Properties

Label 384.5.e.d.257.2
Level $384$
Weight $5$
Character 384.257
Analytic conductor $39.694$
Analytic rank $0$
Dimension $16$
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] = [384,5,Mod(257,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.257");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.e (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: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + 391324 x^{9} + 724325 x^{8} - 2262056 x^{7} + 45109352 x^{6} + \cdots + 21479188203 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 257.2
Root \(-1.10373 + 0.840249i\) of defining polynomial
Character \(\chi\) \(=\) 384.257
Dual form 384.5.e.d.257.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-8.70713 + 2.27724i) q^{3} +28.9306i q^{5} -2.36161 q^{7} +(70.6284 - 39.6564i) q^{9} +O(q^{10})\) \(q+(-8.70713 + 2.27724i) q^{3} +28.9306i q^{5} -2.36161 q^{7} +(70.6284 - 39.6564i) q^{9} -21.1756i q^{11} -259.705 q^{13} +(-65.8818 - 251.903i) q^{15} +350.061i q^{17} -308.733 q^{19} +(20.5629 - 5.37795i) q^{21} -282.768i q^{23} -211.979 q^{25} +(-524.664 + 506.131i) q^{27} +1236.29i q^{29} -158.246 q^{31} +(48.2218 + 184.379i) q^{33} -68.3228i q^{35} -890.346 q^{37} +(2261.28 - 591.409i) q^{39} -1787.97i q^{41} +3446.41 q^{43} +(1147.28 + 2043.32i) q^{45} -3405.32i q^{47} -2395.42 q^{49} +(-797.173 - 3048.03i) q^{51} +580.891i q^{53} +612.622 q^{55} +(2688.18 - 703.058i) q^{57} -3755.71i q^{59} +205.602 q^{61} +(-166.797 + 93.6530i) q^{63} -7513.41i q^{65} -610.202 q^{67} +(643.929 + 2462.10i) q^{69} +3017.17i q^{71} +5237.06 q^{73} +(1845.73 - 482.727i) q^{75} +50.0084i q^{77} -6540.29 q^{79} +(3415.74 - 5601.74i) q^{81} -13003.3i q^{83} -10127.5 q^{85} +(-2815.34 - 10764.6i) q^{87} -5444.47i q^{89} +613.321 q^{91} +(1377.87 - 360.363i) q^{93} -8931.83i q^{95} +1878.51 q^{97} +(-839.748 - 1495.60i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} + 80 q^{7} + 416 q^{15} - 816 q^{19} - 608 q^{21} - 2000 q^{25} - 280 q^{27} + 592 q^{31} - 496 q^{33} - 2240 q^{37} - 16 q^{39} - 368 q^{43} - 800 q^{45} + 3984 q^{49} + 352 q^{51} + 1920 q^{55} + 560 q^{57} + 3520 q^{61} - 816 q^{63} + 3536 q^{67} + 10784 q^{69} + 3680 q^{73} - 5112 q^{75} - 14448 q^{79} - 624 q^{81} + 11136 q^{85} - 14944 q^{87} + 22944 q^{91} - 13760 q^{93} + 3264 q^{97} + 26976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −8.70713 + 2.27724i −0.967459 + 0.253026i
\(4\) 0 0
\(5\) 28.9306i 1.15722i 0.815603 + 0.578612i \(0.196405\pi\)
−0.815603 + 0.578612i \(0.803595\pi\)
\(6\) 0 0
\(7\) −2.36161 −0.0481961 −0.0240981 0.999710i \(-0.507671\pi\)
−0.0240981 + 0.999710i \(0.507671\pi\)
\(8\) 0 0
\(9\) 70.6284 39.6564i 0.871955 0.489585i
\(10\) 0 0
\(11\) 21.1756i 0.175005i −0.996164 0.0875024i \(-0.972111\pi\)
0.996164 0.0875024i \(-0.0278885\pi\)
\(12\) 0 0
\(13\) −259.705 −1.53671 −0.768357 0.640022i \(-0.778925\pi\)
−0.768357 + 0.640022i \(0.778925\pi\)
\(14\) 0 0
\(15\) −65.8818 251.903i −0.292808 1.11957i
\(16\) 0 0
\(17\) 350.061i 1.21129i 0.795737 + 0.605643i \(0.207084\pi\)
−0.795737 + 0.605643i \(0.792916\pi\)
\(18\) 0 0
\(19\) −308.733 −0.855216 −0.427608 0.903964i \(-0.640644\pi\)
−0.427608 + 0.903964i \(0.640644\pi\)
\(20\) 0 0
\(21\) 20.5629 5.37795i 0.0466278 0.0121949i
\(22\) 0 0
\(23\) 282.768i 0.534532i −0.963623 0.267266i \(-0.913880\pi\)
0.963623 0.267266i \(-0.0861203\pi\)
\(24\) 0 0
\(25\) −211.979 −0.339166
\(26\) 0 0
\(27\) −524.664 + 506.131i −0.719703 + 0.694282i
\(28\) 0 0
\(29\) 1236.29i 1.47003i 0.678051 + 0.735015i \(0.262825\pi\)
−0.678051 + 0.735015i \(0.737175\pi\)
\(30\) 0 0
\(31\) −158.246 −0.164668 −0.0823338 0.996605i \(-0.526237\pi\)
−0.0823338 + 0.996605i \(0.526237\pi\)
\(32\) 0 0
\(33\) 48.2218 + 184.379i 0.0442808 + 0.169310i
\(34\) 0 0
\(35\) 68.3228i 0.0557737i
\(36\) 0 0
\(37\) −890.346 −0.650363 −0.325181 0.945652i \(-0.605425\pi\)
−0.325181 + 0.945652i \(0.605425\pi\)
\(38\) 0 0
\(39\) 2261.28 591.409i 1.48671 0.388829i
\(40\) 0 0
\(41\) 1787.97i 1.06364i −0.846858 0.531819i \(-0.821509\pi\)
0.846858 0.531819i \(-0.178491\pi\)
\(42\) 0 0
\(43\) 3446.41 1.86393 0.931965 0.362549i \(-0.118094\pi\)
0.931965 + 0.362549i \(0.118094\pi\)
\(44\) 0 0
\(45\) 1147.28 + 2043.32i 0.566560 + 1.00905i
\(46\) 0 0
\(47\) 3405.32i 1.54157i −0.637097 0.770784i \(-0.719865\pi\)
0.637097 0.770784i \(-0.280135\pi\)
\(48\) 0 0
\(49\) −2395.42 −0.997677
\(50\) 0 0
\(51\) −797.173 3048.03i −0.306487 1.17187i
\(52\) 0 0
\(53\) 580.891i 0.206797i 0.994640 + 0.103398i \(0.0329716\pi\)
−0.994640 + 0.103398i \(0.967028\pi\)
\(54\) 0 0
\(55\) 612.622 0.202520
\(56\) 0 0
\(57\) 2688.18 703.058i 0.827387 0.216392i
\(58\) 0 0
\(59\) 3755.71i 1.07892i −0.842012 0.539458i \(-0.818629\pi\)
0.842012 0.539458i \(-0.181371\pi\)
\(60\) 0 0
