Properties

Label 3420.2.a.h.1.2
Level $3420$
Weight $2$
Character 3420.1
Self dual yes
Analytic conductor $27.309$
Analytic rank $0$
Dimension $2$
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] = [3420,2,Mod(1,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, 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("3420.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3420.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.00000 q^{7} +3.46410 q^{11} -2.73205 q^{13} +3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{23} +1.00000 q^{25} -3.46410 q^{29} -1.46410 q^{31} -2.00000 q^{35} +6.73205 q^{37} +6.00000 q^{41} -4.92820 q^{43} -12.9282 q^{47} -3.00000 q^{49} +10.7321 q^{53} -3.46410 q^{55} -6.92820 q^{59} +12.3923 q^{61} +2.73205 q^{65} +6.73205 q^{67} +2.53590 q^{71} -0.535898 q^{73} +6.92820 q^{77} +2.92820 q^{79} -3.46410 q^{83} -3.46410 q^{85} +15.4641 q^{89} -5.46410 q^{91} -1.00000 q^{95} -16.5885 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{7} - 2 q^{13} + 2 q^{19} + 2 q^{25} + 4 q^{31} - 4 q^{35} + 10 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} - 6 q^{49} + 18 q^{53} + 4 q^{61} + 2 q^{65} + 10 q^{67} + 12 q^{71} - 8 q^{73} - 8 q^{79} + 24 q^{89} - 4 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −2.73205 −0.757735 −0.378867 0.925451i \(-0.623686\pi\)
−0.378867 + 0.925451i \(0.623686\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.73205 1.10674 0.553371 0.832935i \(-0.313341\pi\)
0.553371 + 0.832935i \(0.313341\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.92820 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.9282 −1.88577 −0.942886 0.333115i \(-0.891900\pi\)
−0.942886 + 0.333115i \(0.891900\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.7321 1.47416 0.737080 0.675805i \(-0.236204\pi\)
0.737080 + 0.675805i \(0.236204\pi\)
\(54\) 0 0
\(55\) −3.46410 −0.467099
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.73205 0.338869
\(66\) 0 0
\(67\) 6.73205 0.822451 0.411225 0.911534i \(-0.365101\pi\)
0.411225 + 0.911534i \(0.365101\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −0.535898 −0.0627222 −0.0313611 0.999508i \(-0.509984\pi\)
−0.0313611 + 0.999508i \(0.509984\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 2.92820 0.329449 0.164724 0.986340i \(-0.447327\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.46410 −0.380235 −0.190117 0.981761i \(-0.560887\pi\)
−0.190117 + 0.981761i \(0.560887\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.4641 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(90\) 0 0
\(91\) −5.46410 −0.572793
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −16.5885 −1.68430 −0.842151 0.539241i \(-0.818711\pi\)
−0.842151 + 0.539241i \(0.818711\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.3923 1.63110 0.815548 0.578690i \(-0.196436\pi\)
0.815548 + 0.578690i \(0.196436\pi\)
\(102\) 0 0
\(103\) 13.6603 1.34598 0.672992 0.739649i \(-0.265009\pi\)
0.672992 + 0.739649i \(0.265009\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.66025 0.547197 0.273599 0.961844i \(-0.411786\pi\)
0.273599 + 0.961844i \(0.411786\pi\)
\(108\) 0 0
\(109\) −14.3923 −1.37853 −0.689266 0.724508i \(-0.742067\pi\)
−0.689266 + 0.724508i \(0.742067\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.5885 −1.18422 −0.592111 0.805856i \(-0.701705\pi\)
−0.592111 + 0.805856i \(0.701705\pi\)
\(114\) 0 0
\(115\) −3.46410 −0.323029
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.5885 1.82693 0.913465 0.406917i \(-0.133396\pi\)
0.913465 + 0.406917i \(0.133396\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.9282 1.65376 0.826882 0.562375i \(-0.190112\pi\)
0.826882 + 0.562375i \(0.190112\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −17.3205 −1.47979 −0.739895 0.672722i \(-0.765125\pi\)
