Properties

Label 784.2.bb.b.111.10
Level $784$
Weight $2$
Character 784.111
Analytic conductor $6.260$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(111,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bb (of order \(14\), degree \(6\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 111.10
Character \(\chi\) \(=\) 784.111
Dual form 784.2.bb.b.671.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0314318 - 0.0394142i) q^{3} +(-0.245577 + 0.195841i) q^{5} +(-1.27396 + 2.31884i) q^{7} +(0.666997 - 2.92231i) q^{9} +O(q^{10})\) \(q+(-0.0314318 - 0.0394142i) q^{3} +(-0.245577 + 0.195841i) q^{5} +(-1.27396 + 2.31884i) q^{7} +(0.666997 - 2.92231i) q^{9} +(5.46385 - 1.24709i) q^{11} +(-3.56389 + 0.813435i) q^{13} +(0.0154379 + 0.00352359i) q^{15} +(2.74227 - 5.69439i) q^{17} +1.60672 q^{19} +(0.131438 - 0.0226730i) q^{21} +(2.80789 + 5.83064i) q^{23} +(-1.09065 + 4.77845i) q^{25} +(-0.272406 + 0.131184i) q^{27} +(4.84679 + 2.33409i) q^{29} +6.16469 q^{31} +(-0.220891 - 0.176155i) q^{33} +(-0.141268 - 0.818949i) q^{35} +(0.0632092 + 0.0304399i) q^{37} +(0.144080 + 0.114900i) q^{39} +(5.05633 - 4.03229i) q^{41} +(7.52952 + 6.00459i) q^{43} +(0.408509 + 0.848277i) q^{45} +(-1.42595 - 6.24751i) q^{47} +(-3.75403 - 5.90824i) q^{49} +(-0.310634 + 0.0709002i) q^{51} +(2.92158 - 1.40696i) q^{53} +(-1.09757 + 1.37630i) q^{55} +(-0.0505020 - 0.0633275i) q^{57} +(-6.07214 + 7.61423i) q^{59} +(5.86499 - 12.1788i) q^{61} +(5.92663 + 5.26957i) q^{63} +(0.715906 - 0.897718i) q^{65} +8.32399i q^{67} +(0.141553 - 0.293938i) q^{69} +(-1.95339 - 4.05626i) q^{71} +(-12.2207 - 2.78929i) q^{73} +(0.222620 - 0.107208i) q^{75} +(-4.06895 + 14.2585i) q^{77} -13.3795i q^{79} +(-8.08812 - 3.89503i) q^{81} +(-0.630329 + 2.76165i) q^{83} +(0.441756 + 1.93546i) q^{85} +(-0.0603469 - 0.264397i) q^{87} +(15.9297 + 3.63584i) q^{89} +(2.65404 - 9.30038i) q^{91} +(-0.193767 - 0.242976i) q^{93} +(-0.394573 + 0.314662i) q^{95} -10.0921i q^{97} -16.7988i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 120 q - 24 q^{9} - 14 q^{17} + 16 q^{21} + 40 q^{25} + 32 q^{29} - 62 q^{37} - 28 q^{41} - 60 q^{49} + 14 q^{53} - 34 q^{57} - 112 q^{61} - 32 q^{65} + 112 q^{69} + 42 q^{73} + 66 q^{77} - 44 q^{81} - 12 q^{85} + 28 q^{89} - 58 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{14}\right)\)

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
\(3\) −0.0314318 0.0394142i −0.0181471 0.0227558i 0.772675 0.634802i \(-0.218918\pi\)
−0.790822 + 0.612046i \(0.790347\pi\)
\(4\) 0 0
\(5\) −0.245577 + 0.195841i −0.109825 + 0.0875829i −0.676863 0.736109i \(-0.736661\pi\)
0.567037 + 0.823692i \(0.308090\pi\)
\(6\) 0 0
\(7\) −1.27396 + 2.31884i −0.481513 + 0.876439i
\(8\) 0 0
\(9\) 0.666997 2.92231i 0.222332 0.974102i
\(10\) 0 0
\(11\) 5.46385 1.24709i 1.64741 0.376011i 0.704670 0.709535i \(-0.251095\pi\)
0.942742 + 0.333524i \(0.108238\pi\)
\(12\) 0 0
\(13\) −3.56389 + 0.813435i −0.988446 + 0.225606i −0.686033 0.727571i \(-0.740649\pi\)
−0.302413 + 0.953177i \(0.597792\pi\)
\(14\) 0 0
\(15\) 0.0154379 + 0.00352359i 0.00398604 + 0.000909787i
\(16\) 0 0
\(17\) 2.74227 5.69439i 0.665099 1.38109i −0.246149 0.969232i \(-0.579165\pi\)
0.911247 0.411859i \(-0.135121\pi\)
\(18\) 0 0
\(19\) 1.60672 0.368606 0.184303 0.982869i \(-0.440997\pi\)
0.184303 + 0.982869i \(0.440997\pi\)
\(20\) 0 0
\(21\) 0.131438 0.0226730i 0.0286822 0.00494765i
\(22\) 0 0
\(23\) 2.80789 + 5.83064i 0.585485 + 1.21577i 0.957738 + 0.287641i \(0.0928709\pi\)
−0.372254 + 0.928131i \(0.621415\pi\)
\(24\) 0 0
\(25\) −1.09065 + 4.77845i −0.218130 + 0.955690i
\(26\) 0 0
\(27\) −0.272406 + 0.131184i −0.0524246 + 0.0252463i
\(28\) 0 0
\(29\) 4.84679 + 2.33409i 0.900027 + 0.433430i 0.825899 0.563819i \(-0.190668\pi\)
0.0741280 + 0.997249i \(0.476383\pi\)
\(30\) 0 0
\(31\) 6.16469 1.10721 0.553606 0.832779i \(-0.313252\pi\)
0.553606 + 0.832779i \(0.313252\pi\)
\(32\) 0 0
\(33\) −0.220891 0.176155i −0.0384523 0.0306646i
\(34\) 0 0
\(35\) −0.141268 0.818949i −0.0238786 0.138428i
\(36\) 0 0
\(37\) 0.0632092 + 0.0304399i 0.0103915 + 0.00500430i 0.439072 0.898452i \(-0.355307\pi\)
−0.428681 + 0.903456i \(0.641021\pi\)
\(38\) 0 0
\(39\) 0.144080 + 0.114900i 0.0230713 + 0.0183988i
\(40\) 0 0
\(41\) 5.05633 4.03229i 0.789666 0.629737i −0.143310 0.989678i \(-0.545774\pi\)
0.932975 + 0.359941i \(0.117203\pi\)
\(42\) 0 0
\(43\) 7.52952 + 6.00459i 1.14824 + 0.915692i 0.997342 0.0728576i \(-0.0232119\pi\)
0.150899 + 0.988549i \(0.451783\pi\)
\(44\) 0 0
\(45\) 0.408509 + 0.848277i 0.0608969 + 0.126454i
\(46\) 0 0
\(47\) −1.42595 6.24751i −0.207997 0.911293i −0.965898 0.258924i \(-0.916632\pi\)
0.757901 0.652369i \(-0.226225\pi\)
\(48\) 0 0
\(49\) −3.75403 5.90824i −0.536290 0.844034i
\(50\) 0 0
\(51\) −0.310634 + 0.0709002i −0.0434975 + 0.00992802i
\(52\) 0 0
\(53\) 2.92158 1.40696i 0.401310 0.193261i −0.222331 0.974971i \(-0.571367\pi\)
0.623642 + 0.781710i \(0.285652\pi\)
\(54\) 0 0
\(55\) −1.09757 + 1.37630i −0.147996 + 0.185581i
\(56\) 0 0
\(57\) −0.0505020 0.0633275i −0.00668915 0.00838793i
\(58\) 0 0
\(59\) −6.07214 + 7.61423i −0.790526 + 0.991288i 0.209384 + 0.977834i \(0.432854\pi\)
−0.999909 + 0.0134545i \(0.995717\pi\)
\(60\) 0 0
\(61\) 5.86499 12.1788i 0.750935 1.55933i −0.0760459 0.997104i \(-0.524230\pi\)
0.826981 0.562229i \(-0.190056\pi\)
\(62\) 0 0
\(63\) 5.92663 + 5.26957i 0.746685 + 0.663904i
\(64\) 0 0
\(65\) 0.715906 0.897718i 0.0887973 0.111348i
\(66\) 0 0
\(67\) 8.32399i 1.01694i 0.861080 + 0.508469i \(0.169788\pi\)
−0.861080 + 0.508469i \(0.830212\pi\)
\(68\) 0 0
