# Properties

 Label 6.22.a.a.1.1 Level $6$ Weight $22$ Character 6.1 Self dual yes Analytic conductor $16.769$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6,22,Mod(1,6)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 6.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: $$16.7686406572$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1024.00 q^{2} +59049.0 q^{3} +1.04858e6 q^{4} +2.64446e7 q^{5} -6.04662e7 q^{6} +1.66116e8 q^{7} -1.07374e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$$ $$q-1024.00 q^{2} +59049.0 q^{3} +1.04858e6 q^{4} +2.64446e7 q^{5} -6.04662e7 q^{6} +1.66116e8 q^{7} -1.07374e9 q^{8} +3.48678e9 q^{9} -2.70792e10 q^{10} -1.04879e11 q^{11} +6.19174e10 q^{12} +3.35591e11 q^{13} -1.70103e11 q^{14} +1.56152e12 q^{15} +1.09951e12 q^{16} +1.45961e13 q^{17} -3.57047e12 q^{18} +3.56953e12 q^{19} +2.77291e13 q^{20} +9.80898e12 q^{21} +1.07396e14 q^{22} +2.22369e14 q^{23} -6.34034e13 q^{24} +2.22477e14 q^{25} -3.43646e14 q^{26} +2.05891e14 q^{27} +1.74185e14 q^{28} +2.19411e15 q^{29} -1.59900e15 q^{30} -8.72363e15 q^{31} -1.12590e15 q^{32} -6.19299e15 q^{33} -1.49465e16 q^{34} +4.39286e15 q^{35} +3.65616e15 q^{36} +3.74709e16 q^{37} -3.65520e15 q^{38} +1.98163e16 q^{39} -2.83946e16 q^{40} +8.66167e16 q^{41} -1.00444e16 q^{42} +1.31417e17 q^{43} -1.09973e17 q^{44} +9.22064e16 q^{45} -2.27706e17 q^{46} +3.39041e17 q^{47} +6.49251e16 q^{48} -5.30951e17 q^{49} -2.27817e17 q^{50} +8.61888e17 q^{51} +3.51893e17 q^{52} -1.57149e18 q^{53} -2.10833e17 q^{54} -2.77347e18 q^{55} -1.78366e17 q^{56} +2.10777e17 q^{57} -2.24677e18 q^{58} +5.23298e18 q^{59} +1.63738e18 q^{60} -4.78838e18 q^{61} +8.93299e18 q^{62} +5.79210e17 q^{63} +1.15292e18 q^{64} +8.87456e18 q^{65} +6.34162e18 q^{66} -1.54803e19 q^{67} +1.53052e19 q^{68} +1.31307e19 q^{69} -4.49829e18 q^{70} -1.29309e19 q^{71} -3.74391e18 q^{72} -4.42572e19 q^{73} -3.83702e19 q^{74} +1.31370e19 q^{75} +3.74292e18 q^{76} -1.74220e19 q^{77} -2.02919e19 q^{78} -1.48886e19 q^{79} +2.90761e19 q^{80} +1.21577e19 q^{81} -8.86955e19 q^{82} +3.70851e19 q^{83} +1.02855e19 q^{84} +3.85988e20 q^{85} -1.34571e20 q^{86} +1.29560e20 q^{87} +1.12613e20 q^{88} -1.05572e20 q^{89} -9.44194e19 q^{90} +5.57470e19 q^{91} +2.33171e20 q^{92} -5.15121e20 q^{93} -3.47178e20 q^{94} +9.43946e19 q^{95} -6.64833e19 q^{96} +1.38109e21 q^{97} +5.43694e20 q^{98} -3.65690e20 q^{99} +O(q^{100})$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1024.00 −0.707107
$$3$$ 59049.0 0.577350
$$4$$ 1.04858e6 0.500000
$$5$$ 2.64446e7 1.21102 0.605510 0.795838i $$-0.292969\pi$$
0.605510 + 0.795838i $$0.292969\pi$$
$$6$$ −6.04662e7 −0.408248
$$7$$ 1.66116e8 0.222270 0.111135 0.993805i $$-0.464551\pi$$
0.111135 + 0.993805i $$0.464551\pi$$
$$8$$ −1.07374e9 −0.353553
$$9$$ 3.48678e9 0.333333
$$10$$ −2.70792e10 −0.856320
$$11$$ −1.04879e11 −1.21917 −0.609585 0.792721i $$-0.708664\pi$$
−0.609585 + 0.792721i $$0.708664\pi$$
$$12$$ 6.19174e10 0.288675
$$13$$ 3.35591e11 0.675158 0.337579 0.941297i $$-0.390392\pi$$
0.337579 + 0.941297i $$0.390392\pi$$
$$14$$ −1.70103e11 −0.157169
$$15$$ 1.56152e12 0.699182
$$16$$ 1.09951e12 0.250000
$$17$$ 1.45961e13 1.75600 0.878000 0.478661i $$-0.158878\pi$$
0.878000 + 0.478661i $$0.158878\pi$$
$$18$$ −3.57047e12 −0.235702
$$19$$ 3.56953e12 0.133567 0.0667834 0.997767i $$-0.478726\pi$$
0.0667834 + 0.997767i $$0.478726\pi$$
$$20$$ 2.77291e13 0.605510
$$21$$ 9.80898e12 0.128328
$$22$$ 1.07396e14 0.862084
$$23$$ 2.22369e14 1.11926 0.559632 0.828741i $$-0.310943\pi$$
0.559632 + 0.828741i $$0.310943\pi$$
$$24$$ −6.34034e13 −0.204124
$$25$$ 2.22477e14 0.466568
$$26$$ −3.43646e14 −0.477409
$$27$$ 2.05891e14 0.192450
$$28$$ 1.74185e14 0.111135
$$29$$ 2.19411e15 0.968455 0.484227 0.874942i $$-0.339101\pi$$
0.484227 + 0.874942i $$0.339101\pi$$
$$30$$ −1.59900e15 −0.494397
$$31$$ −8.72363e15 −1.91161 −0.955805 0.294001i $$-0.905013\pi$$
−0.955805 + 0.294001i $$0.905013\pi$$
$$32$$ −1.12590e15 −0.176777
$$33$$ −6.19299e15 −0.703888
$$34$$ −1.49465e16 −1.24168
$$35$$ 4.39286e15 0.269174
$$36$$ 3.65616e15 0.166667
$$37$$ 3.74709e16 1.28108 0.640540 0.767925i $$-0.278711\pi$$
0.640540 + 0.767925i $$0.278711\pi$$
$$38$$ −3.65520e15 −0.0944459
$$39$$ 1.98163e16 0.389803
$$40$$ −2.83946e16 −0.428160
$$41$$ 8.66167e16 1.00779 0.503897 0.863764i $$-0.331899\pi$$
0.503897 + 0.863764i $$0.331899\pi$$
$$42$$ −1.00444e16 −0.0907415
$$43$$ 1.31417e17 0.927326 0.463663 0.886012i $$-0.346535\pi$$
0.463663 + 0.886012i $$0.346535\pi$$
$$44$$ −1.09973e17 −0.609585
$$45$$ 9.22064e16 0.403673
$$46$$ −2.27706e17 −0.791439
$$47$$ 3.39041e17 0.940210 0.470105 0.882610i $$-0.344216\pi$$
0.470105 + 0.882610i $$0.344216\pi$$
$$48$$ 6.49251e16 0.144338
$$49$$ −5.30951e17 −0.950596
$$50$$ −2.27817e17 −0.329914
$$51$$ 8.61888e17 1.01383
$$52$$ 3.51893e17 0.337579
$$53$$ −1.57149e18 −1.23429 −0.617144 0.786850i $$-0.711710\pi$$
−0.617144 + 0.786850i $$0.711710\pi$$
$$54$$ −2.10833e17 −0.136083
$$55$$ −2.77347e18 −1.47644
$$56$$ −1.78366e17 −0.0785845
$$57$$ 2.10777e17 0.0771148
$$58$$ −2.24677e18 −0.684801
$$59$$ 5.23298e18 1.33292 0.666459 0.745542i $$-0.267809\pi$$
0.666459 + 0.745542i $$0.267809\pi$$
$$60$$ 1.63738e18 0.349591
$$61$$ −4.78838e18 −0.859460 −0.429730 0.902957i $$-0.641391\pi$$
−0.429730 + 0.902957i $$0.641391\pi$$
$$62$$ 8.93299e18 1.35171
$$63$$ 5.79210e17 0.0740901
$$64$$ 1.15292e18 0.125000
$$65$$ 8.87456e18 0.817630
$$66$$ 6.34162e18 0.497724
$$67$$ −1.54803e19 −1.03752 −0.518758 0.854921i $$-0.673605\pi$$
−0.518758 + 0.854921i $$0.673605\pi$$