−0.739895 + 0.672722i \(0.765125\pi\)
\(138\) 0 0
\(139\) 11.4641 0.972372 0.486186 0.873855i \(-0.338388\pi\)
0.486186 + 0.873855i \(0.338388\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.46410 −0.791428
\(144\) 0 0
\(145\) 3.46410 0.287678
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.39230 0.359832 0.179916 0.983682i \(-0.442417\pi\)
0.179916 + 0.983682i \(0.442417\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) 23.4641 1.87264 0.936320 0.351149i \(-0.114209\pi\)
0.936320 + 0.351149i \(0.114209\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 7.07180 0.553906 0.276953 0.960883i \(-0.410675\pi\)
0.276953 + 0.960883i \(0.410675\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.7321 0.830471 0.415236 0.909714i \(-0.363699\pi\)
0.415236 + 0.909714i \(0.363699\pi\)
\(168\) 0 0
\(169\) −5.53590 −0.425838
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.80385 −0.289201 −0.144601 0.989490i \(-0.546190\pi\)
−0.144601 + 0.989490i \(0.546190\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.73205 −0.494950
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.92820 −0.501307 −0.250654 0.968077i \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(192\) 0 0
\(193\) −7.80385 −0.561733 −0.280867 0.959747i \(-0.590622\pi\)
−0.280867 + 0.959747i \(0.590622\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.928203 −0.0661317 −0.0330659 0.999453i \(-0.510527\pi\)
−0.0330659 + 0.999453i \(0.510527\pi\)
\(198\) 0 0
\(199\) 26.9282 1.90889 0.954445 0.298387i \(-0.0964487\pi\)
0.954445 + 0.298387i \(0.0964487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 19.3205 1.33008 0.665039 0.746808i \(-0.268415\pi\)
0.665039 + 0.746808i \(0.268415\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.92820 0.336101
\(216\) 0 0
\(217\) −2.92820 −0.198779
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9.46410 −0.636624
\(222\) 0 0
\(223\) −9.66025 −0.646898 −0.323449 0.946246i \(-0.604842\pi\)
−0.323449 + 0.946246i \(0.604842\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.19615 −0.543998 −0.271999 0.962298i \(-0.587685\pi\)
−0.271999 + 0.962298i \(0.587685\pi\)
\(228\) 0 0
\(229\) 17.4641 1.15406 0.577030 0.816723i \(-0.304212\pi\)
0.577030 + 0.816723i \(0.304212\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.928203 0.0608086 0.0304043 0.999538i \(-0.490321\pi\)
0.0304043 + 0.999538i \(0.490321\pi\)
\(234\) 0 0
\(235\) 12.9282 0.843343
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) −11.8564 −0.763738 −0.381869 0.924216i \(-0.624719\pi\)
−0.381869 + 0.924216i \(0.624719\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −2.73205 −0.173836
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.1244 0.943431 0.471716 0.881751i \(-0.343635\pi\)
0.471716 + 0.881751i \(0.343635\pi\)
\(258\) 0 0
\(259\) 13.4641 0.836619
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.53590 −0.526346 −0.263173 0.964749i \(-0.584769\pi\)
−0.263173 + 0.964749i \(0.584769\pi\)
\(264\) 0 0
\(265\) −10.7321 −0.659265
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.3923 −0.633630 −0.316815 0.948487i \(-0.602613\pi\)
−0.316815 + 0.948487i \(0.602613\pi\)
\(270\) 0 0
\(271\) −2.39230 −0.145322 −0.0726611 0.997357i \(-0.523149\pi\)
−0.0726611 + 0.997357i \(0.523149\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410 0.208893
\(276\) 0 0
\(277\) 18.3923 1.10509 0.552543 0.833484i \(-0.313657\pi\)
0.552543 + 0.833484i \(0.313657\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.928203 −0.0553720 −0.0276860 0.999617i \(-0.508814\pi\)
−0.0276860 + 0.999617i \(0.508814\pi\)
\(282\) 0 0
\(283\) 18.3923 1.09331 0.546655 0.837358i \(-0.315901\pi\)
0.546655 + 0.837358i \(0.315901\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.66025 0.330676 0.165338 0.986237i \(-0.447129\pi\)
