Properties

Label 912.2.ci.a.751.1
Level $912$
Weight $2$
Character 912.751
Analytic conductor $7.282$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(79,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ci (of order \(18\), 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: \(7.28235666434\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 751.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 912.751
Dual form 912.2.ci.a.895.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.766044 - 0.642788i) q^{3} +(-0.826352 - 0.300767i) q^{5} +(-0.907604 - 0.524005i) q^{7} +(0.173648 - 0.984808i) q^{9} +O(q^{10})\) \(q+(0.766044 - 0.642788i) q^{3} +(-0.826352 - 0.300767i) q^{5} +(-0.907604 - 0.524005i) q^{7} +(0.173648 - 0.984808i) q^{9} +(-3.79813 + 2.19285i) q^{11} +(-3.64543 + 4.34445i) q^{13} +(-0.826352 + 0.300767i) q^{15} +(-0.539363 - 3.05888i) q^{17} +(-4.28699 - 0.788496i) q^{19} +(-1.03209 + 0.181985i) q^{21} +(-1.78699 - 4.90971i) q^{23} +(-3.23783 - 2.71686i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(6.30200 + 1.11121i) q^{29} +(-3.40033 + 5.88954i) q^{31} +(-1.50000 + 4.12122i) q^{33} +(0.592396 + 0.705990i) q^{35} +9.58186i q^{37} +5.67128i q^{39} +(4.50640 + 5.37051i) q^{41} +(1.63429 - 4.49016i) q^{43} +(-0.439693 + 0.761570i) q^{45} +(-11.1985 - 1.97459i) q^{47} +(-2.95084 - 5.11100i) q^{49} +(-2.37939 - 1.99654i) q^{51} +(1.83022 + 5.02849i) q^{53} +(3.79813 - 0.669713i) q^{55} +(-3.79086 + 2.15160i) q^{57} +(-2.62314 - 14.8766i) q^{59} +(7.56418 - 2.75314i) q^{61} +(-0.673648 + 0.802823i) q^{63} +(4.31908 - 2.49362i) q^{65} +(-0.709141 + 4.02174i) q^{67} +(-4.52481 - 2.61240i) q^{69} +(-3.59240 - 1.30753i) q^{71} +(-10.4474 + 8.76644i) q^{73} -4.22668 q^{75} +4.59627 q^{77} +(-3.54710 + 2.97637i) q^{79} +(-0.939693 - 0.342020i) q^{81} +(6.02481 + 3.47843i) q^{83} +(-0.474308 + 2.68993i) q^{85} +(5.54189 - 3.19961i) q^{87} +(5.71941 - 6.81612i) q^{89} +(5.58512 - 2.03282i) q^{91} +(1.18092 + 6.69734i) q^{93} +(3.30541 + 1.94096i) q^{95} +(3.33022 - 0.587208i) q^{97} +(1.50000 + 4.12122i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{5} - 9 q^{7} - 9 q^{11} - 6 q^{13} - 6 q^{15} - 12 q^{17} - 18 q^{19} + 3 q^{21} - 3 q^{23} - 3 q^{27} - 6 q^{31} - 9 q^{33} - 12 q^{41} + 3 q^{45} - 39 q^{47} - 6 q^{49} - 3 q^{51} - 12 q^{53} + 9 q^{55} + 9 q^{57} - 12 q^{59} + 27 q^{61} - 3 q^{63} + 9 q^{65} - 36 q^{67} - 18 q^{71} - 9 q^{73} - 12 q^{75} + 18 q^{79} + 9 q^{83} + 27 q^{85} + 27 q^{87} + 3 q^{89} + 12 q^{91} + 24 q^{93} + 24 q^{95} - 3 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.766044 0.642788i 0.442276 0.371114i
\(4\) 0 0
\(5\) −0.826352 0.300767i −0.369556 0.134507i 0.150565 0.988600i \(-0.451891\pi\)
−0.520121 + 0.854093i \(0.674113\pi\)
\(6\) 0 0
\(7\) −0.907604 0.524005i −0.343042 0.198055i 0.318574 0.947898i \(-0.396796\pi\)
−0.661616 + 0.749842i \(0.730129\pi\)
\(8\) 0 0
\(9\) 0.173648 0.984808i 0.0578827 0.328269i
\(10\) 0 0
\(11\) −3.79813 + 2.19285i −1.14518 + 0.661170i −0.947708 0.319138i \(-0.896607\pi\)
−0.197472 + 0.980308i \(0.563273\pi\)
\(12\) 0 0
\(13\) −3.64543 + 4.34445i −1.01106 + 1.20493i −0.0323947 + 0.999475i \(0.510313\pi\)
−0.978666 + 0.205460i \(0.934131\pi\)
\(14\) 0 0
\(15\) −0.826352 + 0.300767i −0.213363 + 0.0776578i
\(16\) 0 0
\(17\) −0.539363 3.05888i −0.130815 0.741887i −0.977683 0.210085i \(-0.932626\pi\)
0.846868 0.531802i \(-0.178485\pi\)
\(18\) 0 0
\(19\) −4.28699 0.788496i −0.983503 0.180893i
\(20\) 0 0
\(21\) −1.03209 + 0.181985i −0.225220 + 0.0397124i
\(22\) 0 0
\(23\) −1.78699 4.90971i −0.372613 1.02375i −0.974347 0.225049i \(-0.927746\pi\)
0.601734 0.798696i \(-0.294477\pi\)
\(24\) 0 0
\(25\) −3.23783 2.71686i −0.647565 0.543372i
\(26\) 0 0
\(27\) −0.500000 0.866025i −0.0962250 0.166667i
\(28\) 0 0
\(29\) 6.30200 + 1.11121i 1.17025 + 0.206347i 0.724801 0.688959i \(-0.241932\pi\)
0.445452 + 0.895306i \(0.353043\pi\)
\(30\) 0 0
\(31\) −3.40033 + 5.88954i −0.610717 + 1.05779i 0.380402 + 0.924821i \(0.375785\pi\)
−0.991120 + 0.132972i \(0.957548\pi\)
\(32\) 0 0
\(33\) −1.50000 + 4.12122i −0.261116 + 0.717412i
\(34\) 0 0
\(35\) 0.592396 + 0.705990i 0.100133 + 0.119334i
\(36\) 0 0
\(37\) 9.58186i 1.57525i 0.616156 + 0.787624i \(0.288689\pi\)
−0.616156 + 0.787624i \(0.711311\pi\)
\(38\) 0 0
\(39\) 5.67128i 0.908132i
\(40\) 0 0
\(41\) 4.50640 + 5.37051i 0.703781 + 0.838733i 0.992949 0.118545i \(-0.0378230\pi\)
−0.289168 + 0.957278i \(0.593379\pi\)
\(42\) 0 0
\(43\) 1.63429 4.49016i 0.249226 0.684743i −0.750489 0.660883i \(-0.770182\pi\)
0.999715 0.0238605i \(-0.00759575\pi\)
\(44\) 0 0
\(45\) −0.439693 + 0.761570i −0.0655455 + 0.113528i
\(46\) 0 0
\(47\) −11.1985 1.97459i −1.63346 0.288024i −0.719705 0.694280i \(-0.755723\pi\)
−0.913759 + 0.406257i \(0.866834\pi\)
\(48\) 0 0
\(49\) −2.95084 5.11100i −0.421548 0.730143i
\(50\) 0 0
\(51\) −2.37939 1.99654i −0.333181 0.279572i
\(52\) 0 0
\(53\) 1.83022 + 5.02849i 0.251400 + 0.690717i 0.999628 + 0.0272747i \(0.00868288\pi\)
−0.748228 + 0.663442i \(0.769095\pi\)
\(54\) 0 0
\(55\) 3.79813 0.669713i 0.512140 0.0903041i
\(56\) 0 0
\(57\) −3.79086 + 2.15160i −0.502112 + 0.284986i
\(58\) 0 0
\(59\) −2.62314 14.8766i −0.341504 1.93676i −0.349866 0.936800i \(-0.613773\pi\)
0.00836244 0.999965i \(-0.497338\pi\)
\(60\) 0 0
\(61\) 7.56418 2.75314i 0.968494 0.352503i 0.191137 0.981563i \(-0.438782\pi\)
0.777356 + 0.629060i \(0.216560\pi\)
\(62\) 0 0
\(63\) −0.673648 + 0.802823i −0.0848717 + 0.101146i
\(64\) 0 0
\(65\) 4.31908 2.49362i 0.535716 0.309296i