$$68$$ 1.53052e19 0.878000
$$69$$ 1.31307e19 0.646207
$$70$$ −4.49829e18 −0.190335
$$71$$ −1.29309e19 −0.471429 −0.235715 0.971822i $$-0.575743\pi$$
−0.235715 + 0.971822i $$0.575743\pi$$
$$72$$ −3.74391e18 −0.117851
$$73$$ −4.42572e19 −1.20530 −0.602648 0.798007i $$-0.705888\pi$$
−0.602648 + 0.798007i $$0.705888\pi$$
$$74$$ −3.83702e19 −0.905860
$$75$$ 1.31370e19 0.269373
$$76$$ 3.74292e18 0.0667834
$$77$$ −1.74220e19 −0.270985
$$78$$ −2.02919e19 −0.275632
$$79$$ −1.48886e19 −0.176917 −0.0884583 0.996080i $$-0.528194\pi$$
−0.0884583 + 0.996080i $$0.528194\pi$$
$$80$$ 2.90761e19 0.302755
$$81$$ 1.21577e19 0.111111
$$82$$ −8.86955e19 −0.712617
$$83$$ 3.70851e19 0.262349 0.131174 0.991359i $$-0.458125\pi$$
0.131174 + 0.991359i $$0.458125\pi$$
$$84$$ 1.02855e19 0.0641639
$$85$$ 3.85988e20 2.12655
$$86$$ −1.34571e20 −0.655719
$$87$$ 1.29560e20 0.559138
$$88$$ 1.12613e20 0.431042
$$89$$ −1.05572e20 −0.358884 −0.179442 0.983769i $$-0.557429\pi$$
−0.179442 + 0.983769i $$0.557429\pi$$
$$90$$ −9.44194e19 −0.285440
$$91$$ 5.57470e19 0.150068
$$92$$ 2.33171e20 0.559632
$$93$$ −5.15121e20 −1.10367
$$94$$ −3.47178e20 −0.664829
$$95$$ 9.43946e19 0.161752
$$96$$ −6.64833e19 −0.102062
$$97$$ 1.38109e21 1.90160 0.950801 0.309803i $$-0.100263\pi$$
0.950801 + 0.309803i $$0.100263\pi$$
$$98$$ 5.43694e20 0.672173
$$99$$ −3.65690e20 −0.406390
$$100$$ 2.33284e20 0.233284
$$101$$ −1.49419e21 −1.34596 −0.672979 0.739662i $$-0.734986\pi$$
−0.672979 + 0.739662i $$0.734986\pi$$
$$102$$ −8.82573e20 −0.716884
$$103$$ −1.72821e21 −1.26708 −0.633542 0.773708i $$-0.718400\pi$$
−0.633542 + 0.773708i $$0.718400\pi$$
$$104$$ −3.60338e20 −0.238704
$$105$$ 2.59394e20 0.155408
$$106$$ 1.60921e21 0.872773
$$107$$ −3.21208e21 −1.57854 −0.789272 0.614044i $$-0.789542\pi$$
−0.789272 + 0.614044i $$0.789542\pi$$
$$108$$ 2.15892e20 0.0962250
$$109$$ −5.26985e20 −0.213216 −0.106608 0.994301i $$-0.533999\pi$$
−0.106608 + 0.994301i $$0.533999\pi$$
$$110$$ 2.84003e21 1.04400
$$111$$ 2.21262e21 0.739631
$$112$$ 1.82646e20 0.0555676
$$113$$ 2.65376e20 0.0735425 0.0367713 0.999324i $$-0.488293\pi$$
0.0367713 + 0.999324i $$0.488293\pi$$
$$114$$ −2.15836e20 −0.0545284
$$115$$ 5.88045e21 1.35545
$$116$$ 2.30069e21 0.484227
$$117$$ 1.17013e21 0.225053
$$118$$ −5.35858e21 −0.942515
$$119$$ 2.42465e21 0.390307
$$120$$ −1.67667e21 −0.247198
$$121$$ 3.59930e21 0.486376
$$122$$ 4.90331e21 0.607730
$$123$$ 5.11463e21 0.581850
$$124$$ −9.14739e21 −0.955805
$$125$$ −6.72644e21 −0.645996
$$126$$ −5.93111e20 −0.0523896
$$127$$ −1.95043e22 −1.58559 −0.792797 0.609485i $$-0.791376\pi$$
−0.792797 + 0.609485i $$0.791376\pi$$
$$128$$ −1.18059e21 −0.0883883
$$129$$ 7.76004e21 0.535392
$$130$$ −9.08755e21 −0.578152
$$131$$ −9.92728e21 −0.582749 −0.291375 0.956609i $$-0.594113\pi$$
−0.291375 + 0.956609i $$0.594113\pi$$
$$132$$ −6.49382e21 −0.351944
$$133$$ 5.92956e20 0.0296879
$$134$$ 1.58519e22 0.733635
$$135$$ 5.44470e21 0.233061
$$136$$ −1.56725e22 −0.620840
$$137$$ 4.13768e22 1.51772 0.758858 0.651256i $$-0.225758\pi$$
0.758858 + 0.651256i $$0.225758\pi$$
$$138$$ −1.34458e22 −0.456938
$$139$$ −1.86724e22 −0.588228 −0.294114 0.955770i $$-0.595025\pi$$
−0.294114 + 0.955770i $$0.595025\pi$$
$$140$$ 4.60625e21 0.134587
$$141$$ 2.00200e22 0.542831
$$142$$ 1.32412e22 0.333351
$$143$$ −3.51964e22 −0.823133
$$144$$ 3.83376e21 0.0833333
$$145$$ 5.80222e22 1.17282
$$146$$ 4.53194e22 0.852273
$$147$$ −3.13521e22 −0.548827
$$148$$ 3.92911e22 0.640540
$$149$$ −4.48377e22 −0.681064 −0.340532 0.940233i $$-0.610607\pi$$
−0.340532 + 0.940233i $$0.610607\pi$$
$$150$$ −1.34523e22 −0.190476
$$151$$ 5.27798e22 0.696963 0.348482 0.937316i $$-0.386697\pi$$
0.348482 + 0.937316i $$0.386697\pi$$
$$152$$ −3.83275e21 −0.0472230
$$153$$ 5.08936e22 0.585333
$$154$$ 1.78402e22 0.191616
$$155$$ −2.30692e23 −2.31500
$$156$$ 2.07789e22 0.194901
$$157$$ 4.76649e22 0.418074 0.209037 0.977908i $$-0.432967\pi$$
0.209037 + 0.977908i $$0.432967\pi$$
$$158$$ 1.52459e22 0.125099
$$159$$ −9.27952e22 −0.712616
$$160$$ −2.97739e22 −0.214080
$$161$$ 3.69391e22 0.248779
$$162$$ −1.24494e22 −0.0785674
$$163$$ 4.68583e22 0.277215 0.138607 0.990347i $$-0.455737\pi$$
0.138607 + 0.990347i $$0.455737\pi$$
$$164$$ 9.08242e22 0.503897
$$165$$ −1.63771e23 −0.852422
$$166$$ −3.79751e22 −0.185509
$$167$$ 4.21072e22 0.193123 0.0965613 0.995327i $$-0.469216\pi$$
0.0965613 + 0.995327i $$0.469216\pi$$
$$168$$ −1.05323e22 −0.0453708
$$169$$ −1.34443e23 −0.544161
$$170$$ −3.95252e23 −1.50370
$$171$$ 1.24462e22 0.0445222
$$172$$ 1.37801e23 0.463663
$$173$$ 4.51089e23 1.42816 0.714082 0.700062i $$-0.246844\pi$$
0.714082 + 0.700062i $$0.246844\pi$$
$$174$$ −1.32669e23 −0.395370
$$175$$ 3.69570e22 0.103704
$$176$$ −1.15315e23 −0.304793
$$177$$ 3.09003e23 0.769560
$$178$$ 1.08106e23 0.253769
$$179$$ 2.06595e23 0.457259 0.228629 0.973514i $$-0.426576\pi$$
0.228629 + 0.973514i $$0.426576\pi$$
$$180$$ 9.66855e22 0.201837
$$181$$ 7.17280e23 1.41274 0.706372 0.707841i $$-0.250331\pi$$
0.706372 + 0.707841i $$0.250331\pi$$
$$182$$ −5.70850e22 −0.106114
$$183$$ −2.82749e23 −0.496210
$$184$$ −2.38767e23 −0.395719
$$185$$ 9.90901e23 1.55141
$$186$$ 5.27484e23 0.780412
$$187$$ −1.53083e24 −2.14086
$$188$$ 3.55510e23 0.470105
$$189$$ 3.42018e22 0.0427760
$$190$$ −9.66601e22 −0.114376
$$191$$ −4.48187e23 −0.501890 −0.250945 0.968001i $$-0.580741\pi$$
−0.250945 + 0.968001i $$0.580741\pi$$
$$192$$ 6.80789e22 0.0721688
$$193$$ −7.88064e23 −0.791061 −0.395530 0.918453i $$-0.629439\pi$$
−0.395530 + 0.918453i $$0.629439\pi$$