0.165338 + 0.986237i \(0.447129\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.46410 −0.547323
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.3923 −0.709581
\(306\) 0 0
\(307\) −7.80385 −0.445389 −0.222695 0.974888i \(-0.571485\pi\)
−0.222695 + 0.974888i \(0.571485\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.60770 0.0911640 0.0455820 0.998961i \(-0.485486\pi\)
0.0455820 + 0.998961i \(0.485486\pi\)
\(312\) 0 0
\(313\) 10.7846 0.609582 0.304791 0.952419i \(-0.401413\pi\)
0.304791 + 0.952419i \(0.401413\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.12436 −0.175481 −0.0877406 0.996143i \(-0.527965\pi\)
−0.0877406 + 0.996143i \(0.527965\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.46410 0.192748
\(324\) 0 0
\(325\) −2.73205 −0.151547
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −25.8564 −1.42551
\(330\) 0 0
\(331\) −22.2487 −1.22290 −0.611450 0.791283i \(-0.709413\pi\)
−0.611450 + 0.791283i \(0.709413\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.73205 −0.367811
\(336\) 0 0
\(337\) −17.2679 −0.940645 −0.470323 0.882495i \(-0.655862\pi\)
−0.470323 + 0.882495i \(0.655862\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.07180 −0.274653
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −31.1769 −1.67366 −0.836832 0.547459i \(-0.815595\pi\)
−0.836832 + 0.547459i \(0.815595\pi\)
\(348\) 0 0
\(349\) 20.9282 1.12026 0.560131 0.828404i \(-0.310751\pi\)
0.560131 + 0.828404i \(0.310751\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.60770 −0.0855690 −0.0427845 0.999084i \(-0.513623\pi\)
−0.0427845 + 0.999084i \(0.513623\pi\)
\(354\) 0 0
\(355\) −2.53590 −0.134592
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.53590 0.450507 0.225254 0.974300i \(-0.427679\pi\)
0.225254 + 0.974300i \(0.427679\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.535898 0.0280502
\(366\) 0 0
\(367\) 3.85641 0.201303 0.100651 0.994922i \(-0.467907\pi\)
0.100651 + 0.994922i \(0.467907\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 21.4641 1.11436
\(372\) 0 0
\(373\) −16.5885 −0.858918 −0.429459 0.903086i \(-0.641296\pi\)
−0.429459 + 0.903086i \(0.641296\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.46410 0.487426
\(378\) 0 0
\(379\) 14.9282 0.766810 0.383405 0.923580i \(-0.374751\pi\)
0.383405 + 0.923580i \(0.374751\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.33975 −0.323946 −0.161973 0.986795i \(-0.551786\pi\)
−0.161973 + 0.986795i \(0.551786\pi\)
\(384\) 0 0
\(385\) −6.92820 −0.353094
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.92820 −0.147334
\(396\) 0 0
\(397\) −19.4641 −0.976875 −0.488438 0.872599i \(-0.662433\pi\)
−0.488438 + 0.872599i \(0.662433\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38.7846 −1.93681 −0.968405 0.249381i \(-0.919773\pi\)
−0.968405 + 0.249381i \(0.919773\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.3205 1.15595
\(408\) 0 0
\(409\) −14.3923 −0.711654 −0.355827 0.934552i \(-0.615801\pi\)
−0.355827 + 0.934552i \(0.615801\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 3.46410 0.170046
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) 24.7846 1.19941
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.4641 1.61191 0.805955 0.591977i \(-0.201653\pi\)
0.805955 + 0.591977i \(0.201653\pi\)
\(432\) 0 0
\(433\) −22.3397 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410 0.165710
\(438\) 0 0
\(439\) −31.7128 −1.51357 −0.756785 0.653664i \(-0.773231\pi\)
−0.756785 + 0.653664i \(0.773231\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13.6077 −0.646521 −0.323261 0.946310i \(-0.604779\pi\)
−0.323261 + 0.946310i \(0.604779\pi\)
\(444\) 0 0
\(445\) −15.4641 −0.733069
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.46410 0.163481 0.0817405 0.996654i \(-0.473952\pi\)