\(66\) 0 0
\(67\) −0.709141 + 4.02174i −0.0866353 + 0.491333i 0.910356 + 0.413825i \(0.135808\pi\)
−0.996992 + 0.0775082i \(0.975304\pi\)
\(68\) 0 0
\(69\) −4.52481 2.61240i −0.544724 0.314496i
\(70\) 0 0
\(71\) −3.59240 1.30753i −0.426339 0.155175i 0.119934 0.992782i \(-0.461732\pi\)
−0.546273 + 0.837607i \(0.683954\pi\)
\(72\) 0 0
\(73\) −10.4474 + 8.76644i −1.22278 + 1.02603i −0.224105 + 0.974565i \(0.571946\pi\)
−0.998675 + 0.0514689i \(0.983610\pi\)
\(74\) 0 0
\(75\) −4.22668 −0.488055
\(76\) 0 0
\(77\) 4.59627 0.523793
\(78\) 0 0
\(79\) −3.54710 + 2.97637i −0.399080 + 0.334868i −0.820138 0.572166i \(-0.806103\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(80\) 0 0
\(81\) −0.939693 0.342020i −0.104410 0.0380022i
\(82\) 0 0
\(83\) 6.02481 + 3.47843i 0.661309 + 0.381807i 0.792776 0.609514i \(-0.208635\pi\)
−0.131466 + 0.991321i \(0.541969\pi\)
\(84\) 0 0
\(85\) −0.474308 + 2.68993i −0.0514459 + 0.291764i
\(86\) 0 0
\(87\) 5.54189 3.19961i 0.594153 0.343034i
\(88\) 0 0
\(89\) 5.71941 6.81612i 0.606256 0.722508i −0.372386 0.928078i \(-0.621460\pi\)
0.978642 + 0.205570i \(0.0659048\pi\)
\(90\) 0 0
\(91\) 5.58512 2.03282i 0.585480 0.213097i
\(92\) 0 0
\(93\) 1.18092 + 6.69734i 0.122456 + 0.694482i
\(94\) 0 0
\(95\) 3.30541 + 1.94096i 0.339128 + 0.199138i
\(96\) 0 0
\(97\) 3.33022 0.587208i 0.338133 0.0596219i −0.00200373 0.999998i \(-0.500638\pi\)
0.340137 + 0.940376i \(0.389527\pi\)
\(98\) 0 0
\(99\) 1.50000 + 4.12122i 0.150756 + 0.414198i
\(100\) 0 0
\(101\) −7.74376 6.49778i −0.770532 0.646554i 0.170313 0.985390i \(-0.445522\pi\)
−0.940845 + 0.338837i \(0.889967\pi\)
\(102\) 0 0
\(103\) −3.61334 6.25849i −0.356033 0.616667i 0.631261 0.775570i \(-0.282538\pi\)
−0.987294 + 0.158903i \(0.949204\pi\)
\(104\) 0 0
\(105\) 0.907604 + 0.160035i 0.0885731 + 0.0156178i
\(106\) 0 0
\(107\) −3.00000 + 5.19615i −0.290021 + 0.502331i −0.973814 0.227345i \(-0.926996\pi\)
0.683793 + 0.729676i \(0.260329\pi\)
\(108\) 0 0
\(109\) −2.74123 + 7.53147i −0.262562 + 0.721384i 0.736431 + 0.676513i \(0.236510\pi\)
−0.998993 + 0.0448706i \(0.985712\pi\)
\(110\) 0 0
\(111\) 6.15910 + 7.34013i 0.584596 + 0.696694i
\(112\) 0 0
\(113\) 20.4241i 1.92134i −0.277697 0.960669i \(-0.589571\pi\)
0.277697 0.960669i \(-0.410429\pi\)
\(114\) 0 0
\(115\) 4.59462i 0.428450i
\(116\) 0 0
\(117\) 3.64543 + 4.34445i 0.337020 + 0.401645i
\(118\) 0 0
\(119\) −1.11334 + 3.05888i −0.102060 + 0.280407i
\(120\) 0 0
\(121\) 4.11721 7.13122i 0.374292 0.648293i
\(122\) 0 0
\(123\) 6.90420 + 1.21740i 0.622531 + 0.109769i
\(124\) 0 0
\(125\) 4.05690 + 7.02676i 0.362861 + 0.628493i
\(126\) 0 0
\(127\) 8.32888 + 6.98876i 0.739068 + 0.620152i 0.932587 0.360945i \(-0.117546\pi\)
−0.193519 + 0.981097i \(0.561990\pi\)
\(128\) 0 0
\(129\) −1.63429 4.49016i −0.143891 0.395337i
\(130\) 0 0
\(131\) 4.63429 0.817150i 0.404899 0.0713947i 0.0325103 0.999471i \(-0.489650\pi\)
0.372389 + 0.928077i \(0.378539\pi\)
\(132\) 0 0
\(133\) 3.47771 + 2.96205i 0.301556 + 0.256842i
\(134\) 0 0
\(135\) 0.152704 + 0.866025i 0.0131426 + 0.0745356i
\(136\) 0 0
\(137\) 5.81908 2.11797i 0.497157 0.180950i −0.0812576 0.996693i \(-0.525894\pi\)
0.578415 + 0.815743i \(0.303671\pi\)
\(138\) 0 0
\(139\) 4.28446 5.10602i 0.363403 0.433087i −0.553100 0.833115i \(-0.686555\pi\)
0.916503 + 0.400028i \(0.131000\pi\)
\(140\) 0 0
\(141\) −9.84776 + 5.68561i −0.829331 + 0.478815i
\(142\) 0 0
\(143\) 4.31908 24.4947i 0.361179 2.04835i
\(144\) 0 0
\(145\) −4.87346 2.81369i −0.404718 0.233664i
\(146\) 0 0
\(147\) −5.54576 2.01849i −0.457407 0.166482i
\(148\) 0 0
\(149\) −0.190722 + 0.160035i −0.0156246 + 0.0131106i −0.650566 0.759449i \(-0.725468\pi\)
0.634942 + 0.772560i \(0.281024\pi\)
\(150\) 0 0
\(151\) −15.2199 −1.23858 −0.619288 0.785164i \(-0.712579\pi\)
−0.619288 + 0.785164i \(0.712579\pi\)
\(152\) 0 0
\(153\) −3.10607 −0.251111
\(154\) 0 0
\(155\) 4.58125 3.84413i 0.367975 0.308768i
\(156\) 0 0
\(157\) −10.6912 3.89127i −0.853250 0.310558i −0.121885 0.992544i \(-0.538894\pi\)
−0.731365 + 0.681987i \(0.761116\pi\)
\(158\) 0 0
\(159\) 4.63429 + 2.67561i 0.367523 + 0.212189i
\(160\) 0 0
\(161\) −0.950837 + 5.39246i −0.0749365 + 0.424986i
\(162\) 0 0
\(163\) −6.83157 + 3.94421i −0.535090 + 0.308934i −0.743087 0.669195i \(-0.766639\pi\)
0.207997 + 0.978129i \(0.433306\pi\)
\(164\) 0 0
\(165\) 2.47906 2.95442i 0.192994 0.230002i
\(166\) 0 0
\(167\) 1.65270 0.601535i 0.127890 0.0465482i −0.277282 0.960788i \(-0.589434\pi\)
0.405172 + 0.914240i \(0.367212\pi\)
\(168\) 0 0
\(169\) −3.32770 18.8723i −0.255977 1.45172i
\(170\) 0 0
\(171\) −1.52094 + 4.08494i −0.116310 + 0.312383i
\(172\) 0 0
\(173\) 3.76011 0.663010i 0.285876 0.0504077i −0.0288712 0.999583i \(-0.509191\pi\)
0.314747 + 0.949175i \(0.398080\pi\)
\(174\) 0 0
\(175\) 1.51501 + 4.16247i 0.114524 + 0.314653i
\(176\) 0 0
\(177\) −11.5719 9.70999i −0.869799 0.729848i
\(178\) 0 0
\(179\) 10.0039 + 17.3272i 0.747724 + 1.29510i 0.948911 + 0.315544i \(0.102187\pi\)
−0.201187 + 0.979553i \(0.564480\pi\)
\(180\) 0 0
\(181\) −7.42514 1.30925i −0.551907 0.0973160i −0.109262 0.994013i \(-0.534849\pi\)
−0.442645 + 0.896697i \(0.645960\pi\)
\(182\) 0 0
\(183\) 4.02481 6.97118i 0.297523 0.515325i
\(184\) 0 0
\(185\) 2.88191 7.91799i 0.211882 0.582142i
\(186\) 0 0
\(187\) 8.75624 + 10.4353i 0.640320 + 0.763104i
\(188\) 0 0
\(189\) 1.04801i 0.0762315i
\(190\) 0 0
\(191\) 5.01590i 0.362937i 0.983397 + 0.181469i \(0.0580851\pi\)
−0.983397 + 0.181469i \(0.941915\pi\)
\(192\) 0 0
