Properties

Label 156.3.d.a.53.3
Level $156$
Weight $3$
Character 156.53
Analytic conductor $4.251$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,3,Mod(53,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 156.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.25069212402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.3
Root \(1.12162 + 2.99753i\) of defining polynomial
Character \(\chi\) \(=\) 156.53
Dual form 156.3.d.a.53.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.121622 - 2.99753i) q^{3} -0.347906i q^{5} +10.5746 q^{7} +(-8.97042 + 0.729130i) q^{9} +O(q^{10})\) \(q+(-0.121622 - 2.99753i) q^{3} -0.347906i q^{5} +10.5746 q^{7} +(-8.97042 + 0.729130i) q^{9} -16.7308i q^{11} -3.60555 q^{13} +(-1.04286 + 0.0423129i) q^{15} -27.6952i q^{17} -22.1865 q^{19} +(-1.28610 - 31.6978i) q^{21} +32.7658i q^{23} +24.8790 q^{25} +(3.27659 + 26.8004i) q^{27} +8.31839i q^{29} +48.0555 q^{31} +(-50.1512 + 2.03483i) q^{33} -3.67897i q^{35} +3.53736 q^{37} +(0.438513 + 10.8078i) q^{39} -17.8480i q^{41} +34.1327 q^{43} +(0.253669 + 3.12086i) q^{45} +70.4757i q^{47} +62.8225 q^{49} +(-83.0173 + 3.36834i) q^{51} +26.2604i q^{53} -5.82075 q^{55} +(2.69836 + 66.5047i) q^{57} +54.7886i q^{59} -68.2420 q^{61} +(-94.8587 + 7.71027i) q^{63} +1.25439i q^{65} -12.6589 q^{67} +(98.2166 - 3.98503i) q^{69} -10.4891i q^{71} -11.4595 q^{73} +(-3.02582 - 74.5755i) q^{75} -176.922i q^{77} +52.1738 q^{79} +(79.9367 - 13.0812i) q^{81} +70.6432i q^{83} -9.63532 q^{85} +(24.9347 - 1.01170i) q^{87} +101.408i q^{89} -38.1273 q^{91} +(-5.84459 - 144.048i) q^{93} +7.71881i q^{95} -72.1047 q^{97} +(12.1989 + 150.082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} - 8 q^{7} - 22 q^{9} + 4 q^{15} - 24 q^{19} + 16 q^{21} - 28 q^{25} + 36 q^{27} + 96 q^{31} + 4 q^{33} - 96 q^{37} + 76 q^{43} - 136 q^{45} + 204 q^{49} - 62 q^{51} - 80 q^{55} + 184 q^{57} - 104 q^{61} + 36 q^{63} - 384 q^{67} - 8 q^{73} - 100 q^{75} + 328 q^{79} + 50 q^{81} - 64 q^{85} - 204 q^{87} - 52 q^{91} + 72 q^{93} + 416 q^{97} + 284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.121622 2.99753i −0.0405406 0.999178i
\(4\) 0 0
\(5\) 0.347906i 0.0695812i −0.999395 0.0347906i \(-0.988924\pi\)
0.999395 0.0347906i \(-0.0110764\pi\)
\(6\) 0 0
\(7\) 10.5746 1.51066 0.755330 0.655345i \(-0.227477\pi\)
0.755330 + 0.655345i \(0.227477\pi\)
\(8\) 0 0
\(9\) −8.97042 + 0.729130i −0.996713 + 0.0810145i
\(10\) 0 0
\(11\) 16.7308i 1.52098i −0.649348 0.760491i \(-0.724958\pi\)
0.649348 0.760491i \(-0.275042\pi\)
\(12\) 0 0
\(13\) −3.60555 −0.277350
\(14\) 0 0
\(15\) −1.04286 + 0.0423129i −0.0695239 + 0.00282086i
\(16\) 0 0
\(17\) 27.6952i 1.62913i −0.580072 0.814565i \(-0.696976\pi\)
0.580072 0.814565i \(-0.303024\pi\)
\(18\) 0 0
\(19\) −22.1865 −1.16771 −0.583855 0.811858i \(-0.698456\pi\)
−0.583855 + 0.811858i \(0.698456\pi\)
\(20\) 0 0
\(21\) −1.28610 31.6978i −0.0612430 1.50942i
\(22\) 0 0
\(23\) 32.7658i 1.42460i 0.701875 + 0.712300i \(0.252347\pi\)
−0.701875 + 0.712300i \(0.747653\pi\)
\(24\) 0 0
\(25\) 24.8790 0.995158
\(26\) 0 0
\(27\) 3.27659 + 26.8004i 0.121355 + 0.992609i
\(28\) 0 0
\(29\) 8.31839i 0.286841i 0.989662 + 0.143421i \(0.0458101\pi\)
−0.989662 + 0.143421i \(0.954190\pi\)
\(30\) 0 0
\(31\) 48.0555 1.55018 0.775089 0.631852i \(-0.217705\pi\)
0.775089 + 0.631852i \(0.217705\pi\)
\(32\) 0 0
\(33\) −50.1512 + 2.03483i −1.51973 + 0.0616615i
\(34\) 0 0
\(35\) 3.67897i 0.105113i
\(36\) 0 0
\(37\) 3.53736 0.0956043 0.0478022 0.998857i \(-0.484778\pi\)
0.0478022 + 0.998857i \(0.484778\pi\)
\(38\) 0 0
\(39\) 0.438513 + 10.8078i 0.0112439 + 0.277122i
\(40\) 0 0
\(41\) 17.8480i 0.435316i −0.976025 0.217658i \(-0.930158\pi\)
0.976025 0.217658i \(-0.0698418\pi\)
\(42\) 0 0
\(43\) 34.1327 0.793784 0.396892 0.917865i \(-0.370089\pi\)
0.396892 + 0.917865i \(0.370089\pi\)
\(44\) 0 0
\(45\) 0.253669 + 3.12086i 0.00563708 + 0.0693524i
\(46\) 0 0
\(47\) 70.4757i 1.49948i 0.661730 + 0.749742i \(0.269822\pi\)
−0.661730 + 0.749742i \(0.730178\pi\)
\(48\) 0 0
\(49\) 62.8225 1.28209
\(50\) 0 0
\(51\) −83.0173 + 3.36834i −1.62779 + 0.0660459i
\(52\) 0 0
\(53\) 26.2604i 0.495479i 0.968827 + 0.247740i \(0.0796877\pi\)
−0.968827 + 0.247740i \(0.920312\pi\)
\(54\) 0 0
\(55\) −5.82075 −0.105832
\(56\) 0 0
\(57\) 2.69836 + 66.5047i 0.0473396 + 1.16675i
\(58\) 0 0
\(59\) 54.7886i 0.928621i 0.885672 + 0.464311i \(0.153698\pi\)
−0.885672 + 0.464311i \(0.846302\pi\)
\(60\) 0 0
\(61\) −68.2420 −1.11872 −0.559361 0.828924i \(-0.688953\pi\)
−0.559361 + 0.828924i \(0.688953\pi\)
\(62\) 0 0
\(63\) −94.8587 + 7.71027i −1.50569 + 0.122385i
\(64\) 0 0
\(65\) 1.25439i 0.0192983i
\(66\) 0 0
\(67\) −12.6589 −0.188938 −0.0944692 0.995528i \(-0.530115\pi\)