\(61\) 205.602 0.0552544 0.0276272 0.999618i \(-0.491205\pi\)
0.0276272 + 0.999618i \(0.491205\pi\)
\(62\) 0 0
\(63\) −166.797 + 93.6530i −0.0420249 + 0.0235961i
\(64\) 0 0
\(65\) 7513.41i 1.77832i
\(66\) 0 0
\(67\) −610.202 −0.135933 −0.0679664 0.997688i \(-0.521651\pi\)
−0.0679664 + 0.997688i \(0.521651\pi\)
\(68\) 0 0
\(69\) 643.929 + 2462.10i 0.135251 + 0.517138i
\(70\) 0 0
\(71\) 3017.17i 0.598527i 0.954171 + 0.299263i \(0.0967409\pi\)
−0.954171 + 0.299263i \(0.903259\pi\)
\(72\) 0 0
\(73\) 5237.06 0.982746 0.491373 0.870949i \(-0.336495\pi\)
0.491373 + 0.870949i \(0.336495\pi\)
\(74\) 0 0
\(75\) 1845.73 482.727i 0.328130 0.0858181i
\(76\) 0 0
\(77\) 50.0084i 0.00843455i
\(78\) 0 0
\(79\) −6540.29 −1.04796 −0.523978 0.851732i \(-0.675553\pi\)
−0.523978 + 0.851732i \(0.675553\pi\)
\(80\) 0 0
\(81\) 3415.74 5601.74i 0.520612 0.853793i
\(82\) 0 0
\(83\) 13003.3i 1.88754i −0.330599 0.943771i \(-0.607251\pi\)
0.330599 0.943771i \(-0.392749\pi\)
\(84\) 0 0
\(85\) −10127.5 −1.40173
\(86\) 0 0
\(87\) −2815.34 10764.6i −0.371956 1.42219i
\(88\) 0 0
\(89\) 5444.47i 0.687346i −0.939089 0.343673i \(-0.888329\pi\)
0.939089 0.343673i \(-0.111671\pi\)
\(90\) 0 0
\(91\) 613.321 0.0740636
\(92\) 0 0
\(93\) 1377.87 360.363i 0.159309 0.0416653i
\(94\) 0 0
\(95\) 8931.83i 0.989676i
\(96\) 0 0
\(97\) 1878.51 0.199650 0.0998250 0.995005i \(-0.468172\pi\)
0.0998250 + 0.995005i \(0.468172\pi\)
\(98\) 0 0
\(99\) −839.748 1495.60i −0.0856798 0.152596i
\(100\) 0 0
\(101\) 10462.0i 1.02559i 0.858511 + 0.512795i \(0.171390\pi\)
−0.858511 + 0.512795i \(0.828610\pi\)
\(102\) 0 0
\(103\) 8522.68 0.803345 0.401672 0.915783i \(-0.368429\pi\)
0.401672 + 0.915783i \(0.368429\pi\)
\(104\) 0 0
\(105\) 155.587 + 594.895i 0.0141122 + 0.0539588i
\(106\) 0 0
\(107\) 21110.3i 1.84385i −0.387365 0.921926i \(-0.626615\pi\)
0.387365 0.921926i \(-0.373385\pi\)
\(108\) 0 0
\(109\) 14321.8 1.20544 0.602718 0.797954i \(-0.294084\pi\)
0.602718 + 0.797954i \(0.294084\pi\)
\(110\) 0 0
\(111\) 7752.36 2027.53i 0.629199 0.164559i
\(112\) 0 0
\(113\) 14376.1i 1.12586i −0.826505 0.562929i \(-0.809674\pi\)
0.826505 0.562929i \(-0.190326\pi\)
\(114\) 0 0
\(115\) 8180.63 0.618573
\(116\) 0 0
\(117\) −18342.5 + 10299.0i −1.33995 + 0.752352i
\(118\) 0 0
\(119\) 826.709i 0.0583793i
\(120\) 0 0
\(121\) 14192.6 0.969373
\(122\) 0 0
\(123\) 4071.64 + 15568.1i 0.269128 + 1.02903i
\(124\) 0 0
\(125\) 11948.9i 0.764732i
\(126\) 0 0
\(127\) 19839.4 1.23004 0.615022 0.788510i \(-0.289147\pi\)
0.615022 + 0.788510i \(0.289147\pi\)
\(128\) 0 0
\(129\) −30008.3 + 7848.28i −1.80328 + 0.471623i
\(130\) 0 0
\(131\) 26627.1i 1.55160i 0.630977 + 0.775801i \(0.282654\pi\)
−0.630977 + 0.775801i \(0.717346\pi\)
\(132\) 0 0
\(133\) 729.107 0.0412181
\(134\) 0 0
\(135\) −14642.7 15178.8i −0.803439 0.832858i
\(136\) 0 0
\(137\) 2627.82i 0.140009i 0.997547 + 0.0700044i \(0.0223013\pi\)
−0.997547 + 0.0700044i \(0.977699\pi\)
\(138\) 0 0
\(139\) −18817.8 −0.973958 −0.486979 0.873414i \(-0.661901\pi\)
−0.486979 + 0.873414i \(0.661901\pi\)
\(140\) 0 0
\(141\) 7754.73 + 29650.6i 0.390057 + 1.49140i
\(142\) 0 0
\(143\) 5499.39i 0.268932i
\(144\) 0 0
\(145\) −35766.7 −1.70115
\(146\) 0 0
\(147\) 20857.3 5454.95i 0.965212 0.252439i
\(148\) 0 0
\(149\) 40235.0i 1.81231i −0.422951 0.906153i \(-0.639006\pi\)
0.422951 0.906153i \(-0.360994\pi\)
\(150\) 0 0
\(151\) 3711.55 0.162780 0.0813902 0.996682i \(-0.474064\pi\)
0.0813902 + 0.996682i \(0.474064\pi\)
\(152\) 0 0
\(153\) 13882.2 + 24724.3i 0.593028 + 1.05619i
\(154\) 0 0
\(155\) 4578.14i 0.190557i
\(156\) 0 0
\(157\) 18538.0 0.752079 0.376040 0.926604i \(-0.377286\pi\)
0.376040 + 0.926604i \(0.377286\pi\)
\(158\) 0 0
\(159\) −1322.83 5057.90i −0.0523250 0.200067i
\(160\) 0 0
\(161\) 667.787i 0.0257624i
\(162\) 0 0
\(163\) −18342.9 −0.690387 −0.345194 0.938531i \(-0.612187\pi\)
−0.345194 + 0.938531i \(0.612187\pi\)
\(164\) 0 0
\(165\) −5334.18 + 1395.09i −0.195930 + 0.0512428i
\(166\) 0 0
\(167\) 118.074i 0.00423372i −0.999998 0.00211686i \(-0.999326\pi\)
0.999998 0.00211686i \(-0.000673819\pi\)
\(168\) 0 0
\(169\) 38885.4 1.36149
\(170\) 0 0
\(171\) −21805.3 + 12243.2i −0.745710 + 0.418701i
\(172\) 0 0
\(173\) 11073.2i 0.369982i −0.982740 0.184991i \(-0.940774\pi\)
0.982740 0.184991i \(-0.0592256\pi\)
\(174\) 0 0
\(175\) 500.612 0.0163465
\(176\) 0 0
\(177\) 8552.64 + 32701.4i 0.272994 + 1.04381i
\(178\) 0 0
\(179\) 21979.1i 0.685969i −0.939341 0.342984i \(-0.888562\pi\)
0.939341 0.342984i \(-0.111438\pi\)
\(180\) 0 0
\(181\) −60261.6 −1.83943 −0.919716 0.392584i \(-0.871581\pi\)
−0.919716 + 0.392584i \(0.871581\pi\)
\(182\) 0 0
\(183\) −1790.20 + 468.204i −0.0534564 + 0.0139808i
\(184\) 0 0
\(185\) 25758.2i 0.752615i
\(186\) 0 0
\(187\) 7412.75 0.211981
\(188\) 0 0
\(189\) 1239.05 1195.28i 0.0346869 0.0334617i