\(69\) 0.141553 0.293938i 0.0170410 0.0353860i
\(70\) 0 0
\(71\) −1.95339 4.05626i −0.231825 0.481390i 0.752310 0.658809i \(-0.228939\pi\)
−0.984135 + 0.177419i \(0.943225\pi\)
\(72\) 0 0
\(73\) −12.2207 2.78929i −1.43032 0.326462i −0.563929 0.825823i \(-0.690711\pi\)
−0.866394 + 0.499361i \(0.833568\pi\)
\(74\) 0 0
\(75\) 0.222620 0.107208i 0.0257059 0.0123793i
\(76\) 0 0
\(77\) −4.06895 + 14.2585i −0.463700 + 1.62491i
\(78\) 0 0
\(79\) 13.3795i 1.50532i −0.658412 0.752658i \(-0.728772\pi\)
0.658412 0.752658i \(-0.271228\pi\)
\(80\) 0 0
\(81\) −8.08812 3.89503i −0.898680 0.432781i
\(82\) 0 0
\(83\) −0.630329 + 2.76165i −0.0691877 + 0.303131i −0.997668 0.0682500i \(-0.978258\pi\)
0.928481 + 0.371381i \(0.121116\pi\)
\(84\) 0 0
\(85\) 0.441756 + 1.93546i 0.0479152 + 0.209930i
\(86\) 0 0
\(87\) −0.0603469 0.264397i −0.00646987 0.0283463i
\(88\) 0 0
\(89\) 15.9297 + 3.63584i 1.68854 + 0.385399i 0.955548 0.294837i \(-0.0952652\pi\)
0.732994 + 0.680235i \(0.238122\pi\)
\(90\) 0 0
\(91\) 2.65404 9.30038i 0.278219 0.974945i
\(92\) 0 0
\(93\) −0.193767 0.242976i −0.0200927 0.0251955i
\(94\) 0 0
\(95\) −0.394573 + 0.314662i −0.0404824 + 0.0322836i
\(96\) 0 0
\(97\) 10.0921i 1.02469i −0.858779 0.512347i \(-0.828776\pi\)
0.858779 0.512347i \(-0.171224\pi\)
\(98\) 0 0
\(99\) 16.7988i 1.68835i
\(100\) 0 0
\(101\) −12.7383 + 10.1585i −1.26751 + 1.01081i −0.268645 + 0.963239i \(0.586576\pi\)
−0.998868 + 0.0475682i \(0.984853\pi\)
\(102\) 0 0
\(103\) 1.34044 + 1.68086i 0.132077 + 0.165620i 0.843472 0.537173i \(-0.180508\pi\)
−0.711395 + 0.702792i \(0.751936\pi\)
\(104\) 0 0
\(105\) −0.0278379 + 0.0313090i −0.00271670 + 0.00305544i
\(106\) 0 0
\(107\) −12.7434 2.90860i −1.23195 0.281185i −0.443496 0.896276i \(-0.646262\pi\)
−0.788456 + 0.615092i \(0.789119\pi\)
\(108\) 0 0
\(109\) 4.30811 + 18.8751i 0.412642 + 1.80790i 0.571499 + 0.820602i \(0.306362\pi\)
−0.158857 + 0.987302i \(0.550781\pi\)
\(110\) 0 0
\(111\) −0.000787011 0.00344812i −7.46998e−5 0.000327281i
\(112\) 0 0
\(113\) −1.46554 + 6.42093i −0.137866 + 0.604030i 0.858036 + 0.513590i \(0.171685\pi\)
−0.995902 + 0.0904405i \(0.971172\pi\)
\(114\) 0 0
\(115\) −1.83143 0.881971i −0.170782 0.0822443i
\(116\) 0 0
\(117\) 10.9573i 1.01301i
\(118\) 0 0
\(119\) 9.71081 + 13.6133i 0.890188 + 1.24793i
\(120\) 0 0
\(121\) 18.3877 8.85506i 1.67161 0.805006i
\(122\) 0 0
\(123\) −0.317859 0.0725492i −0.0286604 0.00654154i
\(124\) 0 0
\(125\) −1.34940 2.80207i −0.120694 0.250625i
\(126\) 0 0
\(127\) 0.108509 0.225321i 0.00962862 0.0199940i −0.896099 0.443854i \(-0.853611\pi\)
0.905728 + 0.423860i \(0.139325\pi\)
\(128\) 0 0
\(129\) 0.485505i 0.0427463i
\(130\) 0 0
\(131\) −7.63982 + 9.58003i −0.667494 + 0.837011i −0.994136 0.108137i \(-0.965511\pi\)
0.326642 + 0.945148i \(0.394083\pi\)
\(132\) 0 0
\(133\) −2.04690 + 3.72572i −0.177489 + 0.323061i
\(134\) 0 0
\(135\) 0.0412055 0.0855641i 0.00354641 0.00736419i
\(136\) 0 0
\(137\) 0.0869664 0.109052i 0.00743004 0.00931698i −0.778103 0.628137i \(-0.783818\pi\)
0.785533 + 0.618820i \(0.212389\pi\)
\(138\) 0 0
\(139\) 3.20016 + 4.01288i 0.271434 + 0.340368i 0.898802 0.438356i \(-0.144439\pi\)
−0.627367 + 0.778724i \(0.715867\pi\)
\(140\) 0 0
\(141\) −0.201420 + 0.252573i −0.0169627 + 0.0212705i
\(142\) 0 0
\(143\) −18.4581 + 8.88897i −1.54355 + 0.743333i
\(144\) 0 0
\(145\) −1.64737 + 0.376002i −0.136807 + 0.0312253i
\(146\) 0 0
\(147\) −0.114873 + 0.333669i −0.00947453 + 0.0275205i
\(148\) 0 0
\(149\) −2.04062 8.94052i −0.167174 0.732436i −0.987118 0.159993i \(-0.948853\pi\)
0.819944 0.572443i \(-0.194004\pi\)
\(150\) 0 0
\(151\) −5.01189 10.4073i −0.407862 0.846934i −0.999180 0.0404952i \(-0.987106\pi\)
0.591318 0.806439i \(-0.298608\pi\)
\(152\) 0 0
\(153\) −14.8116 11.8119i −1.19745 0.954935i
\(154\) 0 0
\(155\) −1.51391 + 1.20730i −0.121600 + 0.0969728i
\(156\) 0 0
\(157\) −1.45851 1.16312i −0.116402 0.0928274i 0.563559 0.826076i \(-0.309432\pi\)
−0.679961 + 0.733248i \(0.738003\pi\)
\(158\) 0 0
\(159\) −0.147285 0.0709286i −0.0116804 0.00562501i
\(160\) 0 0
\(161\) −17.0975 0.916984i −1.34747 0.0722685i
\(162\) 0 0
\(163\) −13.2046 10.5303i −1.03426 0.824799i −0.0495147 0.998773i \(-0.515767\pi\)
−0.984750 + 0.173975i \(0.944339\pi\)
\(164\) 0 0
\(165\) 0.0887443 0.00690874
\(166\) 0 0
\(167\) −19.7931 9.53184i −1.53163 0.737596i −0.537249 0.843423i \(-0.680537\pi\)
−0.994385 + 0.105827i \(0.966251\pi\)
\(168\) 0 0
\(169\) 0.327051 0.157499i 0.0251578 0.0121153i
\(170\) 0 0
\(171\) 1.07168 4.69532i 0.0819532 0.359060i
\(172\) 0 0
\(173\) 6.14205 + 12.7541i 0.466971 + 0.969676i 0.992878 + 0.119134i \(0.0380117\pi\)
−0.525907 + 0.850542i \(0.676274\pi\)
\(174\) 0 0
\(175\) −9.69101 8.61662i −0.732572 0.651355i
\(176\) 0 0
\(177\) 0.490967 0.0369033
\(178\) 0 0
\(179\) −1.65780 + 3.44246i −0.123910 + 0.257301i −0.953691 0.300787i \(-0.902751\pi\)
0.829782 + 0.558088i \(0.188465\pi\)
\(180\) 0 0
\(181\) −9.19256 2.09814i −0.683278 0.155954i −0.133228 0.991085i \(-0.542534\pi\)
−0.550050 + 0.835132i \(0.685391\pi\)
\(182\) 0 0
\(183\) −0.664364 + 0.151637i −0.0491112 + 0.0112093i
\(184\) 0 0
\(185\) −0.0214841 + 0.00490361i −0.00157955 + 0.000360521i
\(186\) 0 0
\(187\) 7.88195 34.5331i 0.576386 2.52531i
\(188\) 0 0
\(189\) 0.0428413 0.798789i 0.00311624 0.0581034i
\(190\) 0 0
\(191\) 1.09167 0.870581i 0.0789907 0.0629930i −0.583198 0.812330i \(-0.698199\pi\)
0.662189 + 0.749337i \(0.269628\pi\)
\(192\) 0 0
\(193\) 8.85609 + 11.1052i 0.637476 + 0.799369i 0.990685 0.136175i \(-0.0434810\pi\)
−0.353209 + 0.935544i \(0.614910\pi\)
\(194\) 0 0