$$194$$ −1.41424e24 −1.34464
$$195$$ 5.24034e23 0.472059
$$196$$ −5.56743e23 −0.475298
$$197$$ −5.38083e23 −0.435466 −0.217733 0.976008i $$-0.569866\pi$$
−0.217733 + 0.976008i $$0.569866\pi$$
$$198$$ 3.74466e23 0.287361
$$199$$ −1.82843e24 −1.33083 −0.665414 0.746475i $$-0.731745\pi$$
−0.665414 + 0.746475i $$0.731745\pi$$
$$200$$ −2.38883e23 −0.164957
$$201$$ −9.14098e23 −0.599010
$$202$$ 1.53005e24 0.951736
$$203$$ 3.64476e23 0.215259
$$204$$ 9.03755e23 0.506914
$$205$$ 2.29054e24 1.22046
$$206$$ 1.76969e24 0.895964
$$207$$ 7.75354e23 0.373088
$$208$$ 3.68987e23 0.168790
$$209$$ −3.74368e23 −0.162841
$$210$$ −2.65619e23 −0.109890
$$211$$ −2.17230e24 −0.854974 −0.427487 0.904021i $$-0.640601\pi$$
−0.427487 + 0.904021i $$0.640601\pi$$
$$212$$ −1.64783e24 −0.617144
$$213$$ −7.63557e23 −0.272180
$$214$$ 3.28917e24 1.11620
$$215$$ 3.47526e24 1.12301
$$216$$ −2.21074e23 −0.0680414
$$217$$ −1.44913e24 −0.424894
$$218$$ 5.39632e23 0.150766
$$219$$ −2.61334e24 −0.695878
$$220$$ −2.90820e24 −0.738219
$$221$$ 4.89834e24 1.18558
$$222$$ −2.26572e24 −0.522998
$$223$$ 8.40788e24 1.85134 0.925668 0.378336i $$-0.123504\pi$$
0.925668 + 0.378336i $$0.123504\pi$$
$$224$$ −1.87030e23 −0.0392922
$$225$$ 7.75730e23 0.155523
$$226$$ −2.71745e23 −0.0520024
$$227$$ −6.09701e24 −1.11390 −0.556949 0.830547i $$-0.688028\pi$$
−0.556949 + 0.830547i $$0.688028\pi$$
$$228$$ 2.21016e23 0.0385574
$$229$$ −6.53459e23 −0.108879 −0.0544397 0.998517i $$-0.517337\pi$$
−0.0544397 + 0.998517i $$0.517337\pi$$
$$230$$ −6.02159e24 −0.958448
$$231$$ −1.02875e24 −0.156454
$$232$$ −2.35591e24 −0.342400
$$233$$ −1.09090e25 −1.51547 −0.757735 0.652562i $$-0.773694\pi$$
−0.757735 + 0.652562i $$0.773694\pi$$
$$234$$ −1.19822e24 −0.159136
$$235$$ 8.96579e24 1.13861
$$236$$ 5.48718e24 0.666459
$$237$$ −8.79156e23 −0.102143
$$238$$ −2.48284e24 −0.275989
$$239$$ 5.96515e21 0.000634517 0 0.000317258 1.00000i $$-0.499899\pi$$
0.000317258 1.00000i $$0.499899\pi$$
$$240$$ 1.71691e24 0.174796
$$241$$ 8.17929e24 0.797144 0.398572 0.917137i $$-0.369506\pi$$
0.398572 + 0.917137i $$0.369506\pi$$
$$242$$ −3.68569e24 −0.343920
$$243$$ 7.17898e23 0.0641500
$$244$$ −5.02099e24 −0.429730
$$245$$ −1.40408e25 −1.15119
$$246$$ −5.23738e24 −0.411430
$$247$$ 1.19790e24 0.0901787
$$248$$ 9.36692e24 0.675856
$$249$$ 2.18984e24 0.151467
$$250$$ 6.88787e24 0.456788
$$251$$ 1.37352e25 0.873499 0.436749 0.899583i $$-0.356130\pi$$
0.436749 + 0.899583i $$0.356130\pi$$
$$252$$ 6.07346e23 0.0370451
$$253$$ −2.33218e25 −1.36457
$$254$$ 1.99724e25 1.12118
$$255$$ 2.27922e25 1.22776
$$256$$ 1.20893e24 0.0625000
$$257$$ 4.53006e24 0.224805 0.112403 0.993663i $$-0.464145\pi$$
0.112403 + 0.993663i $$0.464145\pi$$
$$258$$ −7.94628e24 −0.378579
$$259$$ 6.22451e24 0.284746
$$260$$ 9.30565e24 0.408815
$$261$$ 7.65039e24 0.322818
$$262$$ 1.01655e25 0.412066
$$263$$ −5.23061e24 −0.203712 −0.101856 0.994799i $$-0.532478\pi$$
−0.101856 + 0.994799i $$0.532478\pi$$
$$264$$ 6.64967e24 0.248862
$$265$$ −4.15575e25 −1.49475
$$266$$ −6.07186e23 −0.0209925
$$267$$ −6.23392e24 −0.207202
$$268$$ −1.62323e25 −0.518758
$$269$$ −1.15814e25 −0.355927 −0.177964 0.984037i $$-0.556951\pi$$
−0.177964 + 0.984037i $$0.556951\pi$$
$$270$$ −5.57537e24 −0.164799
$$271$$ 1.66790e25 0.474234 0.237117 0.971481i $$-0.423798\pi$$
0.237117 + 0.971481i $$0.423798\pi$$
$$272$$ 1.60486e25 0.439000
$$273$$ 3.29181e24 0.0866416
$$274$$ −4.23698e25 −1.07319
$$275$$ −2.33331e25 −0.568826
$$276$$ 1.37685e25 0.323104
$$277$$ −4.99177e25 −1.12776 −0.563880 0.825857i $$-0.690692\pi$$
−0.563880 + 0.825857i $$0.690692\pi$$
$$278$$ 1.91206e25 0.415940
$$279$$ −3.04174e25 −0.637203
$$280$$ −4.71680e24 −0.0951673
$$281$$ 2.39858e25 0.466162 0.233081 0.972457i $$-0.425119\pi$$
0.233081 + 0.972457i $$0.425119\pi$$
$$282$$ −2.05005e25 −0.383839
$$283$$ −2.68871e25 −0.485050 −0.242525 0.970145i $$-0.577976\pi$$
−0.242525 + 0.970145i $$0.577976\pi$$
$$284$$ −1.35590e25 −0.235715
$$285$$ 5.57391e24 0.0933875
$$286$$ 3.60411e25 0.582043
$$287$$ 1.43884e25 0.224003
$$288$$ −3.92577e24 −0.0589256
$$289$$ 1.43956e26 2.08354
$$290$$ −5.94148e25 −0.829307
$$291$$ 8.15521e25 1.09789
$$292$$ −4.64070e25 −0.602648
$$293$$ 1.29122e25 0.161767 0.0808835 0.996724i $$-0.474226\pi$$
0.0808835 + 0.996724i $$0.474226\pi$$
$$294$$ 3.21046e25 0.388079
$$295$$ 1.38384e26 1.61419
$$296$$ −4.02341e25 −0.452930
$$297$$ −2.15936e25 −0.234629
$$298$$ 4.59138e25 0.481585
$$299$$ 7.46252e25 0.755680
$$300$$ 1.37752e25 0.134687
$$301$$ 2.18304e25 0.206117
$$302$$ −5.40465e25 −0.492827
$$303$$ −8.82304e25 −0.777089
$$304$$ 3.92474e24 0.0333917
$$305$$ −1.26627e26 −1.04082
$$306$$ −5.21151e25 −0.413893
$$307$$ −1.45907e26 −1.11976 −0.559878 0.828575i $$-0.689152\pi$$
−0.559878 + 0.828575i $$0.689152\pi$$
$$308$$ −1.82683e25 −0.135493
$$309$$ −1.02049e26 −0.731552
$$310$$ 2.36229e26 1.63695
$$311$$ −1.00016e25 −0.0670015 −0.0335008 0.999439i $$-0.510666\pi$$
−0.0335008 + 0.999439i $$0.510666\pi$$
$$312$$ −2.12776e25 −0.137816
$$313$$ 2.08348e25 0.130489 0.0652445 0.997869i $$-0.479217\pi$$
0.0652445 + 0.997869i $$0.479217\pi$$
$$314$$ −4.88089e25 −0.295623
$$315$$ 1.53170e25 0.0897246
$$316$$ −1.56118e25 −0.0884583
$$317$$ 1.30949e26 0.717762 0.358881 0.933383i $$-0.383158\pi$$
0.358881 + 0.933383i $$0.383158\pi$$
$$318$$ 9.50223e25 0.503896
$$319$$ −2.30116e26 −1.18071
$$320$$ 3.04885e25 0.151377
$$321$$ −1.89670e26 −0.911373
$$322$$ −3.78256e25 −0.175913
$$323$$ 5.21014e25 0.234543
$$324$$ 1.27482e25 0.0555556
$$325$$ 7.46614e25 0.315007
