Properties

Label 338.6.a.e.1.1
Level $338$
Weight $6$
Character 338.1
Self dual yes
Analytic conductor $54.210$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,6,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.2097310968\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +4.00000 q^{3} +16.0000 q^{4} +68.0000 q^{5} +16.0000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -227.000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +4.00000 q^{3} +16.0000 q^{4} +68.0000 q^{5} +16.0000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -227.000 q^{9} +272.000 q^{10} -390.000 q^{11} +64.0000 q^{12} -328.000 q^{14} +272.000 q^{15} +256.000 q^{16} -1738.00 q^{17} -908.000 q^{18} -1074.00 q^{19} +1088.00 q^{20} -328.000 q^{21} -1560.00 q^{22} -2104.00 q^{23} +256.000 q^{24} +1499.00 q^{25} -1880.00 q^{27} -1312.00 q^{28} -1690.00 q^{29} +1088.00 q^{30} -1430.00 q^{31} +1024.00 q^{32} -1560.00 q^{33} -6952.00 q^{34} -5576.00 q^{35} -3632.00 q^{36} -8852.00 q^{37} -4296.00 q^{38} +4352.00 q^{40} +6760.00 q^{41} -1312.00 q^{42} +16916.0 q^{43} -6240.00 q^{44} -15436.0 q^{45} -8416.00 q^{46} +25158.0 q^{47} +1024.00 q^{48} -10083.0 q^{49} +5996.00 q^{50} -6952.00 q^{51} +38214.0 q^{53} -7520.00 q^{54} -26520.0 q^{55} -5248.00 q^{56} -4296.00 q^{57} -6760.00 q^{58} -21286.0 q^{59} +4352.00 q^{60} -5458.00 q^{61} -5720.00 q^{62} +18614.0 q^{63} +4096.00 q^{64} -6240.00 q^{66} +44542.0 q^{67} -27808.0 q^{68} -8416.00 q^{69} -22304.0 q^{70} -17790.0 q^{71} -14528.0 q^{72} -31064.0 q^{73} -35408.0 q^{74} +5996.00 q^{75} -17184.0 q^{76} +31980.0 q^{77} -45360.0 q^{79} +17408.0 q^{80} +47641.0 q^{81} +27040.0 q^{82} -124546. q^{83} -5248.00 q^{84} -118184. q^{85} +67664.0 q^{86} -6760.00 q^{87} -24960.0 q^{88} +18744.0 q^{89} -61744.0 q^{90} -33664.0 q^{92} -5720.00 q^{93} +100632. q^{94} -73032.0 q^{95} +4096.00 q^{96} -121488. q^{97} -40332.0 q^{98} +88530.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 4.00000 0.256600 0.128300 0.991735i \(-0.459048\pi\)
0.128300 + 0.991735i \(0.459048\pi\)
\(4\) 16.0000 0.500000
\(5\) 68.0000 1.21642 0.608210 0.793776i \(-0.291888\pi\)
0.608210 + 0.793776i \(0.291888\pi\)
\(6\) 16.0000 0.181444
\(7\) −82.0000 −0.632512 −0.316256 0.948674i \(-0.602426\pi\)
−0.316256 + 0.948674i \(0.602426\pi\)
\(8\) 64.0000 0.353553
\(9\) −227.000 −0.934156
\(10\) 272.000 0.860140
\(11\) −390.000 −0.971813 −0.485907 0.874011i \(-0.661511\pi\)
−0.485907 + 0.874011i \(0.661511\pi\)
\(12\) 64.0000 0.128300
\(13\) 0 0
\(14\) −328.000 −0.447254
\(15\) 272.000 0.312134
\(16\) 256.000 0.250000
\(17\) −1738.00 −1.45857 −0.729285 0.684210i \(-0.760147\pi\)
−0.729285 + 0.684210i \(0.760147\pi\)
\(18\) −908.000 −0.660548
\(19\) −1074.00 −0.682528 −0.341264 0.939968i \(-0.610855\pi\)
−0.341264 + 0.939968i \(0.610855\pi\)
\(20\) 1088.00 0.608210
\(21\) −328.000 −0.162303
\(22\) −1560.00 −0.687176
\(23\) −2104.00 −0.829328 −0.414664 0.909975i \(-0.636101\pi\)
−0.414664 + 0.909975i \(0.636101\pi\)
\(24\) 256.000 0.0907218
\(25\) 1499.00 0.479680
\(26\) 0 0
\(27\) −1880.00 −0.496305
\(28\) −1312.00 −0.316256
\(29\) −1690.00 −0.373157 −0.186579 0.982440i \(-0.559740\pi\)
−0.186579 + 0.982440i \(0.559740\pi\)
\(30\) 1088.00 0.220712
\(31\) −1430.00 −0.267259 −0.133629 0.991031i \(-0.542663\pi\)
−0.133629 + 0.991031i \(0.542663\pi\)
\(32\) 1024.00 0.176777
\(33\) −1560.00 −0.249367
\(34\) −6952.00 −1.03137
\(35\) −5576.00 −0.769401
\(36\) −3632.00 −0.467078
\(37\) −8852.00 −1.06301 −0.531505 0.847055i \(-0.678373\pi\)
−0.531505 + 0.847055i \(0.678373\pi\)
\(38\) −4296.00 −0.482620
\(39\) 0 0
\(40\) 4352.00 0.430070
\(41\) 6760.00 0.628040 0.314020 0.949416i \(-0.398324\pi\)
0.314020 + 0.949416i \(0.398324\pi\)
\(42\) −1312.00 −0.114765
\(43\) 16916.0 1.39517 0.697584 0.716503i \(-0.254258\pi\)
0.697584 + 0.716503i \(0.254258\pi\)
\(44\) −6240.00 −0.485907
\(45\) −15436.0 −1.13633
\(46\) −8416.00 −0.586423
\(47\) 25158.0 1.66124 0.830618 0.556842i \(-0.187987\pi\)
0.830618 + 0.556842i \(0.187987\pi\)
\(48\) 1024.00 0.0641500
\(49\) −10083.0 −0.599929
\(50\) 5996.00 0.339185
\(51\) −6952.00 −0.374269
\(52\) 0 0
\(53\) 38214.0 1.86867 0.934335 0.356395i \(-0.115994\pi\)
0.934335 + 0.356395i \(0.115994\pi\)
\(54\) −7520.00 −0.350940
\(55\) −26520.0 −1.18213
\(56\) −5248.00 −0.223627
\(57\) −4296.00 −0.175137
\(58\) −6760.00 −0.263862
\(59\) −21286.0 −0.796093 −0.398047 0.917365i \(-0.630312\pi\)
−0.398047 + 0.917365i \(0.630312\pi\)
\(60\) 4352.00 0.156067
\(61\) −5458.00 −0.187806 −0.0939029 0.995581i \(-0.529934\pi\)
−0.0939029 + 0.995581i \(0.529934\pi\)
\(62\) −5720.00 −0.188980
\(63\) 18614.0 0.590865
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) −6240.00 −0.176329
\(67\) 44542.0 1.21222 0.606112 0.795379i \(-0.292728\pi\)
0.606112 + 0.795379i \(0.292728\pi\)
\(68\) −27808.0 −0.729285
\(69\) −8416.00 −0.212806
\(70\) −22304.0 −0.544049
\(71\) −17790.0 −0.418823 −0.209411 0.977828i \(-0.567155\pi\)
−0.209411 + 0.977828i \(0.567155\pi\)
\(72\) −14528.0 −0.330274
\(73\) −31064.0 −0.682260 −0.341130 0.940016i \(-0.610810\pi\)
−0.341130 + 0.940016i \(0.610810\pi\)