\(193\) 3.82635 + 4.56007i 0.275427 + 0.328241i 0.885970 0.463742i \(-0.153493\pi\)
−0.610544 + 0.791983i \(0.709049\pi\)
\(194\) 0 0
\(195\) 1.70574 4.68647i 0.122150 0.335605i
\(196\) 0 0
\(197\) 12.0594 20.8875i 0.859199 1.48818i −0.0134954 0.999909i \(-0.504296\pi\)
0.872694 0.488267i \(-0.162371\pi\)
\(198\) 0 0
\(199\) 24.4859 + 4.31753i 1.73576 + 0.306061i 0.949949 0.312404i \(-0.101134\pi\)
0.785812 + 0.618466i \(0.212246\pi\)
\(200\) 0 0
\(201\) 2.04189 + 3.53666i 0.144024 + 0.249456i
\(202\) 0 0
\(203\) −5.13744 4.31082i −0.360578 0.302561i
\(204\) 0 0
\(205\) −2.10859 5.79331i −0.147271 0.404622i
\(206\) 0 0
\(207\) −5.14543 + 0.907278i −0.357632 + 0.0630602i
\(208\) 0 0
\(209\) 18.0116 6.40593i 1.24589 0.443107i
\(210\) 0 0
\(211\) −0.440570 2.49860i −0.0303301 0.172010i 0.965880 0.258990i \(-0.0833897\pi\)
−0.996210 + 0.0869795i \(0.972279\pi\)
\(212\) 0 0
\(213\) −3.59240 + 1.30753i −0.246147 + 0.0895902i
\(214\) 0 0
\(215\) −2.70099 + 3.21891i −0.184206 + 0.219528i
\(216\) 0 0
\(217\) 6.17230 3.56358i 0.419003 0.241912i
\(218\) 0 0
\(219\) −2.36824 + 13.4310i −0.160031 + 0.907580i
\(220\) 0 0
\(221\) 15.2554 + 8.80769i 1.02619 + 0.592469i
\(222\) 0 0
\(223\) 20.9884 + 7.63917i 1.40549 + 0.511557i 0.929803 0.368058i \(-0.119977\pi\)
0.475687 + 0.879614i \(0.342199\pi\)
\(224\) 0 0
\(225\) −3.23783 + 2.71686i −0.215855 + 0.181124i
\(226\) 0 0
\(227\) −2.58172 −0.171355 −0.0856773 0.996323i \(-0.527305\pi\)
−0.0856773 + 0.996323i \(0.527305\pi\)
\(228\) 0 0
\(229\) −5.16756 −0.341482 −0.170741 0.985316i \(-0.554616\pi\)
−0.170741 + 0.985316i \(0.554616\pi\)
\(230\) 0 0
\(231\) 3.52094 2.95442i 0.231661 0.194387i
\(232\) 0 0
\(233\) 8.12361 + 2.95675i 0.532195 + 0.193703i 0.594118 0.804378i \(-0.297501\pi\)
−0.0619230 + 0.998081i \(0.519723\pi\)
\(234\) 0 0
\(235\) 8.65998 + 4.99984i 0.564915 + 0.326154i
\(236\) 0 0
\(237\) −0.804063 + 4.56007i −0.0522295 + 0.296208i
\(238\) 0 0
\(239\) −3.04664 + 1.75898i −0.197071 + 0.113779i −0.595288 0.803512i \(-0.702962\pi\)
0.398218 + 0.917291i \(0.369629\pi\)
\(240\) 0 0
\(241\) −9.78611 + 11.6626i −0.630379 + 0.751256i −0.982818 0.184579i \(-0.940908\pi\)
0.352439 + 0.935835i \(0.385352\pi\)
\(242\) 0 0
\(243\) −0.939693 + 0.342020i −0.0602813 + 0.0219406i
\(244\) 0 0
\(245\) 0.901207 + 5.11100i 0.0575760 + 0.326530i
\(246\) 0 0
\(247\) 19.0535 15.7502i 1.21235 1.00216i
\(248\) 0 0
\(249\) 6.85117 1.20805i 0.434175 0.0765568i
\(250\) 0 0
\(251\) −5.95218 16.3535i −0.375698 1.03222i −0.973121 0.230296i \(-0.926031\pi\)
0.597422 0.801927i \(-0.296192\pi\)
\(252\) 0 0
\(253\) 17.5535 + 14.7291i 1.10358 + 0.926013i
\(254\) 0 0
\(255\) 1.36571 + 2.36549i 0.0855244 + 0.148133i
\(256\) 0 0
\(257\) −31.1976 5.50098i −1.94605 0.343141i −0.999806 0.0196932i \(-0.993731\pi\)
−0.946246 0.323448i \(-0.895158\pi\)
\(258\) 0 0
\(259\) 5.02094 8.69653i 0.311986 0.540376i
\(260\) 0 0
\(261\) 2.18866 6.01330i 0.135475 0.372214i
\(262\) 0 0
\(263\) 0.985452 + 1.17442i 0.0607656 + 0.0724176i 0.795571 0.605860i \(-0.207171\pi\)
−0.734805 + 0.678278i \(0.762727\pi\)
\(264\) 0 0
\(265\) 4.70578i 0.289074i
\(266\) 0 0
\(267\) 8.89782i 0.544538i
\(268\) 0 0
\(269\) 14.9932 + 17.8682i 0.914151 + 1.08944i 0.995687 + 0.0927725i \(0.0295729\pi\)
−0.0815362 + 0.996670i \(0.525983\pi\)
\(270\) 0 0
\(271\) 7.75237 21.2995i 0.470923 1.29385i −0.446088 0.894989i \(-0.647183\pi\)
0.917011 0.398862i \(-0.130595\pi\)
\(272\) 0 0
\(273\) 2.97178 5.14728i 0.179860 0.311527i
\(274\) 0 0
\(275\) 18.2554 + 3.21891i 1.10084 + 0.194108i
\(276\) 0 0
\(277\) −10.7947 18.6970i −0.648592 1.12340i −0.983459 0.181130i \(-0.942025\pi\)
0.334867 0.942265i \(-0.391309\pi\)
\(278\) 0 0
\(279\) 5.20961 + 4.37138i 0.311891 + 0.261708i
\(280\) 0 0
\(281\) −9.31299 25.5872i −0.555566 1.52641i −0.826002 0.563667i \(-0.809390\pi\)
0.270436 0.962738i \(-0.412832\pi\)
\(282\) 0 0
\(283\) −25.2788 + 4.45734i −1.50267 + 0.264961i −0.863597 0.504184i \(-0.831794\pi\)
−0.639074 + 0.769145i \(0.720682\pi\)
\(284\) 0 0
\(285\) 3.77972 0.637812i 0.223891 0.0377807i
\(286\) 0 0
\(287\) −1.27584 7.23567i −0.0753107 0.427108i
\(288\) 0 0
\(289\) 6.90895 2.51465i 0.406409 0.147921i
\(290\) 0 0
\(291\) 2.17365 2.59045i 0.127422 0.151855i
\(292\) 0 0
\(293\) −16.7087 + 9.64679i −0.976134 + 0.563571i −0.901101 0.433610i \(-0.857240\pi\)
−0.0750336 + 0.997181i \(0.523906\pi\)
\(294\) 0 0
\(295\) −2.30675 + 13.0822i −0.134304 + 0.761677i
\(296\) 0 0
\(297\) 3.79813 + 2.19285i 0.220390 + 0.127242i
\(298\) 0 0
\(299\) 27.8444 + 10.1345i 1.61028 + 0.586094i
\(300\) 0 0
\(301\) −3.83615 + 3.21891i −0.221112 + 0.185535i
\(302\) 0 0
\(303\) −10.1088 −0.580733
\(304\) 0 0
\(305\) −7.07873 −0.405327
\(306\) 0 0
\(307\) 2.83615 2.37981i 0.161868 0.135823i −0.558256 0.829669i \(-0.688529\pi\)
0.720124 + 0.693845i \(0.244085\pi\)
\(308\) 0 0
\(309\) −6.79086 2.47167i −0.386318 0.140608i
\(310\) 0 0
\(311\) −14.7724 8.52887i −0.837668 0.483628i 0.0188027 0.999823i \(-0.494015\pi\)
−0.856471 + 0.516195i \(0.827348\pi\)
\(312\) 0 0
\(313\) −5.35962 + 30.3959i −0.302944 + 1.71808i 0.330085 + 0.943951i \(0.392923\pi\)
−0.633029 + 0.774128i \(0.718188\pi\)
\(314\) 0 0
\(315\) 0.798133 0.460802i 0.0449697 0.0259633i
\(316\) 0 0
\(317\) −13.9329 + 16.6046i −0.782549 + 0.932605i −0.999046 0.0436780i \(-0.986092\pi\)
0.216497 + 0.976283i \(0.430537\pi\)
\(318\) 0 0
\(319\) −26.3726 + 9.59883i −1.47658 + 0.537431i
\(320\) 0 0
\(321\) 1.04189 + 5.90885i 0.0581526 + 0.329800i
\(322\) 0 0