\(190\) 0 0
\(191\) 64427.5i 1.76606i 0.469321 + 0.883028i \(0.344499\pi\)
−0.469321 + 0.883028i \(0.655501\pi\)
\(192\) 0 0
\(193\) −3486.80 −0.0936080 −0.0468040 0.998904i \(-0.514904\pi\)
−0.0468040 + 0.998904i \(0.514904\pi\)
\(194\) 0 0
\(195\) 17109.8 + 65420.2i 0.449962 + 1.72045i
\(196\) 0 0
\(197\) 37789.8i 0.973737i −0.873475 0.486869i \(-0.838139\pi\)
0.873475 0.486869i \(-0.161861\pi\)
\(198\) 0 0
\(199\) −54178.5 −1.36811 −0.684054 0.729431i \(-0.739785\pi\)
−0.684054 + 0.729431i \(0.739785\pi\)
\(200\) 0 0
\(201\) 5313.11 1389.57i 0.131509 0.0343946i
\(202\) 0 0
\(203\) 2919.65i 0.0708497i
\(204\) 0 0
\(205\) 51727.1 1.23087
\(206\) 0 0
\(207\) −11213.6 19971.4i −0.261699 0.466088i
\(208\) 0 0
\(209\) 6537.60i 0.149667i
\(210\) 0 0
\(211\) −23635.9 −0.530894 −0.265447 0.964126i \(-0.585519\pi\)
−0.265447 + 0.964126i \(0.585519\pi\)
\(212\) 0 0
\(213\) −6870.82 26270.9i −0.151443 0.579050i
\(214\) 0 0
\(215\) 99706.5i 2.15698i
\(216\) 0 0
\(217\) 373.714 0.00793634
\(218\) 0 0
\(219\) −45599.7 + 11926.0i −0.950767 + 0.248661i
\(220\) 0 0
\(221\) 90912.6i 1.86140i
\(222\) 0 0
\(223\) −47287.6 −0.950906 −0.475453 0.879741i \(-0.657716\pi\)
−0.475453 + 0.879741i \(0.657716\pi\)
\(224\) 0 0
\(225\) −14971.7 + 8406.33i −0.295738 + 0.166051i
\(226\) 0 0
\(227\) 48999.1i 0.950902i −0.879742 0.475451i \(-0.842285\pi\)
0.879742 0.475451i \(-0.157715\pi\)
\(228\) 0 0
\(229\) 46911.9 0.894565 0.447283 0.894393i \(-0.352392\pi\)
0.447283 + 0.894393i \(0.352392\pi\)
\(230\) 0 0
\(231\) −113.881 435.430i −0.00213416 0.00816008i
\(232\) 0 0
\(233\) 54299.1i 1.00019i 0.865971 + 0.500093i \(0.166701\pi\)
−0.865971 + 0.500093i \(0.833299\pi\)
\(234\) 0 0
\(235\) 98518.0 1.78394
\(236\) 0 0
\(237\) 56947.2 14893.8i 1.01386 0.265161i
\(238\) 0 0
\(239\) 76506.0i 1.33937i 0.742647 + 0.669683i \(0.233570\pi\)
−0.742647 + 0.669683i \(0.766430\pi\)
\(240\) 0 0
\(241\) −71695.7 −1.23441 −0.617205 0.786803i \(-0.711735\pi\)
−0.617205 + 0.786803i \(0.711735\pi\)
\(242\) 0 0
\(243\) −16984.8 + 56553.5i −0.287639 + 0.957739i
\(244\) 0 0
\(245\) 69301.0i 1.15454i
\(246\) 0 0
\(247\) 80179.3 1.31422
\(248\) 0 0
\(249\) 29611.6 + 113221.i 0.477598 + 1.82612i
\(250\) 0 0
\(251\) 46313.1i 0.735118i 0.930000 + 0.367559i \(0.119806\pi\)
−0.930000 + 0.367559i \(0.880194\pi\)
\(252\) 0 0
\(253\) −5987.77 −0.0935457
\(254\) 0 0
\(255\) 88181.4 23062.7i 1.35611 0.354674i
\(256\) 0 0
\(257\) 72961.7i 1.10466i −0.833625 0.552330i \(-0.813739\pi\)
0.833625 0.552330i \(-0.186261\pi\)
\(258\) 0 0
\(259\) 2102.65 0.0313449
\(260\) 0 0
\(261\) 49027.0 + 87317.5i 0.719705 + 1.28180i
\(262\) 0 0
\(263\) 40584.1i 0.586738i 0.955999 + 0.293369i \(0.0947765\pi\)
−0.955999 + 0.293369i \(0.905224\pi\)
\(264\) 0 0
\(265\) −16805.5 −0.239310
\(266\) 0 0
\(267\) 12398.3 + 47405.7i 0.173917 + 0.664979i
\(268\) 0 0
\(269\) 42063.7i 0.581304i −0.956829 0.290652i \(-0.906128\pi\)
0.956829 0.290652i \(-0.0938722\pi\)
\(270\) 0 0
\(271\) −129290. −1.76046 −0.880228 0.474551i \(-0.842611\pi\)
−0.880228 + 0.474551i \(0.842611\pi\)
\(272\) 0 0
\(273\) −5340.27 + 1396.68i −0.0716535 + 0.0187400i
\(274\) 0 0
\(275\) 4488.78i 0.0593557i
\(276\) 0 0
\(277\) −114230. −1.48874 −0.744372 0.667766i \(-0.767251\pi\)
−0.744372 + 0.667766i \(0.767251\pi\)
\(278\) 0 0
\(279\) −11176.6 + 6275.46i −0.143583 + 0.0806189i
\(280\) 0 0
\(281\) 26580.0i 0.336621i −0.985734 0.168311i \(-0.946169\pi\)
0.985734 0.168311i \(-0.0538312\pi\)
\(282\) 0 0
\(283\) −3293.98 −0.0411290 −0.0205645 0.999789i \(-0.506546\pi\)
−0.0205645 + 0.999789i \(0.506546\pi\)
\(284\) 0 0
\(285\) 20339.9 + 77770.6i 0.250414 + 0.957471i
\(286\) 0 0
\(287\) 4222.50i 0.0512632i
\(288\) 0 0
\(289\) −39022.0 −0.467212
\(290\) 0 0
\(291\) −16356.4 + 4277.81i −0.193153 + 0.0505167i
\(292\) 0 0
\(293\) 4332.67i 0.0504685i 0.999682 + 0.0252342i \(0.00803316\pi\)
−0.999682 + 0.0252342i \(0.991967\pi\)
\(294\) 0 0
\(295\) 108655. 1.24855
\(296\) 0 0
\(297\) 10717.6 + 11110.1i 0.121503 + 0.125951i
\(298\) 0 0
\(299\) 73436.0i 0.821423i
\(300\) 0 0
\(301\) −8139.06 −0.0898342
\(302\) 0 0
\(303\) −23824.6 91094.4i −0.259501 0.992217i
\(304\) 0 0
\(305\) 5948.18i 0.0639417i
\(306\) 0 0
\(307\) −102988. −1.09272 −0.546360 0.837551i \(-0.683987\pi\)
−0.546360 + 0.837551i \(0.683987\pi\)
\(308\) 0 0
\(309\) −74208.2 + 19408.2i −0.777203 + 0.203267i
\(310\) 0 0
\(311\) 97799.9i 1.01115i −0.862781 0.505577i \(-0.831280\pi\)
0.862781 0.505577i \(-0.168720\pi\)
\(312\) 0 0
\(313\) −39759.3 −0.405836 −0.202918 0.979196i \(-0.565042\pi\)
−0.202918 + 0.979196i \(0.565042\pi\)
\(314\) 0 0
\(315\) −2709.44 4825.53i −0.0273060 0.0486322i
\(316\) 0 0
\(317\) 33537.2i 0.333740i −0.985979 0.166870i \(-0.946634\pi\)
0.985979 0.166870i \(-0.0533661\pi\)
\(318\) 0 0