−0.0944692 + 0.995528i \(0.530115\pi\)
\(68\) 0 0
\(69\) 98.2166 3.98503i 1.42343 0.0577541i
\(70\) 0 0
\(71\) 10.4891i 0.147734i −0.997268 0.0738668i \(-0.976466\pi\)
0.997268 0.0738668i \(-0.0235340\pi\)
\(72\) 0 0
\(73\) −11.4595 −0.156979 −0.0784895 0.996915i \(-0.525010\pi\)
−0.0784895 + 0.996915i \(0.525010\pi\)
\(74\) 0 0
\(75\) −3.02582 74.5755i −0.0403443 0.994340i
\(76\) 0 0
\(77\) 176.922i 2.29769i
\(78\) 0 0
\(79\) 52.1738 0.660428 0.330214 0.943906i \(-0.392879\pi\)
0.330214 + 0.943906i \(0.392879\pi\)
\(80\) 0 0
\(81\) 79.9367 13.0812i 0.986873 0.161496i
\(82\) 0 0
\(83\) 70.6432i 0.851123i 0.904929 + 0.425562i \(0.139923\pi\)
−0.904929 + 0.425562i \(0.860077\pi\)
\(84\) 0 0
\(85\) −9.63532 −0.113357
\(86\) 0 0
\(87\) 24.9347 1.01170i 0.286605 0.0116287i
\(88\) 0 0
\(89\) 101.408i 1.13942i 0.821847 + 0.569708i \(0.192944\pi\)
−0.821847 + 0.569708i \(0.807056\pi\)
\(90\) 0 0
\(91\) −38.1273 −0.418982
\(92\) 0 0
\(93\) −5.84459 144.048i −0.0628451 1.54890i
\(94\) 0 0
\(95\) 7.71881i 0.0812506i
\(96\) 0 0
\(97\) −72.1047 −0.743347 −0.371674 0.928363i \(-0.621216\pi\)
−0.371674 + 0.928363i \(0.621216\pi\)
\(98\) 0 0
\(99\) 12.1989 + 150.082i 0.123222 + 1.51598i
\(100\) 0 0
\(101\) 69.9711i 0.692783i −0.938090 0.346392i \(-0.887407\pi\)
0.938090 0.346392i \(-0.112593\pi\)
\(102\) 0 0
\(103\) 141.584 1.37460 0.687301 0.726373i \(-0.258795\pi\)
0.687301 + 0.726373i \(0.258795\pi\)
\(104\) 0 0
\(105\) −11.0278 + 0.447442i −0.105027 + 0.00426136i
\(106\) 0 0
\(107\) 159.940i 1.49477i −0.664391 0.747385i \(-0.731309\pi\)
0.664391 0.747385i \(-0.268691\pi\)
\(108\) 0 0
\(109\) 77.2962 0.709140 0.354570 0.935030i \(-0.384627\pi\)
0.354570 + 0.935030i \(0.384627\pi\)
\(110\) 0 0
\(111\) −0.430220 10.6034i −0.00387585 0.0955257i
\(112\) 0 0
\(113\) 84.4027i 0.746927i −0.927645 0.373463i \(-0.878170\pi\)
0.927645 0.373463i \(-0.121830\pi\)
\(114\) 0 0
\(115\) 11.3994 0.0991253
\(116\) 0 0
\(117\) 32.3433 2.62892i 0.276438 0.0224694i
\(118\) 0 0
\(119\) 292.866i 2.46106i
\(120\) 0 0
\(121\) −158.920 −1.31339
\(122\) 0 0
\(123\) −53.4999 + 2.17070i −0.434958 + 0.0176480i
\(124\) 0 0
\(125\) 17.3532i 0.138825i
\(126\) 0 0
\(127\) −40.9252 −0.322246 −0.161123 0.986934i \(-0.551512\pi\)
−0.161123 + 0.986934i \(0.551512\pi\)
\(128\) 0 0
\(129\) −4.15128 102.314i −0.0321805 0.793132i
\(130\) 0 0
\(131\) 175.720i 1.34138i 0.741739 + 0.670688i \(0.234001\pi\)
−0.741739 + 0.670688i \(0.765999\pi\)
\(132\) 0 0
\(133\) −234.614 −1.76401
\(134\) 0 0
\(135\) 9.32403 1.13994i 0.0690669 0.00844403i
\(136\) 0 0
\(137\) 138.243i 1.00907i 0.863391 + 0.504535i \(0.168336\pi\)
−0.863391 + 0.504535i \(0.831664\pi\)
\(138\) 0 0
\(139\) −27.4576 −0.197537 −0.0987685 0.995110i \(-0.531490\pi\)
−0.0987685 + 0.995110i \(0.531490\pi\)
\(140\) 0 0
\(141\) 211.253 8.57138i 1.49825 0.0607899i
\(142\) 0 0
\(143\) 60.3238i 0.421845i
\(144\) 0 0
\(145\) 2.89402 0.0199587
\(146\) 0 0
\(147\) −7.64058 188.313i −0.0519767 1.28104i
\(148\) 0 0
\(149\) 181.865i 1.22057i 0.792182 + 0.610285i \(0.208945\pi\)
−0.792182 + 0.610285i \(0.791055\pi\)
\(150\) 0 0
\(151\) −87.3314 −0.578353 −0.289177 0.957276i \(-0.593382\pi\)
−0.289177 + 0.957276i \(0.593382\pi\)
\(152\) 0 0
\(153\) 20.1934 + 248.438i 0.131983 + 1.62378i
\(154\) 0 0
\(155\) 16.7188i 0.107863i
\(156\) 0 0
\(157\) −126.402 −0.805107 −0.402553 0.915397i \(-0.631877\pi\)
−0.402553 + 0.915397i \(0.631877\pi\)
\(158\) 0 0
\(159\) 78.7164 3.19383i 0.495072 0.0200870i
\(160\) 0 0
\(161\) 346.486i 2.15209i
\(162\) 0 0
\(163\) 115.838 0.710662 0.355331 0.934741i \(-0.384368\pi\)
0.355331 + 0.934741i \(0.384368\pi\)
\(164\) 0 0
\(165\) 0.707929 + 17.4479i 0.00429048 + 0.105745i
\(166\) 0 0
\(167\) 191.230i 1.14509i −0.819873 0.572546i \(-0.805956\pi\)
0.819873 0.572546i \(-0.194044\pi\)
\(168\) 0 0
\(169\) 13.0000 0.0769231
\(170\) 0 0
\(171\) 199.022 16.1768i 1.16387 0.0946014i
\(172\) 0 0
\(173\) 288.112i 1.66539i −0.553732 0.832695i \(-0.686797\pi\)
0.553732 0.832695i \(-0.313203\pi\)
\(174\) 0 0
\(175\) 263.085 1.50335
\(176\) 0 0
\(177\) 164.231 6.66349i 0.927858 0.0376468i
\(178\) 0 0
\(179\) 79.1969i 0.442441i 0.975224 + 0.221221i \(0.0710041\pi\)
−0.975224 + 0.221221i \(0.928996\pi\)
\(180\) 0 0
\(181\) −109.661 −0.605861 −0.302930 0.953013i \(-0.597965\pi\)
−0.302930 + 0.953013i \(0.597965\pi\)
\(182\) 0 0
\(183\) 8.29971 + 204.558i 0.0453536 + 1.11780i
\(184\) 0 0
\(185\) 1.23067i 0.00665226i
\(186\) 0 0
\(187\) −463.363 −2.47788
\(188\) 0 0
\(189\) 34.6487 + 283.404i 0.183326 + 1.49949i
\(190\) 0 0
\(191\) 231.869i 1.21398i 0.794711 + 0.606988i \(0.207622\pi\)
−0.794711 + 0.606988i \(0.792378\pi\)
\(192\) 0 0
\(193\) −166.643 −0.863438 −0.431719 0.902008i \(-0.642093\pi\)