0.0817405 + 0.996654i \(0.473952\pi\)
\(450\) 0 0
\(451\) 20.7846 0.978709
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.46410 0.256161
\(456\) 0 0
\(457\) 34.7846 1.62716 0.813578 0.581456i \(-0.197517\pi\)
0.813578 + 0.581456i \(0.197517\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.7128 1.01127 0.505633 0.862749i \(-0.331259\pi\)
0.505633 + 0.862749i \(0.331259\pi\)
\(462\) 0 0
\(463\) 4.53590 0.210801 0.105401 0.994430i \(-0.466388\pi\)
0.105401 + 0.994430i \(0.466388\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.3205 1.35679 0.678396 0.734697i \(-0.262676\pi\)
0.678396 + 0.734697i \(0.262676\pi\)
\(468\) 0 0
\(469\) 13.4641 0.621714
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17.0718 −0.784962
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.3923 1.02313 0.511565 0.859244i \(-0.329066\pi\)
0.511565 + 0.859244i \(0.329066\pi\)
\(480\) 0 0
\(481\) −18.3923 −0.838617
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.5885 0.753243
\(486\) 0 0
\(487\) −12.1962 −0.552660 −0.276330 0.961063i \(-0.589118\pi\)
−0.276330 + 0.961063i \(0.589118\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.9282 −0.854218 −0.427109 0.904200i \(-0.640468\pi\)
−0.427109 + 0.904200i \(0.640468\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.07180 0.227501
\(498\) 0 0
\(499\) 23.4641 1.05040 0.525199 0.850980i \(-0.323991\pi\)
0.525199 + 0.850980i \(0.323991\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.4641 −0.689510 −0.344755 0.938693i \(-0.612038\pi\)
−0.344755 + 0.938693i \(0.612038\pi\)
\(504\) 0 0
\(505\) −16.3923 −0.729448
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.32051 0.235827 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(510\) 0 0
\(511\) −1.07180 −0.0474135
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.6603 −0.601943
\(516\) 0 0
\(517\) −44.7846 −1.96962
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.7846 0.647726 0.323863 0.946104i \(-0.395018\pi\)
0.323863 + 0.946104i \(0.395018\pi\)
\(522\) 0 0
\(523\) −37.3731 −1.63421 −0.817105 0.576489i \(-0.804422\pi\)
−0.817105 + 0.576489i \(0.804422\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.07180 −0.220931
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.3923 −0.710030
\(534\) 0 0
\(535\) −5.66025 −0.244714
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) −29.1769 −1.25441 −0.627207 0.778853i \(-0.715802\pi\)
−0.627207 + 0.778853i \(0.715802\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.3923 0.616499
\(546\) 0 0
\(547\) 6.73205 0.287842 0.143921 0.989589i \(-0.454029\pi\)
0.143921 + 0.989589i \(0.454029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 5.85641 0.249040
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.32051 −0.225437 −0.112719 0.993627i \(-0.535956\pi\)
−0.112719 + 0.993627i \(0.535956\pi\)
\(558\) 0 0
\(559\) 13.4641 0.569471
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −44.1962 −1.86265 −0.931323 0.364195i \(-0.881344\pi\)
−0.931323 + 0.364195i \(0.881344\pi\)
\(564\) 0 0
\(565\) 12.5885 0.529600
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 0 0
\(571\) 18.3923 0.769694 0.384847 0.922980i \(-0.374254\pi\)
0.384847 + 0.922980i \(0.374254\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.46410 0.144463
\(576\) 0 0
\(577\) −6.78461 −0.282447 −0.141223 0.989978i \(-0.545104\pi\)
−0.141223 + 0.989978i \(0.545104\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.92820 −0.287430
\(582\) 0 0
\(583\) 37.1769 1.53971
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −43.1769 −1.78210 −0.891051 0.453903i \(-0.850031\pi\)
−0.891051 + 0.453903i \(0.850031\pi\)
\(588\) 0 0
\(589\) −1.46410 −0.0603273
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −31.8564 −1.30819 −0.654093 0.756414i \(-0.726949\pi\)
