# Properties

 Label 16.22.a.b.1.1 Level $16$ Weight $22$ Character 16.1 Self dual yes Analytic conductor $44.716$ 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] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$44.7163750859$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 16.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-59316.0 q^{3} +4.97535e6 q^{5} -1.42743e9 q^{7} -6.94197e9 q^{9} +O(q^{10})$$ $$q-59316.0 q^{3} +4.97535e6 q^{5} -1.42743e9 q^{7} -6.94197e9 q^{9} +1.06768e11 q^{11} -1.50151e11 q^{13} -2.95118e11 q^{15} -1.12040e13 q^{17} -1.10241e13 q^{19} +8.46692e13 q^{21} -1.29503e14 q^{23} -4.52083e14 q^{25} +1.03224e15 q^{27} +2.38237e15 q^{29} +8.78553e14 q^{31} -6.33304e15 q^{33} -7.10194e15 q^{35} +3.11300e16 q^{37} +8.90633e15 q^{39} -2.46129e16 q^{41} +1.33386e17 q^{43} -3.45387e16 q^{45} +1.92524e17 q^{47} +1.47900e18 q^{49} +6.64575e17 q^{51} -5.94166e17 q^{53} +5.31208e17 q^{55} +6.53903e17 q^{57} +2.95595e18 q^{59} +7.98415e18 q^{61} +9.90914e18 q^{63} -7.47052e17 q^{65} -4.83704e18 q^{67} +7.68159e18 q^{69} -8.84902e18 q^{71} +3.66844e19 q^{73} +2.68158e19 q^{75} -1.52403e20 q^{77} -3.38406e19 q^{79} +1.13873e19 q^{81} -2.04215e20 q^{83} -5.57437e19 q^{85} -1.41313e20 q^{87} -4.10241e19 q^{89} +2.14329e20 q^{91} -5.21122e19 q^{93} -5.48485e19 q^{95} -7.27592e20 q^{97} -7.41179e20 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −59316.0 −0.579961 −0.289980 0.957033i $$-0.593649\pi$$
−0.289980 + 0.957033i $$0.593649\pi$$
$$4$$ 0 0
$$5$$ 4.97535e6 0.227845 0.113922 0.993490i $$-0.463659\pi$$
0.113922 + 0.993490i $$0.463659\pi$$
$$6$$ 0 0
$$7$$ −1.42743e9 −1.90996 −0.954980 0.296671i $$-0.904123\pi$$
−0.954980 + 0.296671i $$0.904123\pi$$
$$8$$ 0 0
$$9$$ −6.94197e9 −0.663645
$$10$$ 0 0
$$11$$ 1.06768e11 1.24113 0.620565 0.784155i $$-0.286903\pi$$
0.620565 + 0.784155i $$0.286903\pi$$
$$12$$ 0 0
$$13$$ −1.50151e11 −0.302080 −0.151040 0.988528i $$-0.548262\pi$$
−0.151040 + 0.988528i $$0.548262\pi$$
$$14$$ 0 0
$$15$$ −2.95118e11 −0.132141
$$16$$ 0 0
$$17$$ −1.12040e13 −1.34790 −0.673952 0.738776i $$-0.735404\pi$$
−0.673952 + 0.738776i $$0.735404\pi$$
$$18$$ 0 0
$$19$$ −1.10241e13 −0.412504 −0.206252 0.978499i $$-0.566127\pi$$
−0.206252 + 0.978499i $$0.566127\pi$$
$$20$$ 0 0
$$21$$ 8.46692e13 1.10770
$$22$$ 0 0
$$23$$ −1.29503e14 −0.651834 −0.325917 0.945398i $$-0.605673\pi$$
−0.325917 + 0.945398i $$0.605673\pi$$
$$24$$ 0 0
$$25$$ −4.52083e14 −0.948087
$$26$$ 0 0
$$27$$ 1.03224e15 0.964849
$$28$$ 0 0
$$29$$ 2.38237e15 1.05155 0.525776 0.850623i $$-0.323775\pi$$
0.525776 + 0.850623i $$0.323775\pi$$
$$30$$ 0 0
$$31$$ 8.78553e14 0.192517 0.0962587 0.995356i $$-0.469312\pi$$
0.0962587 + 0.995356i $$0.469312\pi$$
$$32$$ 0 0
$$33$$ −6.33304e15 −0.719807
$$34$$ 0 0
$$35$$ −7.10194e15 −0.435174
$$36$$ 0 0
$$37$$ 3.11300e16 1.06429 0.532146 0.846652i $$-0.321386\pi$$
0.532146 + 0.846652i $$0.321386\pi$$
$$38$$ 0 0
$$39$$ 8.90633e15 0.175194
$$40$$ 0 0
$$41$$ −2.46129e16 −0.286373 −0.143187 0.989696i $$-0.545735\pi$$
−0.143187 + 0.989696i $$0.545735\pi$$
$$42$$ 0 0
$$43$$ 1.33386e17 0.941222 0.470611 0.882341i $$-0.344034\pi$$
0.470611 + 0.882341i $$0.344034\pi$$
$$44$$ 0 0
$$45$$ −3.45387e16 −0.151208
$$46$$ 0 0
$$47$$ 1.92524e17 0.533897 0.266948 0.963711i $$-0.413985\pi$$
0.266948 + 0.963711i $$0.413985\pi$$
$$48$$ 0 0
$$49$$ 1.47900e18 2.64794
$$50$$ 0 0
$$51$$ 6.64575e17 0.781731
$$52$$ 0 0
$$53$$ −5.94166e17 −0.466672 −0.233336 0.972396i $$-0.574964\pi$$
−0.233336 + 0.972396i $$0.574964\pi$$
$$54$$ 0 0
$$55$$ 5.31208e17 0.282785
$$56$$ 0 0
$$57$$ 6.53903e17 0.239236
$$58$$ 0 0
$$59$$ 2.95595e18 0.752925 0.376462 0.926432i $$-0.377140\pi$$
0.376462 + 0.926432i $$0.377140\pi$$
$$60$$ 0 0
$$61$$ 7.98415e18 1.43306 0.716532 0.697554i $$-0.245728\pi$$
0.716532 + 0.697554i $$0.245728\pi$$
$$62$$ 0 0
$$63$$ 9.90914e18 1.26754
$$64$$ 0 0
$$65$$ −7.47052e17 −0.0688272
$$66$$ 0 0
$$67$$ −4.83704e18 −0.324186 −0.162093 0.986775i $$-0.551824\pi$$