−0.431719 + 0.902008i \(0.642093\pi\)
\(194\) 0 0
\(195\) 3.76008 0.152561i 0.0192825 0.000782366i
\(196\) 0 0
\(197\) 67.0787i 0.340501i −0.985401 0.170250i \(-0.945542\pi\)
0.985401 0.170250i \(-0.0544577\pi\)
\(198\) 0 0
\(199\) 94.2428 0.473582 0.236791 0.971561i \(-0.423904\pi\)
0.236791 + 0.971561i \(0.423904\pi\)
\(200\) 0 0
\(201\) 1.53959 + 37.9454i 0.00765967 + 0.188783i
\(202\) 0 0
\(203\) 87.9638i 0.433319i
\(204\) 0 0
\(205\) −6.20941 −0.0302898
\(206\) 0 0
\(207\) −23.8905 293.923i −0.115413 1.41992i
\(208\) 0 0
\(209\) 371.198i 1.77607i
\(210\) 0 0
\(211\) 295.363 1.39982 0.699912 0.714229i \(-0.253222\pi\)
0.699912 + 0.714229i \(0.253222\pi\)
\(212\) 0 0
\(213\) −31.4414 + 1.27570i −0.147612 + 0.00598921i
\(214\) 0 0
\(215\) 11.8750i 0.0552324i
\(216\) 0 0
\(217\) 508.169 2.34179
\(218\) 0 0
\(219\) 1.39372 + 34.3501i 0.00636401 + 0.156850i
\(220\) 0 0
\(221\) 99.8565i 0.451839i
\(222\) 0 0
\(223\) −430.651 −1.93117 −0.965586 0.260084i \(-0.916250\pi\)
−0.965586 + 0.260084i \(0.916250\pi\)
\(224\) 0 0
\(225\) −223.175 + 18.1400i −0.991887 + 0.0806222i
\(226\) 0 0
\(227\) 254.188i 1.11977i −0.828570 0.559886i \(-0.810845\pi\)
0.828570 0.559886i \(-0.189155\pi\)
\(228\) 0 0
\(229\) −116.991 −0.510880 −0.255440 0.966825i \(-0.582220\pi\)
−0.255440 + 0.966825i \(0.582220\pi\)
\(230\) 0 0
\(231\) −530.329 + 21.5175i −2.29580 + 0.0931495i
\(232\) 0 0
\(233\) 122.047i 0.523805i −0.965094 0.261903i \(-0.915650\pi\)
0.965094 0.261903i \(-0.0843500\pi\)
\(234\) 0 0
\(235\) 24.5189 0.104336
\(236\) 0 0
\(237\) −6.34547 156.393i −0.0267741 0.659886i
\(238\) 0 0
\(239\) 110.327i 0.461619i 0.972999 + 0.230809i \(0.0741374\pi\)
−0.972999 + 0.230809i \(0.925863\pi\)
\(240\) 0 0
\(241\) −283.291 −1.17548 −0.587740 0.809050i \(-0.699982\pi\)
−0.587740 + 0.809050i \(0.699982\pi\)
\(242\) 0 0
\(243\) −48.9334 238.022i −0.201372 0.979515i
\(244\) 0 0
\(245\) 21.8563i 0.0892094i
\(246\) 0 0
\(247\) 79.9945 0.323864
\(248\) 0 0
\(249\) 211.755 8.59175i 0.850423 0.0345050i
\(250\) 0 0
\(251\) 470.581i 1.87483i −0.348221 0.937413i \(-0.613214\pi\)
0.348221 0.937413i \(-0.386786\pi\)
\(252\) 0 0
\(253\) 548.199 2.16679
\(254\) 0 0
\(255\) 1.17186 + 28.8822i 0.00459555 + 0.113264i
\(256\) 0 0
\(257\) 136.155i 0.529787i −0.964278 0.264894i \(-0.914663\pi\)
0.964278 0.264894i \(-0.0853369\pi\)
\(258\) 0 0
\(259\) 37.4062 0.144426
\(260\) 0 0
\(261\) −6.06519 74.6194i −0.0232383 0.285898i
\(262\) 0 0
\(263\) 273.452i 1.03974i 0.854245 + 0.519870i \(0.174020\pi\)
−0.854245 + 0.519870i \(0.825980\pi\)
\(264\) 0 0
\(265\) 9.13614 0.0344760
\(266\) 0 0
\(267\) 303.974 12.3334i 1.13848 0.0461926i
\(268\) 0 0
\(269\) 240.819i 0.895237i −0.894225 0.447619i \(-0.852272\pi\)
0.894225 0.447619i \(-0.147728\pi\)
\(270\) 0 0
\(271\) 69.3747 0.255995 0.127998 0.991774i \(-0.459145\pi\)
0.127998 + 0.991774i \(0.459145\pi\)
\(272\) 0 0
\(273\) 4.63711 + 114.288i 0.0169857 + 0.418637i
\(274\) 0 0
\(275\) 416.245i 1.51362i
\(276\) 0 0
\(277\) −2.00717 −0.00724609 −0.00362304 0.999993i \(-0.501153\pi\)
−0.00362304 + 0.999993i \(0.501153\pi\)
\(278\) 0 0
\(279\) −431.078 + 35.0387i −1.54508 + 0.125587i
\(280\) 0 0
\(281\) 216.383i 0.770048i 0.922907 + 0.385024i \(0.125807\pi\)
−0.922907 + 0.385024i \(0.874193\pi\)
\(282\) 0 0
\(283\) 465.997 1.64663 0.823316 0.567584i \(-0.192122\pi\)
0.823316 + 0.567584i \(0.192122\pi\)
\(284\) 0 0
\(285\) 23.1374 0.938774i 0.0811838 0.00329394i
\(286\) 0 0
\(287\) 188.735i 0.657615i
\(288\) 0 0
\(289\) −478.025 −1.65407
\(290\) 0 0
\(291\) 8.76949 + 216.136i 0.0301357 + 0.742736i
\(292\) 0 0
\(293\) 19.5343i 0.0666700i 0.999444 + 0.0333350i \(0.0106128\pi\)
−0.999444 + 0.0333350i \(0.989387\pi\)
\(294\) 0 0
\(295\) 19.0613 0.0646145
\(296\) 0 0
\(297\) 448.393 54.8200i 1.50974 0.184579i
\(298\) 0 0
\(299\) 118.139i 0.395113i
\(300\) 0 0
\(301\) 360.940 1.19914
\(302\) 0 0
\(303\) −209.741 + 8.51001i −0.692214 + 0.0280858i
\(304\) 0 0
\(305\) 23.7418i 0.0778419i
\(306\) 0 0
\(307\) 17.4743 0.0569197 0.0284598 0.999595i \(-0.490940\pi\)
0.0284598 + 0.999595i \(0.490940\pi\)
\(308\) 0 0
\(309\) −17.2197 424.403i −0.0557271 1.37347i
\(310\) 0 0
\(311\) 517.444i 1.66381i −0.554920 0.831903i \(-0.687251\pi\)
0.554920 0.831903i \(-0.312749\pi\)
\(312\) 0 0
\(313\) −221.491 −0.707640 −0.353820 0.935314i \(-0.615117\pi\)
−0.353820 + 0.935314i \(0.615117\pi\)
\(314\) 0 0
\(315\) 2.68245 + 33.0019i 0.00851571 + 0.104768i
\(316\) 0 0
\(317\) 156.440i 0.493501i −0.969079 0.246750i \(-0.920637\pi\)
0.969079 0.246750i \(-0.0793628\pi\)
\(318\) 0 0
\(319\) 139.173 0.436280
\(320\) 0 0
\(321\) −479.427 + 19.4522i −1.49354 + 0.0605988i
\(322\) 0 0
\(323\) 614.460i 1.90235i
\(324\) 0 0