\(319\) 26179.3 0.257262
\(320\) 0 0
\(321\) 48073.1 + 183810.i 0.466543 + 1.78385i
\(322\) 0 0
\(323\) 108076.i 1.03591i
\(324\) 0 0
\(325\) 55051.9 0.521202
\(326\) 0 0
\(327\) −124702. + 32614.1i −1.16621 + 0.305007i
\(328\) 0 0
\(329\) 8042.04i 0.0742976i
\(330\) 0 0
\(331\) −36151.9 −0.329971 −0.164985 0.986296i \(-0.552758\pi\)
−0.164985 + 0.986296i \(0.552758\pi\)
\(332\) 0 0
\(333\) −62883.7 + 35308.0i −0.567087 + 0.318408i
\(334\) 0 0
\(335\) 17653.5i 0.157305i
\(336\) 0 0
\(337\) 212846. 1.87415 0.937077 0.349123i \(-0.113520\pi\)
0.937077 + 0.349123i \(0.113520\pi\)
\(338\) 0 0
\(339\) 32737.8 + 125175.i 0.284872 + 1.08922i
\(340\) 0 0
\(341\) 3350.94i 0.0288176i
\(342\) 0 0
\(343\) 11327.3 0.0962803
\(344\) 0 0
\(345\) −71229.9 + 18629.2i −0.598445 + 0.156515i
\(346\) 0 0
\(347\) 161089.i 1.33785i −0.743329 0.668926i \(-0.766754\pi\)
0.743329 0.668926i \(-0.233246\pi\)
\(348\) 0 0
\(349\) −112340. −0.922327 −0.461164 0.887315i \(-0.652568\pi\)
−0.461164 + 0.887315i \(0.652568\pi\)
\(350\) 0 0
\(351\) 136258. 131445.i 1.10598 1.06691i
\(352\) 0 0
\(353\) 13780.1i 0.110586i −0.998470 0.0552932i \(-0.982391\pi\)
0.998470 0.0552932i \(-0.0176094\pi\)
\(354\) 0 0
\(355\) −87288.6 −0.692629
\(356\) 0 0
\(357\) 1882.61 + 7198.26i 0.0147715 + 0.0564796i
\(358\) 0 0
\(359\) 12567.6i 0.0975132i −0.998811 0.0487566i \(-0.984474\pi\)
0.998811 0.0487566i \(-0.0155259\pi\)
\(360\) 0 0
\(361\) −35005.0 −0.268606
\(362\) 0 0
\(363\) −123577. + 32319.9i −0.937829 + 0.245277i
\(364\) 0 0
\(365\) 151511.i 1.13726i
\(366\) 0 0
\(367\) −169054. −1.25514 −0.627571 0.778559i \(-0.715951\pi\)
−0.627571 + 0.778559i \(0.715951\pi\)
\(368\) 0 0
\(369\) −70904.6 126282.i −0.520741 0.927444i
\(370\) 0 0
\(371\) 1371.84i 0.00996679i
\(372\) 0 0
\(373\) 238003. 1.71066 0.855331 0.518082i \(-0.173354\pi\)
0.855331 + 0.518082i \(0.173354\pi\)
\(374\) 0 0
\(375\) −27210.6 104041.i −0.193497 0.739847i
\(376\) 0 0
\(377\) 321071.i 2.25901i
\(378\) 0 0
\(379\) 203118. 1.41407 0.707034 0.707180i \(-0.250033\pi\)
0.707034 + 0.707180i \(0.250033\pi\)
\(380\) 0 0
\(381\) −172744. + 45179.0i −1.19002 + 0.311234i
\(382\) 0 0
\(383\) 160683.i 1.09540i 0.836676 + 0.547699i \(0.184496\pi\)
−0.836676 + 0.547699i \(0.815504\pi\)
\(384\) 0 0
\(385\) −1446.77 −0.00976066
\(386\) 0 0
\(387\) 243414. 136672.i 1.62526 0.912553i
\(388\) 0 0
\(389\) 38824.4i 0.256570i 0.991737 + 0.128285i \(0.0409472\pi\)
−0.991737 + 0.128285i \(0.959053\pi\)
\(390\) 0 0
\(391\) 98986.0 0.647471
\(392\) 0 0
\(393\) −60636.1 231845.i −0.392596 1.50111i
\(394\) 0 0
\(395\) 189215.i 1.21272i
\(396\) 0 0
\(397\) −100549. −0.637967 −0.318983 0.947760i \(-0.603341\pi\)
−0.318983 + 0.947760i \(0.603341\pi\)
\(398\) 0 0
\(399\) −6348.43 + 1660.35i −0.0398768 + 0.0104293i
\(400\) 0 0
\(401\) 25766.4i 0.160238i 0.996785 + 0.0801189i \(0.0255300\pi\)
−0.996785 + 0.0801189i \(0.974470\pi\)
\(402\) 0 0
\(403\) 41097.1 0.253047
\(404\) 0 0
\(405\) 162062. + 98819.3i 0.988030 + 0.602465i
\(406\) 0 0
\(407\) 18853.6i 0.113817i
\(408\) 0 0
\(409\) 135050. 0.807324 0.403662 0.914908i \(-0.367737\pi\)
0.403662 + 0.914908i \(0.367737\pi\)
\(410\) 0 0
\(411\) −5984.18 22880.8i −0.0354259 0.135453i
\(412\) 0 0
\(413\) 8869.51i 0.0519996i
\(414\) 0 0
\(415\) 376193. 2.18431
\(416\) 0 0
\(417\) 163849. 42852.7i 0.942265 0.246437i
\(418\) 0 0
\(419\) 222868.i 1.26946i −0.772732 0.634732i \(-0.781110\pi\)
0.772732 0.634732i \(-0.218890\pi\)
\(420\) 0 0
\(421\) 77824.4 0.439088 0.219544 0.975603i \(-0.429543\pi\)
0.219544 + 0.975603i \(0.429543\pi\)
\(422\) 0 0
\(423\) −135043. 240512.i −0.754729 1.34418i
\(424\) 0 0
\(425\) 74205.7i 0.410827i
\(426\) 0 0
\(427\) −485.551 −0.00266305
\(428\) 0 0
\(429\) −12523.4 47883.9i −0.0680469 0.260181i
\(430\) 0 0
\(431\) 296695.i 1.59718i −0.601872 0.798592i \(-0.705578\pi\)
0.601872 0.798592i \(-0.294422\pi\)
\(432\) 0 0
\(433\) 155246. 0.828027 0.414014 0.910271i \(-0.364127\pi\)
0.414014 + 0.910271i \(0.364127\pi\)
\(434\) 0 0
\(435\) 311426. 81449.4i 1.64580 0.430437i
\(436\) 0 0
\(437\) 87299.7i 0.457141i
\(438\) 0 0
\(439\) 126457. 0.656165 0.328082 0.944649i \(-0.393598\pi\)
0.328082 + 0.944649i \(0.393598\pi\)
\(440\) 0 0
\(441\) −169185. + 94993.9i −0.869930 + 0.488448i
\(442\) 0 0
\(443\) 145466.i 0.741230i −0.928787 0.370615i \(-0.879147\pi\)
0.928787 0.370615i \(-0.120853\pi\)
\(444\) 0 0
\(445\) 157512. 0.795413
\(446\) 0 0
\(447\) 91624.6 + 350331.i 0.458561 + 1.75333i
\(448\) 0 0
\(449\) 287619.i 1.42668i −0.700820 0.713338i \(-0.747182\pi\)
0.700820 0.713338i \(-0.252818\pi\)
\(450\) 0 0
\(451\) −37861.4 −0.186142
\(452\) 0 0
\(453\) −32317.0 + 8452.09i −0.157483 + 0.0411877i
\(454\) 0 0
\(455\) 17743.7i 0.0857082i
\(456\) 0 0
\(457\) −267619. −1.28140 −0.640701 0.767791i \(-0.721356\pi\)