\(195\) −0.0578851 −0.00414524
\(196\) 0 0
\(197\) −1.19948 −0.0854596 −0.0427298 0.999087i \(-0.513605\pi\)
−0.0427298 + 0.999087i \(0.513605\pi\)
\(198\) 0 0
\(199\) 2.51775 + 3.15716i 0.178478 + 0.223805i 0.863021 0.505168i \(-0.168569\pi\)
−0.684543 + 0.728973i \(0.739998\pi\)
\(200\) 0 0
\(201\) 0.328084 0.261638i 0.0231412 0.0184545i
\(202\) 0 0
\(203\) −11.5870 + 8.26538i −0.813249 + 0.580116i
\(204\) 0 0
\(205\) −0.452031 + 1.98048i −0.0315712 + 0.138322i
\(206\) 0 0
\(207\) 18.9118 4.31648i 1.31446 0.300016i
\(208\) 0 0
\(209\) 8.77886 2.00372i 0.607246 0.138600i
\(210\) 0 0
\(211\) 5.43219 + 1.23986i 0.373967 + 0.0853556i 0.405374 0.914151i \(-0.367141\pi\)
−0.0314063 + 0.999507i \(0.509999\pi\)
\(212\) 0 0
\(213\) −0.0984757 + 0.204487i −0.00674744 + 0.0140112i
\(214\) 0 0
\(215\) −3.02503 −0.206305
\(216\) 0 0
\(217\) −7.85360 + 14.2949i −0.533137 + 0.970403i
\(218\) 0 0
\(219\) 0.274180 + 0.569341i 0.0185274 + 0.0384725i
\(220\) 0 0
\(221\) −5.14115 + 22.5248i −0.345831 + 1.51518i
\(222\) 0 0
\(223\) −22.7167 + 10.9398i −1.52122 + 0.732582i −0.993175 0.116632i \(-0.962790\pi\)
−0.528048 + 0.849215i \(0.677076\pi\)
\(224\) 0 0
\(225\) 13.2366 + 6.37443i 0.882442 + 0.424962i
\(226\) 0 0
\(227\) 15.6441 1.03833 0.519167 0.854673i \(-0.326242\pi\)
0.519167 + 0.854673i \(0.326242\pi\)
\(228\) 0 0
\(229\) −1.94344 1.54984i −0.128426 0.102416i 0.557170 0.830398i \(-0.311887\pi\)
−0.685596 + 0.727982i \(0.740458\pi\)
\(230\) 0 0
\(231\) 0.689883 0.287796i 0.0453910 0.0189356i
\(232\) 0 0
\(233\) −0.873675 0.420740i −0.0572363 0.0275636i 0.405047 0.914296i \(-0.367255\pi\)
−0.462283 + 0.886732i \(0.652970\pi\)
\(234\) 0 0
\(235\) 1.57370 + 1.25499i 0.102657 + 0.0818662i
\(236\) 0 0
\(237\) −0.527344 + 0.420543i −0.0342547 + 0.0273172i
\(238\) 0 0
\(239\) 4.32086 + 3.44577i 0.279493 + 0.222888i 0.753196 0.657797i \(-0.228511\pi\)
−0.473703 + 0.880685i \(0.657083\pi\)
\(240\) 0 0
\(241\) −10.3182 21.4260i −0.664654 1.38017i −0.911576 0.411133i \(-0.865133\pi\)
0.246922 0.969035i \(-0.420581\pi\)
\(242\) 0 0
\(243\) 0.302541 + 1.32552i 0.0194080 + 0.0850319i
\(244\) 0 0
\(245\) 2.07898 + 0.715734i 0.132821 + 0.0457266i
\(246\) 0 0
\(247\) −5.72617 + 1.30696i −0.364347 + 0.0831599i
\(248\) 0 0
\(249\) 0.128661 0.0619598i 0.00815355 0.00392654i
\(250\) 0 0
\(251\) −4.65303 + 5.83471i −0.293697 + 0.368284i −0.906685 0.421808i \(-0.861396\pi\)
0.612989 + 0.790092i \(0.289967\pi\)
\(252\) 0 0
\(253\) 22.6132 + 28.3560i 1.42168 + 1.78273i
\(254\) 0 0
\(255\) 0.0623995 0.0782465i 0.00390761 0.00489999i
\(256\) 0 0
\(257\) 8.22978 17.0893i 0.513360 1.06600i −0.469719 0.882816i \(-0.655645\pi\)
0.983078 0.183186i \(-0.0586410\pi\)
\(258\) 0 0
\(259\) −0.151112 + 0.107793i −0.00938962 + 0.00669790i
\(260\) 0 0
\(261\) 10.0537 12.6070i 0.622310 0.780352i
\(262\) 0 0
\(263\) 13.3186i 0.821259i −0.911802 0.410630i \(-0.865309\pi\)
0.911802 0.410630i \(-0.134691\pi\)
\(264\) 0 0
\(265\) −0.441934 + 0.917684i −0.0271478 + 0.0563729i
\(266\) 0 0
\(267\) −0.357394 0.742136i −0.0218722 0.0454180i
\(268\) 0 0
\(269\) −17.6670 4.03238i −1.07718 0.245858i −0.353105 0.935584i \(-0.614874\pi\)
−0.724071 + 0.689725i \(0.757731\pi\)
\(270\) 0 0
\(271\) 1.72373 0.830106i 0.104709 0.0504253i −0.380796 0.924659i \(-0.624350\pi\)
0.485505 + 0.874234i \(0.338636\pi\)
\(272\) 0 0
\(273\) −0.449988 + 0.187720i −0.0272345 + 0.0113614i
\(274\) 0 0
\(275\) 27.4689i 1.65643i
\(276\) 0 0
\(277\) −15.5068 7.46767i −0.931712 0.448689i −0.0944739 0.995527i \(-0.530117\pi\)
−0.837238 + 0.546839i \(0.815831\pi\)
\(278\) 0 0
\(279\) 4.11183 18.0151i 0.246169 1.07854i
\(280\) 0 0
\(281\) 5.44966 + 23.8765i 0.325099 + 1.42435i 0.828349 + 0.560213i \(0.189281\pi\)
−0.503249 + 0.864141i \(0.667862\pi\)
\(282\) 0 0
\(283\) 2.72170 + 11.9245i 0.161788 + 0.708840i 0.989118 + 0.147124i \(0.0470016\pi\)
−0.827330 + 0.561716i \(0.810141\pi\)
\(284\) 0 0
\(285\) 0.0248043 + 0.00566142i 0.00146928 + 0.000335353i
\(286\) 0 0
\(287\) 2.90864 + 16.8618i 0.171692 + 0.995320i
\(288\) 0 0
\(289\) −14.3066 17.9400i −0.841568 1.05529i
\(290\) 0 0
\(291\) −0.397771 + 0.317212i −0.0233177 + 0.0185953i
\(292\) 0 0
\(293\) 22.0304i 1.28703i −0.765433 0.643516i \(-0.777475\pi\)
0.765433 0.643516i \(-0.222525\pi\)
\(294\) 0 0
\(295\) 3.05906i 0.178105i
\(296\) 0 0
\(297\) −1.32479 + 1.05648i −0.0768719 + 0.0613033i
\(298\) 0 0
\(299\) −14.7498 18.4957i −0.853006 1.06964i
\(300\) 0 0
\(301\) −23.5160 + 9.81011i −1.35544 + 0.565445i
\(302\) 0 0
\(303\) 0.800778 + 0.182772i 0.0460035 + 0.0105000i
\(304\) 0 0
\(305\) 0.944800 + 4.13944i 0.0540991 + 0.237024i
\(306\) 0 0
\(307\) 0.635114 + 2.78261i 0.0362478 + 0.158812i 0.989813 0.142376i \(-0.0454741\pi\)
−0.953565 + 0.301188i \(0.902617\pi\)
\(308\) 0 0
\(309\) 0.0241172 0.105665i 0.00137198 0.00601105i
\(310\) 0 0
\(311\) −8.52819 4.10696i −0.483590 0.232884i 0.176172 0.984359i \(-0.443628\pi\)
−0.659762 + 0.751475i \(0.729343\pi\)
\(312\) 0 0
\(313\) 3.85627i 0.217969i −0.994043 0.108985i \(-0.965240\pi\)
0.994043 0.108985i \(-0.0347599\pi\)
\(314\) 0 0
\(315\) −2.48744 0.133409i −0.140152 0.00751672i
\(316\) 0 0
\(317\) 1.62480 0.782464i 0.0912581 0.0439476i −0.387698 0.921786i \(-0.626730\pi\)
0.478956 + 0.877839i \(0.341015\pi\)
\(318\) 0 0
\(319\) 29.3929 + 6.70875i 1.64569 + 0.375618i
\(320\) 0 0
\(321\) 0.285908 + 0.593694i 0.0159578 + 0.0331368i
\(322\) 0 0
\(323\) 4.40606 9.14927i 0.245160 0.509079i
\(324\) 0 0
\(325\) 17.9171i 0.993859i
\(326\) 0 0
\(327\) 0.608534 0.763078i 0.0336520 0.0421983i
\(328\) 0 0