\(325\) −89.7024 −0.276007
\(326\) 0 0
\(327\) −9.40090 231.698i −0.0287489 0.708557i
\(328\) 0 0
\(329\) 745.254i 2.26521i
\(330\) 0 0
\(331\) 324.729 0.981054 0.490527 0.871426i \(-0.336804\pi\)
0.490527 + 0.871426i \(0.336804\pi\)
\(332\) 0 0
\(333\) −31.7316 + 2.57920i −0.0952901 + 0.00774533i
\(334\) 0 0
\(335\) 4.40410i 0.0131466i
\(336\) 0 0
\(337\) 263.236 0.781117 0.390558 0.920578i \(-0.372282\pi\)
0.390558 + 0.920578i \(0.372282\pi\)
\(338\) 0 0
\(339\) −253.000 + 10.2652i −0.746313 + 0.0302808i
\(340\) 0 0
\(341\) 804.008i 2.35779i
\(342\) 0 0
\(343\) 146.168 0.426145
\(344\) 0 0
\(345\) −1.38642 34.1701i −0.00401860 0.0990438i
\(346\) 0 0
\(347\) 266.003i 0.766579i 0.923628 + 0.383290i \(0.125209\pi\)
−0.923628 + 0.383290i \(0.874791\pi\)
\(348\) 0 0
\(349\) −236.747 −0.678358 −0.339179 0.940722i \(-0.610149\pi\)
−0.339179 + 0.940722i \(0.610149\pi\)
\(350\) 0 0
\(351\) −11.8139 96.6304i −0.0336579 0.275300i
\(352\) 0 0
\(353\) 4.87609i 0.0138133i 0.999976 + 0.00690665i \(0.00219847\pi\)
−0.999976 + 0.00690665i \(0.997802\pi\)
\(354\) 0 0
\(355\) −3.64922 −0.0102795
\(356\) 0 0
\(357\) −877.876 + 35.6189i −2.45904 + 0.0997728i
\(358\) 0 0
\(359\) 359.738i 1.00205i 0.865431 + 0.501027i \(0.167044\pi\)
−0.865431 + 0.501027i \(0.832956\pi\)
\(360\) 0 0
\(361\) 131.240 0.363546
\(362\) 0 0
\(363\) 19.3281 + 476.368i 0.0532455 + 1.31231i
\(364\) 0 0
\(365\) 3.98681i 0.0109228i
\(366\) 0 0
\(367\) 22.7979 0.0621196 0.0310598 0.999518i \(-0.490112\pi\)
0.0310598 + 0.999518i \(0.490112\pi\)
\(368\) 0 0
\(369\) 13.0135 + 160.104i 0.0352669 + 0.433885i
\(370\) 0 0
\(371\) 277.694i 0.748500i
\(372\) 0 0
\(373\) 101.862 0.273089 0.136544 0.990634i \(-0.456400\pi\)
0.136544 + 0.990634i \(0.456400\pi\)
\(374\) 0 0
\(375\) −52.0167 + 2.11052i −0.138711 + 0.00562806i
\(376\) 0 0
\(377\) 29.9924i 0.0795554i
\(378\) 0 0
\(379\) −398.288 −1.05089 −0.525446 0.850827i \(-0.676102\pi\)
−0.525446 + 0.850827i \(0.676102\pi\)
\(380\) 0 0
\(381\) 4.97739 + 122.675i 0.0130640 + 0.321981i
\(382\) 0 0
\(383\) 441.572i 1.15293i 0.817122 + 0.576465i \(0.195568\pi\)
−0.817122 + 0.576465i \(0.804432\pi\)
\(384\) 0 0
\(385\) −61.5521 −0.159876
\(386\) 0 0
\(387\) −306.185 + 24.8872i −0.791175 + 0.0643080i
\(388\) 0 0
\(389\) 193.653i 0.497822i 0.968526 + 0.248911i \(0.0800726\pi\)
−0.968526 + 0.248911i \(0.919927\pi\)
\(390\) 0 0
\(391\) 907.456 2.32086
\(392\) 0 0
\(393\) 526.728 21.3714i 1.34027 0.0543802i
\(394\) 0 0
\(395\) 18.1516i 0.0459534i
\(396\) 0 0
\(397\) −146.313 −0.368547 −0.184274 0.982875i \(-0.558993\pi\)
−0.184274 + 0.982875i \(0.558993\pi\)
\(398\) 0 0
\(399\) 28.5341 + 703.262i 0.0715140 + 1.76256i
\(400\) 0 0
\(401\) 384.050i 0.957731i −0.877888 0.478865i \(-0.841048\pi\)
0.877888 0.478865i \(-0.158952\pi\)
\(402\) 0 0
\(403\) −173.267 −0.429942
\(404\) 0 0
\(405\) −4.55103 27.8105i −0.0112371 0.0686678i
\(406\) 0 0
\(407\) 59.1829i 0.145413i
\(408\) 0 0
\(409\) 148.770 0.363741 0.181870 0.983322i \(-0.441785\pi\)
0.181870 + 0.983322i \(0.441785\pi\)
\(410\) 0 0
\(411\) 414.387 16.8133i 1.00824 0.0409083i
\(412\) 0 0
\(413\) 579.369i 1.40283i
\(414\) 0 0
\(415\) 24.5772 0.0592221
\(416\) 0 0
\(417\) 3.33944 + 82.3052i 0.00800826 + 0.197375i
\(418\) 0 0
\(419\) 609.937i 1.45570i 0.685738 + 0.727848i \(0.259480\pi\)
−0.685738 + 0.727848i \(0.740520\pi\)
\(420\) 0 0
\(421\) 274.485 0.651984 0.325992 0.945372i \(-0.394302\pi\)
0.325992 + 0.945372i \(0.394302\pi\)
\(422\) 0 0
\(423\) −51.3860 632.197i −0.121480 1.49455i
\(424\) 0 0
\(425\) 689.028i 1.62124i
\(426\) 0 0
\(427\) −721.633 −1.69001
\(428\) 0 0
\(429\) 180.823 7.33668i 0.421498 0.0171018i
\(430\) 0 0
\(431\) 503.184i 1.16748i 0.811941 + 0.583740i \(0.198411\pi\)
−0.811941 + 0.583740i \(0.801589\pi\)
\(432\) 0 0
\(433\) 657.164 1.51770 0.758850 0.651265i \(-0.225761\pi\)
0.758850 + 0.651265i \(0.225761\pi\)
\(434\) 0 0
\(435\) −0.351975 8.67491i −0.000809138 0.0199423i
\(436\) 0 0
\(437\) 726.958i 1.66352i
\(438\) 0 0
\(439\) 251.397 0.572659 0.286329 0.958131i \(-0.407565\pi\)
0.286329 + 0.958131i \(0.407565\pi\)
\(440\) 0 0
\(441\) −563.544 + 45.8058i −1.27788 + 0.103868i
\(442\) 0 0
\(443\) 316.752i 0.715016i −0.933910 0.357508i \(-0.883627\pi\)
0.933910 0.357508i \(-0.116373\pi\)
\(444\) 0 0
\(445\) 35.2804 0.0792819
\(446\) 0 0
\(447\) 545.147 22.1187i 1.21957 0.0494826i
\(448\) 0 0
\(449\) 330.174i 0.735355i 0.929953 + 0.367678i \(0.119847\pi\)
−0.929953 + 0.367678i \(0.880153\pi\)
\(450\) 0 0
\(451\) −298.611 −0.662109
\(452\) 0 0
\(453\) 10.6214 + 261.779i 0.0234468 + 0.577878i
\(454\) 0 0
\(455\) 13.2647i 0.0291532i
\(456\) 0 0
\(457\) −726.602 −1.58994 −0.794970 0.606649i \(-0.792513\pi\)