−0.640701 + 0.767791i \(0.721356\pi\)
\(458\) 0 0
\(459\) −177177. 183665.i −0.840973 0.871766i
\(460\) 0 0
\(461\) 85209.3i 0.400945i −0.979699 0.200473i \(-0.935752\pi\)
0.979699 0.200473i \(-0.0642478\pi\)
\(462\) 0 0
\(463\) −239522. −1.11734 −0.558668 0.829391i \(-0.688687\pi\)
−0.558668 + 0.829391i \(0.688687\pi\)
\(464\) 0 0
\(465\) 10425.5 + 39862.5i 0.0482160 + 0.184356i
\(466\) 0 0
\(467\) 225161.i 1.03243i 0.856459 + 0.516214i \(0.172659\pi\)
−0.856459 + 0.516214i \(0.827341\pi\)
\(468\) 0 0
\(469\) 1441.06 0.00655143
\(470\) 0 0
\(471\) −161413. + 42215.4i −0.727606 + 0.190296i
\(472\) 0 0
\(473\) 72979.6i 0.326196i
\(474\) 0 0
\(475\) 65444.9 0.290061
\(476\) 0 0
\(477\) 23036.1 + 41027.4i 0.101245 + 0.180317i
\(478\) 0 0
\(479\) 210999.i 0.919622i −0.888017 0.459811i \(-0.847917\pi\)
0.888017 0.459811i \(-0.152083\pi\)
\(480\) 0 0
\(481\) 231227. 0.999421
\(482\) 0 0
\(483\) −1520.71 5814.51i −0.00651856 0.0249241i
\(484\) 0 0
\(485\) 54346.3i 0.231040i
\(486\) 0 0
\(487\) −346874. −1.46256 −0.731280 0.682077i \(-0.761077\pi\)
−0.731280 + 0.682077i \(0.761077\pi\)
\(488\) 0 0
\(489\) 159714. 41771.1i 0.667922 0.174686i
\(490\) 0 0
\(491\) 46005.6i 0.190831i 0.995438 + 0.0954153i \(0.0304179\pi\)
−0.995438 + 0.0954153i \(0.969582\pi\)
\(492\) 0 0
\(493\) −432779. −1.78063
\(494\) 0 0
\(495\) 43268.5 24294.4i 0.176588 0.0991507i
\(496\) 0 0
\(497\) 7125.39i 0.0288467i
\(498\) 0 0
\(499\) −266428. −1.06999 −0.534994 0.844856i \(-0.679686\pi\)
−0.534994 + 0.844856i \(0.679686\pi\)
\(500\) 0 0
\(501\) 268.883 + 1028.09i 0.00107124 + 0.00409596i
\(502\) 0 0
\(503\) 54796.2i 0.216578i −0.994119 0.108289i \(-0.965463\pi\)
0.994119 0.108289i \(-0.0345372\pi\)
\(504\) 0 0
\(505\) −302673. −1.18684
\(506\) 0 0
\(507\) −338581. + 88551.4i −1.31718 + 0.344492i
\(508\) 0 0
\(509\) 76722.7i 0.296134i 0.988977 + 0.148067i \(0.0473052\pi\)
−0.988977 + 0.148067i \(0.952695\pi\)
\(510\) 0 0
\(511\) −12367.9 −0.0473646
\(512\) 0 0
\(513\) 161981. 156259.i 0.615502 0.593761i
\(514\) 0 0
\(515\) 246566.i 0.929649i
\(516\) 0 0
\(517\) −72109.7 −0.269782
\(518\) 0 0
\(519\) 25216.3 + 96415.8i 0.0936152 + 0.357943i
\(520\) 0 0
\(521\) 400136.i 1.47412i 0.675828 + 0.737060i \(0.263786\pi\)
−0.675828 + 0.737060i \(0.736214\pi\)
\(522\) 0 0
\(523\) 285364. 1.04327 0.521634 0.853169i \(-0.325323\pi\)
0.521634 + 0.853169i \(0.325323\pi\)
\(524\) 0 0
\(525\) −4358.89 + 1140.01i −0.0158146 + 0.00413610i
\(526\) 0 0
\(527\) 55395.7i 0.199460i
\(528\) 0 0
\(529\) 199884. 0.714275
\(530\) 0 0
\(531\) −148938. 265259.i −0.528222 0.940767i
\(532\) 0 0
\(533\) 464345.i 1.63451i
\(534\) 0 0
\(535\) 610732. 2.13375
\(536\) 0 0
\(537\) 50051.7 + 191375.i 0.173568 + 0.663647i
\(538\) 0 0
\(539\) 50724.5i 0.174598i
\(540\) 0 0
\(541\) −321140. −1.09724 −0.548619 0.836073i \(-0.684846\pi\)
−0.548619 + 0.836073i \(0.684846\pi\)
\(542\) 0 0
\(543\) 524706. 137230.i 1.77958 0.465425i
\(544\) 0 0
\(545\) 414338.i 1.39496i
\(546\) 0 0
\(547\) 2543.76 0.00850160 0.00425080 0.999991i \(-0.498647\pi\)
0.00425080 + 0.999991i \(0.498647\pi\)
\(548\) 0 0
\(549\) 14521.3 8153.43i 0.0481794 0.0270518i
\(550\) 0 0
\(551\) 381685.i 1.25719i
\(552\) 0 0
\(553\) 15445.6 0.0505074
\(554\) 0 0
\(555\) 58657.6 + 224280.i 0.190431 + 0.728124i
\(556\) 0 0
\(557\) 265139.i 0.854602i −0.904109 0.427301i \(-0.859464\pi\)
0.904109 0.427301i \(-0.140536\pi\)
\(558\) 0 0
\(559\) −895047. −2.86432
\(560\) 0 0
\(561\) −64543.8 + 16880.6i −0.205083 + 0.0536367i
\(562\) 0 0
\(563\) 77729.4i 0.245227i −0.992454 0.122614i \(-0.960872\pi\)
0.992454 0.122614i \(-0.0391276\pi\)
\(564\) 0 0
\(565\) 415909. 1.30287
\(566\) 0 0
\(567\) −8066.64 + 13229.1i −0.0250915 + 0.0411495i
\(568\) 0 0
\(569\) 147003.i 0.454047i 0.973889 + 0.227023i \(0.0728994\pi\)
−0.973889 + 0.227023i \(0.927101\pi\)
\(570\) 0 0
\(571\) −113294. −0.347484 −0.173742 0.984791i \(-0.555586\pi\)
−0.173742 + 0.984791i \(0.555586\pi\)
\(572\) 0 0
\(573\) −146717. 560979.i −0.446859 1.70859i
\(574\) 0 0
\(575\) 59940.8i 0.181295i
\(576\) 0 0
\(577\) −300139. −0.901512 −0.450756 0.892647i \(-0.648845\pi\)
−0.450756 + 0.892647i \(0.648845\pi\)
\(578\) 0 0
\(579\) 30360.1 7940.28i 0.0905619 0.0236853i
\(580\) 0 0
\(581\) 30708.7i 0.0909722i
\(582\) 0 0
\(583\) 12300.7 0.0361904
\(584\) 0 0
\(585\) −297955. 530660.i −0.870640 1.55062i
\(586\) 0 0
\(587\) 340321.i 0.987673i 0.869555 + 0.493836i \(0.164406\pi\)
−0.869555 + 0.493836i \(0.835594\pi\)
\(588\) 0 0
\(589\) 48855.6 0.140826
\(590\) 0 0
\(591\) 86056.3 + 329041.i 0.246381 + 0.942051i
\(592\) 0 0
\(593\) 457414.i 1.30077i 0.759605 + 0.650385i \(0.225392\pi\)
−0.759605 + 0.650385i \(0.774608\pi\)
\(594\) 0 0
\(595\) 23917.2 0.0675578
\(596\) 0 0
\(597\) 471739. 123377.i 1.32359 0.346168i
\(598\) 0 0