\(74\) −35408.0 −0.751661
\(75\) 5996.00 0.123086
\(76\) −17184.0 −0.341264
\(77\) 31980.0 0.614684
\(78\) 0 0
\(79\) −45360.0 −0.817721 −0.408861 0.912597i \(-0.634074\pi\)
−0.408861 + 0.912597i \(0.634074\pi\)
\(80\) 17408.0 0.304105
\(81\) 47641.0 0.806805
\(82\) 27040.0 0.444091
\(83\) −124546. −1.98442 −0.992212 0.124559i \(-0.960248\pi\)
−0.992212 + 0.124559i \(0.960248\pi\)
\(84\) −5248.00 −0.0811513
\(85\) −118184. −1.77424
\(86\) 67664.0 0.986533
\(87\) −6760.00 −0.0957522
\(88\) −24960.0 −0.343588
\(89\) 18744.0 0.250834 0.125417 0.992104i \(-0.459973\pi\)
0.125417 + 0.992104i \(0.459973\pi\)
\(90\) −61744.0 −0.803505
\(91\) 0 0
\(92\) −33664.0 −0.414664
\(93\) −5720.00 −0.0685786
\(94\) 100632. 1.17467
\(95\) −73032.0 −0.830241
\(96\) 4096.00 0.0453609
\(97\) −121488. −1.31100 −0.655502 0.755193i \(-0.727543\pi\)
−0.655502 + 0.755193i \(0.727543\pi\)
\(98\) −40332.0 −0.424214
\(99\) 88530.0 0.907826
\(100\) 23984.0 0.239840
\(101\) 14218.0 0.138687 0.0693434 0.997593i \(-0.477910\pi\)
0.0693434 + 0.997593i \(0.477910\pi\)
\(102\) −27808.0 −0.264648
\(103\) 62776.0 0.583043 0.291521 0.956564i \(-0.405839\pi\)
0.291521 + 0.956564i \(0.405839\pi\)
\(104\) 0 0
\(105\) −22304.0 −0.197428
\(106\) 152856. 1.32135
\(107\) −79252.0 −0.669192 −0.334596 0.942362i \(-0.608600\pi\)
−0.334596 + 0.942362i \(0.608600\pi\)
\(108\) −30080.0 −0.248152
\(109\) −218084. −1.75816 −0.879078 0.476677i \(-0.841841\pi\)
−0.879078 + 0.476677i \(0.841841\pi\)
\(110\) −106080. −0.835895
\(111\) −35408.0 −0.272768
\(112\) −20992.0 −0.158128
\(113\) 44234.0 0.325882 0.162941 0.986636i \(-0.447902\pi\)
0.162941 + 0.986636i \(0.447902\pi\)
\(114\) −17184.0 −0.123840
\(115\) −143072. −1.00881
\(116\) −27040.0 −0.186579
\(117\) 0 0
\(118\) −85144.0 −0.562923
\(119\) 142516. 0.922563
\(120\) 17408.0 0.110356
\(121\) −8951.00 −0.0555787
\(122\) −21832.0 −0.132799
\(123\) 27040.0 0.161155
\(124\) −22880.0 −0.133629
\(125\) −110568. −0.632928
\(126\) 74456.0 0.417805
\(127\) 310432. 1.70788 0.853940 0.520372i \(-0.174207\pi\)
0.853940 + 0.520372i \(0.174207\pi\)
\(128\) 16384.0 0.0883883
\(129\) 67664.0 0.358000
\(130\) 0 0
\(131\) 310372. 1.58017 0.790086 0.612996i \(-0.210036\pi\)
0.790086 + 0.612996i \(0.210036\pi\)
\(132\) −24960.0 −0.124684
\(133\) 88068.0 0.431707
\(134\) 178168. 0.857171
\(135\) −127840. −0.603716
\(136\) −111232. −0.515683
\(137\) −281032. −1.27925 −0.639623 0.768688i \(-0.720910\pi\)
−0.639623 + 0.768688i \(0.720910\pi\)
\(138\) −33664.0 −0.150476
\(139\) 363820. 1.59716 0.798582 0.601886i \(-0.205584\pi\)
0.798582 + 0.601886i \(0.205584\pi\)
\(140\) −89216.0 −0.384700
\(141\) 100632. 0.426273
\(142\) −71160.0 −0.296152
\(143\) 0 0
\(144\) −58112.0 −0.233539
\(145\) −114920. −0.453916
\(146\) −124256. −0.482431
\(147\) −40332.0 −0.153942
\(148\) −141632. −0.531505
\(149\) −274204. −1.01183 −0.505916 0.862583i \(-0.668845\pi\)
−0.505916 + 0.862583i \(0.668845\pi\)
\(150\) 23984.0 0.0870349
\(151\) 344030. 1.22787 0.613937 0.789355i \(-0.289585\pi\)
0.613937 + 0.789355i \(0.289585\pi\)
\(152\) −68736.0 −0.241310
\(153\) 394526. 1.36253
\(154\) 127920. 0.434647
\(155\) −97240.0 −0.325099
\(156\) 0 0
\(157\) 20518.0 0.0664333 0.0332167 0.999448i \(-0.489425\pi\)
0.0332167 + 0.999448i \(0.489425\pi\)
\(158\) −181440. −0.578216
\(159\) 152856. 0.479501
\(160\) 69632.0 0.215035
\(161\) 172528. 0.524560
\(162\) 190564. 0.570497
\(163\) 36626.0 0.107974 0.0539872 0.998542i \(-0.482807\pi\)
0.0539872 + 0.998542i \(0.482807\pi\)
\(164\) 108160. 0.314020
\(165\) −106080. −0.303336
\(166\) −498184. −1.40320
\(167\) −269442. −0.747608 −0.373804 0.927508i \(-0.621947\pi\)
−0.373804 + 0.927508i \(0.621947\pi\)
\(168\) −20992.0 −0.0573827
\(169\) 0 0
\(170\) −472736. −1.25457
\(171\) 243798. 0.637588
\(172\) 270656. 0.697584
\(173\) −282654. −0.718026 −0.359013 0.933333i \(-0.616887\pi\)
−0.359013 + 0.933333i \(0.616887\pi\)
\(174\) −27040.0 −0.0677070
\(175\) −122918. −0.303403
\(176\) −99840.0 −0.242953
\(177\) −85144.0 −0.204278
\(178\) 74976.0 0.177367
\(179\) −333780. −0.778624 −0.389312 0.921106i \(-0.627287\pi\)
−0.389312 + 0.921106i \(0.627287\pi\)
\(180\) −246976. −0.568164
\(181\) 459938. 1.04352 0.521762 0.853091i \(-0.325275\pi\)
0.521762 + 0.853091i \(0.325275\pi\)
\(182\) 0 0
\(183\) −21832.0 −0.0481910
\(184\) −134656. −0.293212
\(185\) −601936. −1.29307
\(186\) −22880.0 −0.0484924
\(187\) 677820. 1.41746
\(188\) 402528. 0.830618
\(189\) 154160. 0.313919
\(190\) −292128. −0.587069
\(191\) −917088. −1.81898 −0.909489 0.415727i \(-0.863527\pi\)
−0.909489 + 0.415727i \(0.863527\pi\)
\(192\) 16384.0 0.0320750
\(193\) 639056. 1.23494 0.617470 0.786595i \(-0.288158\pi\)
0.617470 + 0.786595i \(0.288158\pi\)
\(194\) −485952. −0.927020
\(195\) 0 0
\(196\) −161328. −0.299964
\(197\) 358292. 0.657766 0.328883 0.944371i \(-0.393328\pi\)
0.328883 + 0.944371i \(0.393328\pi\)
\(198\) 354120. 0.641930
\(199\) −370440. −0.663109 −0.331555 0.943436i \(-0.607573\pi\)
−0.331555 + 0.943436i \(0.607573\pi\)
\(200\) 95936.0 0.169592
\(201\) 178168. 0.311057
\(202\) 56872.0 0.0980664
\(203\) 138580. 0.236026
\(204\) −111232. −0.187135