\(323\) −0.0996702 + 13.5387i −0.00554580 + 0.753311i
\(324\) 0 0
\(325\) 23.6065 4.16247i 1.30945 0.230892i
\(326\) 0 0
\(327\) 2.74123 + 7.53147i 0.151590 + 0.416491i
\(328\) 0 0
\(329\) 9.12907 + 7.66020i 0.503302 + 0.422320i
\(330\) 0 0
\(331\) −1.32383 2.29293i −0.0727640 0.126031i 0.827348 0.561690i \(-0.189849\pi\)
−0.900112 + 0.435659i \(0.856515\pi\)
\(332\) 0 0
\(333\) 9.43629 + 1.66387i 0.517105 + 0.0911796i
\(334\) 0 0
\(335\) 1.79561 3.11008i 0.0981045 0.169922i
\(336\) 0 0
\(337\) 8.05556 22.1325i 0.438814 1.20563i −0.501450 0.865187i \(-0.667200\pi\)
0.940264 0.340446i \(-0.110578\pi\)
\(338\) 0 0
\(339\) −13.1284 15.6458i −0.713034 0.849761i
\(340\) 0 0
\(341\) 29.8257i 1.61515i
\(342\) 0 0
\(343\) 13.5211i 0.730070i
\(344\) 0 0
\(345\) 2.95336 + 3.51968i 0.159004 + 0.189493i
\(346\) 0 0
\(347\) −5.43923 + 14.9442i −0.291993 + 0.802244i 0.703782 + 0.710416i \(0.251493\pi\)
−0.995775 + 0.0918280i \(0.970729\pi\)
\(348\) 0 0
\(349\) −8.73917 + 15.1367i −0.467797 + 0.810248i −0.999323 0.0367938i \(-0.988286\pi\)
0.531526 + 0.847042i \(0.321619\pi\)
\(350\) 0 0
\(351\) 5.58512 + 0.984808i 0.298112 + 0.0525651i
\(352\) 0 0
\(353\) −5.56552 9.63977i −0.296223 0.513073i 0.679046 0.734096i \(-0.262394\pi\)
−0.975269 + 0.221023i \(0.929060\pi\)
\(354\) 0 0
\(355\) 2.57532 + 2.16095i 0.136684 + 0.114691i
\(356\) 0 0
\(357\) 1.11334 + 3.05888i 0.0589242 + 0.161893i
\(358\) 0 0
\(359\) 7.24304 1.27714i 0.382273 0.0674051i 0.0207903 0.999784i \(-0.493382\pi\)
0.361483 + 0.932379i \(0.382271\pi\)
\(360\) 0 0
\(361\) 17.7565 + 6.76055i 0.934555 + 0.355818i
\(362\) 0 0
\(363\) −1.42989 8.10932i −0.0750499 0.425629i
\(364\) 0 0
\(365\) 11.2699 4.10191i 0.589894 0.214704i
\(366\) 0 0
\(367\) 0.682266 0.813093i 0.0356140 0.0424431i −0.747943 0.663762i \(-0.768959\pi\)
0.783557 + 0.621319i \(0.213403\pi\)
\(368\) 0 0
\(369\) 6.07145 3.50535i 0.316067 0.182481i
\(370\) 0 0
\(371\) 0.973841 5.52293i 0.0505593 0.286736i
\(372\) 0 0
\(373\) −16.6723 9.62576i −0.863259 0.498403i 0.00184317 0.999998i \(-0.499413\pi\)
−0.865102 + 0.501595i \(0.832747\pi\)
\(374\) 0 0
\(375\) 7.62449 + 2.77509i 0.393727 + 0.143305i
\(376\) 0 0
\(377\) −27.8011 + 23.3279i −1.43183 + 1.20145i
\(378\) 0 0
\(379\) −21.3651 −1.09745 −0.548727 0.836002i \(-0.684887\pi\)
−0.548727 + 0.836002i \(0.684887\pi\)
\(380\) 0 0
\(381\) 10.8726 0.557019
\(382\) 0 0
\(383\) −12.8027 + 10.7428i −0.654188 + 0.548929i −0.908339 0.418236i \(-0.862649\pi\)
0.254150 + 0.967165i \(0.418204\pi\)
\(384\) 0 0
\(385\) −3.79813 1.38241i −0.193571 0.0704540i
\(386\) 0 0
\(387\) −4.13816 2.38917i −0.210354 0.121448i
\(388\) 0 0
\(389\) −1.17799 + 6.68069i −0.0597262 + 0.338724i −0.999999 0.00169891i \(-0.999459\pi\)
0.940272 + 0.340423i \(0.110570\pi\)
\(390\) 0 0
\(391\) −14.0544 + 8.11430i −0.710760 + 0.410358i
\(392\) 0 0
\(393\) 3.02481 3.60483i 0.152582 0.181840i
\(394\) 0 0
\(395\) 3.82635 1.39268i 0.192525 0.0700732i
\(396\) 0 0
\(397\) 3.76857 + 21.3726i 0.189139 + 1.07266i 0.920521 + 0.390693i \(0.127765\pi\)
−0.731382 + 0.681968i \(0.761124\pi\)
\(398\) 0 0
\(399\) 4.56805 + 0.0336295i 0.228688 + 0.00168358i
\(400\) 0 0
\(401\) 9.63223 1.69842i 0.481010 0.0848151i 0.0721149 0.997396i \(-0.477025\pi\)
0.408895 + 0.912581i \(0.365914\pi\)
\(402\) 0 0
\(403\) −13.1912 36.2425i −0.657100 1.80537i
\(404\) 0 0
\(405\) 0.673648 + 0.565258i 0.0334738 + 0.0280879i
\(406\) 0 0
\(407\) −21.0116 36.3932i −1.04151 1.80394i
\(408\) 0 0
\(409\) −1.51842 0.267738i −0.0750810 0.0132388i 0.135981 0.990711i \(-0.456581\pi\)
−0.211062 + 0.977473i \(0.567692\pi\)
\(410\) 0 0
\(411\) 3.09627 5.36289i 0.152728 0.264532i
\(412\) 0 0
\(413\) −5.41463 + 14.8766i −0.266436 + 0.732028i
\(414\) 0 0
\(415\) −3.93242 4.68647i −0.193035 0.230050i
\(416\) 0 0
\(417\) 6.66544i 0.326408i
\(418\) 0 0
\(419\) 14.1279i 0.690194i 0.938567 + 0.345097i \(0.112154\pi\)
−0.938567 + 0.345097i \(0.887846\pi\)
\(420\) 0 0
\(421\) 10.3285 + 12.3090i 0.503379 + 0.599903i 0.956567 0.291511i \(-0.0941581\pi\)
−0.453189 + 0.891415i \(0.649714\pi\)
\(422\) 0 0
\(423\) −3.88919 + 10.6854i −0.189099 + 0.519544i
\(424\) 0 0
\(425\) −6.56418 + 11.3695i −0.318409 + 0.551501i
\(426\) 0 0
\(427\) −8.30793 1.46491i −0.402049 0.0708921i
\(428\) 0 0
\(429\) −12.4363 21.5403i −0.600430 1.03997i
\(430\) 0 0
\(431\) −10.0155 8.40399i −0.482429 0.404806i 0.368875 0.929479i \(-0.379743\pi\)
−0.851304 + 0.524673i \(0.824188\pi\)
\(432\) 0 0
\(433\) −0.853388 2.34466i −0.0410112 0.112677i 0.917497 0.397744i \(-0.130207\pi\)
−0.958508 + 0.285066i \(0.907984\pi\)
\(434\) 0 0
\(435\) −5.54189 + 0.977185i −0.265713 + 0.0468524i
\(436\) 0 0
\(437\) 3.78952 + 22.4569i 0.181277 + 1.07426i
\(438\) 0 0
\(439\) 1.02616 + 5.81964i 0.0489759 + 0.277756i 0.999454 0.0330348i \(-0.0105172\pi\)
−0.950478 + 0.310791i \(0.899406\pi\)
\(440\) 0 0
\(441\) −5.54576 + 2.01849i −0.264084 + 0.0961186i
\(442\) 0 0
\(443\) −10.4838 + 12.4941i −0.498101 + 0.593613i −0.955258 0.295772i \(-0.904423\pi\)
0.457158 + 0.889386i \(0.348867\pi\)
\(444\) 0 0
\(445\) −6.77631 + 3.91231i −0.321228 + 0.185461i
\(446\) 0 0
\(447\) −0.0432332 + 0.245188i −0.00204486 + 0.0115970i
\(448\) 0 0
\(449\) 12.6218 + 7.28720i 0.595659 + 0.343904i 0.767332 0.641250i \(-0.221584\pi\)
−0.171673 + 0.985154i \(0.554917\pi\)
\(450\) 0 0
\(451\) −28.8926 10.5161i −1.36050 0.495182i
\(452\) 0 0
\(453\) −11.6591 + 9.78315i −0.547792 + 0.459652i
\(454\) 0 0
\(455\) −5.22668 −0.245031
\(456\) 0 0
\(457\) 11.9813 0.560463 0.280232 0.959932i \(-0.409589\pi\)