\(329\) 16.3036 + 4.65255i 0.898846 + 0.256503i
\(330\) 0 0
\(331\) 3.28550 6.82241i 0.180587 0.374993i −0.790950 0.611881i \(-0.790413\pi\)
0.971537 + 0.236888i \(0.0761274\pi\)
\(332\) 0 0
\(333\) 0.131115 0.164413i 0.00718507 0.00900979i
\(334\) 0 0
\(335\) −1.63018 2.04418i −0.0890663 0.111686i
\(336\) 0 0
\(337\) 16.4023 20.5678i 0.893488 1.12040i −0.0986336 0.995124i \(-0.531447\pi\)
0.992122 0.125275i \(-0.0399814\pi\)
\(338\) 0 0
\(339\) 0.299140 0.144058i 0.0162471 0.00782418i
\(340\) 0 0
\(341\) 33.6829 7.68791i 1.82403 0.416324i
\(342\) 0 0
\(343\) 18.4828 1.17811i 0.997975 0.0636122i
\(344\) 0 0
\(345\) 0.0228030 + 0.0999064i 0.00122767 + 0.00537878i
\(346\) 0 0
\(347\) 11.6077 + 24.1035i 0.623132 + 1.29395i 0.938599 + 0.345010i \(0.112124\pi\)
−0.315468 + 0.948936i \(0.602161\pi\)
\(348\) 0 0
\(349\) 17.8354 + 14.2232i 0.954705 + 0.761352i 0.971139 0.238513i \(-0.0766599\pi\)
−0.0164340 + 0.999865i \(0.505231\pi\)
\(350\) 0 0
\(351\) 0.864116 0.689109i 0.0461231 0.0367819i
\(352\) 0 0
\(353\) 3.74884 + 2.98960i 0.199531 + 0.159120i 0.718158 0.695880i \(-0.244985\pi\)
−0.518628 + 0.855000i \(0.673557\pi\)
\(354\) 0 0
\(355\) 1.27409 + 0.613570i 0.0676218 + 0.0325649i
\(356\) 0 0
\(357\) 0.231331 0.810635i 0.0122433 0.0429034i
\(358\) 0 0
\(359\) 2.12333 + 1.69330i 0.112065 + 0.0893687i 0.677919 0.735137i \(-0.262882\pi\)
−0.565854 + 0.824505i \(0.691453\pi\)
\(360\) 0 0
\(361\) −16.4185 −0.864129
\(362\) 0 0
\(363\) −0.926974 0.446407i −0.0486535 0.0234303i
\(364\) 0 0
\(365\) 3.54738 1.70833i 0.185678 0.0894180i
\(366\) 0 0
\(367\) −5.40776 + 23.6929i −0.282283 + 1.23676i 0.612576 + 0.790412i \(0.290133\pi\)
−0.894859 + 0.446350i \(0.852724\pi\)
\(368\) 0 0
\(369\) −8.41102 17.4657i −0.437860 0.909226i
\(370\) 0 0
\(371\) −0.459477 + 8.56710i −0.0238549 + 0.444782i
\(372\) 0 0
\(373\) 23.5459 1.21916 0.609579 0.792725i \(-0.291339\pi\)
0.609579 + 0.792725i \(0.291339\pi\)
\(374\) 0 0
\(375\) −0.0680271 + 0.141260i −0.00351290 + 0.00729462i
\(376\) 0 0
\(377\) −19.1721 4.37590i −0.987412 0.225370i
\(378\) 0 0
\(379\) 23.2827 5.31412i 1.19595 0.272968i 0.422227 0.906490i \(-0.361249\pi\)
0.773724 + 0.633522i \(0.218392\pi\)
\(380\) 0 0
\(381\) −0.0122915 + 0.00280545i −0.000629712 + 0.000143728i
\(382\) 0 0
\(383\) −2.51827 + 11.0332i −0.128677 + 0.563773i 0.868948 + 0.494903i \(0.164796\pi\)
−0.997626 + 0.0688696i \(0.978061\pi\)
\(384\) 0 0
\(385\) −1.79317 4.29844i −0.0913883 0.219069i
\(386\) 0 0
\(387\) 22.5694 17.9985i 1.14727 0.914916i
\(388\) 0 0
\(389\) −0.0281478 0.0352962i −0.00142715 0.00178959i 0.781117 0.624384i \(-0.214650\pi\)
−0.782545 + 0.622595i \(0.786079\pi\)
\(390\) 0 0
\(391\) 40.9019 2.06850
\(392\) 0 0
\(393\) 0.617722 0.0311600
\(394\) 0 0
\(395\) 2.62027 + 3.28571i 0.131840 + 0.165322i
\(396\) 0 0
\(397\) 0.731118 0.583047i 0.0366938 0.0292623i −0.604971 0.796247i \(-0.706815\pi\)
0.641665 + 0.766985i \(0.278244\pi\)
\(398\) 0 0
\(399\) 0.211184 0.0364291i 0.0105724 0.00182373i
\(400\) 0 0
\(401\) 1.30442 5.71505i 0.0651397 0.285396i −0.931858 0.362822i \(-0.881813\pi\)
0.996998 + 0.0774265i \(0.0246703\pi\)
\(402\) 0 0
\(403\) −21.9703 + 5.01458i −1.09442 + 0.249794i
\(404\) 0 0
\(405\) 2.74907 0.627456i 0.136602 0.0311786i
\(406\) 0 0
\(407\) 0.383327 + 0.0874918i 0.0190008 + 0.00433681i
\(408\) 0 0
\(409\) −12.1269 + 25.1817i −0.599636 + 1.24516i 0.351444 + 0.936209i \(0.385691\pi\)
−0.951079 + 0.308947i \(0.900023\pi\)
\(410\) 0 0
\(411\) −0.00703173 −0.000346850
\(412\) 0 0
\(413\) −9.92048 23.7806i −0.488155 1.17017i
\(414\) 0 0
\(415\) −0.386051 0.801644i −0.0189505 0.0393512i
\(416\) 0 0
\(417\) 0.0575776 0.252264i 0.00281959 0.0123534i
\(418\) 0 0
\(419\) −5.30090 + 2.55278i −0.258966 + 0.124712i −0.558863 0.829260i \(-0.688762\pi\)
0.299897 + 0.953972i \(0.403048\pi\)
\(420\) 0 0
\(421\) −14.1212 6.80042i −0.688226 0.331432i 0.0568775 0.998381i \(-0.481886\pi\)
−0.745103 + 0.666949i \(0.767600\pi\)
\(422\) 0 0
\(423\) −19.2082 −0.933937
\(424\) 0 0
\(425\) 24.2195 + 19.3144i 1.17482 + 0.936886i
\(426\) 0 0
\(427\) 20.7689 + 29.1153i 1.00508 + 1.40899i
\(428\) 0 0
\(429\) 0.930524 + 0.448117i 0.0449261 + 0.0216353i
\(430\) 0 0
\(431\) −1.77844 1.41826i −0.0856647 0.0683153i 0.579721 0.814815i \(-0.303162\pi\)
−0.665386 + 0.746500i \(0.731733\pi\)
\(432\) 0 0
\(433\) 5.06413 4.03850i 0.243366 0.194078i −0.494209 0.869343i \(-0.664542\pi\)
0.737575 + 0.675265i \(0.235971\pi\)
\(434\) 0 0
\(435\) 0.0665997 + 0.0531115i 0.00319321 + 0.00254650i
\(436\) 0 0
\(437\) 4.51148 + 9.36819i 0.215813 + 0.448141i
\(438\) 0 0
\(439\) 5.97649 + 26.1847i 0.285242 + 1.24973i 0.890972 + 0.454058i \(0.150024\pi\)
−0.605730 + 0.795670i \(0.707119\pi\)
\(440\) 0 0
\(441\) −19.7696 + 7.02965i −0.941410 + 0.334745i
\(442\) 0 0
\(443\) −22.1954 + 5.06596i −1.05454 + 0.240691i −0.714438 0.699699i \(-0.753317\pi\)
−0.340098 + 0.940390i \(0.610460\pi\)
\(444\) 0 0
\(445\) −4.62401 + 2.22681i −0.219199 + 0.105561i
\(446\) 0 0
\(447\) −0.288243 + 0.361446i −0.0136334 + 0.0170958i
\(448\) 0 0
\(449\) −10.9370 13.7146i −0.516149 0.647230i 0.453637 0.891186i \(-0.350126\pi\)
−0.969786 + 0.243956i \(0.921555\pi\)
\(450\) 0 0
\(451\) 22.5984 28.3375i 1.06412 1.33436i
\(452\) 0 0
\(453\) −0.252663 + 0.524659i −0.0118711 + 0.0246507i
\(454\) 0 0
\(455\) 1.16963 + 2.80373i 0.0548329 + 0.131441i
\(456\) 0 0
\(457\) −0.416580 + 0.522375i −0.0194868 + 0.0244357i −0.791479 0.611196i \(-0.790689\pi\)
0.771993 + 0.635631i \(0.219260\pi\)
\(458\) 0 0
\(459\) 1.91093i 0.0891944i
\(460\) 0 0