\(205\) 459680. 0.763961
\(206\) 251104. 0.412274
\(207\) 477608. 0.774722
\(208\) 0 0
\(209\) 418860. 0.663290
\(210\) −89216.0 −0.139603
\(211\) −177228. −0.274048 −0.137024 0.990568i \(-0.543754\pi\)
−0.137024 + 0.990568i \(0.543754\pi\)
\(212\) 611424. 0.934335
\(213\) −71160.0 −0.107470
\(214\) −317008. −0.473190
\(215\) 1.15029e6 1.69711
\(216\) −120320. −0.175470
\(217\) 117260. 0.169044
\(218\) −872336. −1.24320
\(219\) −124256. −0.175068
\(220\) −424320. −0.591067
\(221\) 0 0
\(222\) −141632. −0.192876
\(223\) −1.11297e6 −1.49872 −0.749359 0.662164i \(-0.769638\pi\)
−0.749359 + 0.662164i \(0.769638\pi\)
\(224\) −83968.0 −0.111813
\(225\) −340273. −0.448096
\(226\) 176936. 0.230433
\(227\) 1.39158e6 1.79244 0.896219 0.443612i \(-0.146303\pi\)
0.896219 + 0.443612i \(0.146303\pi\)
\(228\) −68736.0 −0.0875683
\(229\) −909796. −1.14645 −0.573225 0.819398i \(-0.694308\pi\)
−0.573225 + 0.819398i \(0.694308\pi\)
\(230\) −572288. −0.713337
\(231\) 127920. 0.157728
\(232\) −108160. −0.131931
\(233\) −266154. −0.321176 −0.160588 0.987022i \(-0.551339\pi\)
−0.160588 + 0.987022i \(0.551339\pi\)
\(234\) 0 0
\(235\) 1.71074e6 2.02076
\(236\) −340576. −0.398047
\(237\) −181440. −0.209827
\(238\) 570064. 0.652351
\(239\) −254614. −0.288328 −0.144164 0.989554i \(-0.546049\pi\)
−0.144164 + 0.989554i \(0.546049\pi\)
\(240\) 69632.0 0.0780334
\(241\) 313600. 0.347803 0.173902 0.984763i \(-0.444363\pi\)
0.173902 + 0.984763i \(0.444363\pi\)
\(242\) −35804.0 −0.0393001
\(243\) 647404. 0.703331
\(244\) −87328.0 −0.0939029
\(245\) −685644. −0.729766
\(246\) 108160. 0.113954
\(247\) 0 0
\(248\) −91520.0 −0.0944902
\(249\) −498184. −0.509204
\(250\) −442272. −0.447548
\(251\) 1.07127e6 1.07328 0.536641 0.843811i \(-0.319693\pi\)
0.536641 + 0.843811i \(0.319693\pi\)
\(252\) 297824. 0.295433
\(253\) 820560. 0.805952
\(254\) 1.24173e6 1.20765
\(255\) −472736. −0.455269
\(256\) 65536.0 0.0625000
\(257\) 188382. 0.177913 0.0889563 0.996036i \(-0.471647\pi\)
0.0889563 + 0.996036i \(0.471647\pi\)
\(258\) 270656. 0.253144
\(259\) 725864. 0.672366
\(260\) 0 0
\(261\) 383630. 0.348587
\(262\) 1.24149e6 1.11735
\(263\) −1.48678e6 −1.32543 −0.662714 0.748873i \(-0.730596\pi\)
−0.662714 + 0.748873i \(0.730596\pi\)
\(264\) −99840.0 −0.0881647
\(265\) 2.59855e6 2.27309
\(266\) 352272. 0.305263
\(267\) 74976.0 0.0643642
\(268\) 712672. 0.606112
\(269\) 743990. 0.626883 0.313441 0.949608i \(-0.398518\pi\)
0.313441 + 0.949608i \(0.398518\pi\)
\(270\) −511360. −0.426891
\(271\) 455590. 0.376835 0.188417 0.982089i \(-0.439664\pi\)
0.188417 + 0.982089i \(0.439664\pi\)
\(272\) −444928. −0.364643
\(273\) 0 0
\(274\) −1.12413e6 −0.904564
\(275\) −584610. −0.466159
\(276\) −134656. −0.106403
\(277\) −460198. −0.360367 −0.180184 0.983633i \(-0.557669\pi\)
−0.180184 + 0.983633i \(0.557669\pi\)
\(278\) 1.45528e6 1.12937
\(279\) 324610. 0.249661
\(280\) −356864. −0.272024
\(281\) 49240.0 0.0372008 0.0186004 0.999827i \(-0.494079\pi\)
0.0186004 + 0.999827i \(0.494079\pi\)
\(282\) 402528. 0.301421
\(283\) 544196. 0.403914 0.201957 0.979394i \(-0.435270\pi\)
0.201957 + 0.979394i \(0.435270\pi\)
\(284\) −284640. −0.209411
\(285\) −292128. −0.213040
\(286\) 0 0
\(287\) −554320. −0.397243
\(288\) −232448. −0.165137
\(289\) 1.60079e6 1.12743
\(290\) −459680. −0.320967
\(291\) −485952. −0.336404
\(292\) −497024. −0.341130
\(293\) 1.02504e6 0.697542 0.348771 0.937208i \(-0.386599\pi\)
0.348771 + 0.937208i \(0.386599\pi\)
\(294\) −161328. −0.108853
\(295\) −1.44745e6 −0.968385
\(296\) −566528. −0.375831
\(297\) 733200. 0.482316
\(298\) −1.09682e6 −0.715473
\(299\) 0 0
\(300\) 95936.0 0.0615430
\(301\) −1.38711e6 −0.882461
\(302\) 1.37612e6 0.868238
\(303\) 56872.0 0.0355870
\(304\) −274944. −0.170632
\(305\) −371144. −0.228451
\(306\) 1.57810e6 0.963456
\(307\) −1.57766e6 −0.955362 −0.477681 0.878533i \(-0.658523\pi\)
−0.477681 + 0.878533i \(0.658523\pi\)
\(308\) 511680. 0.307342
\(309\) 251104. 0.149609
\(310\) −388960. −0.229880
\(311\) 330088. 0.193521 0.0967606 0.995308i \(-0.469152\pi\)
0.0967606 + 0.995308i \(0.469152\pi\)
\(312\) 0 0
\(313\) −1.78677e6 −1.03088 −0.515438 0.856927i \(-0.672371\pi\)
−0.515438 + 0.856927i \(0.672371\pi\)
\(314\) 82072.0 0.0469754
\(315\) 1.26575e6 0.718741
\(316\) −725760. −0.408861
\(317\) −182148. −0.101807 −0.0509033 0.998704i \(-0.516210\pi\)
−0.0509033 + 0.998704i \(0.516210\pi\)
\(318\) 611424. 0.339059
\(319\) 659100. 0.362639
\(320\) 278528. 0.152053
\(321\) −317008. −0.171715
\(322\) 690112. 0.370920
\(323\) 1.86661e6 0.995515
\(324\) 762256. 0.403402
\(325\) 0 0
\(326\) 146504. 0.0763494
\(327\) −872336. −0.451143
\(328\) 432640. 0.222046
\(329\) −2.06296e6 −1.05075
\(330\) −424320. −0.214491
\(331\) 216230. 0.108479 0.0542395 0.998528i \(-0.482727\pi\)
0.0542395 + 0.998528i \(0.482727\pi\)
\(332\) −1.99274e6 −0.992212
\(333\) 2.00940e6 0.993017
\(334\) −1.07777e6 −0.528639
\(335\) 3.02886e6 1.47457
\(336\) −83968.0 −0.0405757
\(337\) 2.05314e6 0.984791 0.492396 0.870371i \(-0.336121\pi\)
0.492396 + 0.870371i \(0.336121\pi\)
\(338\) 0 0
\(339\) 176936. 0.0836213
\(340\) −1.89094e6 −0.887118
