.315980
\(738\) 0 0
\(739\) −13.6068 −0.500534 −0.250267 0.968177i \(-0.580518\pi\)
−0.250267 + 0.968177i \(0.580518\pi\)
\(740\) 0 0
\(741\) 26.4819 0.972835
\(742\) 0 0
\(743\) 3.28957 0.120683 0.0603413 0.998178i \(-0.480781\pi\)
0.0603413 + 0.998178i \(0.480781\pi\)
\(744\) 0 0
\(745\) −18.4454 −0.675789
\(746\) 0 0
\(747\) 8.96954 0.328178
\(748\) 0 0
\(749\) −1.63556 −0.0597621
\(750\) 0 0
\(751\) 20.4896 0.747675 0.373837 0.927494i \(-0.378042\pi\)
0.373837 + 0.927494i \(0.378042\pi\)
\(752\) 0 0
\(753\) 28.6735 1.04492
\(754\) 0 0
\(755\) 4.08082 0.148516
\(756\) 0 0
\(757\) 47.7612 1.73591 0.867955 0.496643i \(-0.165434\pi\)
0.867955 + 0.496643i \(0.165434\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 14.1657 0.513505 0.256753 0.966477i \(-0.417347\pi\)
0.256753 + 0.966477i \(0.417347\pi\)
\(762\) 0 0
\(763\) 1.49046 0.0539584
\(764\) 0 0
\(765\) −17.8917 −0.646875
\(766\) 0 0
\(767\) 8.76333 0.316426
\(768\) 0 0
\(769\) 44.9704 1.62167 0.810836 0.585273i \(-0.199013\pi\)
0.810836 + 0.585273i \(0.199013\pi\)
\(770\) 0 0
\(771\) 18.6188 0.670540
\(772\) 0 0
\(773\) −31.2256 −1.12311 −0.561553 0.827441i \(-0.689796\pi\)
−0.561553 + 0.827441i \(0.689796\pi\)
\(774\) 0 0
\(775\) −15.4727 −0.555796
\(776\) 0 0
\(777\) −1.49165 −0.0535127
\(778\) 0 0
\(779\) 33.5515 1.20211
\(780\) 0 0
\(781\) −50.5602 −1.80919
\(782\) 0 0
\(783\) −0.569888 −0.0203661
\(784\) 0 0
\(785\) −17.7125 −0.632187
\(786\) 0 0
\(787\) 6.98462 0.248975 0.124487 0.992221i \(-0.460271\pi\)
0.124487 + 0.992221i \(0.460271\pi\)
\(788\) 0 0
\(789\) 14.3106 0.509470
\(790\) 0 0
\(791\) −1.43916 −0.0511705
\(792\) 0 0
\(793\) 53.6366 1.90469
\(794\) 0 0
\(795\) 13.1041 0.464756
\(796\) 0 0
\(797\) −3.28738 −0.116445 −0.0582225 0.998304i \(-0.518543\pi\)
−0.0582225 + 0.998304i \(0.518543\pi\)
\(798\) 0 0
\(799\) −15.0936 −0.533974
\(800\) 0 0
\(801\) −6.28643 −0.222120
\(802\) 0 0
\(803\) 20.3974 0.719810
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −7.01605 −0.246977
\(808\) 0 0
\(809\) −36.3906 −1.27942 −0.639712 0.768614i \(-0.720946\pi\)
−0.639712 + 0.768614i \(0.720946\pi\)
\(810\) 0 0
\(811\) −50.9938 −1.79063 −0.895317 0.445430i \(-0.853051\pi\)
−0.895317 + 0.445430i \(0.853051\pi\)
\(812\) 0 0
\(813\) 10.6915 0.374967
\(814\) 0 0
\(815\) −6.58641 −0.230712
\(816\) 0 0
\(817\) 46.5778 1.62955
\(818\) 0 0
\(819\) 0.647026 0.0226089
\(820\) 0 0
\(821\) 16.5463 0.577469 0.288734 0.957409i \(-0.406766\pi\)
0.288734 + 0.957409i \(0.406766\pi\)
\(822\) 0 0
\(823\) 8.22811 0.286814 0.143407 0.989664i \(-0.454194\pi\)
0.143407 + 0.989664i \(0.454194\pi\)
\(824\) 0 0
\(825\) 11.8366 0.412098
\(826\) 0 0
\(827\) −22.7771 −0.792036 −0.396018 0.918243i \(-0.629608\pi\)
−0.396018 + 0.918243i \(0.629608\pi\)