\(461\) 5.30594 11.0179i 0.247122 0.513155i −0.740101 0.672495i \(-0.765223\pi\)
0.987224 + 0.159341i \(0.0509368\pi\)
\(462\) 0 0
\(463\) −6.57127 13.6454i −0.305393 0.634155i 0.690634 0.723204i \(-0.257331\pi\)
−0.996027 + 0.0890491i \(0.971617\pi\)
\(464\) 0 0
\(465\) 0.0951696 + 0.0217219i 0.00441339 + 0.00100733i
\(466\) 0 0
\(467\) 26.3418 12.6855i 1.21895 0.587017i 0.289934 0.957047i \(-0.406367\pi\)
0.929020 + 0.370030i \(0.120653\pi\)
\(468\) 0 0
\(469\) −19.3020 10.6045i −0.891284 0.489669i
\(470\) 0 0
\(471\) 0.0940451i 0.00433337i
\(472\) 0 0
\(473\) 48.6284 + 23.4182i 2.23594 + 1.07677i
\(474\) 0 0
\(475\) −1.75237 + 7.67762i −0.0804041 + 0.352274i
\(476\) 0 0
\(477\) −2.16288 9.47620i −0.0990315 0.433885i
\(478\) 0 0
\(479\) 4.61410 + 20.2157i 0.210823 + 0.923678i 0.964009 + 0.265870i \(0.0856591\pi\)
−0.753185 + 0.657808i \(0.771484\pi\)
\(480\) 0 0
\(481\) −0.250032 0.0570681i −0.0114005 0.00260208i
\(482\) 0 0
\(483\) 0.501261 + 0.702705i 0.0228082 + 0.0319742i
\(484\) 0 0
\(485\) 1.97644 + 2.47838i 0.0897457 + 0.112538i
\(486\) 0 0
\(487\) 5.73711 4.57519i 0.259973 0.207322i −0.484825 0.874611i \(-0.661117\pi\)
0.744799 + 0.667289i \(0.232545\pi\)
\(488\) 0 0
\(489\) 0.851436i 0.0385033i
\(490\) 0 0
\(491\) 21.0768i 0.951183i 0.879666 + 0.475592i \(0.157766\pi\)
−0.879666 + 0.475592i \(0.842234\pi\)
\(492\) 0 0
\(493\) 26.5824 21.1988i 1.19721 0.954745i
\(494\) 0 0
\(495\) 3.28991 + 4.12541i 0.147870 + 0.185423i
\(496\) 0 0
\(497\) 11.8944 + 0.637928i 0.533535 + 0.0286150i
\(498\) 0 0
\(499\) −40.5858 9.26345i −1.81687 0.414689i −0.827666 0.561221i \(-0.810332\pi\)
−0.989206 + 0.146532i \(0.953189\pi\)
\(500\) 0 0
\(501\) 0.246442 + 1.07973i 0.0110102 + 0.0482388i
\(502\) 0 0
\(503\) 7.56847 + 33.1596i 0.337461 + 1.47851i 0.804327 + 0.594186i \(0.202526\pi\)
−0.466866 + 0.884328i \(0.654617\pi\)
\(504\) 0 0
\(505\) 1.13880 4.98939i 0.0506757 0.222025i
\(506\) 0 0
\(507\) −0.0164875 0.00793996i −0.000732236 0.000352626i
\(508\) 0 0
\(509\) 3.07456i 0.136277i 0.997676 + 0.0681386i \(0.0217060\pi\)
−0.997676 + 0.0681386i \(0.978294\pi\)
\(510\) 0 0
\(511\) 22.0366 24.7844i 0.974843 1.09640i
\(512\) 0 0
\(513\) −0.437680 + 0.210775i −0.0193240 + 0.00930596i
\(514\) 0 0
\(515\) −0.658362 0.150267i −0.0290109 0.00662155i
\(516\) 0 0
\(517\) −15.5824 32.3571i −0.685312 1.42307i
\(518\) 0 0
\(519\) 0.309637 0.642968i 0.0135916 0.0282232i
\(520\) 0 0
\(521\) 18.5548i 0.812899i 0.913673 + 0.406450i \(0.133233\pi\)
−0.913673 + 0.406450i \(0.866767\pi\)
\(522\) 0 0
\(523\) −1.40143 + 1.75733i −0.0612800 + 0.0768427i −0.811527 0.584315i \(-0.801363\pi\)
0.750247 + 0.661158i \(0.229935\pi\)
\(524\) 0 0
\(525\) −0.0350114 + 0.652799i −0.00152802 + 0.0284905i
\(526\) 0 0
\(527\) 16.9053 35.1041i 0.736405 1.52916i
\(528\) 0 0
\(529\) −11.7718 + 14.7614i −0.511819 + 0.641800i
\(530\) 0 0
\(531\) 18.2010 + 22.8233i 0.789856 + 0.990448i
\(532\) 0 0
\(533\) −14.7402 + 18.4836i −0.638469 + 0.800615i
\(534\) 0 0
\(535\) 3.69911 1.78140i 0.159927 0.0770166i
\(536\) 0 0
\(537\) 0.187789 0.0428617i 0.00810371 0.00184962i
\(538\) 0 0
\(539\) −27.8795 27.6001i −1.20086 1.18882i
\(540\) 0 0
\(541\) −0.0237085 0.103874i −0.00101931 0.00446589i 0.974416 0.224753i \(-0.0721576\pi\)
−0.975435 + 0.220287i \(0.929300\pi\)
\(542\) 0 0
\(543\) 0.206242 + 0.428266i 0.00885069 + 0.0183786i
\(544\) 0 0
\(545\) −4.75449 3.79158i −0.203660 0.162414i
\(546\) 0 0
\(547\) 12.6122 10.0579i 0.539260 0.430046i −0.315609 0.948889i \(-0.602209\pi\)
0.854869 + 0.518844i \(0.173637\pi\)
\(548\) 0 0
\(549\) −31.6782 25.2625i −1.35199 1.07818i
\(550\) 0 0
\(551\) 7.78743 + 3.75023i 0.331756 + 0.159765i
\(552\) 0 0
\(553\) 31.0250 + 17.0451i 1.31932 + 0.724829i
\(554\) 0 0
\(555\) 0.000868557 0 0.000692651i 3.68682e−5 0 2.94014e-5i
\(556\) 0 0
\(557\) 8.94717 0.379104 0.189552 0.981871i \(-0.439296\pi\)
0.189552 + 0.981871i \(0.439296\pi\)
\(558\) 0 0
\(559\) −31.7187 15.2749i −1.34156 0.646061i
\(560\) 0 0
\(561\) −1.60884 + 0.774776i −0.0679252 + 0.0327111i
\(562\) 0 0
\(563\) 3.52247 15.4330i 0.148455 0.650422i −0.844860 0.534987i \(-0.820317\pi\)
0.993315 0.115435i \(-0.0368262\pi\)
\(564\) 0 0
\(565\) −0.897581 1.86385i −0.0377615 0.0784126i
\(566\) 0 0
\(567\) 19.3359 13.7929i 0.812033 0.579248i
\(568\) 0 0
\(569\) 9.18723 0.385149 0.192574 0.981282i \(-0.438316\pi\)
0.192574 + 0.981282i \(0.438316\pi\)
\(570\) 0 0
\(571\) 2.00162 4.15641i 0.0837653 0.173940i −0.854879 0.518828i \(-0.826369\pi\)
0.938644 + 0.344887i \(0.112083\pi\)
\(572\) 0 0
\(573\) −0.0686265 0.0156635i −0.00286691 0.000654354i
\(574\) 0 0
\(575\) −30.9238 + 7.05816i −1.28961 + 0.294346i
\(576\) 0 0
\(577\) −10.1259 + 2.31117i −0.421546 + 0.0962152i −0.428032 0.903764i \(-0.640793\pi\)
0.00648530 + 0.999979i \(0.497936\pi\)
\(578\) 0 0
\(579\) 0.159339 0.698112i 0.00662192 0.0290125i
\(580\) 0 0
\(581\) −5.60081 4.97988i −0.232361 0.206600i
\(582\) 0 0
\(583\) 14.2085 11.3309i 0.588455 0.469277i
\(584\) 0 0
\(585\) −2.14590 2.69087i −0.0887220 0.111254i
\(586\) 0 0
\(587\) −21.7003 −0.895666 −0.447833 0.894117i \(-0.647804\pi\)
−0.447833 + 0.894117i \(0.647804\pi\)
\(588\) 0 0
\(589\) 9.90492 0.408125
\(590\) 0 0
\(591\) 0.0377019 + 0.0472767i 0.00155085 + 0.00194470i
\(592\) 0 0
\(593\) −0.129136 + 0.102983i −0.00530299 + 0.00422900i −0.626138 0.779712i \(-0.715365\pi\)
0.620835 + 0.783941i \(0.286794\pi\)
\(594\) 0 0
\(595\) −5.05081 1.44135i −0.207063 0.0590894i
\(596\) 0 0
\(597\) 0.0452995 0.198470i 0.00185399 0.00812284i
\(598\) 0 0
\(599\) −11.8101 + 2.69558i −0.482548 + 0.110138i −0.456873 0.889532i \(-0.651031\pi\)