\(341\) 557700. 0.259726
\(342\) 975192. 0.450843
\(343\) 2.20498e6 1.01197
\(344\) 1.08262e6 0.493266
\(345\) −572288. −0.258861
\(346\) −1.13062e6 −0.507721
\(347\) 4.28819e6 1.91183 0.955917 0.293637i \(-0.0948658\pi\)
0.955917 + 0.293637i \(0.0948658\pi\)
\(348\) −108160. −0.0478761
\(349\) −3.55152e6 −1.56081 −0.780405 0.625274i \(-0.784987\pi\)
−0.780405 + 0.625274i \(0.784987\pi\)
\(350\) −491672. −0.214539
\(351\) 0 0
\(352\) −399360. −0.171794
\(353\) 2.08678e6 0.891335 0.445667 0.895199i \(-0.352966\pi\)
0.445667 + 0.895199i \(0.352966\pi\)
\(354\) −340576. −0.144446
\(355\) −1.20972e6 −0.509465
\(356\) 299904. 0.125417
\(357\) 570064. 0.236730
\(358\) −1.33512e6 −0.550570
\(359\) 500654. 0.205023 0.102511 0.994732i \(-0.467312\pi\)
0.102511 + 0.994732i \(0.467312\pi\)
\(360\) −987904. −0.401752
\(361\) −1.32262e6 −0.534156
\(362\) 1.83975e6 0.737884
\(363\) −35804.0 −0.0142615
\(364\) 0 0
\(365\) −2.11235e6 −0.829916
\(366\) −87328.0 −0.0340762
\(367\) −1.28027e6 −0.496178 −0.248089 0.968737i \(-0.579802\pi\)
−0.248089 + 0.968737i \(0.579802\pi\)
\(368\) −538624. −0.207332
\(369\) −1.53452e6 −0.586687
\(370\) −2.40774e6 −0.914336
\(371\) −3.13355e6 −1.18196
\(372\) −91520.0 −0.0342893
\(373\) −405666. −0.150972 −0.0754860 0.997147i \(-0.524051\pi\)
−0.0754860 + 0.997147i \(0.524051\pi\)
\(374\) 2.71128e6 1.00229
\(375\) −442272. −0.162409
\(376\) 1.61011e6 0.587336
\(377\) 0 0
\(378\) 616640. 0.221974
\(379\) 4.66217e6 1.66721 0.833604 0.552363i \(-0.186274\pi\)
0.833604 + 0.552363i \(0.186274\pi\)
\(380\) −1.16851e6 −0.415121
\(381\) 1.24173e6 0.438242
\(382\) −3.66835e6 −1.28621
\(383\) −4.35473e6 −1.51692 −0.758462 0.651717i \(-0.774049\pi\)
−0.758462 + 0.651717i \(0.774049\pi\)
\(384\) 65536.0 0.0226805
\(385\) 2.17464e6 0.747714
\(386\) 2.55622e6 0.873234
\(387\) −3.83993e6 −1.30331
\(388\) −1.94381e6 −0.655502
\(389\) −786990. −0.263691 −0.131845 0.991270i \(-0.542090\pi\)
−0.131845 + 0.991270i \(0.542090\pi\)
\(390\) 0 0
\(391\) 3.65675e6 1.20963
\(392\) −645312. −0.212107
\(393\) 1.24149e6 0.405472
\(394\) 1.43317e6 0.465111
\(395\) −3.08448e6 −0.994693
\(396\) 1.41648e6 0.453913
\(397\) 3.97023e6 1.26427 0.632134 0.774859i \(-0.282179\pi\)
0.632134 + 0.774859i \(0.282179\pi\)
\(398\) −1.48176e6 −0.468889
\(399\) 352272. 0.110776
\(400\) 383744. 0.119920
\(401\) 344640. 0.107030 0.0535149 0.998567i \(-0.482958\pi\)
0.0535149 + 0.998567i \(0.482958\pi\)
\(402\) 712672. 0.219950
\(403\) 0 0
\(404\) 227488. 0.0693434
\(405\) 3.23959e6 0.981414
\(406\) 554320. 0.166896
\(407\) 3.45228e6 1.03305
\(408\) −444928. −0.132324
\(409\) −2.55466e6 −0.755137 −0.377568 0.925982i \(-0.623240\pi\)
−0.377568 + 0.925982i \(0.623240\pi\)
\(410\) 1.83872e6 0.540202
\(411\) −1.12413e6 −0.328255
\(412\) 1.00442e6 0.291521
\(413\) 1.74545e6 0.503539
\(414\) 1.91043e6 0.547811
\(415\) −8.46913e6 −2.41390
\(416\) 0 0
\(417\) 1.45528e6 0.409833
\(418\) 1.67544e6 0.469017
\(419\) −2.51894e6 −0.700943 −0.350472 0.936573i \(-0.613979\pi\)
−0.350472 + 0.936573i \(0.613979\pi\)
\(420\) −356864. −0.0987142
\(421\) −4.83670e6 −1.32998 −0.664988 0.746854i \(-0.731563\pi\)
−0.664988 + 0.746854i \(0.731563\pi\)
\(422\) −708912. −0.193781
\(423\) −5.71087e6 −1.55185
\(424\) 2.44570e6 0.660675
\(425\) −2.60526e6 −0.699647
\(426\) −284640. −0.0759927
\(427\) 447556. 0.118789
\(428\) −1.26803e6 −0.334596
\(429\) 0 0
\(430\) 4.60115e6 1.20004
\(431\) −219110. −0.0568158 −0.0284079 0.999596i \(-0.509044\pi\)
−0.0284079 + 0.999596i \(0.509044\pi\)
\(432\) −481280. −0.124076
\(433\) 3.03477e6 0.777867 0.388934 0.921266i \(-0.372844\pi\)
0.388934 + 0.921266i \(0.372844\pi\)
\(434\) 469040. 0.119532
\(435\) −459680. −0.116475
\(436\) −3.48934e6 −0.879078
\(437\) 2.25970e6 0.566039
\(438\) −497024. −0.123792
\(439\) −4.16940e6 −1.03255 −0.516276 0.856422i \(-0.672682\pi\)
−0.516276 + 0.856422i \(0.672682\pi\)
\(440\) −1.69728e6 −0.417948
\(441\) 2.28884e6 0.560427
\(442\) 0 0
\(443\) −6.30548e6 −1.52654 −0.763271 0.646079i \(-0.776408\pi\)
−0.763271 + 0.646079i \(0.776408\pi\)
\(444\) −566528. −0.136384
\(445\) 1.27459e6 0.305120
\(446\) −4.45186e6 −1.05975
\(447\) −1.09682e6 −0.259636
\(448\) −335872. −0.0790640
\(449\) −7.41586e6 −1.73598 −0.867991 0.496579i \(-0.834589\pi\)
−0.867991 + 0.496579i \(0.834589\pi\)
\(450\) −1.36109e6 −0.316852
\(451\) −2.63640e6 −0.610337
\(452\) 707744. 0.162941
\(453\) 1.37612e6 0.315073
\(454\) 5.56633e6 1.26745
\(455\) 0 0
\(456\) −274944. −0.0619202
\(457\) −4.71529e6 −1.05613 −0.528065 0.849204i \(-0.677082\pi\)
−0.528065 + 0.849204i \(0.677082\pi\)
\(458\) −3.63918e6 −0.810663
\(459\) 3.26744e6 0.723896
\(460\) −2.28915e6 −0.504406
\(461\) 3.34566e6 0.733212 0.366606 0.930376i \(-0.380520\pi\)
0.366606 + 0.930376i \(0.380520\pi\)
\(462\) 511680. 0.111530
\(463\) −1.65791e6 −0.359426 −0.179713 0.983719i \(-0.557517\pi\)
−0.179713 + 0.983719i \(0.557517\pi\)
\(464\) −432640. −0.0932893
\(465\) −388960. −0.0834205
\(466\) −1.06462e6 −0.227106
\(467\) −823668. −0.174767 −0.0873836 0.996175i \(-0.527851\pi\)
−0.0873836 + 0.996175i \(0.527851\pi\)
\(468\) 0 0
\(469\) −3.65244e6 −0.766746