−0.162093 + 0.986775i $$0.551824\pi$$
$$68$$ 0 0
$$69$$ 7.68159e18 0.378038
$$70$$ 0 0
$$71$$ −8.84902e18 −0.322613 −0.161307 0.986904i $$-0.551571\pi$$
−0.161307 + 0.986904i $$0.551571\pi$$
$$72$$ 0 0
$$73$$ 3.66844e19 0.999060 0.499530 0.866297i $$-0.333506\pi$$
0.499530 + 0.866297i $$0.333506\pi$$
$$74$$ 0 0
$$75$$ 2.68158e19 0.549853
$$76$$ 0 0
$$77$$ −1.52403e20 −2.37051
$$78$$ 0 0
$$79$$ −3.38406e19 −0.402118 −0.201059 0.979579i $$-0.564438\pi$$
−0.201059 + 0.979579i $$0.564438\pi$$
$$80$$ 0 0
$$81$$ 1.13873e19 0.104071
$$82$$ 0 0
$$83$$ −2.04215e20 −1.44466 −0.722332 0.691547i $$-0.756930\pi$$
−0.722332 + 0.691547i $$0.756930\pi$$
$$84$$ 0 0
$$85$$ −5.57437e19 −0.307112
$$86$$ 0 0
$$87$$ −1.41313e20 −0.609858
$$88$$ 0 0
$$89$$ −4.10241e19 −0.139458 −0.0697290 0.997566i $$-0.522213\pi$$
−0.0697290 + 0.997566i $$0.522213\pi$$
$$90$$ 0 0
$$91$$ 2.14329e20 0.576960
$$92$$ 0 0
$$93$$ −5.21122e19 −0.111653
$$94$$ 0 0
$$95$$ −5.48485e19 −0.0939869
$$96$$ 0 0
$$97$$ −7.27592e20 −1.00181 −0.500905 0.865503i $$-0.666999\pi$$
−0.500905 + 0.865503i $$0.666999\pi$$
$$98$$ 0 0
$$99$$ −7.41179e20 −0.823671
$$100$$ 0 0
$$101$$ 5.93965e20 0.535040 0.267520 0.963552i $$-0.413796\pi$$
0.267520 + 0.963552i $$0.413796\pi$$
$$102$$ 0 0
$$103$$ 6.95712e20 0.510080 0.255040 0.966930i $$-0.417911\pi$$
0.255040 + 0.966930i $$0.417911\pi$$
$$104$$ 0 0
$$105$$ 4.21259e20 0.252384
$$106$$ 0 0
$$107$$ 2.38158e21 1.17041 0.585203 0.810887i $$-0.301015\pi$$
0.585203 + 0.810887i $$0.301015\pi$$
$$108$$ 0 0
$$109$$ 2.01913e21 0.816933 0.408466 0.912773i $$-0.366064\pi$$
0.408466 + 0.912773i $$0.366064\pi$$
$$110$$ 0 0
$$111$$ −1.84651e21 −0.617248
$$112$$ 0 0
$$113$$ 1.81974e21 0.504296 0.252148 0.967689i $$-0.418863\pi$$
0.252148 + 0.967689i $$0.418863\pi$$
$$114$$ 0 0
$$115$$ −6.44322e20 −0.148517
$$116$$ 0 0
$$117$$ 1.04234e21 0.200474
$$118$$ 0 0
$$119$$ 1.59929e22 2.57444
$$120$$ 0 0
$$121$$ 3.99913e21 0.540405
$$122$$ 0 0
$$123$$ 1.45994e21 0.166085
$$124$$ 0 0
$$125$$ −4.62170e21 −0.443861
$$126$$ 0 0
$$127$$ −2.20220e22 −1.79027 −0.895134 0.445797i $$-0.852920\pi$$
−0.895134 + 0.445797i $$0.852920\pi$$
$$128$$ 0 0
$$129$$ −7.91193e21 −0.545872
$$130$$ 0 0
$$131$$ −1.44136e22 −0.846101 −0.423051 0.906106i $$-0.639041\pi$$
−0.423051 + 0.906106i $$0.639041\pi$$
$$132$$ 0 0
$$133$$ 1.57360e22 0.787867
$$134$$ 0 0
$$135$$ 5.13574e21 0.219836
$$136$$ 0 0
$$137$$ −3.57623e22 −1.31177 −0.655887 0.754859i $$-0.727705\pi$$
−0.655887 + 0.754859i $$0.727705\pi$$
$$138$$ 0 0
$$139$$ 2.10431e22 0.662909 0.331454 0.943471i $$-0.392461\pi$$
0.331454 + 0.943471i $$0.392461\pi$$
$$140$$ 0 0
$$141$$ −1.14198e22 −0.309639
$$142$$ 0 0
$$143$$ −1.60313e22 −0.374921
$$144$$ 0 0
$$145$$ 1.18531e22 0.239590
$$146$$ 0 0
$$147$$ −8.77283e22 −1.53570
$$148$$ 0 0
$$149$$ −8.71910e22 −1.32439 −0.662195 0.749332i $$-0.730375\pi$$
−0.662195 + 0.749332i $$0.730375\pi$$
$$150$$ 0 0
$$151$$ 4.00667e22 0.529086 0.264543 0.964374i $$-0.414779\pi$$
0.264543 + 0.964374i $$0.414779\pi$$
$$152$$ 0 0
$$153$$ 7.77776e22 0.894530
$$154$$ 0 0
$$155$$ 4.37111e21 0.0438641
$$156$$ 0 0
$$157$$ −4.60441e22 −0.403857 −0.201929 0.979400i $$-0.564721\pi$$
−0.201929 + 0.979400i $$0.564721\pi$$
$$158$$ 0 0
$$159$$ 3.52436e22 0.270651
$$160$$ 0 0
$$161$$ 1.84856e23 1.24498
$$162$$ 0 0
$$163$$ −5.72127e22 −0.338472 −0.169236 0.985576i $$-0.554130\pi$$
−0.169236 + 0.985576i $$0.554130\pi$$
$$164$$ 0 0
$$165$$ −3.15091e22 −0.164004
$$166$$ 0 0
$$167$$ 1.32913e23 0.609600 0.304800 0.952416i $$-0.401410\pi$$
0.304800 + 0.952416i $$0.401410\pi$$
$$168$$ 0 0
$$169$$ −2.24519e23 −0.908748
$$170$$ 0 0
$$171$$ 7.65286e22 0.273757
$$172$$ 0 0
$$173$$ 5.53136e23 1.75125 0.875624 0.482994i $$-0.160451\pi$$
0.875624 + 0.482994i $$0.160451\pi$$
$$174$$ 0 0
$$175$$ 6.45315e23 1.81081
$$176$$ 0 0
$$177$$ −1.75335e23 −0.436667
$$178$$ 0 0
$$179$$ 5.31479e22 0.117633 0.0588165 0.998269i $$-0.481267\pi$$
0.0588165 + 0.998269i $$0.481267\pi$$
$$180$$ 0 0