−0.654093 + 0.756414i \(0.726949\pi\)
\(594\) 0 0
\(595\) −6.92820 −0.284029
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) 24.6410 1.00513 0.502564 0.864540i \(-0.332390\pi\)
0.502564 + 0.864540i \(0.332390\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −39.9090 −1.61985 −0.809927 0.586530i \(-0.800494\pi\)
−0.809927 + 0.586530i \(0.800494\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.3205 1.42891
\(612\) 0 0
\(613\) 6.39230 0.258183 0.129091 0.991633i \(-0.458794\pi\)
0.129091 + 0.991633i \(0.458794\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.60770 −0.0647234 −0.0323617 0.999476i \(-0.510303\pi\)
−0.0323617 + 0.999476i \(0.510303\pi\)
\(618\) 0 0
\(619\) −2.39230 −0.0961549 −0.0480774 0.998844i \(-0.515309\pi\)
−0.0480774 + 0.998844i \(0.515309\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.9282 1.23911
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.3205 0.929850
\(630\) 0 0
\(631\) 11.4641 0.456379 0.228189 0.973617i \(-0.426719\pi\)
0.228189 + 0.973617i \(0.426719\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20.5885 −0.817028
\(636\) 0 0
\(637\) 8.19615 0.324743
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.9282 0.984605 0.492302 0.870424i \(-0.336155\pi\)
0.492302 + 0.870424i \(0.336155\pi\)
\(642\) 0 0
\(643\) −14.3923 −0.567577 −0.283789 0.958887i \(-0.591591\pi\)
−0.283789 + 0.958887i \(0.591591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.4641 −0.607957 −0.303978 0.952679i \(-0.598315\pi\)
−0.303978 + 0.952679i \(0.598315\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.4641 −1.07475 −0.537377 0.843342i \(-0.680585\pi\)
−0.537377 + 0.843342i \(0.680585\pi\)
\(654\) 0 0
\(655\) −18.9282 −0.739586
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.78461 −0.342200 −0.171100 0.985254i \(-0.554732\pi\)
−0.171100 + 0.985254i \(0.554732\pi\)
\(660\) 0 0
\(661\) 5.21539 0.202855 0.101428 0.994843i \(-0.467659\pi\)
0.101428 + 0.994843i \(0.467659\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 42.9282 1.65722
\(672\) 0 0
\(673\) −50.7321 −1.95558 −0.977788 0.209594i \(-0.932786\pi\)
−0.977788 + 0.209594i \(0.932786\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.9808 −0.652624 −0.326312 0.945262i \(-0.605806\pi\)
−0.326312 + 0.945262i \(0.605806\pi\)
\(678\) 0 0
\(679\) −33.1769 −1.27321
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.6603 0.675751 0.337875 0.941191i \(-0.390292\pi\)
0.337875 + 0.941191i \(0.390292\pi\)
\(684\) 0 0
\(685\) 17.3205 0.661783
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −29.3205 −1.11702
\(690\) 0 0
\(691\) 51.1769 1.94686 0.973431 0.228981i \(-0.0735395\pi\)
0.973431 + 0.228981i \(0.0735395\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.4641 −0.434858
\(696\) 0 0
\(697\) 20.7846 0.787273
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.5359 1.00225 0.501124 0.865376i \(-0.332920\pi\)
0.501124 + 0.865376i \(0.332920\pi\)
\(702\) 0 0
\(703\) 6.73205 0.253904
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32.7846 1.23299
\(708\) 0 0
\(709\) −30.7846 −1.15614 −0.578070 0.815987i \(-0.696194\pi\)
−0.578070 + 0.815987i \(0.696194\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −5.07180 −0.189940
\(714\) 0 0
\(715\) 9.46410 0.353937
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.3205 −0.645946 −0.322973 0.946408i \(-0.604682\pi\)
−0.322973 + 0.946408i \(0.604682\pi\)
\(720\) 0 0
\(721\) 27.3205 1.01747
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) 0.143594 0.00532559 0.00266279 0.999996i \(-0.499152\pi\)
0.00266279 + 0.999996i \(0.499152\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.0718 −0.631423
\(732\) 0 0
\(733\) 10.7846 0.398339 0.199169 0.979965i \(-0.436176\pi\)
0.199169 + 0.979965i \(0.436176\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.3205 0.859022