$$326$$ −4.79829e25 −0.196021
$$327$$ −3.11179e25 −0.123100
$$328$$ −9.30040e25 −0.356309
$$329$$ 5.63201e25 0.208981
$$330$$ 1.67701e26 0.602754
$$331$$ 5.53748e24 0.0192805 0.00964026 0.999954i $$-0.496931\pi$$
0.00964026 + 0.999954i $$0.496931\pi$$
$$332$$ 3.88865e25 0.131174
$$333$$ 1.30653e26 0.427026
$$334$$ −4.31177e25 −0.136558
$$335$$ −4.09370e26 −1.25645
$$336$$ 1.07851e25 0.0320820
$$337$$ −2.35227e26 −0.678223 −0.339112 0.940746i $$-0.610126\pi$$
−0.339112 + 0.940746i $$0.610126\pi$$
$$338$$ 1.37670e26 0.384780
$$339$$ 1.56702e25 0.0424598
$$340$$ 4.04738e26 1.06328
$$341$$ 9.14923e26 2.33058
$$342$$ −1.27449e25 −0.0314820
$$343$$ −1.80983e26 −0.433560
$$344$$ −1.41108e26 −0.327859
$$345$$ 3.47235e26 0.782570
$$346$$ −4.61915e26 −1.00986
$$347$$ −1.60491e26 −0.340401 −0.170200 0.985409i $$-0.554441\pi$$
−0.170200 + 0.985409i $$0.554441\pi$$
$$348$$ 1.35853e26 0.279569
$$349$$ 3.71912e26 0.742631 0.371315 0.928507i $$-0.378907\pi$$
0.371315 + 0.928507i $$0.378907\pi$$
$$350$$ −3.78439e25 −0.0733300
$$351$$ 6.90953e25 0.129934
$$352$$ 1.18083e26 0.215521
$$353$$ 7.03252e26 1.24588 0.622941 0.782269i $$-0.285938\pi$$
0.622941 + 0.782269i $$0.285938\pi$$
$$354$$ −3.16419e26 −0.544161
$$355$$ −3.41952e26 −0.570910
$$356$$ −1.10700e26 −0.179442
$$357$$ 1.43173e26 0.225344
$$358$$ −2.11553e26 −0.323331
$$359$$ −5.01731e26 −0.744696 −0.372348 0.928093i $$-0.621447\pi$$
−0.372348 + 0.928093i $$0.621447\pi$$
$$360$$ −9.90059e25 −0.142720
$$361$$ −7.01468e26 −0.982160
$$362$$ −7.34494e26 −0.998961
$$363$$ 2.12535e26 0.280809
$$364$$ 5.84550e25 0.0750338
$$365$$ −1.17036e27 −1.45964
$$366$$ 2.89535e26 0.350873
$$367$$ 5.69112e26 0.670200 0.335100 0.942183i $$-0.391230\pi$$
0.335100 + 0.942183i $$0.391230\pi$$
$$368$$ 2.44498e26 0.279816
$$369$$ 3.02014e26 0.335931
$$370$$ −1.01468e27 −1.09701
$$371$$ −2.61050e26 −0.274346
$$372$$ −5.40144e26 −0.551834
$$373$$ 2.59366e26 0.257615 0.128807 0.991670i $$-0.458885\pi$$
0.128807 + 0.991670i $$0.458885\pi$$
$$374$$ 1.56757e27 1.51382
$$375$$ −3.97189e26 −0.372966
$$376$$ −3.64043e26 −0.332414
$$377$$ 7.36324e26 0.653860
$$378$$ −3.50226e25 −0.0302472
$$379$$ 1.10384e27 0.927241 0.463621 0.886034i $$-0.346550\pi$$
0.463621 + 0.886034i $$0.346550\pi$$
$$380$$ 9.89799e25 0.0808759
$$381$$ −1.15171e27 −0.915444
$$382$$ 4.58943e26 0.354890
$$383$$ −6.37501e26 −0.479616 −0.239808 0.970820i $$-0.577084\pi$$
−0.239808 + 0.970820i $$0.577084\pi$$
$$384$$ −6.97128e25 −0.0510310
$$385$$ −4.60718e26 −0.328169
$$386$$ 8.06978e26 0.559364
$$387$$ 4.58222e26 0.309109
$$388$$ 1.44818e27 0.950801
$$389$$ 3.61461e25 0.0230989 0.0115494 0.999933i $$-0.496324\pi$$
0.0115494 + 0.999933i $$0.496324\pi$$
$$390$$ −5.36611e26 −0.333796
$$391$$ 3.24573e27 1.96543
$$392$$ 5.70105e26 0.336086
$$393$$ −5.86196e26 −0.336450
$$394$$ 5.50997e26 0.307921
$$395$$ −3.93722e26 −0.214250
$$396$$ −3.83453e26 −0.203195
$$397$$ 6.60817e26 0.341021 0.170511 0.985356i $$-0.445458\pi$$
0.170511 + 0.985356i $$0.445458\pi$$
$$398$$ 1.87232e27 0.941037
$$399$$ 3.50134e25 0.0171403
$$400$$ 2.44616e26 0.116642
$$401$$ −3.16192e27 −1.46871 −0.734354 0.678767i $$-0.762515\pi$$
−0.734354 + 0.678767i $$0.762515\pi$$
$$402$$ 9.36036e26 0.423564
$$403$$ −2.92757e27 −1.29064
$$404$$ −1.56677e27 −0.672979
$$405$$ 3.21504e26 0.134558
$$406$$ −3.73224e26 −0.152211
$$407$$ −3.92990e27 −1.56185
$$408$$ −9.25445e26 −0.358442
$$409$$ 1.26849e27 0.478843 0.239422 0.970916i $$-0.423042\pi$$
0.239422 + 0.970916i $$0.423042\pi$$
$$410$$ −2.34551e27 −0.862993
$$411$$ 2.44326e27 0.876254
$$412$$ −1.81216e27 −0.633542
$$413$$ 8.69282e26 0.296268
$$414$$ −7.93962e26 −0.263813
$$415$$ 9.80698e26 0.317710
$$416$$ −3.77842e26 −0.119352
$$417$$ −1.10259e27 −0.339614
$$418$$ 3.83353e26 0.115146
$$419$$ 4.00666e27 1.17364 0.586821 0.809717i $$-0.300379\pi$$
0.586821 + 0.809717i $$0.300379\pi$$
$$420$$ 2.71994e26 0.0777038
$$421$$ −5.33195e27 −1.48568 −0.742838 0.669471i $$-0.766521\pi$$
−0.742838 + 0.669471i $$0.766521\pi$$
$$422$$ 2.22443e27 0.604558
$$423$$ 1.18216e27 0.313403
$$424$$ 1.68738e27 0.436387
$$425$$ 3.24731e27 0.819294
$$426$$ 7.81882e26 0.192460
$$427$$ −7.95427e26 −0.191033
$$428$$ −3.36811e27 −0.789272
$$429$$ −2.07831e27 −0.475236
$$430$$ −3.55867e27 −0.794088
$$431$$ 2.82207e27 0.614548 0.307274 0.951621i $$-0.400583\pi$$
0.307274 + 0.951621i $$0.400583\pi$$
$$432$$ 2.26380e26 0.0481125
$$433$$ −1.58069e26 −0.0327886 −0.0163943 0.999866i $$-0.505219\pi$$
−0.0163943 + 0.999866i $$0.505219\pi$$
$$434$$ 1.48391e27 0.300446
$$435$$ 3.42616e27 0.677127
$$436$$ −5.52584e26 −0.106608
$$437$$ 7.93754e26 0.149496
$$438$$ 2.67606e27 0.492060
$$439$$ 8.74620e27 1.57015 0.785077 0.619398i $$-0.212623\pi$$
0.785077 + 0.619398i $$0.212623\pi$$
$$440$$ 2.97799e27 0.522000
$$441$$ −1.85131e27 −0.316865
$$442$$ −5.01590e27 −0.838330
$$443$$ 1.81520e27 0.296269 0.148135 0.988967i $$-0.452673\pi$$
0.148135 + 0.988967i $$0.452673\pi$$
$$444$$ 2.32010e27 0.369816
$$445$$ −2.79180e27 −0.434615
$$446$$ −8.60967e27 −1.30909
$$447$$ −2.64762e27 −0.393212
$$448$$ 1.91519e26 0.0277838
$$449$$ 1.06970e28 1.51592 0.757958 0.652304i $$-0.226197\pi$$
0.757958 + 0.652304i $$0.226197\pi$$
$$450$$ −7.94347e26 −0.109971
$$451$$ −9.08426e27 −1.22867
$$452$$ 2.78267e26 0.0367713
$$453$$ 3.11659e27 0.402392
$$454$$ 6.24334e27 0.787644
$$455$$ 1.47421e27 0.181735
$$456$$ −2.26320e26 −0.0272642
$$457$$ −1.59372e28 −1.87626 −0.938129 0.346285i $$-0.887443\pi$$
−0.938129 + 0.346285i $$0.887443\pi$$