$$181$$ 7.59350e23 1.49560 0.747802 0.663921i $$-0.231109\pi$$
0.747802 + 0.663921i $$0.231109\pi$$
$$182$$ 0 0
$$183$$ −4.73588e23 −0.831121
$$184$$ 0 0
$$185$$ 1.54883e23 0.242493
$$186$$ 0 0
$$187$$ −1.19623e24 −1.67292
$$188$$ 0 0
$$189$$ −1.47344e24 −1.84282
$$190$$ 0 0
$$191$$ 9.64674e23 1.08026 0.540132 0.841580i $$-0.318374\pi$$
0.540132 + 0.841580i $$0.318374\pi$$
$$192$$ 0 0
$$193$$ −3.41192e23 −0.342489 −0.171245 0.985229i $$-0.554779\pi$$
−0.171245 + 0.985229i $$0.554779\pi$$
$$194$$ 0 0
$$195$$ 4.43121e22 0.0399171
$$196$$ 0 0
$$197$$ −5.00591e23 −0.405124 −0.202562 0.979269i $$-0.564927\pi$$
−0.202562 + 0.979269i $$0.564927\pi$$
$$198$$ 0 0
$$199$$ 5.09875e23 0.371113 0.185557 0.982634i $$-0.440591\pi$$
0.185557 + 0.982634i $$0.440591\pi$$
$$200$$ 0 0
$$201$$ 2.86914e23 0.188015
$$202$$ 0 0
$$203$$ −3.40066e24 −2.00842
$$204$$ 0 0
$$205$$ −1.22458e23 −0.0652486
$$206$$ 0 0
$$207$$ 8.99004e23 0.432587
$$208$$ 0 0
$$209$$ −1.17702e24 −0.511972
$$210$$ 0 0
$$211$$ −7.29976e23 −0.287305 −0.143652 0.989628i $$-0.545885\pi$$
−0.143652 + 0.989628i $$0.545885\pi$$
$$212$$ 0 0
$$213$$ 5.24888e23 0.187103
$$214$$ 0 0
$$215$$ 6.63643e23 0.214452
$$216$$ 0 0
$$217$$ −1.25407e24 −0.367701
$$218$$ 0 0
$$219$$ −2.17597e24 −0.579416
$$220$$ 0 0
$$221$$ 1.68228e24 0.407174
$$222$$ 0 0
$$223$$ 5.87017e24 1.29256 0.646279 0.763101i $$-0.276324\pi$$
0.646279 + 0.763101i $$0.276324\pi$$
$$224$$ 0 0
$$225$$ 3.13834e24 0.629193
$$226$$ 0 0
$$227$$ −6.13596e24 −1.12101 −0.560507 0.828150i $$-0.689394\pi$$
−0.560507 + 0.828150i $$0.689394\pi$$
$$228$$ 0 0
$$229$$ −3.05217e24 −0.508552 −0.254276 0.967132i $$-0.581837\pi$$
−0.254276 + 0.967132i $$0.581837\pi$$
$$230$$ 0 0
$$231$$ 9.03995e24 1.37480
$$232$$ 0 0
$$233$$ −7.45995e23 −0.103633 −0.0518166 0.998657i $$-0.516501\pi$$
−0.0518166 + 0.998657i $$0.516501\pi$$
$$234$$ 0 0
$$235$$ 9.57874e23 0.121645
$$236$$ 0 0
$$237$$ 2.00729e24 0.233213
$$238$$ 0 0
$$239$$ 1.02561e25 1.09095 0.545473 0.838128i $$-0.316350\pi$$
0.545473 + 0.838128i $$0.316350\pi$$
$$240$$ 0 0
$$241$$ 1.46660e25 1.42933 0.714663 0.699469i $$-0.246580\pi$$
0.714663 + 0.699469i $$0.246580\pi$$
$$242$$ 0 0
$$243$$ −1.14730e25 −1.02521
$$244$$ 0 0
$$245$$ 7.35854e24 0.603320
$$246$$ 0 0
$$247$$ 1.65527e24 0.124609
$$248$$ 0 0
$$249$$ 1.21132e25 0.837848
$$250$$ 0 0
$$251$$ 1.52767e25 0.971526 0.485763 0.874090i $$-0.338542\pi$$
0.485763 + 0.874090i $$0.338542\pi$$
$$252$$ 0 0
$$253$$ −1.38267e25 −0.809011
$$254$$ 0 0
$$255$$ 3.30649e24 0.178113
$$256$$ 0 0
$$257$$ −3.66073e25 −1.81665 −0.908323 0.418270i $$-0.862636\pi$$
−0.908323 + 0.418270i $$0.862636\pi$$
$$258$$ 0 0
$$259$$ −4.44358e25 −2.03276
$$260$$ 0 0
$$261$$ −1.65383e25 −0.697857
$$262$$ 0 0
$$263$$ 1.74317e25 0.678899 0.339449 0.940624i $$-0.389759\pi$$
0.339449 + 0.940624i $$0.389759\pi$$
$$264$$ 0 0
$$265$$ −2.95619e24 −0.106329
$$266$$ 0 0
$$267$$ 2.43338e24 0.0808802
$$268$$ 0 0
$$269$$ 4.63224e25 1.42361 0.711807 0.702375i $$-0.247877\pi$$
0.711807 + 0.702375i $$0.247877\pi$$
$$270$$ 0 0
$$271$$ −1.79840e25 −0.511338 −0.255669 0.966764i $$-0.582296\pi$$
−0.255669 + 0.966764i $$0.582296\pi$$
$$272$$ 0 0
$$273$$ −1.27131e25 −0.334614
$$274$$ 0 0
$$275$$ −4.82680e25 −1.17670
$$276$$ 0 0
$$277$$ −1.07659e25 −0.243227 −0.121614 0.992578i $$-0.538807\pi$$
−0.121614 + 0.992578i $$0.538807\pi$$
$$278$$ 0 0
$$279$$ −6.09888e24 −0.127763
$$280$$ 0 0
$$281$$ −8.52851e25 −1.65751 −0.828756 0.559610i $$-0.810951\pi$$
−0.828756 + 0.559610i $$0.810951\pi$$
$$282$$ 0 0
$$283$$ −8.16776e23 −0.0147348 −0.00736742 0.999973i $$-0.502345\pi$$
−0.00736742 + 0.999973i $$0.502345\pi$$
$$284$$ 0 0
$$285$$ 3.25340e24 0.0545087
$$286$$ 0 0
$$287$$ 3.51331e25 0.546962
$$288$$ 0 0
$$289$$ 5.64373e25 0.816843
$$290$$ 0 0
$$291$$ 4.31579e25 0.581010
$$292$$ 0 0
$$293$$ 5.29341e25 0.663171 0.331586 0.943425i $$-0.392416\pi$$
0.331586 + 0.943425i $$0.392416\pi$$
$$294$$ 0 0
$$295$$ 1.47069e25 0.171550
$$296$$ 0 0
$$297$$ 1.10210e26 1.19750
$$298$$ 0 0
