Properties

Label 338.6.a.b.1.1
Level $338$
Weight $6$
Character 338.1
Self dual yes
Analytic conductor $54.210$
Analytic rank $0$
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: \(0\)
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 \(-0.708112\pi\)
−0.608210 + 0.793776i \(0.708112\pi\)
\(6\) −16.0000 −0.181444
\(7\) 82.0000 0.632512 0.316256 0.948674i \(-0.397574\pi\)
0.316256 + 0.948674i \(0.397574\pi\)
\(8\) −64.0000 −0.353553
\(9\) −227.000 −0.934156
\(10\) 272.000 0.860140
\(11\) 390.000 0.971813 0.485907 0.874011i \(-0.338489\pi\)
0.485907 + 0.874011i \(0.338489\pi\)
\(12\) 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 \(-0.389145\pi\)
0.341264 + 0.939968i \(0.389145\pi\)
\(20\) −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 \(-0.457337\pi\)
0.133629 + 0.991031i \(0.457337\pi\)
\(32\) −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 \(-0.321627\pi\)
0.531505 + 0.847055i \(0.321627\pi\)
\(38\) −4296.00 −0.482620
\(39\) 0 0
\(40\) 4352.00 0.430070
\(41\) −6760.00 −0.628040 −0.314020 0.949416i \(-0.601676\pi\)
−0.314020 + 0.949416i \(0.601676\pi\)
\(42\) −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 \(-0.812013\pi\)
−0.830618 + 0.556842i \(0.812013\pi\)
\(48\) 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 \(-0.369688\pi\)
0.398047 + 0.917365i \(0.369688\pi\)
\(60\) −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 \(-0.707272\pi\)
−0.606112 + 0.795379i \(0.707272\pi\)
\(68\) −27808.0 −0.729285
\(69\) −8416.00 −0.212806
\(70\) 22304.0 0.544049
\(71\) 17790.0 0.418823 0.209411 0.977828i \(-0.432845\pi\)
0.209411 + 0.977828i \(0.432845\pi\)
\(72\) 14528.0 0.330274
\(73\) 31064.0 0.682260 0.341130 0.940016i \(-0.389190\pi\)
0.341130 + 0.940016i \(0.389190\pi\)
\(74\) −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 \(-0.0397516\pi\)
0.992212 + 0.124559i \(0.0397516\pi\)
\(84\) 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 \(-0.540027\pi\)
−0.125417 + 0.992104i \(0.540027\pi\)
\(90\) −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 \(-0.272457\pi\)
0.655502 + 0.755193i \(0.272457\pi\)
\(98\) 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 \(-0.158159\pi\)
0.879078 + 0.476677i \(0.158159\pi\)
\(110\) 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 \(-0.279090\pi\)
0.639623 + 0.768688i \(0.279090\pi\)
\(138\) 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 \(-0.331155\pi\)
0.505916 + 0.862583i \(0.331155\pi\)
\(150\) −23984.0 −0.0870349
\(151\) −344030. −1.22787 −0.613937 0.789355i \(-0.710415\pi\)
−0.613937 + 0.789355i \(0.710415\pi\)
\(152\) −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 \(-0.517193\pi\)
−0.0539872 + 0.998542i \(0.517193\pi\)
\(164\) −108160. −0.314020
\(165\) −106080. −0.303336
\(166\) −498184. −1.40320
\(167\) 269442. 0.747608 0.373804 0.927508i \(-0.378053\pi\)
0.373804 + 0.927508i \(0.378053\pi\)
\(168\) −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 \(-0.711842\pi\)
−0.617470 + 0.786595i \(0.711842\pi\)
\(194\) −485952. −0.927020
\(195\) 0 0
\(196\) −161328. −0.299964
\(197\) −358292. −0.657766 −0.328883 0.944371i \(-0.606672\pi\)
−0.328883 + 0.944371i \(0.606672\pi\)
\(198\) 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 \(-0.230362\pi\)
0.749359 + 0.662164i \(0.230362\pi\)
\(224\) −83968.0 −0.111813
\(225\) −340273. −0.448096
\(226\) −176936. −0.230433
\(227\) −1.39158e6 −1.79244 −0.896219 0.443612i \(-0.853697\pi\)
−0.896219 + 0.443612i \(0.853697\pi\)
\(228\) 68736.0 0.0875683
\(229\) 909796. 1.14645 0.573225 0.819398i \(-0.305692\pi\)
0.573225 + 0.819398i \(0.305692\pi\)
\(230\) −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 \(-0.453951\pi\)
0.144164 + 0.989554i \(0.453951\pi\)
\(240\) −69632.0 −0.0780334
\(241\) −313600. −0.347803 −0.173902 0.984763i \(-0.555637\pi\)
−0.173902 + 0.984763i \(0.555637\pi\)
\(242\) 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 \(-0.560336\pi\)