$$458$$ 6.69142e26 0.0769894
$$459$$ 3.00522e27 0.337942
$$460$$ 6.16610e27 0.677725
$$461$$ 5.33663e27 0.573334 0.286667 0.958030i $$-0.407453\pi$$
0.286667 + 0.958030i $$0.407453\pi$$
$$462$$ 1.05344e27 0.110629
$$463$$ 3.38648e27 0.347655 0.173828 0.984776i $$-0.444386\pi$$
0.173828 + 0.984776i $$0.444386\pi$$
$$464$$ 2.41245e27 0.242114
$$465$$ −1.36222e28 −1.33656
$$466$$ 1.11708e28 1.07160
$$467$$ −1.57577e28 −1.47797 −0.738986 0.673721i $$-0.764695\pi$$
−0.738986 + 0.673721i $$0.764695\pi$$
$$468$$ 1.22698e27 0.112526
$$469$$ −2.57153e27 −0.230609
$$470$$ −9.18097e27 −0.805121
$$471$$ 2.81457e27 0.241375
$$472$$ −5.61887e27 −0.471258
$$473$$ −1.37828e28 −1.13057
$$474$$ 9.00255e26 0.0722259
$$475$$ 7.94139e26 0.0623180
$$476$$ 2.54243e27 0.195153
$$477$$ −5.47946e27 −0.411429
$$478$$ −6.10831e24 −0.000448671 0
$$479$$ −2.12973e28 −1.53039 −0.765193 0.643801i $$-0.777356\pi$$
−0.765193 + 0.643801i $$0.777356\pi$$
$$480$$ −1.75812e27 −0.123599
$$481$$ 1.25749e28 0.864931
$$482$$ −8.37559e27 −0.563666
$$483$$ 2.18121e27 0.143633
$$484$$ 3.77414e27 0.243188
$$485$$ 3.65224e28 2.30288
$$486$$ −7.35128e26 −0.0453609
$$487$$ 1.68508e28 1.01757 0.508786 0.860893i $$-0.330094\pi$$
0.508786 + 0.860893i $$0.330094\pi$$
$$488$$ 5.14149e27 0.303865
$$489$$ 2.76694e27 0.160050
$$490$$ 1.43777e28 0.814014
$$491$$ −1.57697e28 −0.873910 −0.436955 0.899483i $$-0.643943\pi$$
−0.436955 + 0.899483i $$0.643943\pi$$
$$492$$ 5.36308e27 0.290925
$$493$$ 3.20255e28 1.70061
$$494$$ −1.22665e27 −0.0637659
$$495$$ −9.67050e27 −0.492146
$$496$$ −9.59173e27 −0.477903
$$497$$ −2.14803e27 −0.104785
$$498$$ −2.24239e27 −0.107103
$$499$$ 2.57185e28 1.20279 0.601395 0.798952i $$-0.294612\pi$$
0.601395 + 0.798952i $$0.294612\pi$$
$$500$$ −7.05318e27 −0.322998
$$501$$ 2.48639e27 0.111499
$$502$$ −1.40649e28 −0.617657
$$503$$ 2.83510e28 1.21928 0.609642 0.792677i $$-0.291313\pi$$
0.609642 + 0.792677i $$0.291313\pi$$
$$504$$ −6.21922e26 −0.0261948
$$505$$ −3.95132e28 −1.62998
$$506$$ 2.38815e28 0.964899
$$507$$ −7.93872e27 −0.314172
$$508$$ −2.04518e28 −0.792797
$$509$$ 2.41622e28 0.917485 0.458743 0.888569i $$-0.348300\pi$$
0.458743 + 0.888569i $$0.348300\pi$$
$$510$$ −2.33392e28 −0.868160
$$511$$ −7.35182e27 −0.267902
$$512$$ −1.23794e27 −0.0441942
$$513$$ 7.34935e26 0.0257049
$$514$$ −4.63879e27 −0.158961
$$515$$ −4.57017e28 −1.53446
$$516$$ 8.13699e27 0.267696
$$517$$ −3.55582e28 −1.14628
$$518$$ −6.37390e27 −0.201346
$$519$$ 2.66363e28 0.824551
$$520$$ −9.52899e27 −0.289076
$$521$$ 4.35634e28 1.29517 0.647584 0.761994i $$-0.275780\pi$$
0.647584 + 0.761994i $$0.275780\pi$$
$$522$$ −7.83400e27 −0.228267
$$523$$ 1.92361e28 0.549351 0.274676 0.961537i $$-0.411430\pi$$
0.274676 + 0.961537i $$0.411430\pi$$
$$524$$ −1.04095e28 −0.291375
$$525$$ 2.18227e27 0.0598737
$$526$$ 5.35615e27 0.144046
$$527$$ −1.27331e29 −3.35679
$$528$$ −6.80926e27 −0.175972
$$529$$ 9.97650e27 0.252751
$$530$$ 4.25549e28 1.05695
$$531$$ 1.82463e28 0.444306
$$532$$ 6.21759e26 0.0148440
$$533$$ 2.90678e28 0.680420
$$534$$ 6.38354e27 0.146514
$$535$$ −8.49420e28 −1.91165
$$536$$ 1.66219e28 0.366817
$$537$$ 1.21992e28 0.263998
$$538$$ 1.18593e28 0.251678
$$539$$ 5.56855e28 1.15894
$$540$$ 5.70918e27 0.116530
$$541$$ 7.52925e28 1.50723 0.753617 0.657314i $$-0.228307\pi$$
0.753617 + 0.657314i $$0.228307\pi$$
$$542$$ −1.70793e28 −0.335334
$$543$$ 4.23546e28 0.815648
$$544$$ −1.64338e28 −0.310420
$$545$$ −1.39359e28 −0.258209
$$546$$ −3.37081e27 −0.0612649
$$547$$ −6.21957e28 −1.10890 −0.554451 0.832216i $$-0.687072\pi$$
−0.554451 + 0.832216i $$0.687072\pi$$
$$548$$ 4.33867e28 0.758858
$$549$$ −1.66961e28 −0.286487
$$550$$ 2.38931e28 0.402221
$$551$$ 7.83194e27 0.129353
$$552$$ −1.40990e28 −0.228469
$$553$$ −2.47323e27 −0.0393233
$$554$$ 5.11157e28 0.797447
$$555$$ 5.85117e28 0.895708
$$556$$ −1.95795e28 −0.294114
$$557$$ 5.50617e27 0.0811652 0.0405826 0.999176i $$-0.487079\pi$$
0.0405826 + 0.999176i $$0.487079\pi$$
$$558$$ 3.11474e28 0.450571
$$559$$ 4.41024e28 0.626092
$$560$$ 4.83000e27 0.0672935
$$561$$ −9.03937e28 −1.23603
$$562$$ −2.45614e28 −0.329627
$$563$$ −1.21599e29 −1.60174 −0.800868 0.598840i $$-0.795628\pi$$
−0.800868 + 0.598840i $$0.795628\pi$$
$$564$$ 2.09925e28 0.271415
$$565$$ 7.01776e27 0.0890614
$$566$$ 2.75324e28 0.342982
$$567$$ 2.01958e27 0.0246967
$$568$$ 1.38845e28 0.166675
$$569$$ −3.49889e28 −0.412337 −0.206168 0.978517i $$-0.566099\pi$$
−0.206168 + 0.978517i $$0.566099\pi$$
$$570$$ −5.70768e27 −0.0660349
$$571$$ 9.42449e28 1.07048 0.535240 0.844700i $$-0.320221\pi$$
0.535240 + 0.844700i $$0.320221\pi$$
$$572$$ −3.69061e28 −0.411566
$$573$$ −2.64650e28 −0.289766
$$574$$ −1.47337e28 −0.158394
$$575$$ 4.94721e28 0.522213
$$576$$ 4.01999e27 0.0416667
$$577$$ −1.55441e29 −1.58205 −0.791025 0.611784i $$-0.790452\pi$$
−0.791025 + 0.611784i $$0.790452\pi$$
$$578$$ −1.47410e29 −1.47328
$$579$$ −4.65344e28 −0.456719
$$580$$ 6.08407e28 0.586409
$$581$$ 6.16042e27 0.0583124
$$582$$ −8.35094e28 −0.776326
$$583$$ 1.64816e29 1.50481
$$584$$ 4.75208e28 0.426137
$$585$$ 3.09437e28 0.272543
$$586$$ −1.32221e28 −0.114387
$$587$$ 1.17789e29 1.00093 0.500465 0.865757i $$-0.333162\pi$$
0.500465 + 0.865757i $$0.333162\pi$$
$$588$$ −3.28751e28 −0.274413
$$589$$ −3.11392e28 −0.255327
$$590$$ −1.41705e29 −1.14140
$$591$$ −3.17733e28 −0.251416
$$592$$ 4.11997e28 0.320270
$$593$$ 1.16653e29 0.890884 0.445442 0.895311i $$-0.353047\pi$$
0.445442 + 0.895311i $$0.353047\pi$$
$$594$$ 2.21119e28 0.165908
$$595$$ 6.41188e28 0.472669
$$596$$ −4.70158e28 −0.340532