\(470\) 6.84298e6 1.42890
\(471\) 82072.0 0.0170468
\(472\) −1.36230e6 −0.281462
\(473\) −6.59724e6 −1.35584
\(474\) −725760. −0.148370
\(475\) −1.60993e6 −0.327395
\(476\) 2.28026e6 0.461282
\(477\) −8.67458e6 −1.74563
\(478\) −1.01846e6 −0.203879
\(479\) −3.59011e6 −0.714938 −0.357469 0.933925i \(-0.616360\pi\)
−0.357469 + 0.933925i \(0.616360\pi\)
\(480\) 278528. 0.0551780
\(481\) 0 0
\(482\) 1.25440e6 0.245934
\(483\) 690112. 0.134602
\(484\) −143216. −0.0277893
\(485\) −8.26118e6 −1.59473
\(486\) 2.58962e6 0.497330
\(487\) −9.67688e6 −1.84890 −0.924449 0.381306i \(-0.875474\pi\)
−0.924449 + 0.381306i \(0.875474\pi\)
\(488\) −349312. −0.0663994
\(489\) 146504. 0.0277062
\(490\) −2.74258e6 −0.516022
\(491\) −3.45633e6 −0.647011 −0.323506 0.946226i \(-0.604861\pi\)
−0.323506 + 0.946226i \(0.604861\pi\)
\(492\) 432640. 0.0805775
\(493\) 2.93722e6 0.544276
\(494\) 0 0
\(495\) 6.02004e6 1.10430
\(496\) −366080. −0.0668147
\(497\) 1.45878e6 0.264910
\(498\) −1.99274e6 −0.360061
\(499\) 2.09109e6 0.375942 0.187971 0.982175i \(-0.439809\pi\)
0.187971 + 0.982175i \(0.439809\pi\)
\(500\) −1.76909e6 −0.316464
\(501\) −1.07777e6 −0.191836
\(502\) 4.28507e6 0.758925
\(503\) 5.58626e6 0.984468 0.492234 0.870463i \(-0.336180\pi\)
0.492234 + 0.870463i \(0.336180\pi\)
\(504\) 1.19130e6 0.208902
\(505\) 966824. 0.168702
\(506\) 3.28224e6 0.569894
\(507\) 0 0
\(508\) 4.96691e6 0.853940
\(509\) 4.15504e6 0.710854 0.355427 0.934704i \(-0.384335\pi\)
0.355427 + 0.934704i \(0.384335\pi\)
\(510\) −1.89094e6 −0.321924
\(511\) 2.54725e6 0.431538
\(512\) 262144. 0.0441942
\(513\) 2.01912e6 0.338742
\(514\) 753528. 0.125803
\(515\) 4.26877e6 0.709226
\(516\) 1.08262e6 0.179000
\(517\) −9.81162e6 −1.61441
\(518\) 2.90346e6 0.475435
\(519\) −1.13062e6 −0.184245
\(520\) 0 0
\(521\) −9.84416e6 −1.58886 −0.794428 0.607359i \(-0.792229\pi\)
−0.794428 + 0.607359i \(0.792229\pi\)
\(522\) 1.53452e6 0.246488
\(523\) 481324. 0.0769455 0.0384728 0.999260i \(-0.487751\pi\)
0.0384728 + 0.999260i \(0.487751\pi\)
\(524\) 4.96595e6 0.790086
\(525\) −491672. −0.0778533
\(526\) −5.94710e6 −0.937219
\(527\) 2.48534e6 0.389816
\(528\) −399360. −0.0623419
\(529\) −2.00953e6 −0.312216
\(530\) 1.03942e7 1.60732
\(531\) 4.83192e6 0.743676
\(532\) 1.40909e6 0.215853
\(533\) 0 0
\(534\) 299904. 0.0455123
\(535\) −5.38914e6 −0.814019
\(536\) 2.85069e6 0.428586
\(537\) −1.33512e6 −0.199795
\(538\) 2.97596e6 0.443273
\(539\) 3.93237e6 0.583019
\(540\) −2.04544e6 −0.301858
\(541\) −263980. −0.0387773 −0.0193887 0.999812i \(-0.506172\pi\)
−0.0193887 + 0.999812i \(0.506172\pi\)
\(542\) 1.82236e6 0.266462
\(543\) 1.83975e6 0.267769
\(544\) −1.77971e6 −0.257841
\(545\) −1.48297e7 −2.13866
\(546\) 0 0
\(547\) 2.80023e6 0.400152 0.200076 0.979780i \(-0.435881\pi\)
0.200076 + 0.979780i \(0.435881\pi\)
\(548\) −4.49651e6 −0.639623
\(549\) 1.23897e6 0.175440
\(550\) −2.33844e6 −0.329625
\(551\) 1.81506e6 0.254690
\(552\) −538624. −0.0752381
\(553\) 3.71952e6 0.517219
\(554\) −1.84079e6 −0.254818
\(555\) −2.40774e6 −0.331801
\(556\) 5.82112e6 0.798582
\(557\) 2.70983e6 0.370087 0.185043 0.982730i \(-0.440757\pi\)
0.185043 + 0.982730i \(0.440757\pi\)
\(558\) 1.29844e6 0.176537
\(559\) 0 0
\(560\) −1.42746e6 −0.192350
\(561\) 2.71128e6 0.363720
\(562\) 196960. 0.0263049
\(563\) 1.14870e7 1.52733 0.763667 0.645610i \(-0.223397\pi\)
0.763667 + 0.645610i \(0.223397\pi\)
\(564\) 1.61011e6 0.213137
\(565\) 3.00791e6 0.396409
\(566\) 2.17678e6 0.285611
\(567\) −3.90656e6 −0.510314
\(568\) −1.13856e6 −0.148076
\(569\) −7.85065e6 −1.01654 −0.508271 0.861197i \(-0.669715\pi\)
−0.508271 + 0.861197i \(0.669715\pi\)
\(570\) −1.16851e6 −0.150642
\(571\) 6.34071e6 0.813856 0.406928 0.913460i \(-0.366600\pi\)
0.406928 + 0.913460i \(0.366600\pi\)
\(572\) 0 0
\(573\) −3.66835e6 −0.466750
\(574\) −2.21728e6 −0.280893
\(575\) −3.15390e6 −0.397812
\(576\) −929792. −0.116770
\(577\) 7.20867e6 0.901396 0.450698 0.892676i \(-0.351175\pi\)
0.450698 + 0.892676i \(0.351175\pi\)
\(578\) 6.40315e6 0.797212
\(579\) 2.55622e6 0.316886
\(580\) −1.83872e6 −0.226958
\(581\) 1.02128e7 1.25517
\(582\) −1.94381e6 −0.237873
\(583\) −1.49035e7 −1.81600
\(584\) −1.98810e6 −0.241216
\(585\) 0 0
\(586\) 4.10014e6 0.493236
\(587\) 2.48138e6 0.297234 0.148617 0.988895i \(-0.452518\pi\)
0.148617 + 0.988895i \(0.452518\pi\)
\(588\) −645312. −0.0769709
\(589\) 1.53582e6 0.182411
\(590\) −5.78979e6 −0.684751
\(591\) 1.43317e6 0.168783
\(592\) −2.26611e6 −0.265752
\(593\) −1.38811e7 −1.62102 −0.810508 0.585728i \(-0.800809\pi\)
−0.810508 + 0.585728i \(0.800809\pi\)
\(594\) 2.93280e6 0.341049
\(595\) 9.69109e6 1.12223
\(596\) −4.38726e6 −0.505916
\(597\) −1.48176e6 −0.170154
\(598\) 0 0
\(599\) 3.85356e6 0.438829 0.219414 0.975632i \(-0.429585\pi\)
0.219414 + 0.975632i \(0.429585\pi\)
\(600\) 383744. 0.0435175
\(601\) 1.32728e6 0.149892 0.0749458 0.997188i \(-0.476122\pi\)
0.0749458 + 0.997188i \(0.476122\pi\)
\(602\) −5.54845e6 −0.623994
\(603\) −1.01110e7 −1.13241
\(604\) 5.50448e6 0.613937
\(605\) −608668. −0.0676071
\(606\) 227488. 0.0251638
\(607\) 9.73197e6 1.07208 0.536042 0.844191i \(-0.319919\pi\)