$$299$$ 1.94449e25 0.196906
$$300$$ 0 0
$$301$$ −1.90399e26 −1.79769
$$302$$ 0 0
$$303$$ −3.52316e25 −0.310302
$$304$$ 0 0
$$305$$ 3.97239e25 0.326516
$$306$$ 0 0
$$307$$ 1.79951e26 1.38103 0.690514 0.723319i $$-0.257385\pi$$
0.690514 + 0.723319i $$0.257385\pi$$
$$308$$ 0 0
$$309$$ −4.12669e25 −0.295827
$$310$$ 0 0
$$311$$ −2.49771e26 −1.67324 −0.836620 0.547783i $$-0.815472\pi$$
−0.836620 + 0.547783i $$0.815472\pi$$
$$312$$ 0 0
$$313$$ 2.14796e26 1.34527 0.672637 0.739972i $$-0.265161\pi$$
0.672637 + 0.739972i $$0.265161\pi$$
$$314$$ 0 0
$$315$$ 4.93014e25 0.288801
$$316$$ 0 0
$$317$$ 3.06450e26 1.67972 0.839860 0.542803i $$-0.182637\pi$$
0.839860 + 0.542803i $$0.182637\pi$$
$$318$$ 0 0
$$319$$ 2.54361e26 1.30511
$$320$$ 0 0
$$321$$ −1.41266e26 −0.678789
$$322$$ 0 0
$$323$$ 1.23513e26 0.556016
$$324$$ 0 0
$$325$$ 6.78805e25 0.286398
$$326$$ 0 0
$$327$$ −1.19767e26 −0.473789
$$328$$ 0 0
$$329$$ −2.74814e26 −1.01972
$$330$$ 0 0
$$331$$ 1.04905e26 0.365261 0.182631 0.983182i $$-0.441539\pi$$
0.182631 + 0.983182i $$0.441539\pi$$
$$332$$ 0 0
$$333$$ −2.16103e26 −0.706313
$$334$$ 0 0
$$335$$ −2.40660e25 −0.0738640
$$336$$ 0 0
$$337$$ −1.95001e26 −0.562242 −0.281121 0.959672i $$-0.590706\pi$$
−0.281121 + 0.959672i $$0.590706\pi$$
$$338$$ 0 0
$$339$$ −1.07940e26 −0.292472
$$340$$ 0 0
$$341$$ 9.38012e25 0.238939
$$342$$ 0 0
$$343$$ −1.31388e27 −3.14751
$$344$$ 0 0
$$345$$ 3.82186e25 0.0861339
$$346$$ 0 0
$$347$$ 5.59947e26 1.18765 0.593824 0.804595i $$-0.297618\pi$$
0.593824 + 0.804595i $$0.297618\pi$$
$$348$$ 0 0
$$349$$ −2.09819e26 −0.418966 −0.209483 0.977812i $$-0.567178\pi$$
−0.209483 + 0.977812i $$0.567178\pi$$
$$350$$ 0 0
$$351$$ −1.54991e26 −0.291462
$$352$$ 0 0
$$353$$ 5.72422e26 1.01410 0.507051 0.861916i $$-0.330736\pi$$
0.507051 + 0.861916i $$0.330736\pi$$
$$354$$ 0 0
$$355$$ −4.40270e25 −0.0735057
$$356$$ 0 0
$$357$$ −9.48632e26 −1.49307
$$358$$ 0 0
$$359$$ 4.84990e26 0.719848 0.359924 0.932982i $$-0.382803\pi$$
0.359924 + 0.932982i $$0.382803\pi$$
$$360$$ 0 0
$$361$$ −5.92680e26 −0.829840
$$362$$ 0 0
$$363$$ −2.37213e26 −0.313414
$$364$$ 0 0
$$365$$ 1.82518e26 0.227630
$$366$$ 0 0
$$367$$ −6.65755e26 −0.784008 −0.392004 0.919963i $$-0.628218\pi$$
−0.392004 + 0.919963i $$0.628218\pi$$
$$368$$ 0 0
$$369$$ 1.70862e26 0.190050
$$370$$ 0 0
$$371$$ 8.48128e26 0.891324
$$372$$ 0 0
$$373$$ 1.46537e27 1.45547 0.727735 0.685858i $$-0.240573\pi$$
0.727735 + 0.685858i $$0.240573\pi$$
$$374$$ 0 0
$$375$$ 2.74141e26 0.257422
$$376$$ 0 0
$$377$$ −3.57714e26 −0.317652
$$378$$ 0 0
$$379$$ 1.80711e27 1.51800 0.759002 0.651088i $$-0.225687\pi$$
0.759002 + 0.651088i $$0.225687\pi$$
$$380$$ 0 0
$$381$$ 1.30626e27 1.03829
$$382$$ 0 0
$$383$$ −9.69700e26 −0.729542 −0.364771 0.931097i $$-0.618853\pi$$
−0.364771 + 0.931097i $$0.618853\pi$$
$$384$$ 0 0
$$385$$ −7.58260e26 −0.540108
$$386$$ 0 0
$$387$$ −9.25962e26 −0.624637
$$388$$ 0 0
$$389$$ 3.19483e25 0.0204163 0.0102081 0.999948i $$-0.496751\pi$$
0.0102081 + 0.999948i $$0.496751\pi$$
$$390$$ 0 0
$$391$$ 1.45095e27 0.878609
$$392$$ 0 0
$$393$$ 8.54954e26 0.490706
$$394$$ 0 0
$$395$$ −1.68369e26 −0.0916204
$$396$$ 0 0
$$397$$ −5.23867e26 −0.270346 −0.135173 0.990822i $$-0.543159\pi$$
−0.135173 + 0.990822i $$0.543159\pi$$
$$398$$ 0 0
$$399$$ −9.33398e26 −0.456932
$$400$$ 0 0
$$401$$ 1.65135e27 0.767049 0.383525 0.923531i $$-0.374710\pi$$
0.383525 + 0.923531i $$0.374710\pi$$
$$402$$ 0 0
$$403$$ −1.31915e26 −0.0581557
$$404$$ 0 0
$$405$$ 5.66558e25 0.0237119
$$406$$ 0 0
$$407$$ 3.32369e27 1.32093
$$408$$ 0 0
$$409$$ 1.41506e27 0.534172 0.267086 0.963673i $$-0.413939\pi$$
0.267086 + 0.963673i $$0.413939\pi$$
$$410$$ 0 0
$$411$$ 2.12127e27 0.760778
$$412$$ 0 0
$$413$$ −4.21941e27 −1.43806
$$414$$ 0 0
$$415$$ −1.01604e27 −0.329159
$$416$$ 0 0
$$417$$ −1.24819e27 −0.384461
$$418$$ 0 0
$$419$$ −1.80515e27 −0.528769 −0.264385 0.964417i $$-0.585169\pi$$
−0.264385 + 0.964417i $$0.585169\pi$$
$$420$$ 0 0
$$421$$ −1.98386e27 −0.552775 −0.276388 0.961046i $$-0.589137\pi$$