0.280232 + 0.959932i \(0.409589\pi\)
\(458\) 0 0
\(459\) −2.37939 + 1.99654i −0.111060 + 0.0931906i
\(460\) 0 0
\(461\) −28.0719 10.2173i −1.30744 0.475869i −0.408027 0.912970i \(-0.633783\pi\)
−0.899413 + 0.437101i \(0.856005\pi\)
\(462\) 0 0
\(463\) −11.5079 6.64409i −0.534818 0.308777i 0.208158 0.978095i \(-0.433253\pi\)
−0.742976 + 0.669318i \(0.766586\pi\)
\(464\) 0 0
\(465\) 1.03849 5.88954i 0.0481586 0.273121i
\(466\) 0 0
\(467\) −19.3571 + 11.1758i −0.895740 + 0.517156i −0.875816 0.482646i \(-0.839676\pi\)
−0.0199241 + 0.999801i \(0.506342\pi\)
\(468\) 0 0
\(469\) 2.75103 3.27855i 0.127031 0.151389i
\(470\) 0 0
\(471\) −10.6912 + 3.89127i −0.492624 + 0.179300i
\(472\) 0 0
\(473\) 3.63903 + 20.6380i 0.167323 + 0.948935i
\(474\) 0 0
\(475\) 11.7383 + 14.2002i 0.538590 + 0.651548i
\(476\) 0 0
\(477\) 5.26991 0.929228i 0.241293 0.0425464i
\(478\) 0 0
\(479\) −4.16234 11.4359i −0.190182 0.522521i 0.807552 0.589796i \(-0.200792\pi\)
−0.997734 + 0.0672746i \(0.978570\pi\)
\(480\) 0 0
\(481\) −41.6279 34.9300i −1.89807 1.59267i
\(482\) 0 0
\(483\) 2.73783 + 4.74205i 0.124575 + 0.215771i
\(484\) 0 0
\(485\) −2.92855 0.516382i −0.132979 0.0234477i
\(486\) 0 0
\(487\) 4.92380 8.52827i 0.223119 0.386453i −0.732635 0.680622i \(-0.761710\pi\)
0.955753 + 0.294169i \(0.0950429\pi\)
\(488\) 0 0
\(489\) −2.69800 + 7.41268i −0.122008 + 0.335213i
\(490\) 0 0
\(491\) 25.2973 + 30.1481i 1.14165 + 1.36056i 0.923018 + 0.384757i \(0.125715\pi\)
0.218631 + 0.975808i \(0.429841\pi\)
\(492\) 0 0
\(493\) 19.8764i 0.895189i
\(494\) 0 0
\(495\) 3.85673i 0.173347i
\(496\) 0 0
\(497\) 2.57532 + 3.06915i 0.115519 + 0.137670i
\(498\) 0 0
\(499\) −5.79544 + 15.9229i −0.259440 + 0.712805i 0.739762 + 0.672868i \(0.234938\pi\)
−0.999202 + 0.0399367i \(0.987284\pi\)
\(500\) 0 0
\(501\) 0.879385 1.52314i 0.0392880 0.0680489i
\(502\) 0 0
\(503\) −12.4547 2.19610i −0.555328 0.0979193i −0.111061 0.993814i \(-0.535425\pi\)
−0.444267 + 0.895894i \(0.646536\pi\)
\(504\) 0 0
\(505\) 4.44475 + 7.69852i 0.197789 + 0.342580i
\(506\) 0 0
\(507\) −14.6800 12.3180i −0.651964 0.547062i
\(508\) 0 0
\(509\) −0.703678 1.93334i −0.0311900 0.0856937i 0.923120 0.384511i \(-0.125630\pi\)
−0.954310 + 0.298818i \(0.903408\pi\)
\(510\) 0 0
\(511\) 14.0758 2.48194i 0.622676 0.109795i
\(512\) 0 0
\(513\) 1.46064 + 4.10689i 0.0644887 + 0.181324i
\(514\) 0 0
\(515\) 1.10354 + 6.25849i 0.0486278 + 0.275782i
\(516\) 0 0
\(517\) 46.8632 17.0568i 2.06104 0.750158i
\(518\) 0 0
\(519\) 2.45424 2.92485i 0.107729 0.128387i
\(520\) 0 0
\(521\) −25.8460 + 14.9222i −1.13233 + 0.653753i −0.944521 0.328451i \(-0.893473\pi\)
−0.187813 + 0.982205i \(0.560140\pi\)
\(522\) 0 0
\(523\) 6.18644 35.0851i 0.270514 1.53416i −0.482345 0.875981i \(-0.660215\pi\)
0.752860 0.658181i \(-0.228674\pi\)
\(524\) 0 0
\(525\) 3.83615 + 2.21480i 0.167423 + 0.0966619i
\(526\) 0 0
\(527\) 19.8494 + 7.22460i 0.864654 + 0.314708i
\(528\) 0 0
\(529\) −3.29292 + 2.76309i −0.143170 + 0.120134i
\(530\) 0 0
\(531\) −15.1061 −0.655547
\(532\) 0 0
\(533\) −39.7597 −1.72218
\(534\) 0 0
\(535\) 4.04189 3.39155i 0.174746 0.146629i
\(536\) 0 0
\(537\) 18.8011 + 6.84305i 0.811328 + 0.295299i
\(538\) 0 0
\(539\) 22.4153 + 12.9415i 0.965497 + 0.557430i
\(540\) 0 0
\(541\) −1.92097 + 10.8944i −0.0825889 + 0.468385i 0.915262 + 0.402859i \(0.131984\pi\)
−0.997851 + 0.0655259i \(0.979128\pi\)
\(542\) 0 0
\(543\) −6.52956 + 3.76984i −0.280210 + 0.161780i
\(544\) 0 0
\(545\) 4.53044 5.39917i 0.194063 0.231275i
\(546\) 0 0
\(547\) −25.0398 + 9.11375i −1.07063 + 0.389676i −0.816411 0.577472i \(-0.804039\pi\)
−0.254215 + 0.967148i \(0.581817\pi\)
\(548\) 0 0
\(549\) −1.39780 7.92734i −0.0596568 0.338331i
\(550\) 0 0
\(551\) −26.1404 9.73286i −1.11362 0.414634i
\(552\) 0 0
\(553\) 4.77900 0.842667i 0.203224 0.0358338i
\(554\) 0 0
\(555\) −2.88191 7.91799i −0.122330 0.336100i
\(556\) 0 0
\(557\) −22.6655 19.0186i −0.960368 0.805844i 0.0206450 0.999787i \(-0.493428\pi\)
−0.981013 + 0.193942i \(0.937872\pi\)
\(558\) 0 0
\(559\) 13.5496 + 23.4686i 0.573088 + 0.992618i
\(560\) 0 0
\(561\) 13.4153 + 2.36549i 0.566396 + 0.0998709i
\(562\) 0 0
\(563\) −6.79473 + 11.7688i −0.286364 + 0.495997i −0.972939 0.231062i \(-0.925780\pi\)
0.686575 + 0.727059i \(0.259113\pi\)
\(564\) 0 0
\(565\) −6.14290 + 16.8775i −0.258434 + 0.710041i
\(566\) 0 0
\(567\) 0.673648 + 0.802823i 0.0282906 + 0.0337154i
\(568\) 0 0
\(569\) 28.9965i 1.21560i 0.794091 + 0.607799i \(0.207947\pi\)
−0.794091 + 0.607799i \(0.792053\pi\)
\(570\) 0 0
\(571\) 16.3619i 0.684725i −0.939568 0.342362i \(-0.888773\pi\)
0.939568 0.342362i \(-0.111227\pi\)
\(572\) 0 0
\(573\) 3.22416 + 3.84240i 0.134691 + 0.160519i
\(574\) 0 0
\(575\) −7.55303 + 20.7518i −0.314983 + 0.865409i
\(576\) 0 0
\(577\) −1.09920 + 1.90388i −0.0457604 + 0.0792594i −0.887998 0.459847i \(-0.847904\pi\)
0.842238 + 0.539106i \(0.181238\pi\)
\(578\) 0 0
\(579\) 5.86231 + 1.03368i 0.243629 + 0.0429584i
\(580\) 0 0
\(581\) −3.64543 6.31407i −0.151238 0.261952i
\(582\) 0 0
\(583\) −17.9782 15.0855i −0.744580 0.624777i
\(584\) 0 0
\(585\) −1.70574 4.68647i −0.0705235 0.193762i
\(586\) 0 0
\(587\) −25.9231 + 4.57094i −1.06996 + 0.188663i −0.680772 0.732495i \(-0.738356\pi\)
−0.389188 + 0.921158i \(0.627244\pi\)
\(588\) 0 0
\(589\) 19.2211 22.5673i 0.791990 0.929868i
\(590\) 0 0
\(591\) −4.18820 23.7524i −0.172279 0.977045i
\(592\) 0 0
\(593\) −5.84167 + 2.12619i −0.239889 + 0.0873123i −0.459167 0.888350i \(-0.651852\pi\)
0.219278 + 0.975662i \(0.429630\pi\)