0.536042 + 0.844191i \(0.319919\pi\)
\(608\) −1.09978e6 −0.120655
\(609\) 554320. 0.0605644
\(610\) −1.48458e6 −0.161539
\(611\) 0 0
\(612\) 6.31242e6 0.681267
\(613\) −1.40465e7 −1.50979 −0.754894 0.655846i \(-0.772312\pi\)
−0.754894 + 0.655846i \(0.772312\pi\)
\(614\) −6.31065e6 −0.675543
\(615\) 1.83872e6 0.196032
\(616\) 2.04672e6 0.217323
\(617\) −3.72561e6 −0.393989 −0.196995 0.980405i \(-0.563118\pi\)
−0.196995 + 0.980405i \(0.563118\pi\)
\(618\) 1.00442e6 0.105789
\(619\) 8.96911e6 0.940855 0.470428 0.882439i \(-0.344100\pi\)
0.470428 + 0.882439i \(0.344100\pi\)
\(620\) −1.55584e6 −0.162550
\(621\) 3.95552e6 0.411599
\(622\) 1.32035e6 0.136840
\(623\) −1.53701e6 −0.158656
\(624\) 0 0
\(625\) −1.22030e7 −1.24959
\(626\) −7.14706e6 −0.728940
\(627\) 1.67544e6 0.170200
\(628\) 328288. 0.0332167
\(629\) 1.53848e7 1.55047
\(630\) 5.06301e6 0.508226
\(631\) 1.72189e7 1.72160 0.860800 0.508943i \(-0.169964\pi\)
0.860800 + 0.508943i \(0.169964\pi\)
\(632\) −2.90304e6 −0.289108
\(633\) −708912. −0.0703207
\(634\) −728592. −0.0719882
\(635\) 2.11094e7 2.07750
\(636\) 2.44570e6 0.239751
\(637\) 0 0
\(638\) 2.63640e6 0.256425
\(639\) 4.03833e6 0.391246
\(640\) 1.11411e6 0.107517
\(641\) −8.51692e6 −0.818724 −0.409362 0.912372i \(-0.634249\pi\)
−0.409362 + 0.912372i \(0.634249\pi\)
\(642\) −1.26803e6 −0.121421
\(643\) 8.14145e6 0.776559 0.388280 0.921542i \(-0.373069\pi\)
0.388280 + 0.921542i \(0.373069\pi\)
\(644\) 2.76045e6 0.262280
\(645\) 4.60115e6 0.435479
\(646\) 7.46645e6 0.703935
\(647\) 2.39391e6 0.224826 0.112413 0.993662i \(-0.464142\pi\)
0.112413 + 0.993662i \(0.464142\pi\)
\(648\) 3.04902e6 0.285248
\(649\) 8.30154e6 0.773654
\(650\) 0 0
\(651\) 469040. 0.0433768
\(652\) 586016. 0.0539872
\(653\) 1.17900e7 1.08201 0.541003 0.841020i \(-0.318045\pi\)
0.541003 + 0.841020i \(0.318045\pi\)
\(654\) −3.48934e6 −0.319006
\(655\) 2.11053e7 1.92215
\(656\) 1.73056e6 0.157010
\(657\) 7.05153e6 0.637338
\(658\) −8.25182e6 −0.742994
\(659\) 4.84562e6 0.434646 0.217323 0.976100i \(-0.430267\pi\)
0.217323 + 0.976100i \(0.430267\pi\)
\(660\) −1.69728e6 −0.151668
\(661\) −1.14461e7 −1.01895 −0.509476 0.860485i \(-0.670161\pi\)
−0.509476 + 0.860485i \(0.670161\pi\)
\(662\) 864920. 0.0767063
\(663\) 0 0
\(664\) −7.97094e6 −0.701600
\(665\) 5.98862e6 0.525137
\(666\) 8.03762e6 0.702169
\(667\) 3.55576e6 0.309470
\(668\) −4.31107e6 −0.373804
\(669\) −4.45186e6 −0.384571
\(670\) 1.21154e7 1.04268
\(671\) 2.12862e6 0.182512
\(672\) −335872. −0.0286913
\(673\) 5.34001e6 0.454469 0.227234 0.973840i \(-0.427032\pi\)
0.227234 + 0.973840i \(0.427032\pi\)
\(674\) 8.21257e6 0.696353
\(675\) −2.81812e6 −0.238067
\(676\) 0 0
\(677\) −7.06132e6 −0.592126 −0.296063 0.955168i \(-0.595674\pi\)
−0.296063 + 0.955168i \(0.595674\pi\)
\(678\) 707744. 0.0591292
\(679\) 9.96202e6 0.829226
\(680\) −7.56378e6 −0.627287
\(681\) 5.56633e6 0.459940
\(682\) 2.23080e6 0.183654
\(683\) 3.50035e6 0.287117 0.143559 0.989642i \(-0.454145\pi\)
0.143559 + 0.989642i \(0.454145\pi\)
\(684\) 3.90077e6 0.318794
\(685\) −1.91102e7 −1.55610
\(686\) 8.81992e6 0.715574
\(687\) −3.63918e6 −0.294179
\(688\) 4.33050e6 0.348792
\(689\) 0 0
\(690\) −2.28915e6 −0.183042
\(691\) 302510. 0.0241015 0.0120508 0.999927i \(-0.496164\pi\)
0.0120508 + 0.999927i \(0.496164\pi\)
\(692\) −4.52246e6 −0.359013
\(693\) −7.25946e6 −0.574211
\(694\) 1.71528e7 1.35187
\(695\) 2.47398e7 1.94282
\(696\) −432640. −0.0338535
\(697\) −1.17489e7 −0.916040
\(698\) −1.42061e7 −1.10366
\(699\) −1.06462e6 −0.0824138
\(700\) −1.96669e6 −0.151702
\(701\) 1.03212e7 0.793294 0.396647 0.917971i \(-0.370174\pi\)
0.396647 + 0.917971i \(0.370174\pi\)
\(702\) 0 0
\(703\) 9.50705e6 0.725533
\(704\) −1.59744e6 −0.121477
\(705\) 6.84298e6 0.518528
\(706\) 8.34714e6 0.630269
\(707\) −1.16588e6 −0.0877211
\(708\) −1.36230e6 −0.102139
\(709\) −5.27524e6 −0.394118 −0.197059 0.980392i \(-0.563139\pi\)
−0.197059 + 0.980392i \(0.563139\pi\)
\(710\) −4.83888e6 −0.360246
\(711\) 1.02967e7 0.763880
\(712\) 1.19962e6 0.0886834
\(713\) 3.00872e6 0.221645
\(714\) 2.28026e6 0.167393
\(715\) 0 0
\(716\) −5.34048e6 −0.389312
\(717\) −1.01846e6 −0.0739851
\(718\) 2.00262e6 0.144973
\(719\) 5.02216e6 0.362300 0.181150 0.983455i \(-0.442018\pi\)
0.181150 + 0.983455i \(0.442018\pi\)
\(720\) −3.95162e6 −0.284082
\(721\) −5.14763e6 −0.368782
\(722\) −5.29049e6 −0.377705
\(723\) 1.25440e6 0.0892463
\(724\) 7.35901e6 0.521762
\(725\) −2.53331e6 −0.178996
\(726\) −143216. −0.0100844
\(727\) −8.80441e6 −0.617823 −0.308912 0.951091i \(-0.599965\pi\)
−0.308912 + 0.951091i \(0.599965\pi\)
\(728\) 0 0
\(729\) −8.98715e6 −0.626330
\(730\) −8.44941e6 −0.586839
\(731\) −2.94000e7 −2.03495
\(732\) −349312. −0.0240955
\(733\) 3.05052e6 0.209708 0.104854 0.994488i \(-0.466563\pi\)
0.104854 + 0.994488i \(0.466563\pi\)
\(734\) −5.12109e6 −0.350850
\(735\) −2.74258e6 −0.187258
\(736\) −2.15450e6 −0.146606
\(737\) −1.73714e7 −1.17806
\(738\) −6.13808e6 −0.414851
\(739\) −7.62605e6 −0.513675 −0.256837 0.966455i \(-0.582680\pi\)
−0.256837 + 0.966455i \(0.582680\pi\)
\(740\) −9.63098e6 −0.646533
\(741\) 0 0