−0.276388 + 0.961046i $$0.589137\pi$$
$$422$$ 0 0
$$423$$ −1.33650e27 −0.354318
$$424$$ 0 0
$$425$$ 5.06513e27 1.27793
$$426$$ 0 0
$$427$$ −1.13968e28 −2.73709
$$428$$ 0 0
$$429$$ 9.50910e26 0.217439
$$430$$ 0 0
$$431$$ 3.46591e27 0.754756 0.377378 0.926059i $$-0.376826\pi$$
0.377378 + 0.926059i $$0.376826\pi$$
$$432$$ 0 0
$$433$$ −5.88060e27 −1.21983 −0.609915 0.792467i $$-0.708796\pi$$
−0.609915 + 0.792467i $$0.708796\pi$$
$$434$$ 0 0
$$435$$ −7.03080e26 −0.138953
$$436$$ 0 0
$$437$$ 1.42765e27 0.268884
$$438$$ 0 0
$$439$$ 6.95231e26 0.124811 0.0624053 0.998051i $$-0.480123\pi$$
0.0624053 + 0.998051i $$0.480123\pi$$
$$440$$ 0 0
$$441$$ −1.02672e28 −1.75730
$$442$$ 0 0
$$443$$ 4.77606e26 0.0779526 0.0389763 0.999240i $$-0.487590\pi$$
0.0389763 + 0.999240i $$0.487590\pi$$
$$444$$ 0 0
$$445$$ −2.04109e26 −0.0317747
$$446$$ 0 0
$$447$$ 5.17182e27 0.768094
$$448$$ 0 0
$$449$$ 6.02792e27 0.854241 0.427121 0.904195i $$-0.359528\pi$$
0.427121 + 0.904195i $$0.359528\pi$$
$$450$$ 0 0
$$451$$ −2.62787e27 −0.355427
$$452$$ 0 0
$$453$$ −2.37660e27 −0.306849
$$454$$ 0 0
$$455$$ 1.06636e27 0.131457
$$456$$ 0 0
$$457$$ −3.92994e27 −0.462664 −0.231332 0.972875i $$-0.574308\pi$$
−0.231332 + 0.972875i $$0.574308\pi$$
$$458$$ 0 0
$$459$$ −1.15652e28 −1.30052
$$460$$ 0 0
$$461$$ 1.19082e28 1.27934 0.639672 0.768648i $$-0.279070\pi$$
0.639672 + 0.768648i $$0.279070\pi$$
$$462$$ 0 0
$$463$$ 6.15811e27 0.632189 0.316095 0.948728i $$-0.397628\pi$$
0.316095 + 0.948728i $$0.397628\pi$$
$$464$$ 0 0
$$465$$ −2.59277e26 −0.0254394
$$466$$ 0 0
$$467$$ 1.30147e28 1.22069 0.610346 0.792135i $$-0.291030\pi$$
0.610346 + 0.792135i $$0.291030\pi$$
$$468$$ 0 0
$$469$$ 6.90452e27 0.619182
$$470$$ 0 0
$$471$$ 2.73115e27 0.234221
$$472$$ 0 0
$$473$$ 1.42414e28 1.16818
$$474$$ 0 0
$$475$$ 4.98379e27 0.391090
$$476$$ 0 0
$$477$$ 4.12468e27 0.309705
$$478$$ 0 0
$$479$$ 1.70133e28 1.22255 0.611273 0.791420i $$-0.290658\pi$$
0.611273 + 0.791420i $$0.290658\pi$$
$$480$$ 0 0
$$481$$ −4.67419e27 −0.321501
$$482$$ 0 0
$$483$$ −1.09649e28 −0.722038
$$484$$ 0 0
$$485$$ −3.62003e27 −0.228257
$$486$$ 0 0
$$487$$ 2.45443e27 0.148216 0.0741082 0.997250i $$-0.476389\pi$$
0.0741082 + 0.997250i $$0.476389\pi$$
$$488$$ 0 0
$$489$$ 3.39363e27 0.196301
$$490$$ 0 0
$$491$$ −3.07994e28 −1.70682 −0.853408 0.521243i $$-0.825468\pi$$
−0.853408 + 0.521243i $$0.825468\pi$$
$$492$$ 0 0
$$493$$ −2.66920e28 −1.41739
$$494$$ 0 0
$$495$$ −3.68763e27 −0.187669
$$496$$ 0 0
$$497$$ 1.26313e28 0.616178
$$498$$ 0 0
$$499$$ 2.07463e28 0.970252 0.485126 0.874444i $$-0.338774\pi$$
0.485126 + 0.874444i $$0.338774\pi$$
$$500$$ 0 0
$$501$$ −7.88388e27 −0.353544
$$502$$ 0 0
$$503$$ 3.54838e28 1.52604 0.763022 0.646373i $$-0.223715\pi$$
0.763022 + 0.646373i $$0.223715\pi$$
$$504$$ 0 0
$$505$$ 2.95518e27 0.121906
$$506$$ 0 0
$$507$$ 1.33176e28 0.527038
$$508$$ 0 0
$$509$$ 2.04071e28 0.774900 0.387450 0.921891i $$-0.373356\pi$$
0.387450 + 0.921891i $$0.373356\pi$$
$$510$$ 0 0
$$511$$ −5.23643e28 −1.90816
$$512$$ 0 0
$$513$$ −1.13794e28 −0.398005
$$514$$ 0 0
$$515$$ 3.46141e27 0.116219
$$516$$ 0 0
$$517$$ 2.05554e28 0.662636
$$518$$ 0 0
$$519$$ −3.28098e28 −1.01566
$$520$$ 0 0
$$521$$ −2.64349e28 −0.785927 −0.392963 0.919554i $$-0.628550\pi$$
−0.392963 + 0.919554i $$0.628550\pi$$
$$522$$ 0 0
$$523$$ −3.70305e28 −1.05753 −0.528763 0.848769i $$-0.677344\pi$$
−0.528763 + 0.848769i $$0.677344\pi$$
$$524$$ 0 0
$$525$$ −3.82775e28 −1.05020
$$526$$ 0 0
$$527$$ −9.84329e27 −0.259495
$$528$$ 0 0
$$529$$ −2.27006e28 −0.575112
$$530$$ 0 0
$$531$$ −2.05201e28 −0.499675
$$532$$ 0 0
$$533$$ 3.69564e27 0.0865077
$$534$$ 0 0
$$535$$ 1.18492e28 0.266670
$$536$$ 0 0
$$537$$ −3.15252e27 −0.0682225
$$538$$ 0 0
$$539$$ 1.57910e29 3.28645
$$540$$ 0 0
$$541$$ −9.59401e27 −0.192056 −0.0960282 0.995379i $$-0.530614\pi$$
−0.0960282 + 0.995379i $$0.530614\pi$$
$$542$$ 0 0
$$543$$ −4.50416e28 −0.867392
$$544$$ 0 0
$$545$$ 1.00459e28 0.186134
$$546$$ 0 0