−0.0256749 + 0.999670i \(0.508173\pi\)
\(600\) 0 0
\(601\) 5.42057 1.23721i 0.221110 0.0504669i −0.110531 0.993873i \(-0.535255\pi\)
0.331641 + 0.943406i \(0.392398\pi\)
\(602\) 0 0
\(603\) 24.3253 + 5.55208i 0.990601 + 0.226098i
\(604\) 0 0
\(605\) −2.78142 + 5.77568i −0.113081 + 0.234815i
\(606\) 0 0
\(607\) −15.9049 −0.645561 −0.322781 0.946474i \(-0.604618\pi\)
−0.322781 + 0.946474i \(0.604618\pi\)
\(608\) 0 0
\(609\) 0.689974 + 0.196898i 0.0279592 + 0.00797869i
\(610\) 0 0
\(611\) 10.1639 + 21.1055i 0.411187 + 0.853838i
\(612\) 0 0
\(613\) 9.44921 41.3997i 0.381650 1.67212i −0.310663 0.950520i \(-0.600551\pi\)
0.692313 0.721598i \(-0.256592\pi\)
\(614\) 0 0
\(615\) 0.0922670 0.0444334i 0.00372056 0.00179173i
\(616\) 0 0
\(617\) −2.36694 1.13986i −0.0952892 0.0458889i 0.385632 0.922653i \(-0.373983\pi\)
−0.480921 + 0.876764i \(0.659698\pi\)
\(618\) 0 0
\(619\) −31.0549 −1.24820 −0.624100 0.781344i \(-0.714534\pi\)
−0.624100 + 0.781344i \(0.714534\pi\)
\(620\) 0 0
\(621\) −1.52977 1.21995i −0.0613876 0.0489550i
\(622\) 0 0
\(623\) −28.7248 + 32.3064i −1.15083 + 1.29433i
\(624\) 0 0
\(625\) −21.1996 10.2092i −0.847985 0.408368i
\(626\) 0 0
\(627\) −0.354910 0.283031i −0.0141737 0.0113032i
\(628\) 0 0
\(629\) 0.346674 0.276463i 0.0138228 0.0110233i
\(630\) 0 0
\(631\) −26.4409 21.0859i −1.05259 0.839416i −0.0652280 0.997870i \(-0.520777\pi\)
−0.987366 + 0.158454i \(0.949349\pi\)
\(632\) 0 0
\(633\) −0.121875 0.253077i −0.00484411 0.0100589i
\(634\) 0 0
\(635\) 0.0174799 + 0.0765843i 0.000693668 + 0.00303916i
\(636\) 0 0
\(637\) 18.1849 + 18.0027i 0.720513 + 0.713291i
\(638\) 0 0
\(639\) −13.1565 + 3.00289i −0.520465 + 0.118793i
\(640\) 0 0
\(641\) −9.36480 + 4.50985i −0.369887 + 0.178128i −0.609588 0.792719i \(-0.708665\pi\)
0.239700 + 0.970847i \(0.422951\pi\)
\(642\) 0 0
\(643\) −12.8331 + 16.0922i −0.506089 + 0.634616i −0.967591 0.252524i \(-0.918739\pi\)
0.461502 + 0.887139i \(0.347311\pi\)
\(644\) 0 0
\(645\) 0.0950820 + 0.119229i 0.00374385 + 0.00469464i
\(646\) 0 0
\(647\) −25.2828 + 31.7036i −0.993969 + 1.24640i −0.0248784 + 0.999690i \(0.507920\pi\)
−0.969090 + 0.246707i \(0.920652\pi\)
\(648\) 0 0
\(649\) −23.6817 + 49.1755i −0.929586 + 1.93031i
\(650\) 0 0
\(651\) 0.810276 0.139772i 0.0317572 0.00547809i
\(652\) 0 0
\(653\) −8.60685 + 10.7927i −0.336812 + 0.422349i −0.921178 0.389141i \(-0.872772\pi\)
0.584366 + 0.811490i \(0.301343\pi\)
\(654\) 0 0
\(655\) 3.84883i 0.150386i
\(656\) 0 0
\(657\) −16.3023 + 33.8521i −0.636015 + 1.32070i
\(658\) 0 0
\(659\) 14.3786 + 29.8575i 0.560112 + 1.16308i 0.968208 + 0.250145i \(0.0804784\pi\)
−0.408097 + 0.912939i \(0.633807\pi\)
\(660\) 0 0
\(661\) 19.2715 + 4.39860i 0.749576 + 0.171086i 0.580209 0.814467i \(-0.302971\pi\)
0.169366 + 0.985553i \(0.445828\pi\)
\(662\) 0 0
\(663\) 1.04939 0.505361i 0.0407551 0.0196266i
\(664\) 0 0
\(665\) −0.226978 1.31582i −0.00880182 0.0510253i
\(666\) 0 0
\(667\) 34.8137i 1.34799i
\(668\) 0 0
\(669\) 1.14521 + 0.551504i 0.0442764 + 0.0213224i
\(670\) 0 0
\(671\) 16.8574 73.8572i 0.650773 2.85122i
\(672\) 0 0
\(673\) 2.73446 + 11.9805i 0.105406 + 0.461813i 0.999892 + 0.0147168i \(0.00468468\pi\)
−0.894486 + 0.447096i \(0.852458\pi\)
\(674\) 0 0
\(675\) −0.329756 1.44475i −0.0126923 0.0556086i
\(676\) 0 0
\(677\) −37.2628 8.50500i −1.43213 0.326874i −0.565053 0.825054i \(-0.691144\pi\)
−0.867073 + 0.498181i \(0.834002\pi\)
\(678\) 0 0
\(679\) 23.4019 + 12.8569i 0.898082 + 0.493404i
\(680\) 0 0
\(681\) −0.491721 0.616599i −0.0188428 0.0236281i
\(682\) 0 0
\(683\) 7.11436 5.67352i 0.272224 0.217091i −0.477857 0.878438i \(-0.658586\pi\)
0.750080 + 0.661347i \(0.230015\pi\)
\(684\) 0 0
\(685\) 0.0438124i 0.00167399i
\(686\) 0 0
\(687\) 0.125313i 0.00478100i
\(688\) 0 0
\(689\) −9.26774 + 7.39077i −0.353073 + 0.281566i
\(690\) 0 0
\(691\) −25.3483 31.7858i −0.964296 1.20919i −0.977855 0.209281i \(-0.932888\pi\)
0.0135591 0.999908i \(-0.495684\pi\)
\(692\) 0 0
\(693\) 38.9538 + 21.4011i 1.47973 + 0.812961i
\(694\) 0 0
\(695\) −1.57178 0.358747i −0.0596208 0.0136081i
\(696\) 0 0
\(697\) −9.09557 39.8503i −0.344519 1.50944i
\(698\) 0 0
\(699\) 0.0108780 + 0.0476598i 0.000411445 + 0.00180266i
\(700\) 0 0
\(701\) −0.218861 + 0.958894i −0.00826628 + 0.0362169i −0.978893 0.204374i \(-0.934484\pi\)
0.970627 + 0.240591i \(0.0773413\pi\)
\(702\) 0 0
\(703\) 0.101559 + 0.0489084i 0.00383038 + 0.00184462i
\(704\) 0 0
\(705\) 0.101473i 0.00382168i
\(706\) 0 0
\(707\) −7.32771 42.4797i −0.275587 1.59761i
\(708\) 0 0
\(709\) 14.3106 6.89160i 0.537444 0.258820i −0.145413 0.989371i \(-0.546451\pi\)
0.682858 + 0.730551i \(0.260737\pi\)
\(710\) 0 0
\(711\) −39.0991 8.92411i −1.46633 0.334680i
\(712\) 0 0
\(713\) 17.3098 + 35.9441i 0.648255 + 1.34612i
\(714\) 0 0
\(715\) 2.79207 5.79779i 0.104417 0.216825i
\(716\) 0 0
\(717\) 0.278610i 0.0104049i
\(718\) 0 0
\(719\) 12.9657 16.2585i 0.483540 0.606340i −0.478889 0.877876i \(-0.658960\pi\)
0.962428 + 0.271536i \(0.0875317\pi\)
\(720\) 0 0
\(721\) −5.60530 + 0.966909i −0.208752 + 0.0360096i
\(722\) 0 0
\(723\) −0.520168 + 1.08014i −0.0193453 + 0.0401708i
\(724\) 0 0
\(725\) −16.4395 + 20.6145i −0.610548 + 0.765602i
\(726\) 0 0
\(727\) −23.5326 29.5089i −0.872775 1.09443i −0.994795 0.101898i \(-0.967509\pi\)
0.122020 0.992528i \(-0.461063\pi\)
\(728\) 0 0
\(729\) −16.7487 + 21.0022i −0.620323 + 0.777860i
\(730\) 0 0
\(731\) 54.8405 26.4098i 2.02835 0.976801i
\(732\) 0 0
\(733\) 1.61153 0.367821i 0.0595232 0.0135858i −0.192656 0.981266i \(-0.561710\pi\)