\(594\) 0 0
\(595\) 1.84002 2.19285i 0.0754336 0.0898982i
\(596\) 0 0
\(597\) 21.5326 12.4318i 0.881269 0.508801i
\(598\) 0 0
\(599\) 1.34952 7.65350i 0.0551398 0.312713i −0.944746 0.327802i \(-0.893692\pi\)
0.999886 + 0.0150889i \(0.00480313\pi\)
\(600\) 0 0
\(601\) 1.34090 + 0.774169i 0.0546964 + 0.0315790i 0.527099 0.849804i \(-0.323280\pi\)
−0.472402 + 0.881383i \(0.656613\pi\)
\(602\) 0 0
\(603\) 3.83750 + 1.39673i 0.156275 + 0.0568794i
\(604\) 0 0
\(605\) −5.54710 + 4.65457i −0.225522 + 0.189235i
\(606\) 0 0
\(607\) −2.46110 −0.0998931 −0.0499466 0.998752i \(-0.515905\pi\)
−0.0499466 + 0.998752i \(0.515905\pi\)
\(608\) 0 0
\(609\) −6.70645 −0.271759
\(610\) 0 0
\(611\) 49.4017 41.4530i 1.99858 1.67701i
\(612\) 0 0
\(613\) −12.1686 4.42901i −0.491485 0.178886i 0.0843753 0.996434i \(-0.473111\pi\)
−0.575860 + 0.817548i \(0.695333\pi\)
\(614\) 0 0
\(615\) −5.33915 3.08256i −0.215295 0.124301i
\(616\) 0 0
\(617\) −7.56805 + 42.9205i −0.304678 + 1.72792i 0.320338 + 0.947303i \(0.396204\pi\)
−0.625016 + 0.780612i \(0.714907\pi\)
\(618\) 0 0
\(619\) 22.0956 12.7569i 0.888095 0.512742i 0.0147762 0.999891i \(-0.495296\pi\)
0.873319 + 0.487149i \(0.161963\pi\)
\(620\) 0 0
\(621\) −3.35844 + 4.00243i −0.134770 + 0.160612i
\(622\) 0 0
\(623\) −8.76264 + 3.18934i −0.351068 + 0.127778i
\(624\) 0 0
\(625\) 2.43077 + 13.7856i 0.0972308 + 0.551423i
\(626\) 0 0
\(627\) 9.68004 16.4849i 0.386584 0.658342i
\(628\) 0 0
\(629\) 29.3097 5.16810i 1.16866 0.206066i
\(630\) 0 0
\(631\) 8.07057 + 22.1737i 0.321284 + 0.882722i 0.990234 + 0.139414i \(0.0445219\pi\)
−0.668950 + 0.743308i \(0.733256\pi\)
\(632\) 0 0
\(633\) −1.94356 1.63084i −0.0772497 0.0648202i
\(634\) 0 0
\(635\) −4.78059 8.28023i −0.189712 0.328591i
\(636\) 0 0
\(637\) 32.9616 + 5.81201i 1.30598 + 0.230280i
\(638\) 0 0
\(639\) −1.91147 + 3.31077i −0.0756167 + 0.130972i
\(640\) 0 0
\(641\) −9.47272 + 26.0261i −0.374150 + 1.02797i 0.599591 + 0.800307i \(0.295330\pi\)
−0.973740 + 0.227661i \(0.926892\pi\)
\(642\) 0 0
\(643\) 15.3833 + 18.3331i 0.606656 + 0.722985i 0.978715 0.205224i \(-0.0657924\pi\)
−0.372059 + 0.928209i \(0.621348\pi\)
\(644\) 0 0
\(645\) 4.20199i 0.165453i
\(646\) 0 0
\(647\) 2.85382i 0.112195i −0.998425 0.0560975i \(-0.982134\pi\)
0.998425 0.0560975i \(-0.0178658\pi\)
\(648\) 0 0
\(649\) 42.5852 + 50.7510i 1.67161 + 1.99215i
\(650\) 0 0
\(651\) 2.43763 6.69734i 0.0955384 0.262490i
\(652\) 0 0
\(653\) 3.06283 5.30498i 0.119858 0.207600i −0.799853 0.600196i \(-0.795089\pi\)
0.919711 + 0.392596i \(0.128423\pi\)
\(654\) 0 0
\(655\) −4.07532 0.718589i −0.159236 0.0280776i
\(656\) 0 0
\(657\) 6.81908 + 11.8110i 0.266038 + 0.460791i
\(658\) 0 0
\(659\) 35.0009 + 29.3693i 1.36344 + 1.14406i 0.974902 + 0.222633i \(0.0714650\pi\)
0.388540 + 0.921432i \(0.372979\pi\)
\(660\) 0 0
\(661\) −11.3858 31.2822i −0.442856 1.21674i −0.937606 0.347700i \(-0.886963\pi\)
0.494750 0.869035i \(-0.335260\pi\)
\(662\) 0 0
\(663\) 17.3478 3.05888i 0.673731 0.118797i
\(664\) 0 0
\(665\) −1.98293 3.49367i −0.0768946 0.135479i
\(666\) 0 0
\(667\) −5.80587 32.9267i −0.224804 1.27493i
\(668\) 0 0
\(669\) 20.9884 7.63917i 0.811460 0.295347i
\(670\) 0 0
\(671\) −22.6925 + 27.0439i −0.876036 + 1.04402i
\(672\) 0 0
\(673\) 22.1716 12.8008i 0.854652 0.493434i −0.00756578 0.999971i \(-0.502408\pi\)
0.862218 + 0.506538i \(0.169075\pi\)
\(674\) 0 0
\(675\) −0.733956 + 4.16247i −0.0282500 + 0.160213i
\(676\) 0 0
\(677\) −18.0831 10.4403i −0.694989 0.401252i 0.110490 0.993877i \(-0.464758\pi\)
−0.805478 + 0.592625i \(0.798091\pi\)
\(678\) 0 0
\(679\) −3.33022 1.21210i −0.127802 0.0465162i
\(680\) 0 0
\(681\) −1.97771 + 1.65950i −0.0757861 + 0.0635921i
\(682\) 0 0
\(683\) −3.65808 −0.139973 −0.0699863 0.997548i \(-0.522296\pi\)
−0.0699863 + 0.997548i \(0.522296\pi\)
\(684\) 0 0
\(685\) −5.44562 −0.208067
\(686\) 0 0
\(687\) −3.95858 + 3.32164i −0.151029 + 0.126729i
\(688\) 0 0
\(689\) −28.5180 10.3797i −1.08645 0.395435i
\(690\) 0 0
\(691\) −38.9097 22.4645i −1.48020 0.854591i −0.480447 0.877024i \(-0.659526\pi\)
−0.999748 + 0.0224324i \(0.992859\pi\)
\(692\) 0 0
\(693\) 0.798133 4.52644i 0.0303186 0.171945i
\(694\) 0 0
\(695\) −5.07620 + 2.93075i −0.192551 + 0.111169i
\(696\) 0 0
\(697\) 13.9972 16.6812i 0.530181 0.631845i
\(698\) 0 0
\(699\) 8.12361 2.95675i 0.307263 0.111835i
\(700\) 0 0
\(701\) 1.28905 + 7.31056i 0.0486867 + 0.276116i 0.999426 0.0338754i \(-0.0107849\pi\)
−0.950739 + 0.309991i \(0.899674\pi\)
\(702\) 0 0
\(703\) 7.55525 41.0773i 0.284952 1.54926i
\(704\) 0 0
\(705\) 9.84776 1.73643i 0.370888 0.0653976i
\(706\) 0 0
\(707\) 3.62339 + 9.95518i 0.136272 + 0.374403i
\(708\) 0 0
\(709\) −38.9673 32.6974i −1.46345 1.22798i −0.921970 0.387260i \(-0.873421\pi\)
−0.541475 0.840717i \(-0.682134\pi\)
\(710\) 0 0
\(711\) 2.31521 + 4.01006i 0.0868271 + 0.150389i
\(712\) 0 0
\(713\) 34.9923 + 6.17009i 1.31047 + 0.231072i
\(714\) 0 0
\(715\) −10.9363 + 18.9422i −0.408994 + 0.708398i
\(716\) 0 0
\(717\) −1.20321 + 3.30579i −0.0449347 + 0.123457i
\(718\) 0 0
\(719\) 24.6246 + 29.3465i 0.918344 + 1.09444i 0.995245 + 0.0974015i \(0.0310531\pi\)
−0.0769008 + 0.997039i \(0.524502\pi\)
\(720\) 0 0
\(721\) 7.57364i 0.282057i
\(722\) 0 0
\(723\) 15.2245i 0.566205i
\(724\) 0 0
\(725\) −17.3858 20.7196i −0.645692 0.769505i
\(726\) 0 0
\(727\) −13.8712 + 38.1109i −0.514456 + 1.41346i 0.362093 + 0.932142i \(0.382062\pi\)
−0.876549 + 0.481313i \(0.840160\pi\)
\(728\) 0 0
\(729\) −0.500000 + 0.866025i −0.0185185 + 0.0320750i
\(730\) 0 0