$$547$$ −7.60716e28 −1.35630 −0.678150 0.734924i $$-0.737218\pi$$
−0.678150 + 0.734924i $$0.737218\pi$$
$$548$$ 0 0
$$549$$ −5.54257e28 −0.951046
$$550$$ 0 0
$$551$$ −2.62634e28 −0.433769
$$552$$ 0 0
$$553$$ 4.83050e28 0.768029
$$554$$ 0 0
$$555$$ −9.18702e27 −0.140637
$$556$$ 0 0
$$557$$ 4.97908e28 0.733956 0.366978 0.930230i $$-0.380392\pi$$
0.366978 + 0.930230i $$0.380392\pi$$
$$558$$ 0 0
$$559$$ −2.00280e28 −0.284324
$$560$$ 0 0
$$561$$ 7.09553e28 0.970230
$$562$$ 0 0
$$563$$ 8.49629e28 1.11916 0.559579 0.828777i $$-0.310963\pi$$
0.559579 + 0.828777i $$0.310963\pi$$
$$564$$ 0 0
$$565$$ 9.05383e27 0.114901
$$566$$ 0 0
$$567$$ −1.62545e28 −0.198771
$$568$$ 0 0
$$569$$ 5.57042e28 0.656461 0.328230 0.944598i $$-0.393548\pi$$
0.328230 + 0.944598i $$0.393548\pi$$
$$570$$ 0 0
$$571$$ 1.58127e29 1.79608 0.898041 0.439911i $$-0.144990\pi$$
0.898041 + 0.439911i $$0.144990\pi$$
$$572$$ 0 0
$$573$$ −5.72206e28 −0.626511
$$574$$ 0 0
$$575$$ 5.85460e28 0.617995
$$576$$ 0 0
$$577$$ −1.14412e29 −1.16446 −0.582229 0.813025i $$-0.697819\pi$$
−0.582229 + 0.813025i $$0.697819\pi$$
$$578$$ 0 0
$$579$$ 2.02381e28 0.198630
$$580$$ 0 0
$$581$$ 2.91501e29 2.75925
$$582$$ 0 0
$$583$$ −6.34379e28 −0.579200
$$584$$ 0 0
$$585$$ 5.18601e27 0.0456769
$$586$$ 0 0
$$587$$ −3.03657e28 −0.258038 −0.129019 0.991642i $$-0.541183\pi$$
−0.129019 + 0.991642i $$0.541183\pi$$
$$588$$ 0 0
$$589$$ −9.68522e27 −0.0794143
$$590$$ 0 0
$$591$$ 2.96931e28 0.234956
$$592$$ 0 0
$$593$$ −1.80310e29 −1.37704 −0.688519 0.725218i $$-0.741739\pi$$
−0.688519 + 0.725218i $$0.741739\pi$$
$$594$$ 0 0
$$595$$ 7.95700e28 0.586572
$$596$$ 0 0
$$597$$ −3.02437e28 −0.215231
$$598$$ 0 0
$$599$$ −2.43184e29 −1.67091 −0.835455 0.549558i $$-0.814796\pi$$
−0.835455 + 0.549558i $$0.814796\pi$$
$$600$$ 0 0
$$601$$ −1.53373e29 −1.01758 −0.508789 0.860891i $$-0.669907\pi$$
−0.508789 + 0.860891i $$0.669907\pi$$
$$602$$ 0 0
$$603$$ 3.35786e28 0.215145
$$604$$ 0 0
$$605$$ 1.98971e28 0.123128
$$606$$ 0 0
$$607$$ 2.90773e28 0.173809 0.0869047 0.996217i $$-0.472302\pi$$
0.0869047 + 0.996217i $$0.472302\pi$$
$$608$$ 0 0
$$609$$ 2.01713e29 1.16480
$$610$$ 0 0
$$611$$ −2.89076e28 −0.161280
$$612$$ 0 0
$$613$$ −1.21635e29 −0.655726 −0.327863 0.944725i $$-0.606329\pi$$
−0.327863 + 0.944725i $$0.606329\pi$$
$$614$$ 0 0
$$615$$ 7.26371e27 0.0378416
$$616$$ 0 0
$$617$$ 8.98199e28 0.452250 0.226125 0.974098i $$-0.427394\pi$$
0.226125 + 0.974098i $$0.427394\pi$$
$$618$$ 0 0
$$619$$ −3.27118e29 −1.59203 −0.796017 0.605275i $$-0.793063\pi$$
−0.796017 + 0.605275i $$0.793063\pi$$
$$620$$ 0 0
$$621$$ −1.33677e29 −0.628922
$$622$$ 0 0
$$623$$ 5.85588e28 0.266359
$$624$$ 0 0
$$625$$ 1.92575e29 0.846956
$$626$$ 0 0
$$627$$ 6.98158e28 0.296924
$$628$$ 0 0
$$629$$ −3.48780e29 −1.43456
$$630$$ 0 0
$$631$$ −9.03181e28 −0.359308 −0.179654 0.983730i $$-0.557498\pi$$
−0.179654 + 0.983730i $$0.557498\pi$$
$$632$$ 0 0
$$633$$ 4.32993e28 0.166626
$$634$$ 0 0
$$635$$ −1.09567e29 −0.407903
$$636$$ 0 0
$$637$$ −2.22072e29 −0.799891
$$638$$ 0 0
$$639$$ 6.14296e28 0.214101
$$640$$ 0 0
$$641$$ −3.16031e29 −1.06591 −0.532955 0.846144i $$-0.678918\pi$$
−0.532955 + 0.846144i $$0.678918\pi$$
$$642$$ 0 0
$$643$$ 1.84895e29 0.603546 0.301773 0.953380i $$-0.402422\pi$$
0.301773 + 0.953380i $$0.402422\pi$$
$$644$$ 0 0
$$645$$ −3.93646e28 −0.124374
$$646$$ 0 0
$$647$$ 1.55883e29 0.476765 0.238383 0.971171i $$-0.423383\pi$$
0.238383 + 0.971171i $$0.423383\pi$$
$$648$$ 0 0
$$649$$ 3.15601e29 0.934478
$$650$$ 0 0
$$651$$ 7.43864e28 0.213252
$$652$$ 0 0
$$653$$ 5.45219e29 1.51350 0.756751 0.653704i $$-0.226786\pi$$
0.756751 + 0.653704i $$0.226786\pi$$
$$654$$ 0 0
$$655$$ −7.17125e28 −0.192780
$$656$$ 0 0
$$657$$ −2.54662e29 −0.663022
$$658$$ 0 0
$$659$$ −3.88443e29 −0.979558 −0.489779 0.871847i $$-0.662923\pi$$
−0.489779 + 0.871847i $$0.662923\pi$$
$$660$$ 0 0
$$661$$ 5.22430e29 1.27618 0.638091 0.769961i $$-0.279724\pi$$
0.638091 + 0.769961i $$0.279724\pi$$
$$662$$ 0 0
$$663$$ −9.97864e28 −0.236145
$$664$$ 0 0