$$597$$ −1.07967e29 −0.768353
$$598$$ −7.64162e28 −0.534346
$$599$$ −5.27398e28 −0.362374 −0.181187 0.983449i $$-0.557994\pi$$
−0.181187 + 0.983449i $$0.557994\pi$$
$$600$$ −1.41058e28 −0.0952378
$$601$$ −5.00756e28 −0.332234 −0.166117 0.986106i $$-0.553123\pi$$
−0.166117 + 0.986106i $$0.553123\pi$$
$$602$$ −2.23544e28 −0.145747
$$603$$ −5.39766e28 −0.345839
$$604$$ 5.53436e28 0.348482
$$605$$ 9.51820e28 0.589011
$$606$$ 9.03480e28 0.549485
$$607$$ −6.44426e28 −0.385205 −0.192603 0.981277i $$-0.561693\pi$$
−0.192603 + 0.981277i $$0.561693\pi$$
$$608$$ −4.01893e27 −0.0236115
$$609$$ 2.15220e28 0.124280
$$610$$ 1.29666e29 0.735973
$$611$$ 1.13779e29 0.634791
$$612$$ 5.33658e28 0.292667
$$613$$ 1.46854e29 0.791680 0.395840 0.918319i $$-0.370453\pi$$
0.395840 + 0.918319i $$0.370453\pi$$
$$614$$ 1.49409e29 0.791788
$$615$$ 1.35254e29 0.704631
$$616$$ 1.87068e28 0.0958078
$$617$$ −2.87513e29 −1.44765 −0.723823 0.689985i $$-0.757617\pi$$
−0.723823 + 0.689985i $$0.757617\pi$$
$$618$$ 1.04498e29 0.517285
$$619$$ 3.83570e29 1.86678 0.933388 0.358869i $$-0.116837\pi$$
0.933388 + 0.358869i $$0.116837\pi$$
$$620$$ −2.41899e29 −1.15750
$$621$$ 4.57839e28 0.215402
$$622$$ 1.02416e28 0.0473772
$$623$$ −1.75372e28 −0.0797692
$$624$$ 2.17883e28 0.0974507
$$625$$ −2.83963e29 −1.24888
$$626$$ −2.13349e28 −0.0922697
$$627$$ −2.21060e28 −0.0940160
$$628$$ 4.99803e28 0.209037
$$629$$ 5.46931e29 2.24957
$$630$$ −1.56846e28 −0.0634449
$$631$$ −4.12646e29 −1.64161 −0.820803 0.571211i $$-0.806474\pi$$
−0.820803 + 0.571211i $$0.806474\pi$$
$$632$$ 1.59865e28 0.0625495
$$633$$ −1.28272e29 −0.493620
$$634$$ −1.34092e29 −0.507535
$$635$$ −5.15784e29 −1.92019
$$636$$ −9.73028e28 −0.356308
$$637$$ −1.78183e29 −0.641803
$$638$$ 2.35638e29 0.834889
$$639$$ −4.50873e28 −0.157143
$$640$$ −3.12202e28 −0.107040
$$641$$ −4.45897e29 −1.50392 −0.751960 0.659208i $$-0.770891\pi$$
−0.751960 + 0.659208i $$0.770891\pi$$
$$642$$ 1.94222e29 0.644438
$$643$$ −1.19395e29 −0.389736 −0.194868 0.980829i $$-0.562428\pi$$
−0.194868 + 0.980829i $$0.562428\pi$$
$$644$$ 3.87334e28 0.124390
$$645$$ 2.05211e29 0.648370
$$646$$ −5.33518e28 −0.165847
$$647$$ 5.06908e29 1.55037 0.775183 0.631737i $$-0.217658\pi$$
0.775183 + 0.631737i $$0.217658\pi$$
$$648$$ −1.30542e28 −0.0392837
$$649$$ −5.48829e29 −1.62505
$$650$$ −7.64532e28 −0.222744
$$651$$ −8.55698e28 −0.245313
$$652$$ 4.91345e28 0.138607
$$653$$ 2.19335e29 0.608862 0.304431 0.952534i $$-0.401534\pi$$
0.304431 + 0.952534i $$0.401534\pi$$
$$654$$ 3.18648e28 0.0870451
$$655$$ −2.62523e29 −0.705721
$$656$$ 9.52361e28 0.251948
$$657$$ −1.54315e29 −0.401765
$$658$$ −5.76718e28 −0.147772
$$659$$ 2.49016e29 0.627957 0.313979 0.949430i $$-0.398338\pi$$
0.313979 + 0.949430i $$0.398338\pi$$
$$660$$ −1.71726e29 −0.426211
$$661$$ 3.02171e29 0.738137 0.369069 0.929402i $$-0.379677\pi$$
0.369069 + 0.929402i $$0.379677\pi$$
$$662$$ −5.67038e27 −0.0136334
$$663$$ 2.89242e29 0.684494
$$664$$ −3.98198e28 −0.0927543
$$665$$ 1.56804e28 0.0359527
$$666$$ −1.33789e29 −0.301953
$$667$$ 4.87903e29 1.08396
$$668$$ 4.41526e28 0.0965613
$$669$$ 4.96477e29 1.06887
$$670$$ 4.19195e29 0.888446
$$671$$ 5.02200e29 1.04783
$$672$$ −1.10439e28 −0.0226854
$$673$$ 1.15417e29 0.233406 0.116703 0.993167i $$-0.462767\pi$$
0.116703 + 0.993167i $$0.462767\pi$$
$$674$$ 2.40872e29 0.479576
$$675$$ 4.58061e28 0.0897911
$$676$$ −1.40974e29 −0.272081
$$677$$ −2.04966e29 −0.389494 −0.194747 0.980853i $$-0.562389\pi$$
−0.194747 + 0.980853i $$0.562389\pi$$
$$678$$ −1.60463e28 −0.0300236
$$679$$ 2.29421e29 0.422670
$$680$$ −4.14452e29 −0.751849
$$681$$ −3.60022e29 −0.643109
$$682$$ −9.36881e29 −1.64797
$$683$$ −7.56950e29 −1.31114 −0.655571 0.755134i $$-0.727572\pi$$
−0.655571 + 0.755134i $$0.727572\pi$$
$$684$$ 1.30508e28 0.0222611
$$685$$ 1.09419e30 1.83798
$$686$$ 1.85326e29 0.306573
$$687$$ −3.85861e28 −0.0628616
$$688$$ 1.44494e29 0.231832
$$689$$ −5.27380e29 −0.833339
$$690$$ −3.55569e29 −0.553360
$$691$$ −4.98998e29 −0.764855 −0.382427 0.923986i $$-0.624912\pi$$
−0.382427 + 0.923986i $$0.624912\pi$$
$$692$$ 4.73001e29 0.714082
$$693$$ −6.07468e28 −0.0903285
$$694$$ 1.64342e29 0.240700
$$695$$ −4.93784e29 −0.712356
$$696$$ −1.39114e29 −0.197685
$$697$$ 1.26427e30 1.76968
$$698$$ −3.80838e29 −0.525119
$$699$$ −6.44165e29 −0.874957
$$700$$ 3.87522e28 0.0518522
$$701$$ 4.27981e29 0.564137 0.282069 0.959394i $$-0.408979\pi$$
0.282069 + 0.959394i $$0.408979\pi$$
$$702$$ −7.07536e28 −0.0918774
$$703$$ 1.33753e29 0.171110
$$704$$ −1.20917e29 −0.152396
$$705$$ 5.29421e29 0.657378
$$706$$ −7.20130e29 −0.880971
$$707$$ −2.48209e29 −0.299167
$$708$$ 3.24013e29 0.384780
$$709$$ 3.31006e29 0.387303 0.193651 0.981070i $$-0.437967\pi$$
0.193651 + 0.981070i $$0.437967\pi$$
$$710$$ 3.50159e29 0.403694
$$711$$ −5.19133e28 −0.0589722
$$712$$ 1.13357e29 0.126885
$$713$$ −1.93987e30 −2.13960
$$714$$ −1.46609e29 −0.159342
$$715$$ −9.30753e29 −0.996830
$$716$$ 2.16630e29 0.228629
$$717$$ 3.52236e26 0.000366338 0
$$718$$ 5.13772e29 0.526580
$$719$$ −1.36160e30 −1.37530 −0.687648 0.726044i $$-0.741357\pi$$
−0.687648 + 0.726044i $$0.741357\pi$$
$$720$$ 1.01382e29 0.100918
$$721$$ −2.87083e29 −0.281636
$$722$$ 7.18303e29 0.694492
$$723$$ 4.82979e29 0.460231
$$724$$ 7.52122e29 0.706372
$$725$$ 4.88139e29 0.451850
$$726$$ −2.17636e29 −0.198562
$$727$$ −5.58942e29 −0.502638 −0.251319 0.967904i $$-0.580864\pi$$
−0.251319 + 0.967904i $$0.580864\pi$$
$$728$$ −5.98579e28 −0.0530569
$$729$$ 4.23912e28 0.0370370
$$730$$ 1.19845e30 1.03212