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.8756 −0.765853 −0.382927 0.923779i \(-0.625084\pi\)
−0.382927 + 0.923779i \(0.625084\pi\)
\(744\) 0 0
\(745\) −4.39230 −0.160922
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 11.3205 0.413642
\(750\) 0 0
\(751\) −25.4641 −0.929198 −0.464599 0.885521i \(-0.653802\pi\)
−0.464599 + 0.885521i \(0.653802\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.39230 0.305427
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19.6077 −0.710778 −0.355389 0.934718i \(-0.615652\pi\)
−0.355389 + 0.934718i \(0.615652\pi\)
\(762\) 0 0
\(763\) −28.7846 −1.04207
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.9282 0.683458
\(768\) 0 0
\(769\) −30.5359 −1.10115 −0.550576 0.834785i \(-0.685592\pi\)
−0.550576 + 0.834785i \(0.685592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22.0526 −0.793175 −0.396588 0.917997i \(-0.629806\pi\)
−0.396588 + 0.917997i \(0.629806\pi\)
\(774\) 0 0
\(775\) −1.46410 −0.0525921
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.78461 0.314338
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −23.4641 −0.837470
\(786\) 0 0
\(787\) −12.1962 −0.434746 −0.217373 0.976089i \(-0.569749\pi\)
−0.217373 + 0.976089i \(0.569749\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −25.1769 −0.895188
\(792\) 0 0
\(793\) −33.8564 −1.20228
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.66025 0.200496 0.100248 0.994962i \(-0.468036\pi\)
0.100248 + 0.994962i \(0.468036\pi\)
\(798\) 0 0
\(799\) −44.7846 −1.58437
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.85641 −0.0655112
\(804\) 0 0
\(805\) −6.92820 −0.244187
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.71281 −0.341484 −0.170742 0.985316i \(-0.554617\pi\)
−0.170742 + 0.985316i \(0.554617\pi\)
\(810\) 0 0
\(811\) 15.6077 0.548060 0.274030 0.961721i \(-0.411643\pi\)
0.274030 + 0.961721i \(0.411643\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.07180 −0.247714
\(816\) 0 0
\(817\) −4.92820 −0.172416
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16.1436 0.563415 0.281708 0.959500i \(-0.409099\pi\)
0.281708 + 0.959500i \(0.409099\pi\)
\(822\) 0 0
\(823\) −39.5692 −1.37930 −0.689648 0.724145i \(-0.742235\pi\)
−0.689648 + 0.724145i \(0.742235\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.6603 1.03139 0.515694 0.856773i \(-0.327534\pi\)
0.515694 + 0.856773i \(0.327534\pi\)
\(828\) 0 0
\(829\) 8.24871 0.286490 0.143245 0.989687i \(-0.454246\pi\)
0.143245 + 0.989687i \(0.454246\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10.3923 −0.360072
\(834\) 0 0
\(835\) −10.7321 −0.371398
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18.9282 −0.653474 −0.326737 0.945115i \(-0.605949\pi\)
−0.326737 + 0.945115i \(0.605949\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.53590 0.190441
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 23.3205 0.799417
\(852\) 0 0
\(853\) −44.6410 −1.52848 −0.764240 0.644932i \(-0.776886\pi\)
−0.764240 + 0.644932i \(0.776886\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.80385 −0.129937 −0.0649685 0.997887i \(-0.520695\pi\)
−0.0649685 + 0.997887i \(0.520695\pi\)
\(858\) 0 0
\(859\) 47.7128 1.62794 0.813970 0.580907i \(-0.197302\pi\)
0.813970 + 0.580907i \(0.197302\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37.2679 1.26862 0.634308 0.773081i \(-0.281285\pi\)
0.634308 + 0.773081i \(0.281285\pi\)
\(864\) 0 0
\(865\) 3.80385 0.129335
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.1436 0.344098
\(870\) 0 0
\(871\) −18.3923 −0.623199
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 0.980762 0.0331180 0.0165590 0.999863i \(-0.494729\pi\)
0.0165590 + 0.999863i \(0.494729\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.679492 0.0228927 0.0114463 0.999934i \(-0.496356\pi\)