\(599\) 527946.i 1.47142i 0.677297 + 0.735709i \(0.263151\pi\)
−0.677297 + 0.735709i \(0.736849\pi\)
\(600\) 0 0
\(601\) −229399. −0.635101 −0.317551 0.948241i \(-0.602860\pi\)
−0.317551 + 0.948241i \(0.602860\pi\)
\(602\) 0 0
\(603\) −43097.6 + 24198.4i −0.118527 + 0.0665507i
\(604\) 0 0
\(605\) 410600.i 1.12178i
\(606\) 0 0
\(607\) 287358. 0.779912 0.389956 0.920834i \(-0.372490\pi\)
0.389956 + 0.920834i \(0.372490\pi\)
\(608\) 0 0
\(609\) 6648.73 + 25421.7i 0.0179268 + 0.0685442i
\(610\) 0 0
\(611\) 884378.i 2.36895i
\(612\) 0 0
\(613\) −282301. −0.751262 −0.375631 0.926769i \(-0.622574\pi\)
−0.375631 + 0.926769i \(0.622574\pi\)
\(614\) 0 0
\(615\) −450395. + 117795.i −1.19081 + 0.311442i
\(616\) 0 0
\(617\) 284331.i 0.746885i 0.927653 + 0.373442i \(0.121823\pi\)
−0.927653 + 0.373442i \(0.878177\pi\)
\(618\) 0 0
\(619\) 448233. 1.16983 0.584915 0.811094i \(-0.301128\pi\)
0.584915 + 0.811094i \(0.301128\pi\)
\(620\) 0 0
\(621\) 143118. + 148358.i 0.371116 + 0.384705i
\(622\) 0 0
\(623\) 12857.7i 0.0331274i
\(624\) 0 0
\(625\) −478177. −1.22413
\(626\) 0 0
\(627\) −14887.7 56923.7i −0.0378697 0.144797i
\(628\) 0 0
\(629\) 311676.i 0.787775i
\(630\) 0 0
\(631\) 564174. 1.41695 0.708475 0.705736i \(-0.249383\pi\)
0.708475 + 0.705736i \(0.249383\pi\)
\(632\) 0 0
\(633\) 205801. 53824.6i 0.513618 0.134330i
\(634\) 0 0
\(635\) 573965.i 1.42344i
\(636\) 0 0
\(637\) 622102. 1.53314
\(638\) 0 0
\(639\) 119650. + 213098.i 0.293030 + 0.521889i
\(640\) 0 0
\(641\) 376472.i 0.916257i 0.888886 + 0.458128i \(0.151480\pi\)
−0.888886 + 0.458128i \(0.848520\pi\)
\(642\) 0 0
\(643\) −531024. −1.28438 −0.642188 0.766547i \(-0.721973\pi\)
−0.642188 + 0.766547i \(0.721973\pi\)
\(644\) 0 0
\(645\) −227055. 868158.i −0.545774 2.08679i
\(646\) 0 0
\(647\) 482001.i 1.15144i 0.817649 + 0.575718i \(0.195277\pi\)
−0.817649 + 0.575718i \(0.804723\pi\)
\(648\) 0 0
\(649\) −79529.2 −0.188815
\(650\) 0 0
\(651\) −3253.98 + 851.036i −0.00767809 + 0.00200810i
\(652\) 0 0
\(653\) 47193.4i 0.110676i −0.998468 0.0553382i \(-0.982376\pi\)
0.998468 0.0553382i \(-0.0176237\pi\)
\(654\) 0 0
\(655\) −770336. −1.79555
\(656\) 0 0
\(657\) 369885. 207683.i 0.856911 0.481138i
\(658\) 0 0
\(659\) 344237.i 0.792660i −0.918108 0.396330i \(-0.870284\pi\)
0.918108 0.396330i \(-0.129716\pi\)
\(660\) 0 0
\(661\) −23725.6 −0.0543018 −0.0271509 0.999631i \(-0.508643\pi\)
−0.0271509 + 0.999631i \(0.508643\pi\)
\(662\) 0 0
\(663\) 207029. + 791588.i 0.470983 + 1.80083i
\(664\) 0 0
\(665\) 21093.5i 0.0476985i
\(666\) 0 0
\(667\) 349584. 0.785778
\(668\) 0 0
\(669\) 411739. 107685.i 0.919962 0.240604i
\(670\) 0 0
\(671\) 4353.73i 0.00966979i
\(672\) 0 0
\(673\) 201011. 0.443804 0.221902 0.975069i \(-0.428774\pi\)
0.221902 + 0.975069i \(0.428774\pi\)
\(674\) 0 0
\(675\) 111218. 107289.i 0.244099 0.235477i
\(676\) 0 0
\(677\) 726242.i 1.58454i −0.610169 0.792272i \(-0.708898\pi\)
0.610169 0.792272i \(-0.291102\pi\)
\(678\) 0 0
\(679\) −4436.30 −0.00962235
\(680\) 0 0
\(681\) 111582. + 426641.i 0.240603 + 0.919960i
\(682\) 0 0
\(683\) 316498.i 0.678468i −0.940702 0.339234i \(-0.889832\pi\)
0.940702 0.339234i \(-0.110168\pi\)
\(684\) 0 0
\(685\) −76024.5 −0.162021
\(686\) 0 0
\(687\) −408468. + 106829.i −0.865455 + 0.226349i
\(688\) 0 0
\(689\) 150860.i 0.317787i
\(690\) 0 0
\(691\) −299153. −0.626524 −0.313262 0.949667i \(-0.601422\pi\)
−0.313262 + 0.949667i \(0.601422\pi\)
\(692\) 0 0
\(693\) 1983.16 + 3532.02i 0.00412943 + 0.00735455i
\(694\) 0 0
\(695\) 544411.i 1.12709i
\(696\) 0 0
\(697\) 625901. 1.28837
\(698\) 0 0
\(699\) −123652. 472790.i −0.253074 0.967640i
\(700\) 0 0
\(701\) 185762.i 0.378024i 0.981975 + 0.189012i \(0.0605286\pi\)
−0.981975 + 0.189012i \(0.939471\pi\)
\(702\) 0 0
\(703\) 274879. 0.556200
\(704\) 0 0
\(705\) −857810. + 224349.i −1.72589 + 0.451383i
\(706\) 0 0
\(707\) 24707.3i 0.0494295i
\(708\) 0 0
\(709\) −279476. −0.555970 −0.277985 0.960585i \(-0.589667\pi\)
−0.277985 + 0.960585i \(0.589667\pi\)
\(710\) 0 0
\(711\) −461930. + 259365.i −0.913771 + 0.513064i
\(712\) 0 0
\(713\) 44746.7i 0.0880202i
\(714\) 0 0
\(715\) −159101. −0.311215
\(716\) 0 0
\(717\) −174222. 666148.i −0.338895 1.29578i
\(718\) 0 0
\(719\) 252618.i 0.488659i −0.969692 0.244330i \(-0.921432\pi\)
0.969692 0.244330i \(-0.0785679\pi\)
\(720\) 0 0
\(721\) −20127.3 −0.0387181
\(722\) 0 0
\(723\) 624264. 163268.i 1.19424 0.312338i
\(724\) 0 0
\(725\) 262069.i 0.498585i
\(726\) 0 0
\(727\) −478683. −0.905689 −0.452844 0.891590i \(-0.649591\pi\)
−0.452844 + 0.891590i \(0.649591\pi\)
\(728\) 0 0
\(729\) 19103.0 531098.i 0.0359457 0.999354i
\(730\) 0 0
\(731\) 1.20645e6i 2.25775i
\(732\) 0 0
\(733\) 554036. 1.03117 0.515585 0.856838i \(-0.327575\pi\)
0.515585 + 0.856838i \(0.327575\pi\)
\(734\) 0 0
\(735\) 157815. + 603413.i 0.292128 + 1.11697i