\(828\) 0 0
\(829\) −5.13824 −0.178459 −0.0892293 0.996011i \(-0.528440\pi\)
−0.0892293 + 0.996011i \(0.528440\pi\)
\(830\) 0 0
\(831\) 22.1540 0.768514
\(832\) 0 0
\(833\) −46.4108 −1.60804
\(834\) 0 0
\(835\) −27.6977 −0.958519
\(836\) 0 0
\(837\) −6.95948 −0.240555
\(838\) 0 0
\(839\) 10.2732 0.354671 0.177336 0.984150i \(-0.443252\pi\)
0.177336 + 0.984150i \(0.443252\pi\)
\(840\) 0 0
\(841\) −28.6752 −0.988801
\(842\) 0 0
\(843\) −7.06273 −0.243253
\(844\) 0 0
\(845\) −4.73084 −0.162746
\(846\) 0 0
\(847\) 2.92111 0.100371
\(848\) 0 0
\(849\) −26.9335 −0.924355
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −31.2764 −1.07089 −0.535443 0.844572i \(-0.679855\pi\)
−0.535443 + 0.844572i \(0.679855\pi\)
\(854\) 0 0
\(855\) −18.5254 −0.633556
\(856\) 0 0
\(857\) 18.7082 0.639060 0.319530 0.947576i \(-0.396475\pi\)
0.319530 + 0.947576i \(0.396475\pi\)
\(858\) 0 0
\(859\) 41.3968 1.41244 0.706221 0.707991i \(-0.250398\pi\)
0.706221 + 0.707991i \(0.250398\pi\)
\(860\) 0 0
\(861\) 0.819758 0.0279373
\(862\) 0 0
\(863\) −7.83474 −0.266698 −0.133349 0.991069i \(-0.542573\pi\)
−0.133349 + 0.991069i \(0.542573\pi\)
\(864\) 0 0
\(865\) 8.10187 0.275472
\(866\) 0 0
\(867\) 27.3168 0.927729
\(868\) 0 0
\(869\) −13.9802 −0.474246
\(870\) 0 0
\(871\) 6.19016 0.209746
\(872\) 0 0
\(873\) 13.2783 0.449404
\(874\) 0 0
\(875\) 1.25683 0.0424887
\(876\) 0 0
\(877\) 38.0372 1.28443 0.642213 0.766526i \(-0.278016\pi\)
0.642213 + 0.766526i \(0.278016\pi\)
\(878\) 0 0
\(879\) −28.5308 −0.962320
\(880\) 0 0
\(881\) −12.6784 −0.427147 −0.213573 0.976927i \(-0.568510\pi\)
−0.213573 + 0.976927i \(0.568510\pi\)
\(882\) 0 0
\(883\) 3.53058 0.118813 0.0594067 0.998234i \(-0.481079\pi\)
0.0594067 + 0.998234i \(0.481079\pi\)
\(884\) 0 0
\(885\) −6.13041 −0.206071
\(886\) 0 0
\(887\) 40.2404 1.35114 0.675569 0.737296i \(-0.263898\pi\)
0.675569 + 0.737296i \(0.263898\pi\)
\(888\) 0 0
\(889\) −2.55413 −0.0856627
\(890\) 0 0
\(891\) 5.32400 0.178361
\(892\) 0 0
\(893\) −15.6283 −0.522980
\(894\) 0 0
\(895\) 36.3154 1.21389
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.96613 0.132278
\(900\) 0 0
\(901\) −32.4584 −1.08134
\(902\) 0 0
\(903\) 1.13803 0.0378712
\(904\) 0 0
\(905\) 37.3568 1.24178
\(906\) 0 0
\(907\) 25.8219 0.857403 0.428701 0.903446i \(-0.358971\pi\)
0.428701 + 0.903446i \(0.358971\pi\)
\(908\) 0 0
\(909\) −7.26663 −0.241019
\(910\) 0 0
\(911\) 14.8460 0.491870 0.245935 0.969286i \(-0.420905\pi\)
0.245935 + 0.969286i \(0.420905\pi\)
\(912\) 0 0
\(913\) 47.7538 1.58042
\(914\) 0 0
\(915\) −37.5216 −1.24043
\(916\) 0 0
\(917\) −0.296343 −0.00978612
\(918\) 0 0
\(919\) 25.3485 0.836171 0.418085 0.908408i \(-0.362701\pi\)
0.418085 + 0.908408i \(0.362701\pi\)
\(920\) 0 0