−0.188417 + 0.982089i \(0.560336\pi\)
\(272\) −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 \(-0.505921\pi\)
−0.0186004 + 0.999827i \(0.505921\pi\)
\(282\) 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 \(-0.613401\pi\)
−0.348771 + 0.937208i \(0.613401\pi\)
\(294\) 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 \(-0.341477\pi\)
0.477681 + 0.878533i \(0.341477\pi\)
\(308\) 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 \(-0.483790\pi\)
0.0509033 + 0.998704i \(0.483790\pi\)
\(318\) −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 \(-0.517273\pi\)
−0.0542395 + 0.998528i \(0.517273\pi\)
\(332\) 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 \(-0.215013\pi\)
0.780405 + 0.625274i \(0.215013\pi\)
\(350\) −491672. −0.214539
\(351\) 0 0
\(352\) −399360. −0.171794
\(353\) −2.08678e6 −0.891335 −0.445667 0.895199i \(-0.647034\pi\)
−0.445667 + 0.895199i \(0.647034\pi\)
\(354\) −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 \(-0.532688\pi\)
−0.102511 + 0.994732i \(0.532688\pi\)
\(360\) −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 \(-0.813726\pi\)
−0.833604 + 0.552363i \(0.813726\pi\)
\(380\) −1.16851e6 −0.415121
\(381\) 1.24173e6 0.438242
\(382\) 3.66835e6 1.28621
\(383\) 4.35473e6 1.51692 0.758462 0.651717i \(-0.225951\pi\)
0.758462 + 0.651717i \(0.225951\pi\)
\(384\) −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 \(-0.717821\pi\)
−0.632134 + 0.774859i \(0.717821\pi\)
\(398\) 1.48176e6 0.468889
\(399\) 352272. 0.110776
\(400\) 383744. 0.119920
\(401\) −344640. −0.107030 −0.0535149 0.998567i \(-0.517042\pi\)
−0.0535149 + 0.998567i \(0.517042\pi\)
\(402\) 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 \(-0.376760\pi\)
0.377568 + 0.925982i \(0.376760\pi\)
\(410\) −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 \(-0.268437\pi\)
0.664988 + 0.746854i \(0.268437\pi\)
\(422\) 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 \(-0.490956\pi\)
0.0284079 + 0.999596i \(0.490956\pi\)
\(432\) −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 \(-0.165411\pi\)
0.867991 + 0.496579i \(0.165411\pi\)
\(450\) 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 \(-0.322918\pi\)
0.528065 + 0.849204i \(0.322918\pi\)
\(458\) −3.63918e6 −0.810663
\(459\) 3.26744e6 0.723896
\(460\) 2.28915e6 0.504406
\(461\) −3.34566e6 −0.733212 −0.366606 0.930376i \(-0.619480\pi\)
−0.366606 + 0.930376i \(0.619480\pi\)
\(462\) −511680. −0.111530
\(463\) 1.65791e6 0.359426 0.179713 0.983719i \(-0.442483\pi\)
0.179713 + 0.983719i \(0.442483\pi\)
\(464\) −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 \(-0.383640\pi\)
0.357469 + 0.933925i \(0.383640\pi\)
\(480\) 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 \(-0.124526\pi\)
0.924449 + 0.381306i \(0.124526\pi\)
\(488\) 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 \(-0.560191\pi\)
−0.187971 + 0.982175i \(0.560191\pi\)
\(500\) 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 \(-0.615665\pi\)
−0.355427 + 0.934704i \(0.615665\pi\)
\(510\) −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 \(-0.493828\pi\)
0.0193887 + 0.999812i \(0.493828\pi\)
\(542\) 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 \(-0.559243\pi\)
−0.185043 + 0.982730i \(0.559243\pi\)
\(558\) 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 \(-0.648825\pi\)
−0.450698 + 0.892676i \(0.648825\pi\)
\(578\) −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 \(-0.547482\pi\)
−0.148617 + 0.988895i \(0.547482\pi\)
\(588\) −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 \(-0.199191\pi\)
0.810508 + 0.585728i \(0.199191\pi\)
\(594\) 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 \(-0.227688\pi\)
0.754894 + 0.655846i \(0.227688\pi\)
\(614\) −6.31065e6 −0.675543
\(615\) 1.83872e6 0.196032
\(616\) −2.04672e6 −0.217323
\(617\) 3.72561e6 0.393989 0.196995 0.980405i \(-0.436882\pi\)