0.252179 + 0.967681i \(0.418853\pi\)
\(734\) 0 0
\(735\) −0.0371360 0.104438i −0.00136978 0.00385226i
\(736\) 0 0
\(737\) 10.3807 + 45.4810i 0.382380 + 1.67531i
\(738\) 0 0
\(739\) −0.0706938 0.146797i −0.00260051 0.00540002i 0.899665 0.436582i \(-0.143811\pi\)
−0.902265 + 0.431182i \(0.858097\pi\)
\(740\) 0 0
\(741\) 0.231497 + 0.184612i 0.00850424 + 0.00678190i
\(742\) 0 0
\(743\) −19.5359 + 15.5794i −0.716704 + 0.571553i −0.912492 0.409094i \(-0.865845\pi\)
0.195788 + 0.980646i \(0.437274\pi\)
\(744\) 0 0
\(745\) 2.25205 + 1.79595i 0.0825088 + 0.0657986i
\(746\) 0 0
\(747\) 7.64997 + 3.68403i 0.279898 + 0.134792i
\(748\) 0 0
\(749\) 22.9792 25.8445i 0.839642 0.944336i
\(750\) 0 0
\(751\) 40.5585 + 32.3444i 1.48000 + 1.18026i 0.941209 + 0.337824i \(0.109691\pi\)
0.538793 + 0.842438i \(0.318880\pi\)
\(752\) 0 0
\(753\) 0.376224 0.0137103
\(754\) 0 0
\(755\) 3.26898 + 1.57426i 0.118971 + 0.0572932i
\(756\) 0 0
\(757\) −5.97197 + 2.87595i −0.217055 + 0.104528i −0.539252 0.842145i \(-0.681293\pi\)
0.322197 + 0.946673i \(0.395579\pi\)
\(758\) 0 0
\(759\) 0.406858 1.78256i 0.0147680 0.0647028i
\(760\) 0 0
\(761\) −13.2484 27.5106i −0.480254 0.997257i −0.990536 0.137250i \(-0.956174\pi\)
0.510283 0.860007i \(-0.329541\pi\)
\(762\) 0 0
\(763\) −49.2566 14.0563i −1.78321 0.508874i
\(764\) 0 0
\(765\) 5.95066 0.215147
\(766\) 0 0
\(767\) 15.4468 32.0756i 0.557751 1.15818i
\(768\) 0 0
\(769\) −15.9284 3.63556i −0.574394 0.131102i −0.0745503 0.997217i \(-0.523752\pi\)
−0.499844 + 0.866116i \(0.666609\pi\)
\(770\) 0 0
\(771\) −0.932238 + 0.212777i −0.0335737 + 0.00766299i
\(772\) 0 0
\(773\) −6.22545 + 1.42092i −0.223914 + 0.0511069i −0.333006 0.942925i \(-0.608063\pi\)
0.109092 + 0.994032i \(0.465206\pi\)
\(774\) 0 0
\(775\) −6.72352 + 29.4577i −0.241516 + 1.05815i
\(776\) 0 0
\(777\) 0.00899827 + 0.00256783i 0.000322811 + 9.21204e-5i
\(778\) 0 0
\(779\) 8.12409 6.47875i 0.291076 0.232125i
\(780\) 0 0
\(781\) −15.7315 19.7267i −0.562919 0.705878i
\(782\) 0 0
\(783\) −1.62649 −0.0581260
\(784\) 0 0
\(785\) 0.585965 0.0209140
\(786\) 0 0
\(787\) −17.7022 22.1979i −0.631017 0.791270i 0.358831 0.933403i \(-0.383175\pi\)
−0.989848 + 0.142133i \(0.954604\pi\)
\(788\) 0 0
\(789\) −0.524942 + 0.418627i −0.0186884 + 0.0149035i
\(790\) 0 0
\(791\) −13.0221 11.5784i −0.463011 0.411680i
\(792\) 0 0
\(793\) −10.9956 + 48.1747i −0.390463 + 1.71073i
\(794\) 0 0
\(795\) 0.0500606 0.0114260i 0.00177547 0.000405238i
\(796\) 0 0
\(797\) 20.0487 4.57599i 0.710162 0.162090i 0.147844 0.989011i \(-0.452767\pi\)
0.562318 + 0.826921i \(0.309910\pi\)
\(798\) 0 0
\(799\) −39.4861 9.01244i −1.39692 0.318837i
\(800\) 0 0
\(801\) 21.2501 44.1263i 0.750835 1.55913i
\(802\) 0 0
\(803\) −70.2504 −2.47908
\(804\) 0 0
\(805\) 4.37833 3.12320i 0.154316 0.110078i
\(806\) 0 0
\(807\) 0.396372 + 0.823076i 0.0139530 + 0.0289736i
\(808\) 0 0
\(809\) 0.240503 1.05371i 0.00845563 0.0370465i −0.970524 0.241004i \(-0.922523\pi\)
0.978980 + 0.203957i \(0.0653804\pi\)
\(810\) 0 0
\(811\) 27.2502 13.1230i 0.956885 0.460812i 0.110790 0.993844i \(-0.464662\pi\)
0.846095 + 0.533032i \(0.178948\pi\)
\(812\) 0 0
\(813\) −0.0868979 0.0418478i −0.00304764 0.00146767i
\(814\) 0 0
\(815\) 5.30502 0.185827
\(816\) 0 0
\(817\) 12.0978 + 9.64769i 0.423249 + 0.337530i
\(818\) 0 0
\(819\) −25.4083 13.9593i −0.887838 0.487776i
\(820\) 0 0
\(821\) −46.5733 22.4285i −1.62542 0.782761i −0.999997 0.00247511i \(-0.999212\pi\)
−0.625423 0.780286i \(-0.715074\pi\)
\(822\) 0 0
\(823\) 0.641606 + 0.511664i 0.0223650 + 0.0178355i 0.634609 0.772834i \(-0.281161\pi\)
−0.612244 + 0.790669i \(0.709733\pi\)
\(824\) 0 0
\(825\) 1.08266 0.863395i 0.0376935 0.0300596i
\(826\) 0 0
\(827\) −8.03129 6.40474i −0.279275 0.222715i 0.473827 0.880618i \(-0.342872\pi\)
−0.753102 + 0.657903i \(0.771444\pi\)
\(828\) 0 0
\(829\) 13.2908 + 27.5986i 0.461608 + 0.958540i 0.993723 + 0.111867i \(0.0356832\pi\)
−0.532115 + 0.846672i \(0.678603\pi\)
\(830\) 0 0
\(831\) 0.193073 + 0.845910i 0.00669764 + 0.0293443i
\(832\) 0 0
\(833\) −43.9383 + 5.17491i −1.52237 + 0.179300i
\(834\) 0 0
\(835\) 6.72746 1.53550i 0.232813 0.0531381i
\(836\) 0 0
\(837\) −1.67930 + 0.808708i −0.0580451 + 0.0279530i
\(838\) 0 0
\(839\) −26.4827 + 33.2082i −0.914283 + 1.14647i 0.0745162 + 0.997220i \(0.476259\pi\)
−0.988799 + 0.149254i \(0.952313\pi\)
\(840\) 0 0
\(841\) −0.0378030 0.0474035i −0.00130355 0.00163460i
\(842\) 0 0
\(843\) 0.769782 0.965276i 0.0265127 0.0332459i
\(844\) 0 0
\(845\) −0.0494714 + 0.102728i −0.00170187 + 0.00353396i
\(846\) 0 0
\(847\) −2.89184 + 53.9192i −0.0993647 + 1.85269i
\(848\) 0 0
\(849\) 0.384448 0.482083i 0.0131942 0.0165450i
\(850\) 0 0
\(851\) 0.454022i 0.0155637i
\(852\) 0 0
\(853\) −18.1125 + 37.6109i −0.620159 + 1.28777i 0.320123 + 0.947376i \(0.396276\pi\)
−0.940282 + 0.340397i \(0.889438\pi\)
\(854\) 0 0
\(855\) 0.656359 + 1.36294i 0.0224470 + 0.0466117i
\(856\) 0 0
\(857\) 44.3935 + 10.1325i 1.51645 + 0.346120i 0.898104 0.439783i \(-0.144945\pi\)
0.618349 + 0.785904i \(0.287802\pi\)
\(858\) 0 0
\(859\) 24.4278 11.7638i 0.833466 0.401376i 0.0320521 0.999486i \(-0.489796\pi\)
0.801414 + 0.598110i \(0.204081\pi\)
\(860\) 0 0
\(861\) 0.573170 0.644638i 0.0195336 0.0219692i
\(862\) 0 0
\(863\) 11.6599i 0.396909i 0.980110 + 0.198455i \(0.0635922\pi\)
−0.980110 + 0.198455i \(0.936408\pi\)
\(864\) 0 0
\(865\) −4.00612 1.92925i −0.136212 0.0655964i
\(866\) 0 0
\(867\) −0.257406 + 1.12777i −0.00874197 + 0.0383011i
\(868\) 0 0
\(869\) −16.6854 73.1037i −0.566015 2.47987i
\(870\) 0 0
\(871\) −6.77103 29.6658i −0.229428 1.00519i