\(921\) −11.6303 −0.383230
\(922\) 0 0
\(923\) −36.4853 −1.20093
\(924\) 0 0
\(925\) −19.6916 −0.647456
\(926\) 0 0
\(927\) 4.28252 0.140656
\(928\) 0 0
\(929\) 45.2024 1.48304 0.741521 0.670929i \(-0.234105\pi\)
0.741521 + 0.670929i \(0.234105\pi\)
\(930\) 0 0
\(931\) −48.0548 −1.57493
\(932\) 0 0
\(933\) 26.2524 0.859465
\(934\) 0 0
\(935\) −95.2552 −3.11518
\(936\) 0 0
\(937\) −52.7763 −1.72413 −0.862064 0.506800i \(-0.830828\pi\)
−0.862064 + 0.506800i \(0.830828\pi\)
\(938\) 0 0
\(939\) 25.6165 0.835962
\(940\) 0 0
\(941\) 35.3014 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −0.452628 −0.0147240
\(946\) 0 0
\(947\) 59.7208 1.94066 0.970332 0.241775i \(-0.0777296\pi\)
0.970332 + 0.241775i \(0.0777296\pi\)
\(948\) 0 0
\(949\) 14.7192 0.477806
\(950\) 0 0
\(951\) 20.1963 0.654910
\(952\) 0 0
\(953\) −54.1571 −1.75432 −0.877161 0.480197i \(-0.840565\pi\)
−0.877161 + 0.480197i \(0.840565\pi\)
\(954\) 0 0
\(955\) 34.3962 1.11304
\(956\) 0 0
\(957\) −3.03408 −0.0980780
\(958\) 0 0
\(959\) −2.47339 −0.0798700
\(960\) 0 0
\(961\) 17.4344 0.562399
\(962\) 0 0
\(963\) −9.71161 −0.312952
\(964\) 0 0
\(965\) −43.3729 −1.39622
\(966\) 0 0
\(967\) −13.5723 −0.436457 −0.218229 0.975898i \(-0.570028\pi\)
−0.218229 + 0.975898i \(0.570028\pi\)
\(968\) 0 0
\(969\) 45.8866 1.47409
\(970\) 0 0
\(971\) 3.68048 0.118112 0.0590561 0.998255i \(-0.481191\pi\)
0.0590561 + 0.998255i \(0.481191\pi\)
\(972\) 0 0
\(973\) 3.76451 0.120685
\(974\) 0 0
\(975\) 8.54154 0.273548
\(976\) 0 0
\(977\) −22.0511 −0.705477 −0.352738 0.935722i \(-0.614749\pi\)
−0.352738 + 0.935722i \(0.614749\pi\)
\(978\) 0 0
\(979\) −33.4690 −1.06967
\(980\) 0 0
\(981\) 8.85005 0.282560
\(982\) 0 0
\(983\) 27.2107 0.867887 0.433944 0.900940i \(-0.357122\pi\)
0.433944 + 0.900940i \(0.357122\pi\)
\(984\) 0 0
\(985\) −35.4086 −1.12821
\(986\) 0 0
\(987\) −0.381843 −0.0121542
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 14.5200 0.461244 0.230622 0.973043i \(-0.425924\pi\)
0.230622 + 0.973043i \(0.425924\pi\)
\(992\) 0 0
\(993\) −21.5420 −0.683616
\(994\) 0 0
\(995\) −24.8768 −0.788649
\(996\) 0 0
\(997\) −5.08045 −0.160899 −0.0804497 0.996759i \(-0.525636\pi\)
−0.0804497 + 0.996759i \(0.525636\pi\)
\(998\) 0 0
\(999\) −8.85711 −0.280226
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 6348.2.a.t.1.2 10
23.17 odd 22 276.2.i.a.13.2 20
23.19 odd 22 276.2.i.a.85.2 yes 20
23.22 odd 2 6348.2.a.s.1.9 10
69.17 even 22 828.2.q.c.289.1 20
69.65 even 22 828.2.q.c.361.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
276.2.i.a.13.2 20 23.17 odd 22
276.2.i.a.85.2 yes 20 23.19 odd 22
828.2.q.c.289.1 20 69.17 even 22
828.2.q.c.361.1 20 69.65 even 22
6348.2.a.s.1.9 10 23.22 odd 2
6348.2.a.t.1.2 10 1.1 even 1 trivial