0.196995 + 0.980405i \(0.436882\pi\)
\(618\) −1.00442e6 −0.105789
\(619\) −8.96911e6 −0.940855 −0.470428 0.882439i \(-0.655900\pi\)
−0.470428 + 0.882439i \(0.655900\pi\)
\(620\) −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 \(-0.830036\pi\)
−0.860800 + 0.508943i \(0.830036\pi\)
\(632\) 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 \(-0.626931\pi\)
−0.388280 + 0.921542i \(0.626931\pi\)
\(644\) −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 \(-0.329839\pi\)
0.509476 + 0.860485i \(0.329839\pi\)
\(662\) 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 \(-0.545855\pi\)
−0.143559 + 0.989642i \(0.545855\pi\)
\(684\) −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 \(-0.503836\pi\)
−0.0120508 + 0.999927i \(0.503836\pi\)
\(692\) −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 \(-0.436861\pi\)
0.197059 + 0.980392i \(0.436861\pi\)
\(710\) 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 \(-0.533437\pi\)
−0.104854 + 0.994488i \(0.533437\pi\)
\(734\) 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 \(-0.417320\pi\)
0.256837 + 0.966455i \(0.417320\pi\)
\(740\) −9.63098e6 −0.646533
\(741\) 0 0
\(742\) −1.25342e7 −0.835770
\(743\) 2.18236e7 1.45029 0.725146 0.688595i \(-0.241772\pi\)
0.725146 + 0.688595i \(0.241772\pi\)
\(744\) −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 \(-0.570107\pi\)
−0.218472 + 0.975843i \(0.570107\pi\)
\(762\) −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 \(-0.195922\pi\)
0.816481 + 0.577372i \(0.195922\pi\)
\(770\) 8.69856e6 0.528714
\(771\) 753528. 0.0456524
\(772\) −1.02249e7 −0.617470
\(773\) −710244. −0.0427522 −0.0213761 0.999772i \(-0.506805\pi\)
−0.0213761 + 0.999772i \(0.506805\pi\)
\(774\) 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 \(-0.547675\pi\)
−0.149215 + 0.988805i \(0.547675\pi\)
\(788\) −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 \(-0.312577\pi\)
0.555369 + 0.831604i \(0.312577\pi\)
\(812\) −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 \(-0.684537\pi\)
−0.547808 + 0.836604i \(0.684537\pi\)
\(822\) −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 \(-0.358538\pi\)
0.429932 + 0.902861i \(0.358538\pi\)
\(828\) 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 \(-0.505661\pi\)
−0.0177821 + 0.999842i \(0.505661\pi\)
\(840\) 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 \(-0.421915\pi\)
0.242859 + 0.970062i \(0.421915\pi\)
\(854\) 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 \(-0.355174\pi\)
0.439448 + 0.898268i \(0.355174\pi\)
\(864\) 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 \(-0.180015\pi\)
0.844303 + 0.535866i \(0.180015\pi\)
\(878\) 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 \(-0.644387\pi\)
−0.438210 + 0.898873i \(0.644387\pi\)
\(930\) 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 \(-0.234378\pi\)
0.740944 + 0.671567i \(0.234378\pi\)
\(942\) −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 \(-0.561723\pi\)
−0.192697 + 0.981258i \(0.561723\pi\)
\(948\) −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 \(-0.566341\pi\)
−0.206910 + 0.978360i \(0.566341\pi\)
\(968\) 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 \(-0.785877\pi\)
−0.782150 + 0.623090i \(0.785877\pi\)
\(978\) 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 \(-0.606477\pi\)
−0.328304 + 0.944572i \(0.606477\pi\)
\(984\) 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.b.1.1 1
13.5 odd 4 26.6.b.b.25.2 yes 2
13.8 odd 4 26.6.b.b.25.1 2
13.12 even 2 338.6.a.e.1.1 1
39.5 even 4 234.6.b.a.181.1 2
39.8 even 4 234.6.b.a.181.2 2
52.31 even 4 208.6.f.a.129.2 2
52.47 even 4 208.6.f.a.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.b.25.1 2 13.8 odd 4
26.6.b.b.25.2 yes 2 13.5 odd 4
208.6.f.a.129.1 2 52.47 even 4
208.6.f.a.129.2 2 52.31 even 4
234.6.b.a.181.1 2 39.5 even 4
234.6.b.a.181.2 2 39.8 even 4
338.6.a.b.1.1 1 1.1 even 1 trivial
338.6.a.e.1.1 1 13.12 even 2