$$731$$ 1.91818e30 1.62838
$$732$$ −2.96484e29 −0.248105
$$733$$ 1.20616e30 0.994975 0.497488 0.867471i $$-0.334256\pi$$
0.497488 + 0.867471i $$0.334256\pi$$
$$734$$ −5.82771e29 −0.473903
$$735$$ −8.29093e29 −0.664640
$$736$$ −2.50366e29 −0.197860
$$737$$ 1.62356e30 1.26491
$$738$$ −3.09262e29 −0.237539
$$739$$ −1.99646e30 −1.51180 −0.755901 0.654686i $$-0.772801\pi$$
−0.755901 + 0.654686i $$0.772801\pi$$
$$740$$ 1.03903e30 0.775706
$$741$$ 7.07350e28 0.0520647
$$742$$ 2.67315e29 0.193992
$$743$$ 1.25058e30 0.894803 0.447402 0.894333i $$-0.352350\pi$$
0.447402 + 0.894333i $$0.352350\pi$$
$$744$$ 5.53107e29 0.390206
$$745$$ −1.18571e30 −0.824782
$$746$$ −2.65591e29 −0.182161
$$747$$ 1.29308e29 0.0874496
$$748$$ −1.60519e30 −1.07043
$$749$$ −5.33577e29 −0.350864
$$750$$ 4.06722e29 0.263727
$$751$$ 1.17402e30 0.750685 0.375343 0.926886i $$-0.377525\pi$$
0.375343 + 0.926886i $$0.377525\pi$$
$$752$$ 3.72780e29 0.235053
$$753$$ 8.11053e29 0.504315
$$754$$ −7.53996e29 −0.462349
$$755$$ 1.39574e30 0.844036
$$756$$ 3.58632e28 0.0213880
$$757$$ 5.86936e29 0.345210 0.172605 0.984991i $$-0.444782\pi$$
0.172605 + 0.984991i $$0.444782\pi$$
$$758$$ −1.13033e30 −0.655658
$$759$$ −1.37713e30 −0.787837
$$760$$ −1.01355e29 −0.0571879
$$761$$ −1.13068e30 −0.629217 −0.314609 0.949221i $$-0.601873\pi$$
−0.314609 + 0.949221i $$0.601873\pi$$
$$762$$ 1.17935e30 0.647316
$$763$$ −8.75405e28 −0.0473916
$$764$$ −4.69958e29 −0.250945
$$765$$ 1.34586e30 0.708850
$$766$$ 6.52801e29 0.339140
$$767$$ 1.75614e30 0.899930
$$768$$ 7.13859e28 0.0360844
$$769$$ −3.28732e30 −1.63914 −0.819569 0.572981i $$-0.805787\pi$$
−0.819569 + 0.572981i $$0.805787\pi$$
$$770$$ 4.71775e29 0.232050
$$771$$ 2.67496e29 0.129791
$$772$$ −8.26345e29 −0.395530
$$773$$ −2.96976e30 −1.40229 −0.701144 0.713020i $$-0.747327\pi$$
−0.701144 + 0.713020i $$0.747327\pi$$
$$774$$ −4.69220e29 −0.218573
$$775$$ −1.94081e30 −0.891897
$$776$$ −1.48294e30 −0.672318
$$777$$ 3.67551e29 0.164398
$$778$$ −3.70136e28 −0.0163334
$$779$$ 3.09181e29 0.134608
$$780$$ 5.49489e29 0.236029
$$781$$ 1.35618e30 0.574752
$$782$$ −3.32363e30 −1.38977
$$783$$ 4.51748e29 0.186379
$$784$$ −5.83787e29 −0.237649
$$785$$ 1.26048e30 0.506295
$$786$$ 6.00265e29 0.237906
$$787$$ 7.59025e29 0.296839 0.148420 0.988924i $$-0.452581\pi$$
0.148420 + 0.988924i $$0.452581\pi$$
$$788$$ −5.64221e29 −0.217733
$$789$$ −3.08862e29 −0.117613
$$790$$ 4.03171e29 0.151497
$$791$$ 4.40832e28 0.0163463
$$792$$ 3.92656e29 0.143681
$$793$$ −1.60694e30 −0.580272
$$794$$ −6.76677e29 −0.241138
$$795$$ −2.45393e30 −0.862992
$$796$$ −1.91725e30 −0.665414
$$797$$ 4.13590e30 1.41664 0.708318 0.705894i $$-0.249454\pi$$
0.708318 + 0.705894i $$0.249454\pi$$
$$798$$ −3.58538e28 −0.0121200
$$799$$ 4.94869e30 1.65101
$$800$$ −2.50487e29 −0.0824784
$$801$$ −3.68107e29 −0.119628
$$802$$ 3.23781e30 1.03853
$$803$$ 4.64164e30 1.46946
$$804$$ −9.58501e29 −0.299505
$$805$$ 9.76837e29 0.301277
$$806$$ 2.99784e30 0.912620
$$807$$ −6.83868e29 −0.205495
$$808$$ 1.60437e30 0.475868
$$809$$ −6.22694e30 −1.82312 −0.911560 0.411166i $$-0.865122\pi$$
−0.911560 + 0.411166i $$0.865122\pi$$
$$810$$ −3.29220e29 −0.0951467
$$811$$ 1.69279e30 0.482930 0.241465 0.970409i $$-0.422372\pi$$
0.241465 + 0.970409i $$0.422372\pi$$
$$812$$ 3.82181e29 0.107629
$$813$$ 9.84878e29 0.273799
$$814$$ 4.02422e30 1.10440
$$815$$ 1.23915e30 0.335713
$$816$$ 9.47656e29 0.253457
$$817$$ 4.69097e29 0.123860
$$818$$ −1.29894e30 −0.338593
$$819$$ 1.94378e29 0.0500226
$$820$$ 2.40181e30 0.610228
$$821$$ −8.06422e29 −0.202283 −0.101141 0.994872i $$-0.532249\pi$$
−0.101141 + 0.994872i $$0.532249\pi$$
$$822$$ −2.50189e30 −0.619605
$$823$$ −2.36474e30 −0.578210 −0.289105 0.957297i $$-0.593358\pi$$
−0.289105 + 0.957297i $$0.593358\pi$$
$$824$$ 1.85565e30 0.447982
$$825$$ −1.37780e30 −0.328412
$$826$$ −8.90145e29 −0.209493
$$827$$ −1.05624e30 −0.245445 −0.122722 0.992441i $$-0.539162\pi$$
−0.122722 + 0.992441i $$0.539162\pi$$
$$828$$ 8.13017e29 0.186544
$$829$$ −3.64602e30 −0.826032 −0.413016 0.910724i $$-0.635525\pi$$
−0.413016 + 0.910724i $$0.635525\pi$$
$$830$$ −1.00423e30 −0.224655
$$831$$ −2.94759e30 −0.651113
$$832$$ 3.86910e29 0.0843948
$$833$$ −7.74984e30 −1.66925
$$834$$ 1.12905e30 0.240143
$$835$$ 1.11351e30 0.233875
$$836$$ −3.92553e29 −0.0814203
$$837$$ −1.79612e30 −0.367890
$$838$$ −4.10282e30 −0.829890
$$839$$ −6.80400e29 −0.135914 −0.0679569 0.997688i $$-0.521648\pi$$
−0.0679569 + 0.997688i $$0.521648\pi$$
$$840$$ −2.78522e29 −0.0549449
$$841$$ −3.18725e29 −0.0620953
$$842$$ 5.45992e30 1.05053
$$843$$ 1.41634e30 0.269139
$$844$$ −2.27782e30 −0.427487
$$845$$ −3.55528e30 −0.658990
$$846$$ −1.21054e30 −0.221610
$$847$$ 5.97902e29 0.108107
$$848$$ −1.72788e30 −0.308572
$$849$$ −1.58766e30 −0.280044
$$850$$ −3.32524e30 −0.579328
$$851$$ 8.33237e30 1.43387
$$852$$ −8.00648e29 −0.136090
$$853$$ 9.57477e30 1.60755 0.803773 0.594936i $$-0.202823\pi$$
0.803773 + 0.594936i $$0.202823\pi$$
$$854$$ 8.14517e29 0.135080
$$855$$ 3.29134e29 0.0539173
$$856$$ 3.44894e30 0.558100
$$857$$ −4.04955e30 −0.647304 −0.323652 0.946176i $$-0.604911\pi$$
−0.323652 + 0.946176i $$0.604911\pi$$
$$858$$ 2.12819e30 0.336043
$$859$$ −9.44318e30 −1.47296 −0.736479 0.676461i $$-0.763513\pi$$
−0.736479 + 0.676461i $$0.763513\pi$$
$$860$$ 3.64408e30 0.561505
$$861$$ 8.49622e29 0.129328
$$862$$ −2.88980e30 −0.434551
$$863$$ −5.03037e30 −0.747285 −0.373642 0.927573i $$-0.621891\pi$$
−0.373642 + 0.927573i $$0.621891\pi$$
$$864$$ −2.31813e29 −0.0340207
$$865$$ 1.19288e31 1.72953