$$665$$ 7.82922e28 0.179511
$$666$$ 0 0
$$667$$ −3.08524e29 −0.685437
$$668$$ 0 0
$$669$$ −3.48195e29 −0.749633
$$670$$ 0 0
$$671$$ 8.52451e29 1.77862
$$672$$ 0 0
$$673$$ −3.52401e29 −0.712655 −0.356327 0.934361i $$-0.615971\pi$$
−0.356327 + 0.934361i $$0.615971\pi$$
$$674$$ 0 0
$$675$$ −4.66656e29 −0.914761
$$676$$ 0 0
$$677$$ 4.45535e29 0.846644 0.423322 0.905979i $$-0.360864\pi$$
0.423322 + 0.905979i $$0.360864\pi$$
$$678$$ 0 0
$$679$$ 1.03858e30 1.91341
$$680$$ 0 0
$$681$$ 3.63961e29 0.650144
$$682$$ 0 0
$$683$$ 6.32620e29 1.09579 0.547893 0.836548i $$-0.315430\pi$$
0.547893 + 0.836548i $$0.315430\pi$$
$$684$$ 0 0
$$685$$ −1.77930e29 −0.298881
$$686$$ 0 0
$$687$$ 1.81042e29 0.294940
$$688$$ 0 0
$$689$$ 8.92144e28 0.140972
$$690$$ 0 0
$$691$$ −3.28853e29 −0.504061 −0.252030 0.967719i $$-0.581098\pi$$
−0.252030 + 0.967719i $$0.581098\pi$$
$$692$$ 0 0
$$693$$ 1.05798e30 1.57318
$$694$$ 0 0
$$695$$ 1.04697e29 0.151040
$$696$$ 0 0
$$697$$ 2.75763e29 0.386004
$$698$$ 0 0
$$699$$ 4.42494e28 0.0601032
$$700$$ 0 0
$$701$$ −1.17358e30 −1.54694 −0.773471 0.633832i $$-0.781481\pi$$
−0.773471 + 0.633832i $$0.781481\pi$$
$$702$$ 0 0
$$703$$ −3.43179e29 −0.439025
$$704$$ 0 0
$$705$$ −5.68173e28 −0.0705496
$$706$$ 0 0
$$707$$ −8.47841e29 −1.02190
$$708$$ 0 0
$$709$$ 4.17850e28 0.0488917 0.0244458 0.999701i $$-0.492218\pi$$
0.0244458 + 0.999701i $$0.492218\pi$$
$$710$$ 0 0
$$711$$ 2.34920e29 0.266864
$$712$$ 0 0
$$713$$ −1.13775e29 −0.125489
$$714$$ 0 0
$$715$$ −7.97611e28 −0.0854236
$$716$$ 0 0
$$717$$ −6.08350e29 −0.632706
$$718$$ 0 0
$$719$$ −1.11149e30 −1.12267 −0.561336 0.827588i $$-0.689713\pi$$
−0.561336 + 0.827588i $$0.689713\pi$$
$$720$$ 0 0
$$721$$ −9.93077e29 −0.974233
$$722$$ 0 0
$$723$$ −8.69926e29 −0.828954
$$724$$ 0 0
$$725$$ −1.07703e30 −0.996962
$$726$$ 0 0
$$727$$ 1.37984e30 1.24084 0.620421 0.784269i $$-0.286962\pi$$
0.620421 + 0.784269i $$0.286962\pi$$
$$728$$ 0 0
$$729$$ 5.61417e29 0.490509
$$730$$ 0 0
$$731$$ −1.49446e30 −1.26868
$$732$$ 0 0
$$733$$ 2.24957e30 1.85570 0.927849 0.372956i $$-0.121656\pi$$
0.927849 + 0.372956i $$0.121656\pi$$
$$734$$ 0 0
$$735$$ −4.36479e29 −0.349902
$$736$$ 0 0
$$737$$ −5.16441e29 −0.402357
$$738$$ 0 0
$$739$$ 1.72254e30 1.30438 0.652188 0.758057i $$-0.273851\pi$$
0.652188 + 0.758057i $$0.273851\pi$$
$$740$$ 0 0
$$741$$ −9.81839e28 −0.0722685
$$742$$ 0 0
$$743$$ −1.06638e30 −0.763006 −0.381503 0.924368i $$-0.624593\pi$$
−0.381503 + 0.924368i $$0.624593\pi$$
$$744$$ 0 0
$$745$$ −4.33806e29 −0.301755
$$746$$ 0 0
$$747$$ 1.41765e30 0.958744
$$748$$ 0 0
$$749$$ −3.39954e30 −2.23543
$$750$$ 0 0
$$751$$ −4.90593e29 −0.313691 −0.156846 0.987623i $$-0.550132\pi$$
−0.156846 + 0.987623i $$0.550132\pi$$
$$752$$ 0 0
$$753$$ −9.06151e29 −0.563447
$$754$$ 0 0
$$755$$ 1.99346e29 0.120549
$$756$$ 0 0
$$757$$ 1.80527e30 1.06178 0.530891 0.847440i $$-0.321857\pi$$
0.530891 + 0.847440i $$0.321857\pi$$
$$758$$ 0 0
$$759$$ 8.20147e29 0.469195
$$760$$ 0 0
$$761$$ 1.03009e30 0.573240 0.286620 0.958044i $$-0.407468\pi$$
0.286620 + 0.958044i $$0.407468\pi$$
$$762$$ 0 0
$$763$$ −2.88216e30 −1.56031
$$764$$ 0 0
$$765$$ 3.86971e29 0.203814
$$766$$ 0 0
$$767$$ −4.43838e29 −0.227443
$$768$$ 0 0
$$769$$ 7.98485e29 0.398144 0.199072 0.979985i $$-0.436207\pi$$
0.199072 + 0.979985i $$0.436207\pi$$
$$770$$ 0 0
$$771$$ 2.17140e30 1.05358
$$772$$ 0 0
$$773$$ 2.56868e30 1.21290 0.606450 0.795122i $$-0.292593\pi$$
0.606450 + 0.795122i $$0.292593\pi$$
$$774$$ 0 0
$$775$$ −3.97179e29 −0.182523
$$776$$ 0 0
$$777$$ 2.63575e30 1.17892
$$778$$ 0 0
$$779$$ 2.71334e29 0.118130
$$780$$ 0 0
$$781$$ −9.44791e29 −0.400405
$$782$$ 0 0
$$783$$ 2.45917e30 1.01459
$$784$$ 0 0
$$785$$ −2.29086e29 −0.0920166
$$786$$ 0 0
$$787$$ 3.43359e30 1.34281 0.671403 0.741092i $$-0.265692\pi$$
0.671403 + 0.741092i $$0.265692\pi$$
$$788$$ 0 0
$$789$$ −1.03398e30 −0.393735
$$790$$ 0 0
$$791$$ −2.59754e30 −0.963184
$$792$$ 0 0
$$793$$ −1.19882e30 −0.432900
$$794$$ 0 0
$$795$$ 1.75349e29 0.0616664
$$796$$ 0 0