\(736\) 0 0
\(737\) 12921.4i 0.0237889i
\(738\) 0 0
\(739\) −268187. −0.491077 −0.245538 0.969387i \(-0.578965\pi\)
−0.245538 + 0.969387i \(0.578965\pi\)
\(740\) 0 0
\(741\) −698132. + 182587.i −1.27146 + 0.332533i
\(742\) 0 0
\(743\) 487745.i 0.883517i −0.897134 0.441758i \(-0.854355\pi\)
0.897134 0.441758i \(-0.145645\pi\)
\(744\) 0 0
\(745\) 1.16402e6 2.09724
\(746\) 0 0
\(747\) −515664. 918401.i −0.924113 1.64585i
\(748\) 0 0
\(749\) 49854.2i 0.0888665i
\(750\) 0 0
\(751\) −256091. −0.454061 −0.227031 0.973888i \(-0.572902\pi\)
−0.227031 + 0.973888i \(0.572902\pi\)
\(752\) 0 0
\(753\) −105466. 403255.i −0.186004 0.711197i
\(754\) 0 0
\(755\) 107377.i 0.188373i
\(756\) 0 0
\(757\) 138602. 0.241867 0.120934 0.992661i \(-0.461411\pi\)
0.120934 + 0.992661i \(0.461411\pi\)
\(758\) 0 0
\(759\) 52136.3 13635.6i 0.0905016 0.0236695i
\(760\) 0 0
\(761\) 487206.i 0.841286i 0.907226 + 0.420643i \(0.138195\pi\)
−0.907226 + 0.420643i \(0.861805\pi\)
\(762\) 0 0
\(763\) −33822.5 −0.0580973
\(764\) 0 0
\(765\) −715288. + 401620.i −1.22224 + 0.686266i
\(766\) 0 0
\(767\) 975374.i 1.65798i
\(768\) 0 0
\(769\) 35520.3 0.0600654 0.0300327 0.999549i \(-0.490439\pi\)
0.0300327 + 0.999549i \(0.490439\pi\)
\(770\) 0 0
\(771\) 166151. + 635288.i 0.279508 + 1.06871i
\(772\) 0 0
\(773\) 813847.i 1.36202i 0.732274 + 0.681010i \(0.238459\pi\)
−0.732274 + 0.681010i \(0.761541\pi\)
\(774\) 0 0
\(775\) 33544.8 0.0558497
\(776\) 0 0
\(777\) −18308.1 + 4788.23i −0.0303250 + 0.00793110i
\(778\) 0 0
\(779\) 552006.i 0.909639i
\(780\) 0 0
\(781\) 63890.4 0.104745
\(782\) 0 0
\(783\) −625728. 648639.i −1.02061 1.05799i
\(784\) 0 0
\(785\) 536315.i 0.870324i
\(786\) 0 0
\(787\) 100644. 0.162494 0.0812470 0.996694i \(-0.474110\pi\)
0.0812470 + 0.996694i \(0.474110\pi\)
\(788\) 0 0
\(789\) −92419.6 353371.i −0.148460 0.567645i
\(790\) 0 0
\(791\) 33950.7i 0.0542620i
\(792\) 0 0
\(793\) −53395.7 −0.0849102
\(794\) 0 0
\(795\) 146328. 38270.2i 0.231523 0.0605517i
\(796\) 0 0
\(797\) 603149.i 0.949529i −0.880113 0.474764i \(-0.842533\pi\)
0.880113 0.474764i \(-0.157467\pi\)
\(798\) 0 0
\(799\) 1.19207e6 1.86728
\(800\) 0 0
\(801\) −215908. 384534.i −0.336515 0.599335i
\(802\) 0 0
\(803\) 110898.i 0.171985i
\(804\) 0 0
\(805\) −19319.5 −0.0298128
\(806\) 0 0
\(807\) 95789.1 + 366255.i 0.147085 + 0.562388i
\(808\) 0 0
\(809\) 443718.i 0.677970i 0.940792 + 0.338985i \(0.110084\pi\)
−0.940792 + 0.338985i \(0.889916\pi\)
\(810\) 0 0
\(811\) −782028. −1.18900 −0.594498 0.804097i \(-0.702649\pi\)
−0.594498 + 0.804097i \(0.702649\pi\)
\(812\) 0 0
\(813\) 1.12574e6 294423.i 1.70317 0.445442i
\(814\) 0 0
\(815\) 530671.i 0.798933i
\(816\) 0 0
\(817\) −1.06402e6 −1.59406
\(818\) 0 0
\(819\) 43317.8 24322.1i 0.0645802 0.0362605i
\(820\) 0 0
\(821\) 1.21496e6i 1.80251i 0.433292 + 0.901254i \(0.357352\pi\)
−0.433292 + 0.901254i \(0.642648\pi\)
\(822\) 0 0
\(823\) 126836. 0.187260 0.0936299 0.995607i \(-0.470153\pi\)
0.0936299 + 0.995607i \(0.470153\pi\)
\(824\) 0 0
\(825\) −10222.0 39084.4i −0.0150186 0.0574243i
\(826\) 0 0
\(827\) 1.11286e6i 1.62715i −0.581458 0.813577i \(-0.697517\pi\)
0.581458 0.813577i \(-0.302483\pi\)
\(828\) 0 0
\(829\) 759496. 1.10514 0.552569 0.833467i \(-0.313648\pi\)
0.552569 + 0.833467i \(0.313648\pi\)
\(830\) 0 0
\(831\) 994614. 260128.i 1.44030 0.376691i
\(832\) 0 0
\(833\) 838545.i 1.20847i
\(834\) 0 0
\(835\) 3415.96 0.00489937
\(836\) 0 0
\(837\) 83025.7 80093.1i 0.118512 0.114326i
\(838\) 0 0
\(839\) 965454.i 1.37154i −0.727820 0.685769i \(-0.759466\pi\)
0.727820 0.685769i \(-0.240534\pi\)
\(840\) 0 0
\(841\) −821144. −1.16099
\(842\) 0 0
\(843\) 60528.9 + 231435.i 0.0851741 + 0.325668i
\(844\) 0 0
\(845\) 1.12498e6i 1.57555i
\(846\) 0 0
\(847\) −33517.4 −0.0467200
\(848\) 0 0
\(849\) 28681.1 7501.17i 0.0397906 0.0104067i
\(850\) 0 0
\(851\) 251761.i 0.347640i
\(852\) 0 0
\(853\) −1.09387e6 −1.50337 −0.751686 0.659522i \(-0.770759\pi\)
−0.751686 + 0.659522i \(0.770759\pi\)
\(854\) 0 0
\(855\) −354204. 630840.i −0.484531 0.862953i
\(856\) 0 0
\(857\) 1.10568e6i 1.50545i 0.658333 + 0.752727i \(0.271262\pi\)
−0.658333 + 0.752727i \(0.728738\pi\)
\(858\) 0 0
\(859\) 780160. 1.05730 0.528649 0.848841i \(-0.322699\pi\)
0.528649 + 0.848841i \(0.322699\pi\)
\(860\) 0 0
\(861\) −9615.63 36765.8i −0.0129709 0.0495950i
\(862\) 0 0
\(863\) 610965.i 0.820341i −0.912009 0.410170i \(-0.865469\pi\)
0.912009 0.410170i \(-0.134531\pi\)
\(864\) 0 0
\(865\) 320354. 0.428152
\(866\) 0 0
\(867\) 339770. 88862.4i 0.452009 0.118217i
\(868\) 0 0
\(869\) 138494.i 0.183397i
\(870\) 0 0
\(871\) 158472. 0.208890
\(872\) 0 0
\(873\) 132676. 74494.9i 0.174086 0.0977457i
\(874\) 0 0
\(875\) 28218.7i 0.0368571i
\(876\) 0 0
\(877\) −141276. −0.183684 −0.0918419 0.995774i \(-0.529275\pi\)