−0.794970 + 0.606649i \(0.792513\pi\)
\(458\) 0 0
\(459\) 742.244 90.7459i 1.61709 0.197703i
\(460\) 0 0
\(461\) 108.280i 0.234881i 0.993080 + 0.117441i \(0.0374690\pi\)
−0.993080 + 0.117441i \(0.962531\pi\)
\(462\) 0 0
\(463\) −708.111 −1.52940 −0.764699 0.644388i \(-0.777112\pi\)
−0.764699 + 0.644388i \(0.777112\pi\)
\(464\) 0 0
\(465\) −50.1151 + 2.03337i −0.107774 + 0.00437283i
\(466\) 0 0
\(467\) 700.234i 1.49943i 0.661761 + 0.749715i \(0.269809\pi\)
−0.661761 + 0.749715i \(0.730191\pi\)
\(468\) 0 0
\(469\) −133.863 −0.285422
\(470\) 0 0
\(471\) 15.3732 + 378.894i 0.0326395 + 0.804445i
\(472\) 0 0
\(473\) 571.068i 1.20733i
\(474\) 0 0
\(475\) −551.977 −1.16206
\(476\) 0 0
\(477\) −19.1472 235.567i −0.0401410 0.493850i
\(478\) 0 0
\(479\) 519.311i 1.08416i −0.840328 0.542078i \(-0.817638\pi\)
0.840328 0.542078i \(-0.182362\pi\)
\(480\) 0 0
\(481\) −12.7541 −0.0265159
\(482\) 0 0
\(483\) 1038.60 42.1402i 2.15032 0.0872468i
\(484\) 0 0
\(485\) 25.0856i 0.0517229i
\(486\) 0 0
\(487\) −114.400 −0.234907 −0.117454 0.993078i \(-0.537473\pi\)
−0.117454 + 0.993078i \(0.537473\pi\)
\(488\) 0 0
\(489\) −14.0884 347.228i −0.0288106 0.710077i
\(490\) 0 0
\(491\) 41.6858i 0.0848998i −0.999099 0.0424499i \(-0.986484\pi\)
0.999099 0.0424499i \(-0.0135163\pi\)
\(492\) 0 0
\(493\) 230.380 0.467301
\(494\) 0 0
\(495\) 52.2145 4.24408i 0.105484 0.00857390i
\(496\) 0 0
\(497\) 110.918i 0.223175i
\(498\) 0 0
\(499\) −148.957 −0.298512 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(500\) 0 0
\(501\) −573.219 + 23.2578i −1.14415 + 0.0464227i
\(502\) 0 0
\(503\) 41.2672i 0.0820421i −0.999158 0.0410210i \(-0.986939\pi\)
0.999158 0.0410210i \(-0.0130611\pi\)
\(504\) 0 0
\(505\) −24.3434 −0.0482047
\(506\) 0 0
\(507\) −1.58108 38.9679i −0.00311850 0.0768598i
\(508\) 0 0
\(509\) 815.589i 1.60234i 0.598440 + 0.801168i \(0.295788\pi\)
−0.598440 + 0.801168i \(0.704212\pi\)
\(510\) 0 0
\(511\) −121.179 −0.237142
\(512\) 0 0
\(513\) −72.6960 594.608i −0.141708 1.15908i
\(514\) 0 0
\(515\) 49.2579i 0.0956464i
\(516\) 0 0
\(517\) 1179.12 2.28069
\(518\) 0 0
\(519\) −863.627 + 35.0407i −1.66402 + 0.0675158i
\(520\) 0 0
\(521\) 424.001i 0.813821i −0.913468 0.406910i \(-0.866606\pi\)
0.913468 0.406910i \(-0.133394\pi\)
\(522\) 0 0
\(523\) −911.760 −1.74333 −0.871663 0.490105i \(-0.836958\pi\)
−0.871663 + 0.490105i \(0.836958\pi\)
\(524\) 0 0
\(525\) −31.9969 788.608i −0.0609465 1.50211i
\(526\) 0 0
\(527\) 1330.91i 2.52544i
\(528\) 0 0
\(529\) −544.598 −1.02949
\(530\) 0 0
\(531\) −39.9481 491.477i −0.0752317 0.925569i
\(532\) 0 0
\(533\) 64.3518i 0.120735i
\(534\) 0 0
\(535\) −55.6442 −0.104008
\(536\) 0 0
\(537\) 237.396 9.63207i 0.442077 0.0179368i
\(538\) 0 0
\(539\) 1051.07i 1.95004i
\(540\) 0 0
\(541\) 471.354 0.871264 0.435632 0.900125i \(-0.356525\pi\)
0.435632 + 0.900125i \(0.356525\pi\)
\(542\) 0 0
\(543\) 13.3371 + 328.712i 0.0245619 + 0.605363i
\(544\) 0 0
\(545\) 26.8918i 0.0493428i
\(546\) 0 0
\(547\) −891.112 −1.62909 −0.814545 0.580101i \(-0.803013\pi\)
−0.814545 + 0.580101i \(0.803013\pi\)
\(548\) 0 0
\(549\) 612.159 49.7573i 1.11504 0.0906326i
\(550\) 0 0
\(551\) 184.556i 0.334947i
\(552\) 0 0
\(553\) 551.718 0.997682
\(554\) 0 0
\(555\) −3.68897 + 0.149676i −0.00664679 + 0.000269686i
\(556\) 0 0
\(557\) 56.0496i 0.100628i 0.998733 + 0.0503138i \(0.0160221\pi\)
−0.998733 + 0.0503138i \(0.983978\pi\)
\(558\) 0 0
\(559\) −123.067 −0.220156
\(560\) 0 0
\(561\) 56.3550 + 1388.95i 0.100455 + 2.47584i
\(562\) 0 0
\(563\) 156.015i 0.277113i −0.990355 0.138557i \(-0.955754\pi\)
0.990355 0.138557i \(-0.0442463\pi\)
\(564\) 0 0
\(565\) −29.3642 −0.0519720
\(566\) 0 0
\(567\) 845.300 138.329i 1.49083 0.243966i
\(568\) 0 0
\(569\) 484.828i 0.852071i −0.904706 0.426035i \(-0.859910\pi\)
0.904706 0.426035i \(-0.140090\pi\)
\(570\) 0 0
\(571\) 899.488 1.57529 0.787643 0.616133i \(-0.211301\pi\)
0.787643 + 0.616133i \(0.211301\pi\)
\(572\) 0 0
\(573\) 695.036 28.2004i 1.21298 0.0492153i
\(574\) 0 0
\(575\) 815.179i 1.41770i
\(576\) 0 0
\(577\) 507.027 0.878730 0.439365 0.898309i \(-0.355203\pi\)
0.439365 + 0.898309i \(0.355203\pi\)
\(578\) 0 0
\(579\) 20.2675 + 499.519i 0.0350042 + 0.862728i
\(580\) 0 0
\(581\) 747.025i 1.28576i
\(582\) 0 0
\(583\) 439.358 0.753615
\(584\) 0 0
\(585\) −0.914615 11.2524i −0.00156344 0.0192349i
\(586\) 0 0
\(587\) 693.001i 1.18058i 0.807191 + 0.590291i \(0.200987\pi\)
−0.807191 + 0.590291i \(0.799013\pi\)
\(588\) 0 0
\(589\) −1066.18 −1.81016
\(590\) 0 0
\(591\) −201.071 + 8.15822i −0.340221 + 0.0138041i
\(592\) 0 0
\(593\) 995.564i 1.67886i −0.543467 0.839430i \(-0.682889\pi\)
0.543467 0.839430i \(-0.317111\pi\)
\(594\) 0 0
\(595\) −101.890 −0.171243
\(596\) 0 0
\(597\) −11.4620 282.496i −0.0191993 0.473193i