0.0114463 + 0.999934i \(0.496356\pi\)
\(882\) 0 0
\(883\) −22.0000 −0.740359 −0.370179 0.928960i \(-0.620704\pi\)
−0.370179 + 0.928960i \(0.620704\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.4115 −0.786083 −0.393041 0.919521i \(-0.628577\pi\)
−0.393041 + 0.919521i \(0.628577\pi\)
\(888\) 0 0
\(889\) 41.1769 1.38103
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.9282 −0.432626
\(894\) 0 0
\(895\) 20.7846 0.694753
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.07180 0.169154
\(900\) 0 0
\(901\) 37.1769 1.23854
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) 7.41154 0.246096 0.123048 0.992401i \(-0.460733\pi\)
0.123048 + 0.992401i \(0.460733\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.5359 −0.481596 −0.240798 0.970575i \(-0.577409\pi\)
−0.240798 + 0.970575i \(0.577409\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 37.8564 1.25013
\(918\) 0 0
\(919\) 44.4974 1.46783 0.733917 0.679239i \(-0.237690\pi\)
0.733917 + 0.679239i \(0.237690\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.92820 −0.228045
\(924\) 0 0
\(925\) 6.73205 0.221348
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 47.5692 1.56070 0.780348 0.625346i \(-0.215042\pi\)
0.780348 + 0.625346i \(0.215042\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 51.1769 1.67188 0.835938 0.548823i \(-0.184924\pi\)
0.835938 + 0.548823i \(0.184924\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −52.6410 −1.71605 −0.858024 0.513610i \(-0.828308\pi\)
−0.858024 + 0.513610i \(0.828308\pi\)
\(942\) 0 0
\(943\) 20.7846 0.676840
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.67949 −0.217054 −0.108527 0.994093i \(-0.534613\pi\)
−0.108527 + 0.994093i \(0.534613\pi\)
\(948\) 0 0
\(949\) 1.46410 0.0475267
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −60.5885 −1.96265 −0.981326 0.192350i \(-0.938389\pi\)
−0.981326 + 0.192350i \(0.938389\pi\)
\(954\) 0 0
\(955\) 6.92820 0.224191
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −34.6410 −1.11862
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.80385 0.251215
\(966\) 0 0
\(967\) −45.3205 −1.45741 −0.728705 0.684828i \(-0.759877\pi\)
−0.728705 + 0.684828i \(0.759877\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 14.5359 0.466479 0.233240 0.972419i \(-0.425067\pi\)
0.233240 + 0.972419i \(0.425067\pi\)
\(972\) 0 0
\(973\) 22.9282 0.735044
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32.1962 −1.03005 −0.515023 0.857176i \(-0.672217\pi\)
−0.515023 + 0.857176i \(0.672217\pi\)
\(978\) 0 0
\(979\) 53.5692 1.71208
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.44486 0.0779790 0.0389895 0.999240i \(-0.487586\pi\)
0.0389895 + 0.999240i \(0.487586\pi\)
\(984\) 0 0
\(985\) 0.928203 0.0295750
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.0718 −0.542852
\(990\) 0 0
\(991\) −8.39230 −0.266590 −0.133295 0.991076i \(-0.542556\pi\)
−0.133295 + 0.991076i \(0.542556\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −26.9282 −0.853681
\(996\) 0 0
\(997\) 30.3923 0.962534 0.481267 0.876574i \(-0.340177\pi\)
0.481267 + 0.876574i \(0.340177\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3420.2.a.h.1.2 2
3.2 odd 2 380.2.a.d.1.2 2
12.11 even 2 1520.2.a.l.1.1 2
15.2 even 4 1900.2.c.e.1749.1 4
15.8 even 4 1900.2.c.e.1749.4 4
15.14 odd 2 1900.2.a.d.1.1 2
24.5 odd 2 6080.2.a.z.1.1 2
24.11 even 2 6080.2.a.bj.1.2 2
57.56 even 2 7220.2.a.h.1.1 2
60.59 even 2 7600.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.2 2 3.2 odd 2
1520.2.a.l.1.1 2 12.11 even 2
1900.2.a.d.1.1 2 15.14 odd 2
1900.2.c.e.1749.1 4 15.2 even 4
1900.2.c.e.1749.4 4 15.8 even 4
3420.2.a.h.1.2 2 1.1 even 1 trivial
6080.2.a.z.1.1 2 24.5 odd 2
6080.2.a.bj.1.2 2 24.11 even 2
7220.2.a.h.1.1 2 57.56 even 2
7600.2.a.bf.1.2 2 60.59 even 2