−0.0918419 + 0.995774i \(0.529275\pi\)
\(878\) 0 0
\(879\) −9866.51 37725.1i −0.0127699 0.0488262i
\(880\) 0 0
\(881\) 410601.i 0.529016i −0.964384 0.264508i \(-0.914791\pi\)
0.964384 0.264508i \(-0.0852095\pi\)
\(882\) 0 0
\(883\) 609192. 0.781327 0.390663 0.920534i \(-0.372246\pi\)
0.390663 + 0.920534i \(0.372246\pi\)
\(884\) 0 0
\(885\) −946072. + 247433.i −1.20792 + 0.315915i
\(886\) 0 0
\(887\) 797723.i 1.01392i −0.861969 0.506961i \(-0.830769\pi\)
0.861969 0.506961i \(-0.169231\pi\)
\(888\) 0 0
\(889\) −46852.9 −0.0592834
\(890\) 0 0
\(891\) −118620. 72330.2i −0.149418 0.0911096i
\(892\) 0 0
\(893\) 1.05134e6i 1.31837i
\(894\) 0 0
\(895\) 635869. 0.793819
\(896\) 0 0
\(897\) −167231. 639417.i −0.207842 0.794693i
\(898\) 0 0
\(899\) 195638.i 0.242066i
\(900\) 0 0
\(901\) −203348. −0.250490
\(902\) 0 0
\(903\) 70867.9 18534.6i 0.0869109 0.0227304i
\(904\) 0 0
\(905\) 1.74340e6i 2.12863i
\(906\) 0 0
\(907\) 71090.1 0.0864161 0.0432081 0.999066i \(-0.486242\pi\)
0.0432081 + 0.999066i \(0.486242\pi\)
\(908\) 0 0
\(909\) 414887. + 738917.i 0.502114 + 0.894269i
\(910\) 0 0
\(911\) 4036.23i 0.00486339i −0.999997 0.00243169i \(-0.999226\pi\)
0.999997 0.00243169i \(-0.000774033\pi\)
\(912\) 0 0
\(913\) −275352. −0.330329
\(914\) 0 0
\(915\) −13545.4 51791.6i −0.0161789 0.0618610i
\(916\) 0 0
\(917\) 62882.7i 0.0747812i
\(918\) 0 0
\(919\) 604713. 0.716009 0.358004 0.933720i \(-0.383457\pi\)
0.358004 + 0.933720i \(0.383457\pi\)
\(920\) 0 0
\(921\) 896728. 234528.i 1.05716 0.276487i
\(922\) 0 0
\(923\) 783574.i 0.919764i
\(924\) 0 0
\(925\) 188735. 0.220581
\(926\) 0 0
\(927\) 601943. 337979.i 0.700481 0.393306i
\(928\) 0 0
\(929\) 136984.i 0.158722i 0.996846 + 0.0793612i \(0.0252880\pi\)
−0.996846 + 0.0793612i \(0.974712\pi\)
\(930\) 0 0
\(931\) 739546. 0.853229
\(932\) 0 0
\(933\) 222714. + 851557.i 0.255849 + 0.978251i
\(934\) 0 0
\(935\) 214455.i 0.245309i
\(936\) 0 0
\(937\) 1.68479e6 1.91897 0.959483 0.281768i \(-0.0909209\pi\)
0.959483 + 0.281768i \(0.0909209\pi\)
\(938\) 0 0
\(939\) 346190. 90541.5i 0.392630 0.102687i
\(940\) 0 0
\(941\) 347602.i 0.392557i 0.980548 + 0.196278i \(0.0628857\pi\)
−0.980548 + 0.196278i \(0.937114\pi\)
\(942\) 0 0
\(943\) −505581. −0.568548
\(944\) 0 0
\(945\) 34580.3 + 35846.5i 0.0387226 + 0.0401405i
\(946\) 0 0
\(947\) 67955.1i 0.0757743i 0.999282 + 0.0378871i \(0.0120627\pi\)
−0.999282 + 0.0378871i \(0.987937\pi\)
\(948\) 0 0
\(949\) −1.36009e6 −1.51020
\(950\) 0 0
\(951\) 76372.2 + 292013.i 0.0844451 + 0.322880i
\(952\) 0 0
\(953\) 760779.i 0.837669i −0.908063 0.418835i \(-0.862439\pi\)
0.908063 0.418835i \(-0.137561\pi\)
\(954\) 0 0
\(955\) −1.86392e6 −2.04372
\(956\) 0 0
\(957\) −227946. + 59616.4i −0.248891 + 0.0650941i
\(958\) 0 0
\(959\) 6205.90i 0.00674788i
\(960\) 0 0
\(961\) −898479. −0.972885
\(962\) 0 0
\(963\) −837158. 1.49098e6i −0.902723 1.60776i
\(964\) 0 0
\(965\) 100875.i 0.108325i
\(966\) 0 0
\(967\) −1.63961e6 −1.75343 −0.876716 0.481009i \(-0.840270\pi\)
−0.876716 + 0.481009i \(0.840270\pi\)
\(968\) 0 0
\(969\) 246114. + 941028.i 0.262113 + 1.00220i
\(970\) 0 0
\(971\) 487147.i 0.516680i −0.966054 0.258340i \(-0.916825\pi\)
0.966054 0.258340i \(-0.0831755\pi\)
\(972\) 0 0
\(973\) 44440.4 0.0469410
\(974\) 0 0
\(975\) −479344. + 125366.i −0.504241 + 0.131878i
\(976\) 0 0
\(977\) 1.10091e6i 1.15336i −0.816971 0.576678i \(-0.804349\pi\)
0.816971 0.576678i \(-0.195651\pi\)
\(978\) 0 0
\(979\) −115290. −0.120289
\(980\) 0 0
\(981\) 1.01152e6 567951.i 1.05109 0.590164i
\(982\) 0 0
\(983\) 812256.i 0.840593i −0.907387 0.420297i \(-0.861926\pi\)
0.907387 0.420297i \(-0.138074\pi\)
\(984\) 0 0
\(985\) 1.09328e6 1.12683
\(986\) 0 0
\(987\) −18313.6 70023.2i −0.0187992 0.0718799i
\(988\) 0 0
\(989\) 974532.i 0.996330i
\(990\) 0 0
\(991\) −1.01281e6 −1.03129 −0.515646 0.856802i \(-0.672448\pi\)
−0.515646 + 0.856802i \(0.672448\pi\)
\(992\) 0 0
\(993\) 314780. 82326.5i 0.319233 0.0834913i
\(994\) 0 0
\(995\) 1.56741e6i 1.58321i
\(996\) 0 0
\(997\) 571619. 0.575064 0.287532 0.957771i \(-0.407165\pi\)
0.287532 + 0.957771i \(0.407165\pi\)
\(998\) 0 0
\(999\) 467132. 450632.i 0.468068 0.451535i
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 384.5.e.d.257.2 yes 16
3.2 odd 2 inner 384.5.e.d.257.1 yes 16
4.3 odd 2 384.5.e.a.257.15 16
8.3 odd 2 384.5.e.c.257.2 yes 16
8.5 even 2 384.5.e.b.257.15 yes 16
12.11 even 2 384.5.e.a.257.16 yes 16
24.5 odd 2 384.5.e.b.257.16 yes 16
24.11 even 2 384.5.e.c.257.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.e.a.257.15 16 4.3 odd 2
384.5.e.a.257.16 yes 16 12.11 even 2
384.5.e.b.257.15 yes 16 8.5 even 2
384.5.e.b.257.16 yes 16 24.5 odd 2
384.5.e.c.257.1 yes 16 24.11 even 2
384.5.e.c.257.2 yes 16 8.3 odd 2
384.5.e.d.257.1 yes 16 3.2 odd 2 inner
384.5.e.d.257.2 yes 16 1.1 even 1 trivial