\(742\) −1.25342e7 −0.835770
\(743\) −2.18236e7 −1.45029 −0.725146 0.688595i \(-0.758228\pi\)
−0.725146 + 0.688595i \(0.758228\pi\)
\(744\) −366080. −0.0242462
\(745\) −1.86459e7 −1.23081
\(746\) −1.62266e6 −0.106753
\(747\) 2.82719e7 1.85376
\(748\) 1.08451e7 0.708729
\(749\) 6.49866e6 0.423272
\(750\) −1.76909e6 −0.114841
\(751\) −1.69030e7 −1.09361 −0.546807 0.837259i \(-0.684157\pi\)
−0.546807 + 0.837259i \(0.684157\pi\)
\(752\) 6.44045e6 0.415309
\(753\) 4.28507e6 0.275404
\(754\) 0 0
\(755\) 2.33940e7 1.49361
\(756\) 2.46656e6 0.156959
\(757\) −8.90252e6 −0.564642 −0.282321 0.959320i \(-0.591104\pi\)
−0.282321 + 0.959320i \(0.591104\pi\)
\(758\) 1.86487e7 1.17889
\(759\) 3.28224e6 0.206807
\(760\) −4.67405e6 −0.293535
\(761\) 6.98052e6 0.436944 0.218472 0.975843i \(-0.429893\pi\)
0.218472 + 0.975843i \(0.429893\pi\)
\(762\) 4.96691e6 0.309884
\(763\) 1.78829e7 1.11206
\(764\) −1.46734e7 −0.909489
\(765\) 2.68278e7 1.65741
\(766\) −1.74189e7 −1.07263
\(767\) 0 0
\(768\) 262144. 0.0160375
\(769\) −2.67789e7 −1.63296 −0.816481 0.577372i \(-0.804078\pi\)
−0.816481 + 0.577372i \(0.804078\pi\)
\(770\) 8.69856e6 0.528714
\(771\) 753528. 0.0456524
\(772\) 1.02249e7 0.617470
\(773\) 710244. 0.0427522 0.0213761 0.999772i \(-0.493195\pi\)
0.0213761 + 0.999772i \(0.493195\pi\)
\(774\) −1.53597e7 −0.921576
\(775\) −2.14357e6 −0.128199
\(776\) −7.77523e6 −0.463510
\(777\) 2.90346e6 0.172529
\(778\) −3.14796e6 −0.186458
\(779\) −7.26024e6 −0.428654
\(780\) 0 0
\(781\) 6.93810e6 0.407017
\(782\) 1.46270e7 0.855340
\(783\) 3.17720e6 0.185200
\(784\) −2.58125e6 −0.149982
\(785\) 1.39522e6 0.0808109
\(786\) 4.96595e6 0.286712
\(787\) 5.18538e6 0.298431 0.149215 0.988805i \(-0.452325\pi\)
0.149215 + 0.988805i \(0.452325\pi\)
\(788\) 5.73267e6 0.328883
\(789\) −5.94710e6 −0.340105
\(790\) −1.23379e7 −0.703354
\(791\) −3.62719e6 −0.206124
\(792\) 5.66592e6 0.320965
\(793\) 0 0
\(794\) 1.58809e7 0.893973
\(795\) 1.03942e7 0.583275
\(796\) −5.92704e6 −0.331555
\(797\) 2.93628e6 0.163739 0.0818695 0.996643i \(-0.473911\pi\)
0.0818695 + 0.996643i \(0.473911\pi\)
\(798\) 1.40909e6 0.0783305
\(799\) −4.37246e7 −2.42303
\(800\) 1.53498e6 0.0847962
\(801\) −4.25489e6 −0.234319
\(802\) 1.37856e6 0.0756815
\(803\) 1.21150e7 0.663030
\(804\) 2.85069e6 0.155528
\(805\) 1.17319e7 0.638085
\(806\) 0 0
\(807\) 2.97596e6 0.160858
\(808\) 909952. 0.0490332
\(809\) −1.25821e7 −0.675900 −0.337950 0.941164i \(-0.609733\pi\)
−0.337950 + 0.941164i \(0.609733\pi\)
\(810\) 1.29584e7 0.693964
\(811\) −2.08048e7 −1.11074 −0.555369 0.831604i \(-0.687423\pi\)
−0.555369 + 0.831604i \(0.687423\pi\)
\(812\) 2.21728e6 0.118013
\(813\) 1.82236e6 0.0966958
\(814\) 1.38091e7 0.730474
\(815\) 2.49057e6 0.131342
\(816\) −1.77971e6 −0.0935674
\(817\) −1.81678e7 −0.952241
\(818\) −1.02187e7 −0.533962
\(819\) 0 0
\(820\) 7.35488e6 0.381980
\(821\) 2.11600e7 1.09562 0.547808 0.836604i \(-0.315463\pi\)
0.547808 + 0.836604i \(0.315463\pi\)
\(822\) −4.49651e6 −0.232111
\(823\) 2.20857e7 1.13661 0.568306 0.822817i \(-0.307599\pi\)
0.568306 + 0.822817i \(0.307599\pi\)
\(824\) 4.01766e6 0.206137
\(825\) −2.33844e6 −0.119617
\(826\) 6.98181e6 0.356056
\(827\) −1.69119e7 −0.859864 −0.429932 0.902861i \(-0.641462\pi\)
−0.429932 + 0.902861i \(0.641462\pi\)
\(828\) 7.64173e6 0.387361
\(829\) 2.34520e7 1.18521 0.592604 0.805494i \(-0.298100\pi\)
0.592604 + 0.805494i \(0.298100\pi\)
\(830\) −3.38765e7 −1.70688
\(831\) −1.84079e6 −0.0924703
\(832\) 0 0
\(833\) 1.75243e7 0.875038
\(834\) 5.82112e6 0.289795
\(835\) −1.83221e7 −0.909406
\(836\) 6.70176e6 0.331645
\(837\) 2.68840e6 0.132642
\(838\) −1.00758e7 −0.495642
\(839\) 725134. 0.0355642 0.0177821 0.999842i \(-0.494339\pi\)
0.0177821 + 0.999842i \(0.494339\pi\)
\(840\) −1.42746e6 −0.0698015
\(841\) −1.76550e7 −0.860754
\(842\) −1.93468e7 −0.940435
\(843\) 196960. 0.00954573
\(844\) −2.83565e6 −0.137024
\(845\) 0 0
\(846\) −2.28435e7 −1.09733
\(847\) 733982. 0.0351542
\(848\) 9.78278e6 0.467168
\(849\) 2.17678e6 0.103644
\(850\) −1.04210e7 −0.494725
\(851\) 1.86246e7 0.881583
\(852\) −1.13856e6 −0.0537350
\(853\) −1.03218e7 −0.485719 −0.242859 0.970062i \(-0.578085\pi\)
−0.242859 + 0.970062i \(0.578085\pi\)
\(854\) 1.79022e6 0.0839968
\(855\) 1.65783e7 0.775575
\(856\) −5.07213e6 −0.236595
\(857\) 3.71067e7 1.72584 0.862919 0.505343i \(-0.168634\pi\)
0.862919 + 0.505343i \(0.168634\pi\)
\(858\) 0 0
\(859\) 3.47061e7 1.60481 0.802405 0.596780i \(-0.203554\pi\)
0.802405 + 0.596780i \(0.203554\pi\)
\(860\) 1.84046e7 0.848556
\(861\) −2.21728e6 −0.101932
\(862\) −876440. −0.0401748
\(863\) −1.92294e7 −0.878897 −0.439448 0.898268i \(-0.644826\pi\)
−0.439448 + 0.898268i \(0.644826\pi\)
\(864\) −1.92512e6 −0.0877351
\(865\) −1.92205e7 −0.873421
\(866\) 1.21391e7 0.550035
\(867\) 6.40315e6 0.289298
\(868\) 1.87616e6 0.0845222
\(869\) 1.76904e7 0.794673
\(870\) −1.83872e6 −0.0823602
\(871\) 0 0
\(872\) −1.39574e7 −0.621602
\(873\) 2.75778e7 1.22468
\(874\) 9.03878e6 0.400250
\(875\) 9.06658e6 0.400335
\(876\) −1.98810e6 −0.0875341
\(877\) −3.84616e7 −1.68861 −0.844303 0.535866i \(-0.819985\pi\)