\(731\) −14.6163 2.57725i −0.540605 0.0953232i
\(732\) 0 0
\(733\) −4.55391 7.88760i −0.168203 0.291335i 0.769585 0.638544i \(-0.220463\pi\)
−0.937788 + 0.347209i \(0.887130\pi\)
\(734\) 0 0
\(735\) 3.97565 + 3.33597i 0.146644 + 0.123049i
\(736\) 0 0
\(737\) −6.12567 16.8301i −0.225642 0.619946i
\(738\) 0 0
\(739\) 34.0087 5.99665i 1.25103 0.220590i 0.491393 0.870938i \(-0.336488\pi\)
0.759637 + 0.650348i \(0.225377\pi\)
\(740\) 0 0
\(741\) 4.47178 24.3127i 0.164275 0.893150i
\(742\) 0 0
\(743\) −2.02094 11.4613i −0.0741413 0.420476i −0.999176 0.0405929i \(-0.987075\pi\)
0.925035 0.379883i \(-0.124036\pi\)
\(744\) 0 0
\(745\) 0.205737 0.0748822i 0.00753762 0.00274347i
\(746\) 0 0
\(747\) 4.47178 5.32926i 0.163614 0.194987i
\(748\) 0 0
\(749\) 5.44562 3.14403i 0.198979 0.114880i
\(750\) 0 0
\(751\) −0.805874 + 4.57034i −0.0294068 + 0.166774i −0.995974 0.0896375i \(-0.971429\pi\)
0.966568 + 0.256412i \(0.0825402\pi\)
\(752\) 0 0
\(753\) −15.0715 8.70151i −0.549234 0.317101i
\(754\) 0 0
\(755\) 12.5770 + 4.57764i 0.457723 + 0.166597i
\(756\) 0 0
\(757\) −40.7131 + 34.1623i −1.47974 + 1.24165i −0.573282 + 0.819358i \(0.694330\pi\)
−0.906460 + 0.422292i \(0.861226\pi\)
\(758\) 0 0
\(759\) 22.9145 0.831742
\(760\) 0 0
\(761\) 19.7324 0.715298 0.357649 0.933856i \(-0.383578\pi\)
0.357649 + 0.933856i \(0.383578\pi\)
\(762\) 0 0
\(763\) 6.43448 5.39917i 0.232944 0.195463i
\(764\) 0 0
\(765\) 2.56670 + 0.934204i 0.0927994 + 0.0337762i
\(766\) 0 0
\(767\) 74.1931 + 42.8354i 2.67896 + 1.54670i
\(768\) 0 0
\(769\) 4.18155 23.7148i 0.150791 0.855176i −0.811743 0.584014i \(-0.801481\pi\)
0.962534 0.271161i \(-0.0874077\pi\)
\(770\) 0 0
\(771\) −27.4347 + 15.8394i −0.988036 + 0.570443i
\(772\) 0 0
\(773\) −26.9247 + 32.0876i −0.968415 + 1.15411i 0.0196079 + 0.999808i \(0.493758\pi\)
−0.988023 + 0.154305i \(0.950686\pi\)
\(774\) 0 0
\(775\) 27.0107 9.83110i 0.970254 0.353144i
\(776\) 0 0
\(777\) −1.74376 9.88933i −0.0625569 0.354778i
\(778\) 0 0
\(779\) −15.0842 26.5766i −0.540449 0.952206i
\(780\) 0 0
\(781\) 16.5116 2.91144i 0.590832 0.104180i
\(782\) 0 0
\(783\) −2.18866 6.01330i −0.0782164 0.214898i
\(784\) 0 0
\(785\) 7.66431 + 6.43112i 0.273551 + 0.229537i
\(786\) 0 0
\(787\) −8.76011 15.1730i −0.312264 0.540858i 0.666588 0.745426i \(-0.267754\pi\)
−0.978852 + 0.204569i \(0.934421\pi\)
\(788\) 0 0
\(789\) 1.50980 + 0.266219i 0.0537503 + 0.00947763i
\(790\) 0 0
\(791\) −10.7023 + 18.5370i −0.380531 + 0.659099i
\(792\) 0 0
\(793\) −15.6138 + 42.8986i −0.554463 + 1.52337i
\(794\) 0 0
\(795\) −3.02481 3.60483i −0.107279 0.127850i
\(796\) 0 0
\(797\) 39.9751i 1.41599i −0.706216 0.707996i \(-0.749599\pi\)
0.706216 0.707996i \(-0.250401\pi\)
\(798\) 0 0
\(799\) 35.3198i 1.24952i
\(800\) 0 0
\(801\) −5.71941 6.81612i −0.202085 0.240836i
\(802\) 0 0
\(803\) 20.4572 56.2058i 0.721920 1.98346i
\(804\) 0 0
\(805\) 2.40760 4.17009i 0.0848569 0.146976i
\(806\) 0 0
\(807\) 22.9709 + 4.05039i 0.808614 + 0.142580i
\(808\) 0 0
\(809\) −19.8418 34.3671i −0.697602 1.20828i −0.969296 0.245898i \(-0.920917\pi\)
0.271694 0.962384i \(-0.412416\pi\)
\(810\) 0 0
\(811\) 31.4918 + 26.4248i 1.10583 + 0.927900i 0.997803 0.0662480i \(-0.0211028\pi\)
0.108025 + 0.994148i \(0.465547\pi\)
\(812\) 0 0
\(813\) −7.75237 21.2995i −0.271888 0.747005i
\(814\) 0 0
\(815\) 6.83157 1.20459i 0.239299 0.0421949i
\(816\) 0 0
\(817\) −10.5466 + 17.9606i −0.368980 + 0.628364i
\(818\) 0 0
\(819\) −1.03209 5.85327i −0.0360641 0.204530i
\(820\) 0 0
\(821\) 24.0993 8.77141i 0.841070 0.306124i 0.114676 0.993403i \(-0.463417\pi\)
0.726394 + 0.687279i \(0.241195\pi\)
\(822\) 0 0
\(823\) −2.80928 + 3.34797i −0.0979253 + 0.116703i −0.812781 0.582569i \(-0.802048\pi\)
0.714856 + 0.699271i \(0.246492\pi\)
\(824\) 0 0
\(825\) 16.0535 9.26849i 0.558911 0.322687i
\(826\) 0 0
\(827\) −3.64125 + 20.6506i −0.126619 + 0.718091i 0.853714 + 0.520742i \(0.174344\pi\)
−0.980333 + 0.197349i \(0.936767\pi\)
\(828\) 0 0
\(829\) 28.6038 + 16.5144i 0.993453 + 0.573570i 0.906305 0.422625i \(-0.138891\pi\)
0.0871481 + 0.996195i \(0.472225\pi\)
\(830\) 0 0
\(831\) −20.2875 7.38403i −0.703764 0.256149i
\(832\) 0 0
\(833\) −14.0424 + 11.7829i −0.486539 + 0.408255i
\(834\) 0 0
\(835\) −1.54664 −0.0535236
\(836\) 0 0
\(837\) 6.80066 0.235065
\(838\) 0 0
\(839\) 26.9702 22.6307i 0.931114 0.781298i −0.0449028 0.998991i \(-0.514298\pi\)
0.976017 + 0.217694i \(0.0698534\pi\)
\(840\) 0 0
\(841\) 11.2294 + 4.08716i 0.387220 + 0.140936i
\(842\) 0 0
\(843\) −23.5813 13.6147i −0.812183 0.468914i
\(844\) 0 0
\(845\) −2.92633 + 16.5960i −0.100669 + 0.570921i
\(846\) 0 0
\(847\) −7.47359 + 4.31488i −0.256796 + 0.148261i
\(848\) 0 0
\(849\) −16.4996 + 19.6634i −0.566264 + 0.674848i
\(850\) 0 0
\(851\) 47.0442 17.1227i 1.61265 0.586958i
\(852\) 0 0
\(853\) −8.14755 46.2070i −0.278967 1.58210i −0.726073 0.687618i \(-0.758657\pi\)
0.447106 0.894481i \(-0.352455\pi\)
\(854\) 0 0
\(855\) 2.48545 2.91815i 0.0850007 0.0997985i
\(856\) 0 0
\(857\) 14.3846 2.53639i 0.491369 0.0866415i 0.0775266 0.996990i \(-0.475298\pi\)
0.413842 + 0.910349i \(0.364187\pi\)
\(858\) 0 0
\(859\) 12.1009 + 33.2468i 0.412876 + 1.13437i 0.955654 + 0.294491i \(0.0951501\pi\)
−0.542779 + 0.839876i \(0.682628\pi\)
\(860\) 0 0
\(861\) −5.62836 4.72275i −0.191814 0.160951i
\(862\) 0 0
\(863\) −16.8366 29.1619i −0.573125 0.992682i −0.996243 0.0866070i \(-0.972398\pi\)
0.423117 0.906075i \(-0.360936\pi\)
\(864\) 0 0
\(865\) −3.30659 0.583041i −0.112427 0.0198240i
\(866\) 0 0
\(867\) 3.67617 6.36732i 0.124849 0.216246i