\(598\) 0 0
\(599\) 83.0301i 0.138615i −0.997595 0.0693073i \(-0.977921\pi\)
0.997595 0.0693073i \(-0.0220789\pi\)
\(600\) 0 0
\(601\) −717.072 −1.19313 −0.596566 0.802564i \(-0.703469\pi\)
−0.596566 + 0.802564i \(0.703469\pi\)
\(602\) 0 0
\(603\) 113.555 9.22997i 0.188317 0.0153067i
\(604\) 0 0
\(605\) 55.2892i 0.0913871i
\(606\) 0 0
\(607\) 553.785 0.912331 0.456166 0.889895i \(-0.349222\pi\)
0.456166 + 0.889895i \(0.349222\pi\)
\(608\) 0 0
\(609\) 263.674 10.6983i 0.432963 0.0175670i
\(610\) 0 0
\(611\) 254.104i 0.415882i
\(612\) 0 0
\(613\) −945.189 −1.54191 −0.770954 0.636891i \(-0.780220\pi\)
−0.770954 + 0.636891i \(0.780220\pi\)
\(614\) 0 0
\(615\) 0.755199 + 18.6129i 0.00122797 + 0.0302649i
\(616\) 0 0
\(617\) 44.3256i 0.0718404i −0.999355 0.0359202i \(-0.988564\pi\)
0.999355 0.0359202i \(-0.0114362\pi\)
\(618\) 0 0
\(619\) −1049.94 −1.69619 −0.848094 0.529846i \(-0.822250\pi\)
−0.848094 + 0.529846i \(0.822250\pi\)
\(620\) 0 0
\(621\) −878.138 + 107.360i −1.41407 + 0.172883i
\(622\) 0 0
\(623\) 1072.35i 1.72127i
\(624\) 0 0
\(625\) 615.937 0.985499
\(626\) 0 0
\(627\) 1112.68 45.1457i 1.77461 0.0720027i
\(628\) 0 0
\(629\) 97.9680i 0.155752i
\(630\) 0 0
\(631\) 613.583 0.972398 0.486199 0.873848i \(-0.338383\pi\)
0.486199 + 0.873848i \(0.338383\pi\)
\(632\) 0 0
\(633\) −35.9225 885.361i −0.0567497 1.39867i
\(634\) 0 0
\(635\) 14.2381i 0.0224222i
\(636\) 0 0
\(637\) −226.510 −0.355588
\(638\) 0 0
\(639\) 7.64791 + 94.0915i 0.0119686 + 0.147248i
\(640\) 0 0
\(641\) 1125.37i 1.75565i 0.478979 + 0.877826i \(0.341007\pi\)
−0.478979 + 0.877826i \(0.658993\pi\)
\(642\) 0 0
\(643\) −560.059 −0.871009 −0.435505 0.900187i \(-0.643430\pi\)
−0.435505 + 0.900187i \(0.643430\pi\)
\(644\) 0 0
\(645\) −35.5956 + 1.44425i −0.0551870 + 0.00223915i
\(646\) 0 0
\(647\) 675.328i 1.04378i −0.853012 0.521892i \(-0.825226\pi\)
0.853012 0.521892i \(-0.174774\pi\)
\(648\) 0 0
\(649\) 916.658 1.41242
\(650\) 0 0
\(651\) −61.8043 1523.25i −0.0949375 2.33987i
\(652\) 0 0
\(653\) 563.974i 0.863666i 0.901954 + 0.431833i \(0.142133\pi\)
−0.901954 + 0.431833i \(0.857867\pi\)
\(654\) 0 0
\(655\) 61.1341 0.0933345
\(656\) 0 0
\(657\) 102.796 8.35544i 0.156463 0.0127176i
\(658\) 0 0
\(659\) 564.784i 0.857031i 0.903534 + 0.428516i \(0.140963\pi\)
−0.903534 + 0.428516i \(0.859037\pi\)
\(660\) 0 0
\(661\) −607.478 −0.919028 −0.459514 0.888171i \(-0.651976\pi\)
−0.459514 + 0.888171i \(0.651976\pi\)
\(662\) 0 0
\(663\) 299.323 12.1447i 0.451468 0.0183178i
\(664\) 0 0
\(665\) 81.6234i 0.122742i
\(666\) 0 0
\(667\) −272.559 −0.408634
\(668\) 0 0
\(669\) 52.3765 + 1290.89i 0.0782908 + 1.92958i
\(670\) 0 0
\(671\) 1141.74i 1.70156i
\(672\) 0 0
\(673\) −798.788 −1.18691 −0.593454 0.804868i \(-0.702236\pi\)
−0.593454 + 0.804868i \(0.702236\pi\)
\(674\) 0 0
\(675\) 81.5181 + 666.767i 0.120768 + 0.987803i
\(676\) 0 0
\(677\) 391.116i 0.577719i −0.957371 0.288860i \(-0.906724\pi\)
0.957371 0.288860i \(-0.0932761\pi\)
\(678\) 0 0
\(679\) −762.479 −1.12294
\(680\) 0 0
\(681\) −761.938 + 30.9148i −1.11885 + 0.0453962i
\(682\) 0 0
\(683\) 1130.48i 1.65518i −0.561337 0.827588i \(-0.689713\pi\)
0.561337 0.827588i \(-0.310287\pi\)
\(684\) 0 0
\(685\) 48.0954 0.0702123
\(686\) 0 0
\(687\) 14.2287 + 350.686i 0.0207113 + 0.510460i
\(688\) 0 0
\(689\) 94.6832i 0.137421i
\(690\) 0 0
\(691\) 350.909 0.507828 0.253914 0.967227i \(-0.418282\pi\)
0.253914 + 0.967227i \(0.418282\pi\)
\(692\) 0 0
\(693\) 128.999 + 1587.06i 0.186146 + 2.29013i
\(694\) 0 0
\(695\) 9.55267i 0.0137448i
\(696\) 0 0
\(697\) −494.303 −0.709187
\(698\) 0 0
\(699\) −365.839 + 14.8435i −0.523375 + 0.0212354i
\(700\) 0 0
\(701\) 1005.60i 1.43452i 0.696808 + 0.717258i \(0.254603\pi\)
−0.696808 + 0.717258i \(0.745397\pi\)
\(702\) 0 0
\(703\) −78.4816 −0.111638
\(704\) 0 0
\(705\) −2.98203 73.4963i −0.00422983 0.104250i
\(706\) 0 0
\(707\) 739.918i 1.04656i
\(708\) 0 0
\(709\) 895.241 1.26268 0.631341 0.775506i \(-0.282505\pi\)
0.631341 + 0.775506i \(0.282505\pi\)
\(710\) 0 0
\(711\) −468.021 + 38.0415i −0.658258 + 0.0535043i
\(712\) 0 0
\(713\) 1574.58i 2.20838i
\(714\) 0 0
\(715\) 20.9870 0.0293524
\(716\) 0 0
\(717\) 330.709 13.4181i 0.461239 0.0187143i
\(718\) 0 0
\(719\) 456.136i 0.634404i 0.948358 + 0.317202i \(0.102743\pi\)
−0.948358 + 0.317202i \(0.897257\pi\)
\(720\) 0 0
\(721\) 1497.20 2.07656
\(722\) 0 0
\(723\) 34.4543 + 849.174i 0.0476546 + 1.17451i
\(724\) 0 0
\(725\) 206.953i 0.285452i
\(726\) 0 0
\(727\) −426.245 −0.586307 −0.293153 0.956065i \(-0.594705\pi\)
−0.293153 + 0.956065i \(0.594705\pi\)
\(728\) 0 0
\(729\) −707.528 + 175.628i −0.970546 + 0.240916i
\(730\) 0 0
\(731\) 945.313i 1.29318i
\(732\) 0 0