\(872\) 0 0
\(873\) −29.4921 6.73138i −0.998156 0.227823i
\(874\) 0 0
\(875\) 8.21664 + 0.440681i 0.277773 + 0.0148977i
\(876\) 0 0
\(877\) −16.9222 21.2198i −0.571423 0.716542i 0.409201 0.912444i \(-0.365808\pi\)
−0.980623 + 0.195903i \(0.937236\pi\)
\(878\) 0 0
\(879\) −0.868312 + 0.692456i −0.0292874 + 0.0233560i
\(880\) 0 0
\(881\) 11.5301i 0.388457i −0.980956 0.194229i \(-0.937780\pi\)
0.980956 0.194229i \(-0.0622204\pi\)
\(882\) 0 0
\(883\) 39.0945i 1.31563i 0.753178 + 0.657817i \(0.228520\pi\)
−0.753178 + 0.657817i \(0.771480\pi\)
\(884\) 0 0
\(885\) −0.120570 + 0.0961516i −0.00405293 + 0.00323210i
\(886\) 0 0
\(887\) 5.76733 + 7.23200i 0.193648 + 0.242827i 0.869171 0.494512i \(-0.164653\pi\)
−0.675523 + 0.737339i \(0.736082\pi\)
\(888\) 0 0
\(889\) 0.384247 + 0.538666i 0.0128872 + 0.0180663i
\(890\) 0 0
\(891\) −49.0497 11.1953i −1.64323 0.375056i
\(892\) 0 0
\(893\) −2.29111 10.0380i −0.0766689 0.335908i
\(894\) 0 0
\(895\) −0.267057 1.17005i −0.00892674 0.0391106i
\(896\) 0 0
\(897\) −0.265380 + 1.16271i −0.00886079 + 0.0388217i
\(898\) 0 0
\(899\) 29.8790 + 14.3890i 0.996520 + 0.479899i
\(900\) 0 0
\(901\) 20.4949i 0.682784i
\(902\) 0 0
\(903\) 1.12581 + 0.618516i 0.0374646 + 0.0205829i
\(904\) 0 0
\(905\) 2.66839 1.28503i 0.0887002 0.0427157i
\(906\) 0 0
\(907\) −13.7819 3.14563i −0.457621 0.104449i −0.0125046 0.999922i \(-0.503980\pi\)
−0.445116 + 0.895473i \(0.646838\pi\)
\(908\) 0 0
\(909\) 21.1898 + 44.0010i 0.702821 + 1.45942i
\(910\) 0 0
\(911\) 2.62723 5.45551i 0.0870441 0.180749i −0.852903 0.522070i \(-0.825160\pi\)
0.939947 + 0.341321i \(0.110874\pi\)
\(912\) 0 0
\(913\) 15.8753i 0.525397i
\(914\) 0 0
\(915\) 0.133456 0.167349i 0.00441192 0.00553237i
\(916\) 0 0
\(917\) −12.4817 29.9201i −0.412182 0.988050i
\(918\) 0 0
\(919\) 14.7113 30.5482i 0.485279 1.00769i −0.504276 0.863542i \(-0.668241\pi\)
0.989556 0.144151i \(-0.0460450\pi\)
\(920\) 0 0
\(921\) 0.0897118 0.112495i 0.00295610 0.00370684i
\(922\) 0 0
\(923\) 10.2612 + 12.8671i 0.337751 + 0.423526i
\(924\) 0 0
\(925\) −0.214395 + 0.268843i −0.00704926 + 0.00883949i
\(926\) 0 0
\(927\) 5.80604 2.79604i 0.190695 0.0918341i
\(928\) 0 0
\(929\) −4.01269 + 0.915869i −0.131652 + 0.0300487i −0.287839 0.957679i \(-0.592937\pi\)
0.156187 + 0.987727i \(0.450080\pi\)
\(930\) 0 0
\(931\) −6.03167 9.49287i −0.197680 0.311116i
\(932\) 0 0
\(933\) 0.106184 + 0.465221i 0.00347630 + 0.0152307i
\(934\) 0 0
\(935\) 4.82738 + 10.0242i 0.157872 + 0.327825i
\(936\) 0 0
\(937\) 44.4091 + 35.4151i 1.45078 + 1.15696i 0.957986 + 0.286815i \(0.0925965\pi\)
0.492796 + 0.870145i \(0.335975\pi\)
\(938\) 0 0
\(939\) −0.151992 + 0.121209i −0.00496006 + 0.00395552i
\(940\) 0 0
\(941\) 18.1740 + 14.4933i 0.592454 + 0.472466i 0.873231 0.487307i \(-0.162021\pi\)
−0.280777 + 0.959773i \(0.590592\pi\)
\(942\) 0 0
\(943\) 37.7084 + 18.1594i 1.22795 + 0.591351i
\(944\) 0 0
\(945\) 0.145915 + 0.204555i 0.00474662 + 0.00665416i
\(946\) 0 0
\(947\) −37.0550 29.5504i −1.20412 0.960258i −0.204299 0.978909i \(-0.565491\pi\)
−0.999826 + 0.0186511i \(0.994063\pi\)
\(948\) 0 0
\(949\) 45.8221 1.48745
\(950\) 0 0
\(951\) −0.0819107 0.0394461i −0.00265614 0.00127913i
\(952\) 0 0
\(953\) 7.97101 3.83864i 0.258206 0.124346i −0.300303 0.953844i \(-0.597088\pi\)
0.558509 + 0.829498i \(0.311373\pi\)
\(954\) 0 0
\(955\) −0.0975945 + 0.427590i −0.00315808 + 0.0138365i
\(956\) 0 0
\(957\) −0.659452 1.36937i −0.0213171 0.0442654i
\(958\) 0 0
\(959\) 0.142083 + 0.340590i 0.00458810 + 0.0109982i
\(960\) 0 0
\(961\) 7.00342 0.225917
\(962\) 0 0
\(963\) −16.9996 + 35.3001i −0.547806 + 1.13753i
\(964\) 0 0
\(965\) −4.34971 0.992793i −0.140022 0.0319591i
\(966\) 0 0
\(967\) 25.0464 5.71668i 0.805438 0.183836i 0.200070 0.979782i \(-0.435883\pi\)
0.605368 + 0.795946i \(0.293026\pi\)
\(968\) 0 0
\(969\) −0.499102 + 0.113917i −0.0160335 + 0.00365953i
\(970\) 0 0
\(971\) 4.81406 21.0918i 0.154490 0.676867i −0.837056 0.547117i \(-0.815725\pi\)
0.991547 0.129750i \(-0.0414175\pi\)
\(972\) 0 0
\(973\) −13.3821 + 2.30840i −0.429011 + 0.0740040i
\(974\) 0 0
\(975\) −0.706187 + 0.563165i −0.0226161 + 0.0180357i
\(976\) 0 0
\(977\) 14.7340 + 18.4758i 0.471381 + 0.591093i 0.959509 0.281679i \(-0.0908914\pi\)
−0.488128 + 0.872772i \(0.662320\pi\)
\(978\) 0 0
\(979\) 91.5715 2.92664
\(980\) 0 0
\(981\) 58.0322 1.85283
\(982\) 0 0
\(983\) 1.70683 + 2.14029i 0.0544393 + 0.0682648i 0.808306 0.588763i \(-0.200385\pi\)
−0.753866 + 0.657028i \(0.771813\pi\)
\(984\) 0 0
\(985\) 0.294566 0.234908i 0.00938564 0.00748480i
\(986\) 0 0
\(987\) −0.329074 0.788831i −0.0104746 0.0251088i
\(988\) 0 0
\(989\) −13.8686 + 60.7621i −0.440994 + 1.93212i
\(990\) 0 0
\(991\) −4.17217 + 0.952270i −0.132533 + 0.0302498i −0.288273 0.957548i \(-0.593081\pi\)
0.155740 + 0.987798i \(0.450224\pi\)
\(992\) 0 0
\(993\) −0.372169 + 0.0849451i −0.0118104 + 0.00269565i
\(994\) 0 0
\(995\) −1.23660 0.282247i −0.0392030 0.00894782i
\(996\) 0 0
\(997\) −5.43635 + 11.2887i −0.172171 + 0.357517i −0.969143 0.246500i \(-0.920720\pi\)
0.796972 + 0.604016i \(0.206434\pi\)
\(998\) 0 0
\(999\) −0.0212118 −0.000671111
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 784.2.bb.b.111.10 120
4.3 odd 2 inner 784.2.bb.b.111.11 yes 120
49.34 odd 14 inner 784.2.bb.b.671.11 yes 120
196.83 even 14 inner 784.2.bb.b.671.10 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
784.2.bb.b.111.10 120 1.1 even 1 trivial
784.2.bb.b.111.11 yes 120 4.3 odd 2 inner
784.2.bb.b.671.10 yes 120 196.83 even 14 inner
784.2.bb.b.671.11 yes 120 49.34 odd 14 inner