\(868\) 0 0
\(869\) 6.94562 19.0829i 0.235614 0.647344i
\(870\) 0 0
\(871\) −14.8871 17.7418i −0.504431 0.601157i
\(872\) 0 0
\(873\) 3.38160i 0.114450i
\(874\) 0 0
\(875\) 8.50336i 0.287466i
\(876\) 0 0
\(877\) −5.80587 6.91917i −0.196050 0.233644i 0.659059 0.752091i \(-0.270955\pi\)
−0.855110 + 0.518447i \(0.826510\pi\)
\(878\) 0 0
\(879\) −6.59879 + 18.1300i −0.222572 + 0.611511i
\(880\) 0 0
\(881\) −6.43330 + 11.1428i −0.216743 + 0.375410i −0.953810 0.300409i \(-0.902877\pi\)
0.737067 + 0.675819i \(0.236210\pi\)
\(882\) 0 0
\(883\) 14.7078 + 2.59338i 0.494957 + 0.0872742i 0.415554 0.909568i \(-0.363588\pi\)
0.0794025 + 0.996843i \(0.474699\pi\)
\(884\) 0 0
\(885\) 6.64203 + 11.5043i 0.223269 + 0.386714i
\(886\) 0 0
\(887\) −13.9185 11.6790i −0.467337 0.392143i 0.378485 0.925608i \(-0.376445\pi\)
−0.845822 + 0.533465i \(0.820890\pi\)
\(888\) 0 0
\(889\) −3.89717 10.7074i −0.130707 0.359115i
\(890\) 0 0
\(891\) 4.31908 0.761570i 0.144695 0.0255136i
\(892\) 0 0
\(893\) 46.4507 + 17.2950i 1.55441 + 0.578755i
\(894\) 0 0
\(895\) −3.05525 17.3272i −0.102126 0.579185i
\(896\) 0 0
\(897\) 27.8444 10.1345i 0.929696 0.338382i
\(898\) 0 0
\(899\) −27.9734 + 33.3374i −0.932966 + 1.11187i
\(900\) 0 0
\(901\) 14.3944 8.31061i 0.479547 0.276867i
\(902\) 0 0
\(903\) −0.869585 + 4.93166i −0.0289380 + 0.164115i
\(904\) 0 0
\(905\) 5.74200 + 3.31515i 0.190871 + 0.110199i
\(906\) 0 0
\(907\) 15.6186 + 5.68469i 0.518606 + 0.188757i 0.588044 0.808829i \(-0.299898\pi\)
−0.0694379 + 0.997586i \(0.522121\pi\)
\(908\) 0 0
\(909\) −7.74376 + 6.49778i −0.256844 + 0.215518i
\(910\) 0 0
\(911\) 9.37464 0.310596 0.155298 0.987868i \(-0.450366\pi\)
0.155298 + 0.987868i \(0.450366\pi\)
\(912\) 0 0
\(913\) −30.5107 −1.00976
\(914\) 0 0
\(915\) −5.42262 + 4.55012i −0.179266 + 0.150422i
\(916\) 0 0
\(917\) −4.63429 1.68674i −0.153038 0.0557011i
\(918\) 0 0
\(919\) 34.1072 + 19.6918i 1.12509 + 0.649572i 0.942696 0.333653i \(-0.108281\pi\)
0.182396 + 0.983225i \(0.441615\pi\)
\(920\) 0 0
\(921\) 0.642903 3.64609i 0.0211844 0.120143i
\(922\) 0 0
\(923\) 18.7763 10.8405i 0.618030 0.356820i
\(924\) 0 0
\(925\) 26.0326 31.0244i 0.855945 1.02008i
\(926\) 0 0
\(927\) −6.79086 + 2.47167i −0.223041 + 0.0811803i
\(928\) 0 0
\(929\) −9.04804 51.3140i −0.296856 1.68356i −0.659560 0.751652i \(-0.729257\pi\)
0.362703 0.931905i \(-0.381854\pi\)
\(930\) 0 0
\(931\) 8.62020 + 24.2375i 0.282516 + 0.794353i
\(932\) 0 0
\(933\) −16.7986 + 2.96205i −0.549961 + 0.0969730i
\(934\) 0 0
\(935\) −4.09714 11.2568i −0.133991 0.368137i
\(936\) 0 0
\(937\) −2.21024 1.85461i −0.0722053 0.0605874i 0.605971 0.795487i \(-0.292785\pi\)
−0.678176 + 0.734900i \(0.737229\pi\)
\(938\) 0 0
\(939\) 15.4324 + 26.7297i 0.503618 + 0.872292i
\(940\) 0 0
\(941\) −30.5424 5.38544i −0.995652 0.175560i −0.347999 0.937495i \(-0.613139\pi\)
−0.647654 + 0.761935i \(0.724250\pi\)
\(942\) 0 0
\(943\) 18.3148 31.7222i 0.596412 1.03302i
\(944\) 0 0
\(945\) 0.315207 0.866025i 0.0102537 0.0281718i
\(946\) 0 0
\(947\) 33.4749 + 39.8939i 1.08779 + 1.29638i 0.952154 + 0.305618i \(0.0988630\pi\)
0.135635 + 0.990759i \(0.456693\pi\)
\(948\) 0 0
\(949\) 77.3458i 2.51075i
\(950\) 0 0
\(951\) 21.6757i 0.702883i
\(952\) 0 0
\(953\) 5.62418 + 6.70264i 0.182185 + 0.217120i 0.849406 0.527740i \(-0.176961\pi\)
−0.667221 + 0.744860i \(0.732516\pi\)
\(954\) 0 0
\(955\) 1.50862 4.14489i 0.0488177 0.134126i
\(956\) 0 0
\(957\) −14.0326 + 24.3051i −0.453608 + 0.785672i
\(958\) 0 0
\(959\) −6.39124 1.12695i −0.206384 0.0363911i
\(960\) 0 0
\(961\) −7.62449 13.2060i −0.245951 0.426000i
\(962\) 0 0
\(963\) 4.59627 + 3.85673i 0.148113 + 0.124281i
\(964\) 0 0
\(965\) −1.79039 4.91906i −0.0576348 0.158350i
\(966\) 0 0
\(967\) 45.9320 8.09905i 1.47707 0.260448i 0.623666 0.781691i \(-0.285643\pi\)
0.853408 + 0.521243i \(0.174532\pi\)
\(968\) 0 0
\(969\) 8.62613 + 10.4353i 0.277111 + 0.335230i
\(970\) 0 0
\(971\) 5.25372 + 29.7953i 0.168600 + 0.956177i 0.945275 + 0.326276i \(0.105794\pi\)
−0.776675 + 0.629902i \(0.783095\pi\)
\(972\) 0 0
\(973\) −6.56418 + 2.38917i −0.210438 + 0.0765931i
\(974\) 0 0
\(975\) 15.4081 18.3626i 0.493453 0.588075i
\(976\) 0 0
\(977\) 7.31150 4.22130i 0.233916 0.135051i −0.378462 0.925617i \(-0.623547\pi\)
0.612377 + 0.790566i \(0.290213\pi\)
\(978\) 0 0
\(979\) −6.77631 + 38.4304i −0.216572 + 1.22824i
\(980\) 0 0
\(981\) 6.94104 + 4.00741i 0.221610 + 0.127947i
\(982\) 0 0
\(983\) −12.2892 4.47291i −0.391965 0.142664i 0.138516 0.990360i \(-0.455767\pi\)
−0.530481 + 0.847697i \(0.677989\pi\)
\(984\) 0 0
\(985\) −16.2476 + 13.6334i −0.517692 + 0.434396i
\(986\) 0 0
\(987\) 11.9172 0.379327
\(988\) 0 0
\(989\) −24.9659 −0.793868
\(990\) 0 0
\(991\) 12.7556 10.7032i 0.405195 0.339999i −0.417302 0.908768i \(-0.637024\pi\)
0.822498 + 0.568768i \(0.192580\pi\)
\(992\) 0 0
\(993\) −2.48798 0.905550i −0.0789536 0.0287368i
\(994\) 0 0
\(995\) −18.9354 10.9324i −0.600293 0.346579i
\(996\) 0 0
\(997\) −0.364533 + 2.06737i −0.0115449 + 0.0654742i −0.990036 0.140815i \(-0.955028\pi\)
0.978491 + 0.206289i \(0.0661388\pi\)
\(998\) 0 0
\(999\) 8.29813 4.79093i 0.262541 0.151578i
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 912.2.ci.a.751.1 6
4.3 odd 2 912.2.ci.b.751.1 yes 6
19.2 odd 18 912.2.ci.b.895.1 yes 6
76.59 even 18 inner 912.2.ci.a.895.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.ci.a.751.1 6 1.1 even 1 trivial
912.2.ci.a.895.1 yes 6 76.59 even 18 inner
912.2.ci.b.751.1 yes 6 4.3 odd 2
912.2.ci.b.895.1 yes 6 19.2 odd 18