$$866$$ 1.61863e29 0.0231851
$$867$$ 8.50043e30 1.20293
$$868$$ −1.51953e30 −0.212447
$$869$$ 1.56150e30 0.215692
$$870$$ −3.50838e30 −0.478801
$$871$$ −5.19506e30 −0.700487
$$872$$ 5.65846e29 0.0753832
$$873$$ 4.81557e30 0.633867
$$874$$ −8.12804e29 −0.105710
$$875$$ −1.11737e30 −0.143586
$$876$$ −2.74029e30 −0.347939
$$877$$ 1.56535e30 0.196388 0.0981942 0.995167i $$-0.468693\pi$$
0.0981942 + 0.995167i $$0.468693\pi$$
$$878$$ −8.95611e30 −1.11027
$$879$$ 7.62452e29 0.0933963
$$880$$ −3.04946e30 −0.369110
$$881$$ 4.04777e30 0.484138 0.242069 0.970259i $$-0.422174\pi$$
0.242069 + 0.970259i $$0.422174\pi$$
$$882$$ 1.89574e30 0.224058
$$883$$ 6.12581e29 0.0715445 0.0357723 0.999360i $$-0.488611\pi$$
0.0357723 + 0.999360i $$0.488611\pi$$
$$884$$ 5.13628e30 0.592789
$$885$$ 8.17143e30 0.931953
$$886$$ −1.85877e30 −0.209494
$$887$$ 9.40602e30 1.04763 0.523815 0.851832i $$-0.324508\pi$$
0.523815 + 0.851832i $$0.324508\pi$$
$$888$$ −2.37578e30 −0.261499
$$889$$ −3.23998e30 −0.352431
$$890$$ 2.85881e30 0.307319
$$891$$ −1.27508e30 −0.135463
$$892$$ 8.81630e30 0.925668
$$893$$ 1.21022e30 0.125581
$$894$$ 2.71117e30 0.278043
$$895$$ 5.46330e30 0.553749
$$896$$ −1.96115e29 −0.0196461
$$897$$ 4.40654e30 0.436292
$$898$$ −1.09537e31 −1.07191
$$899$$ −1.91406e31 −1.85131
$$900$$ 8.13411e29 0.0777614
$$901$$ −2.29378e31 −2.16741
$$902$$ 9.30228e30 0.868802
$$903$$ 1.28907e30 0.119002
$$904$$ −2.84946e29 −0.0260012
$$905$$ 1.89681e31 1.71086
$$906$$ −3.19139e30 −0.284534
$$907$$ −2.95076e30 −0.260051 −0.130025 0.991511i $$-0.541506\pi$$
−0.130025 + 0.991511i $$0.541506\pi$$
$$908$$ −6.39318e30 −0.556949
$$909$$ −5.20992e30 −0.448653
$$910$$ −1.50959e30 −0.128506
$$911$$ 5.31915e30 0.447610 0.223805 0.974634i $$-0.428152\pi$$
0.223805 + 0.974634i $$0.428152\pi$$
$$912$$ 2.31752e29 0.0192787
$$913$$ −3.88944e30 −0.319848
$$914$$ 1.63197e31 1.32672
$$915$$ −7.47718e30 −0.600920
$$916$$ −6.85202e29 −0.0544397
$$917$$ −1.64908e30 −0.129528
$$918$$ −3.07734e30 −0.238961
$$919$$ 3.46199e30 0.265774 0.132887 0.991131i $$-0.457575\pi$$
0.132887 + 0.991131i $$0.457575\pi$$
$$920$$ −6.31409e30 −0.479224
$$921$$ −8.61567e30 −0.646492
$$922$$ −5.46471e30 −0.405408
$$923$$ −4.33950e30 −0.318289
$$924$$ −1.07873e30 −0.0782268
$$925$$ 8.33641e30 0.597711
$$926$$ −3.46776e30 −0.245829
$$927$$ −6.02590e30 −0.422362
$$928$$ −2.47035e30 −0.171200
$$929$$ −8.04165e30 −0.551036 −0.275518 0.961296i $$-0.588849\pi$$
−0.275518 + 0.961296i $$0.588849\pi$$
$$930$$ 1.39491e31 0.945094
$$931$$ −1.89525e30 −0.126968
$$932$$ −1.14389e31 −0.757735
$$933$$ −5.90584e29 −0.0386834
$$934$$ 1.61359e31 1.04508
$$935$$ −4.04820e31 −2.59263
$$936$$ −1.25642e30 −0.0795682
$$937$$ 5.36884e30 0.336213 0.168106 0.985769i $$-0.446235\pi$$
0.168106 + 0.985769i $$0.446235\pi$$
$$938$$ 2.63324e30 0.163065
$$939$$ 1.23028e30 0.0753379
$$940$$ 9.40131e30 0.569306
$$941$$ 3.46189e30 0.207311 0.103655 0.994613i $$-0.466946\pi$$
0.103655 + 0.994613i $$0.466946\pi$$
$$942$$ −2.88212e30 −0.170678
$$943$$ 1.92609e31 1.12799
$$944$$ 5.75373e30 0.333229
$$945$$ 9.04451e29 0.0518025
$$946$$ 1.41136e31 0.799433
$$947$$ 1.53479e31 0.859753 0.429877 0.902888i $$-0.358557\pi$$
0.429877 + 0.902888i $$0.358557\pi$$
$$948$$ −9.21862e29 −0.0510714
$$949$$ −1.48523e31 −0.813766
$$950$$ −8.13198e29 −0.0440655
$$951$$ 7.73242e30 0.414400
$$952$$ −2.60345e30 −0.137994
$$953$$ 3.39202e31 1.77821 0.889105 0.457703i $$-0.151328\pi$$
0.889105 + 0.457703i $$0.151328\pi$$
$$954$$ 5.61097e30 0.290924
$$955$$ −1.18521e31 −0.607799
$$956$$ 6.25491e27 0.000317258 0
$$957$$ −1.35881e31 −0.681684
$$958$$ 2.18084e31 1.08215
$$959$$ 6.87334e30 0.337343
$$960$$ 1.80031e30 0.0873978
$$961$$ 5.52762e31 2.65425
$$962$$ −1.28767e31 −0.611599
$$963$$ −1.11998e31 −0.526181
$$964$$ 8.57661e30 0.398572
$$965$$ −2.08400e31 −0.957990
$$966$$ −2.23356e30 −0.101564
$$967$$ −5.24173e30 −0.235775 −0.117887 0.993027i $$-0.537612\pi$$
−0.117887 + 0.993027i $$0.537612\pi$$
$$968$$ −3.86472e30 −0.171960
$$969$$ 3.07653e30 0.135414
$$970$$ −3.73989e31 −1.62838
$$971$$ −1.66001e31 −0.715004 −0.357502 0.933912i $$-0.616371\pi$$
−0.357502 + 0.933912i $$0.616371\pi$$
$$972$$ 7.52771e29 0.0320750
$$973$$ −3.10179e30 −0.130746
$$974$$ −1.72552e31 −0.719533
$$975$$ 4.40868e30 0.181870
$$976$$ −5.26488e30 −0.214865
$$977$$ 4.02046e31 1.62324 0.811621 0.584185i $$-0.198586\pi$$
0.811621 + 0.584185i $$0.198586\pi$$
$$978$$ −2.83334e30 −0.113173
$$979$$ 1.10723e31 0.437540
$$980$$ −1.47228e31 −0.575595
$$981$$ −1.83748e30 −0.0710720
$$982$$ 1.61481e31 0.617948
$$983$$ 1.31702e31 0.498633 0.249317 0.968422i $$-0.419794\pi$$
0.249317 + 0.968422i $$0.419794\pi$$
$$984$$ −5.49179e30 −0.205715
$$985$$ −1.42294e31 −0.527357
$$986$$ −3.27942e31 −1.20251
$$987$$ 3.32565e30 0.120655
$$988$$ 1.25609e30 0.0450893
$$989$$ 2.92231e31 1.03792
$$990$$ 9.90259e30 0.348000
$$991$$ −4.21282e31 −1.46487 −0.732436 0.680836i $$-0.761616\pi$$
−0.732436 + 0.680836i $$0.761616\pi$$
$$992$$ 9.82193e30 0.337928
$$993$$ 3.26983e29 0.0111316
$$994$$ 2.19958e30 0.0740940
$$995$$ −4.83521e31 −1.61166
$$996$$ 2.29621e30 0.0757336
$$997$$ −5.77509e31 −1.88477 −0.942387 0.334524i $$-0.891424\pi$$
−0.942387 + 0.334524i $$0.891424\pi$$
$$998$$ −2.63357e31 −0.850501
$$999$$ 7.71492e30 0.246544
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6.22.a.a.1.1 1
3.2 odd 2 18.22.a.d.1.1 1
4.3 odd 2 48.22.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
6.22.a.a.1.1 1 1.1 even 1 trivial
18.22.a.d.1.1 1 3.2 odd 2
48.22.a.c.1.1 1 4.3 odd 2