\(733\) −923.521 −1.25992 −0.629960 0.776628i \(-0.716929\pi\)
−0.629960 + 0.776628i \(0.716929\pi\)
\(734\) 0 0
\(735\) −65.5150 + 2.65820i −0.0891361 + 0.00361660i
\(736\) 0 0
\(737\) 211.793i 0.287372i
\(738\) 0 0
\(739\) 1128.09 1.52651 0.763254 0.646099i \(-0.223601\pi\)
0.763254 + 0.646099i \(0.223601\pi\)
\(740\) 0 0
\(741\) −9.72907 239.786i −0.0131296 0.323598i
\(742\) 0 0
\(743\) 506.769i 0.682058i −0.940053 0.341029i \(-0.889225\pi\)
0.940053 0.341029i \(-0.110775\pi\)
\(744\) 0 0
\(745\) 63.2719 0.0849287
\(746\) 0 0
\(747\) −51.5081 633.699i −0.0689533 0.848325i
\(748\) 0 0
\(749\) 1691.31i 2.25809i
\(750\) 0 0
\(751\) 1167.71 1.55488 0.777438 0.628960i \(-0.216519\pi\)
0.777438 + 0.628960i \(0.216519\pi\)
\(752\) 0 0
\(753\) −1410.58 + 57.2329i −1.87328 + 0.0760065i
\(754\) 0 0
\(755\) 30.3831i 0.0402425i
\(756\) 0 0
\(757\) 670.756 0.886071 0.443036 0.896504i \(-0.353901\pi\)
0.443036 + 0.896504i \(0.353901\pi\)
\(758\) 0 0
\(759\) −66.6728 1643.24i −0.0878430 2.16501i
\(760\) 0 0
\(761\) 92.8836i 0.122055i 0.998136 + 0.0610273i \(0.0194377\pi\)
−0.998136 + 0.0610273i \(0.980562\pi\)
\(762\) 0 0
\(763\) 817.378 1.07127
\(764\) 0 0
\(765\) 86.4329 7.02541i 0.112984 0.00918354i
\(766\) 0 0
\(767\) 197.543i 0.257553i
\(768\) 0 0
\(769\) 865.644 1.12567 0.562837 0.826568i \(-0.309710\pi\)
0.562837 + 0.826568i \(0.309710\pi\)
\(770\) 0 0
\(771\) −408.130 + 16.5594i −0.529352 + 0.0214779i
\(772\) 0 0
\(773\) 900.561i 1.16502i 0.812823 + 0.582510i \(0.197929\pi\)
−0.812823 + 0.582510i \(0.802071\pi\)
\(774\) 0 0
\(775\) 1195.57 1.54267
\(776\) 0 0
\(777\) −4.54941 112.126i −0.00585509 0.144307i
\(778\) 0 0
\(779\) 395.984i 0.508323i
\(780\) 0 0
\(781\) −175.491 −0.224700
\(782\) 0 0
\(783\) −222.937 + 27.2560i −0.284721 + 0.0348096i
\(784\) 0 0
\(785\) 43.9759i 0.0560203i
\(786\) 0 0
\(787\) −1280.16 −1.62663 −0.813315 0.581823i \(-0.802340\pi\)
−0.813315 + 0.581823i \(0.802340\pi\)
\(788\) 0 0
\(789\) 819.680 33.2576i 1.03889 0.0421516i
\(790\) 0 0
\(791\) 892.526i 1.12835i
\(792\) 0 0
\(793\) 246.050 0.310277
\(794\) 0 0
\(795\) −1.11115 27.3859i −0.00139768 0.0344477i
\(796\) 0 0
\(797\) 697.579i 0.875256i 0.899156 + 0.437628i \(0.144181\pi\)
−0.899156 + 0.437628i \(0.855819\pi\)
\(798\) 0 0
\(799\) 1951.84 2.44285
\(800\) 0 0
\(801\) −73.9396 909.672i −0.0923092 1.13567i
\(802\) 0 0
\(803\) 191.726i 0.238762i
\(804\) 0 0
\(805\) 120.544 0.149745
\(806\) 0 0
\(807\) −721.862 + 29.2888i −0.894501 + 0.0362934i
\(808\) 0 0
\(809\) 156.783i 0.193799i 0.995294 + 0.0968993i \(0.0308925\pi\)
−0.995294 + 0.0968993i \(0.969108\pi\)
\(810\) 0 0
\(811\) −708.896 −0.874101 −0.437050 0.899437i \(-0.643977\pi\)
−0.437050 + 0.899437i \(0.643977\pi\)
\(812\) 0 0
\(813\) −8.43746 207.953i −0.0103782 0.255785i
\(814\) 0 0
\(815\) 40.3007i 0.0494487i
\(816\) 0 0
\(817\) −757.285 −0.926910
\(818\) 0 0
\(819\) 342.018 27.7998i 0.417604 0.0339436i
\(820\) 0 0
\(821\) 904.304i 1.10147i 0.834681 + 0.550733i \(0.185652\pi\)
−0.834681 + 0.550733i \(0.814348\pi\)
\(822\) 0 0
\(823\) −917.496 −1.11482 −0.557409 0.830238i \(-0.688205\pi\)
−0.557409 + 0.830238i \(0.688205\pi\)
\(824\) 0 0
\(825\) −1247.71 + 50.6244i −1.51237 + 0.0613630i
\(826\) 0 0
\(827\) 373.879i 0.452091i 0.974117 + 0.226045i \(0.0725798\pi\)
−0.974117 + 0.226045i \(0.927420\pi\)
\(828\) 0 0
\(829\) −473.628 −0.571325 −0.285662 0.958330i \(-0.592214\pi\)
−0.285662 + 0.958330i \(0.592214\pi\)
\(830\) 0 0
\(831\) 0.244115 + 6.01655i 0.000293761 + 0.00724013i
\(832\) 0 0
\(833\) 1739.88i 2.08869i
\(834\) 0 0
\(835\) −66.5301 −0.0796768
\(836\) 0 0
\(837\) 157.458 + 1287.91i 0.188122 + 1.53872i
\(838\) 0 0
\(839\) 447.115i 0.532915i −0.963847 0.266457i \(-0.914147\pi\)
0.963847 0.266457i \(-0.0858531\pi\)
\(840\) 0 0
\(841\) 771.804 0.917722
\(842\) 0 0
\(843\) 648.616 26.3169i 0.769415 0.0312182i
\(844\) 0 0
\(845\) 4.52277i 0.00535240i
\(846\) 0 0
\(847\) −1680.52 −1.98408
\(848\) 0 0
\(849\) −56.6753 1396.84i −0.0667554 1.64528i
\(850\) 0 0
\(851\) 115.904i 0.136198i
\(852\) 0 0
\(853\) 407.791 0.478067 0.239033 0.971011i \(-0.423169\pi\)
0.239033 + 0.971011i \(0.423169\pi\)
\(854\) 0 0
\(855\) −5.62801 69.2409i −0.00658247 0.0809835i
\(856\) 0 0
\(857\) 969.943i 1.13179i −0.824478 0.565894i \(-0.808531\pi\)
0.824478 0.565894i \(-0.191469\pi\)
\(858\) 0 0
\(859\) 501.577 0.583908 0.291954 0.956432i \(-0.405695\pi\)
0.291954 + 0.956432i \(0.405695\pi\)
\(860\) 0 0
\(861\) −565.741 + 22.9543i −0.657074 + 0.0266601i
\(862\) 0 0
\(863\) 809.427i 0.937922i −0.883219 0.468961i \(-0.844629\pi\)
0.883219 0.468961i \(-0.155371\pi\)
\(864\) 0 0
\(865\) −100.236 −0.115880
\(866\) 0 0
\(867\) 58.1382 + 1432.90i 0.0670568 + 1.65271i
\(868\) 0 0
\(869\) 872.911i 1.00450i