−0.844303 + 0.535866i \(0.819985\pi\)
\(878\) −1.66776e7 −0.730125
\(879\) 4.10014e6 0.178989
\(880\) −6.78912e6 −0.295534
\(881\) −3.29337e7 −1.42955 −0.714777 0.699353i \(-0.753472\pi\)
−0.714777 + 0.699353i \(0.753472\pi\)
\(882\) 9.15536e6 0.396282
\(883\) −2.67529e7 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(884\) 0 0
\(885\) −5.78979e6 −0.248488
\(886\) −2.52219e7 −1.07943
\(887\) −1.05284e7 −0.449317 −0.224659 0.974438i \(-0.572127\pi\)
−0.224659 + 0.974438i \(0.572127\pi\)
\(888\) −2.26611e6 −0.0964382
\(889\) −2.54554e7 −1.08025
\(890\) 5.09837e6 0.215753
\(891\) −1.85800e7 −0.784063
\(892\) −1.78075e7 −0.749359
\(893\) −2.70197e7 −1.13384
\(894\) −4.38726e6 −0.183590
\(895\) −2.26970e7 −0.947134
\(896\) −1.34349e6 −0.0559067
\(897\) 0 0
\(898\) −2.96634e7 −1.22753
\(899\) 2.41670e6 0.0997295
\(900\) −5.44437e6 −0.224048
\(901\) −6.64159e7 −2.72559
\(902\) −1.05456e7 −0.431574
\(903\) −5.54845e6 −0.226439
\(904\) 2.83098e6 0.115217
\(905\) 3.12758e7 1.26937
\(906\) 5.50448e6 0.222790
\(907\) 2.53255e7 1.02221 0.511104 0.859519i \(-0.329237\pi\)
0.511104 + 0.859519i \(0.329237\pi\)
\(908\) 2.22653e7 0.896219
\(909\) −3.22749e6 −0.129555
\(910\) 0 0
\(911\) 6.02395e6 0.240484 0.120242 0.992745i \(-0.461633\pi\)
0.120242 + 0.992745i \(0.461633\pi\)
\(912\) −1.09978e6 −0.0437842
\(913\) 4.85729e7 1.92849
\(914\) −1.88612e7 −0.746797
\(915\) −1.48458e6 −0.0586205
\(916\) −1.45567e7 −0.573225
\(917\) −2.54505e7 −0.999478
\(918\) 1.30698e7 0.511871
\(919\) 9.75228e6 0.380906 0.190453 0.981696i \(-0.439004\pi\)
0.190453 + 0.981696i \(0.439004\pi\)
\(920\) −9.15661e6 −0.356669
\(921\) −6.31065e6 −0.245146
\(922\) 1.33826e7 0.518459
\(923\) 0 0
\(924\) 2.04672e6 0.0788639
\(925\) −1.32691e7 −0.509904
\(926\) −6.63166e6 −0.254153
\(927\) −1.42502e7 −0.544653
\(928\) −1.73056e6 −0.0659655
\(929\) 2.30543e7 0.876419 0.438210 0.898873i \(-0.355613\pi\)
0.438210 + 0.898873i \(0.355613\pi\)
\(930\) −1.55584e6 −0.0589872
\(931\) 1.08291e7 0.409468
\(932\) −4.25846e6 −0.160588
\(933\) 1.32035e6 0.0496576
\(934\) −3.29467e6 −0.123579
\(935\) 4.60918e7 1.72423
\(936\) 0 0
\(937\) −4.03783e7 −1.50245 −0.751223 0.660049i \(-0.770536\pi\)
−0.751223 + 0.660049i \(0.770536\pi\)
\(938\) −1.46098e7 −0.542171
\(939\) −7.14706e6 −0.264523
\(940\) 2.73719e7 1.01038
\(941\) −4.02522e7 −1.48189 −0.740944 0.671567i \(-0.765622\pi\)
−0.740944 + 0.671567i \(0.765622\pi\)
\(942\) 328288. 0.0120539
\(943\) −1.42230e7 −0.520851
\(944\) −5.44922e6 −0.199023
\(945\) 1.04829e7 0.381857
\(946\) −2.63890e7 −0.958726
\(947\) 1.06360e7 0.385393 0.192697 0.981258i \(-0.438277\pi\)
0.192697 + 0.981258i \(0.438277\pi\)
\(948\) −2.90304e6 −0.104914
\(949\) 0 0
\(950\) −6.43970e6 −0.231503
\(951\) −728592. −0.0261236
\(952\) 9.12102e6 0.326175
\(953\) −90234.0 −0.00321838 −0.00160919 0.999999i \(-0.500512\pi\)
−0.00160919 + 0.999999i \(0.500512\pi\)
\(954\) −3.46983e7 −1.23435
\(955\) −6.23620e7 −2.21264
\(956\) −4.07382e6 −0.144164
\(957\) 2.63640e6 0.0930532
\(958\) −1.43604e7 −0.505538
\(959\) 2.30446e7 0.809139
\(960\) 1.11411e6 0.0390167
\(961\) −2.65843e7 −0.928573
\(962\) 0 0
\(963\) 1.79902e7 0.625130
\(964\) 5.01760e6 0.173902
\(965\) 4.34558e7 1.50221
\(966\) 2.76045e6 0.0951780
\(967\) 1.20331e7 0.413821 0.206910 0.978360i \(-0.433659\pi\)
0.206910 + 0.978360i \(0.433659\pi\)
\(968\) −572864. −0.0196500
\(969\) 7.46645e6 0.255449
\(970\) −3.30447e7 −1.12765
\(971\) 1.84061e7 0.626489 0.313245 0.949672i \(-0.398584\pi\)
0.313245 + 0.949672i \(0.398584\pi\)
\(972\) 1.03585e7 0.351665
\(973\) −2.98332e7 −1.01023
\(974\) −3.87075e7 −1.30737
\(975\) 0 0
\(976\) −1.39725e6 −0.0469514
\(977\) 4.66720e7 1.56430 0.782150 0.623090i \(-0.214123\pi\)
0.782150 + 0.623090i \(0.214123\pi\)
\(978\) 586016. 0.0195913
\(979\) −7.31016e6 −0.243764
\(980\) −1.09703e7 −0.364883
\(981\) 4.95051e7 1.64239
\(982\) −1.38253e7 −0.457506
\(983\) 1.98925e7 0.656608 0.328304 0.944572i \(-0.393523\pi\)
0.328304 + 0.944572i \(0.393523\pi\)
\(984\) 1.73056e6 0.0569769
\(985\) 2.43639e7 0.800121
\(986\) 1.17489e7 0.384861
\(987\) −8.25182e6 −0.269623
\(988\) 0 0
\(989\) −3.55913e7 −1.15705
\(990\) 2.40802e7 0.780857
\(991\) −4.58344e7 −1.48254 −0.741271 0.671206i \(-0.765777\pi\)
−0.741271 + 0.671206i \(0.765777\pi\)
\(992\) −1.46432e6 −0.0472451
\(993\) 864920. 0.0278357
\(994\) 5.83512e6 0.187320
\(995\) −2.51899e7 −0.806620
\(996\) −7.97094e6 −0.254602
\(997\) 2.51716e7 0.801999 0.400999 0.916078i \(-0.368663\pi\)
0.400999 + 0.916078i \(0.368663\pi\)
\(998\) 8.36434e6 0.265831
\(999\) 1.66418e7 0.527577
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 338.6.a.e.1.1 1
13.5 odd 4 26.6.b.b.25.1 2
13.8 odd 4 26.6.b.b.25.2 yes 2
13.12 even 2 338.6.a.b.1.1 1
39.5 even 4 234.6.b.a.181.2 2
39.8 even 4 234.6.b.a.181.1 2
52.31 even 4 208.6.f.a.129.1 2
52.47 even 4 208.6.f.a.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.b.25.1 2 13.5 odd 4
26.6.b.b.25.2 yes 2 13.8 odd 4
208.6.f.a.129.1 2 52.31 even 4
208.6.f.a.129.2 2 52.47 even 4
234.6.b.a.181.1 2 39.8 even 4
234.6.b.a.181.2 2 39.5 even 4
338.6.a.b.1.1 1 13.12 even 2
338.6.a.e.1.1 1 1.1 even 1 trivial