\(870\) 0 0
\(871\) 45.6422 0.0524021
\(872\) 0 0
\(873\) 646.809 52.5737i 0.740904 0.0602219i
\(874\) 0 0
\(875\) 183.503i 0.209718i
\(876\) 0 0
\(877\) 554.476 0.632242 0.316121 0.948719i \(-0.397619\pi\)
0.316121 + 0.948719i \(0.397619\pi\)
\(878\) 0 0
\(879\) 58.5548 2.37580i 0.0666152 0.00270284i
\(880\) 0 0
\(881\) 1167.11i 1.32476i −0.749168 0.662380i \(-0.769546\pi\)
0.749168 0.662380i \(-0.230454\pi\)
\(882\) 0 0
\(883\) −502.216 −0.568761 −0.284381 0.958711i \(-0.591788\pi\)
−0.284381 + 0.958711i \(0.591788\pi\)
\(884\) 0 0
\(885\) −2.31827 57.1368i −0.00261951 0.0645614i
\(886\) 0 0
\(887\) 251.594i 0.283646i −0.989892 0.141823i \(-0.954704\pi\)
0.989892 0.141823i \(-0.0452964\pi\)
\(888\) 0 0
\(889\) −432.768 −0.486803
\(890\) 0 0
\(891\) −218.859 1337.41i −0.245633 1.50102i
\(892\) 0 0
\(893\) 1563.61i 1.75096i
\(894\) 0 0
\(895\) 27.5531 0.0307856
\(896\) 0 0
\(897\) −354.125 + 14.3682i −0.394788 + 0.0160181i
\(898\) 0 0
\(899\) 399.745i 0.444655i
\(900\) 0 0
\(901\) 727.287 0.807200
\(902\) 0 0
\(903\) −43.8982 1081.93i −0.0486137 1.19815i
\(904\) 0 0
\(905\) 38.1516i 0.0421565i
\(906\) 0 0
\(907\) 5.05962 0.00557842 0.00278921 0.999996i \(-0.499112\pi\)
0.00278921 + 0.999996i \(0.499112\pi\)
\(908\) 0 0
\(909\) 51.0181 + 627.670i 0.0561255 + 0.690506i
\(910\) 0 0
\(911\) 52.1811i 0.0572789i −0.999590 0.0286395i \(-0.990883\pi\)
0.999590 0.0286395i \(-0.00911747\pi\)
\(912\) 0 0
\(913\) 1181.92 1.29454
\(914\) 0 0
\(915\) 71.1668 2.88752i 0.0777779 0.00315575i
\(916\) 0 0
\(917\) 1858.18i 2.02636i
\(918\) 0 0
\(919\) −417.586 −0.454392 −0.227196 0.973849i \(-0.572956\pi\)
−0.227196 + 0.973849i \(0.572956\pi\)
\(920\) 0 0
\(921\) −2.12526 52.3799i −0.00230756 0.0568729i
\(922\) 0 0
\(923\) 37.8190i 0.0409740i
\(924\) 0 0
\(925\) 88.0059 0.0951415
\(926\) 0 0
\(927\) −1270.07 + 103.233i −1.37008 + 0.111363i
\(928\) 0 0
\(929\) 591.593i 0.636807i −0.947955 0.318403i \(-0.896853\pi\)
0.947955 0.318403i \(-0.103147\pi\)
\(930\) 0 0
\(931\) −1393.81 −1.49711
\(932\) 0 0
\(933\) −1551.06 + 62.9324i −1.66244 + 0.0674517i
\(934\) 0 0
\(935\) 161.207i 0.172414i
\(936\) 0 0
\(937\) 415.023 0.442927 0.221464 0.975169i \(-0.428917\pi\)
0.221464 + 0.975169i \(0.428917\pi\)
\(938\) 0 0
\(939\) 26.9382 + 663.928i 0.0286881 + 0.707058i
\(940\) 0 0
\(941\) 192.556i 0.204629i 0.994752 + 0.102314i \(0.0326248\pi\)
−0.994752 + 0.102314i \(0.967375\pi\)
\(942\) 0 0
\(943\) 584.803 0.620152
\(944\) 0 0
\(945\) 98.5980 12.0545i 0.104337 0.0127561i
\(946\) 0 0
\(947\) 149.806i 0.158190i −0.996867 0.0790951i \(-0.974797\pi\)
0.996867 0.0790951i \(-0.0252031\pi\)
\(948\) 0 0
\(949\) 41.3177 0.0435381
\(950\) 0 0
\(951\) −468.933 + 19.0265i −0.493095 + 0.0200068i
\(952\) 0 0
\(953\) 231.170i 0.242571i −0.992618 0.121286i \(-0.961298\pi\)
0.992618 0.121286i \(-0.0387017\pi\)
\(954\) 0 0
\(955\) 80.6687 0.0844699
\(956\) 0 0
\(957\) −16.9265 417.177i −0.0176870 0.435922i
\(958\) 0 0
\(959\) 1461.86i 1.52436i
\(960\) 0 0
\(961\) 1348.33 1.40305
\(962\) 0 0
\(963\) 116.617 + 1434.73i 0.121098 + 1.48986i
\(964\) 0 0
\(965\) 57.9762i 0.0600790i
\(966\) 0 0
\(967\) 403.483 0.417252 0.208626 0.977995i \(-0.433101\pi\)
0.208626 + 0.977995i \(0.433101\pi\)
\(968\) 0 0
\(969\) 1841.86 74.7316i 1.90079 0.0771224i
\(970\) 0 0
\(971\) 121.687i 0.125321i 0.998035 + 0.0626605i \(0.0199585\pi\)
−0.998035 + 0.0626605i \(0.980041\pi\)
\(972\) 0 0
\(973\) −290.354 −0.298411
\(974\) 0 0
\(975\) 10.9098 + 268.886i 0.0111895 + 0.275780i
\(976\) 0 0
\(977\) 1825.82i 1.86880i 0.356223 + 0.934401i \(0.384064\pi\)
−0.356223 + 0.934401i \(0.615936\pi\)
\(978\) 0 0
\(979\) 1696.64 1.73303
\(980\) 0 0
\(981\) −693.379 + 56.3590i −0.706809 + 0.0574506i
\(982\) 0 0
\(983\) 71.7807i 0.0730220i 0.999333 + 0.0365110i \(0.0116244\pi\)
−0.999333 + 0.0365110i \(0.988376\pi\)
\(984\) 0 0
\(985\) −23.3371 −0.0236924
\(986\) 0 0
\(987\) 2233.92 90.6390i 2.26335 0.0918329i
\(988\) 0 0
\(989\) 1118.39i 1.13083i
\(990\) 0 0
\(991\) −1104.62 −1.11465 −0.557324 0.830295i \(-0.688172\pi\)
−0.557324 + 0.830295i \(0.688172\pi\)
\(992\) 0 0
\(993\) −39.4941 973.385i −0.0397725 0.980247i
\(994\) 0 0
\(995\) 32.7876i 0.0329524i
\(996\) 0 0
\(997\) 204.813 0.205429 0.102714 0.994711i \(-0.467247\pi\)
0.102714 + 0.994711i \(0.467247\pi\)
\(998\) 0 0
\(999\) 11.5905 + 94.8028i 0.0116021 + 0.0948977i
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 156.3.d.a.53.3 8
3.2 odd 2 inner 156.3.d.a.53.4 yes 8
4.3 odd 2 624.3.f.a.209.6 8
12.11 even 2 624.3.f.a.209.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.3.d.a.53.3 8 1.1 even 1 trivial
156.3.d.a.53.4 yes 8 3.2 odd 2 inner
624.3.f.a.209.5 8 12.11 even 2
624.3.f.a.209.6 8 4.3 odd 2