$$797$$ −3.29683e30 −1.12923 −0.564617 0.825353i $$-0.690976\pi$$
−0.564617 + 0.825353i $$0.690976\pi$$
$$798$$ 0 0
$$799$$ −2.15704e30 −0.719641
$$800$$ 0 0
$$801$$ 2.84788e29 0.0925507
$$802$$ 0 0
$$803$$ 3.91672e30 1.23996
$$804$$ 0 0
$$805$$ 9.19722e29 0.283661
$$806$$ 0 0
$$807$$ −2.74766e30 −0.825640
$$808$$ 0 0
$$809$$ 5.34854e30 1.56594 0.782971 0.622059i $$-0.213703\pi$$
0.782971 + 0.622059i $$0.213703\pi$$
$$810$$ 0 0
$$811$$ 5.96622e30 1.70208 0.851041 0.525099i $$-0.175972\pi$$
0.851041 + 0.525099i $$0.175972\pi$$
$$812$$ 0 0
$$813$$ 1.06674e30 0.296556
$$814$$ 0 0
$$815$$ −2.84653e29 −0.0771190
$$816$$ 0 0
$$817$$ −1.47046e30 −0.388258
$$818$$ 0 0
$$819$$ −1.48786e30 −0.382897
$$820$$ 0 0
$$821$$ 6.73645e30 1.68977 0.844886 0.534947i $$-0.179668\pi$$
0.844886 + 0.534947i $$0.179668\pi$$
$$822$$ 0 0
$$823$$ 2.93477e29 0.0717588 0.0358794 0.999356i $$-0.488577\pi$$
0.0358794 + 0.999356i $$0.488577\pi$$
$$824$$ 0 0
$$825$$ 2.86306e30 0.682440
$$826$$ 0 0
$$827$$ −6.88636e30 −1.60023 −0.800114 0.599847i $$-0.795228\pi$$
−0.800114 + 0.599847i $$0.795228\pi$$
$$828$$ 0 0
$$829$$ −4.58006e30 −1.03764 −0.518822 0.854882i $$-0.673629\pi$$
−0.518822 + 0.854882i $$0.673629\pi$$
$$830$$ 0 0
$$831$$ 6.38590e29 0.141062
$$832$$ 0 0
$$833$$ −1.65707e31 −3.56917
$$834$$ 0 0
$$835$$ 6.61290e29 0.138894
$$836$$ 0 0
$$837$$ 9.06874e29 0.185750
$$838$$ 0 0
$$839$$ −6.83025e30 −1.36438 −0.682191 0.731174i $$-0.738973\pi$$
−0.682191 + 0.731174i $$0.738973\pi$$
$$840$$ 0 0
$$841$$ 5.42848e29 0.105760
$$842$$ 0 0
$$843$$ 5.05877e30 0.961292
$$844$$ 0 0
$$845$$ −1.11706e30 −0.207053
$$846$$ 0 0
$$847$$ −5.70847e30 −1.03215
$$848$$ 0 0
$$849$$ 4.84479e28 0.00854563
$$850$$ 0 0
$$851$$ −4.03142e30 −0.693742
$$852$$ 0 0
$$853$$ 6.29163e30 1.05633 0.528164 0.849143i $$-0.322881\pi$$
0.528164 + 0.849143i $$0.322881\pi$$
$$854$$ 0 0
$$855$$ 3.80757e29 0.0623740
$$856$$ 0 0
$$857$$ −7.23134e30 −1.15590 −0.577950 0.816072i $$-0.696147\pi$$
−0.577950 + 0.816072i $$0.696147\pi$$
$$858$$ 0 0
$$859$$ 3.32473e30 0.518594 0.259297 0.965798i $$-0.416509\pi$$
0.259297 + 0.965798i $$0.416509\pi$$
$$860$$ 0 0
$$861$$ −2.08396e30 −0.317216
$$862$$ 0 0
$$863$$ 3.34330e30 0.496664 0.248332 0.968675i $$-0.420118\pi$$
0.248332 + 0.968675i $$0.420118\pi$$
$$864$$ 0 0
$$865$$ 2.75204e30 0.399012
$$866$$ 0 0
$$867$$ −3.34763e30 −0.473737
$$868$$ 0 0
$$869$$ −3.61309e30 −0.499081
$$870$$ 0 0
$$871$$ 7.26285e29 0.0979301
$$872$$ 0 0
$$873$$ 5.05092e30 0.664846
$$874$$ 0 0
$$875$$ 6.59714e30 0.847756
$$876$$ 0 0
$$877$$ −1.00355e31 −1.25905 −0.629523 0.776982i $$-0.716749\pi$$
−0.629523 + 0.776982i $$0.716749\pi$$
$$878$$ 0 0
$$879$$ −3.13984e30 −0.384613
$$880$$ 0 0
$$881$$ −2.14507e29 −0.0256563 −0.0128281 0.999918i $$-0.504083\pi$$
−0.0128281 + 0.999918i $$0.504083\pi$$
$$882$$ 0 0
$$883$$ −8.37555e30 −0.978196 −0.489098 0.872229i $$-0.662674\pi$$
−0.489098 + 0.872229i $$0.662674\pi$$
$$884$$ 0 0
$$885$$ −8.72355e29 −0.0994922
$$886$$ 0 0
$$887$$ 6.39937e30 0.712754 0.356377 0.934342i $$-0.384012\pi$$
0.356377 + 0.934342i $$0.384012\pi$$
$$888$$ 0 0
$$889$$ 3.14348e31 3.41934
$$890$$ 0 0
$$891$$ 1.21580e30 0.129165
$$892$$ 0 0
$$893$$ −2.12240e30 −0.220235
$$894$$ 0 0
$$895$$ 2.64429e29 0.0268020
$$896$$ 0 0
$$897$$ −1.15340e30 −0.114198
$$898$$ 0 0
$$899$$ 2.09304e30 0.202442
$$900$$ 0 0
$$901$$ 6.65703e30 0.629028
$$902$$ 0 0
$$903$$ 1.12937e31 1.04259
$$904$$ 0 0
$$905$$ 3.77803e30 0.340765
$$906$$ 0 0
$$907$$ −3.33701e30 −0.294091 −0.147045 0.989130i $$-0.546976\pi$$
−0.147045 + 0.989130i $$0.546976\pi$$
$$908$$ 0 0
$$909$$ −4.12328e30 −0.355077
$$910$$ 0 0
$$911$$ 5.75477e30 0.484267 0.242134 0.970243i $$-0.422153\pi$$
0.242134 + 0.970243i $$0.422153\pi$$
$$912$$ 0 0
$$913$$ −2.18036e31 −1.79302
$$914$$ 0 0
$$915$$ −2.35627e30 −0.189366
$$916$$ 0 0
$$917$$ 2.05743e31 1.61602
$$918$$ 0 0
$$919$$ 2.25413e31 1.73048 0.865239 0.501359i $$-0.167166\pi$$
0.865239 + 0.501359i $$0.167166\pi$$
$$920$$ 0 0
$$921$$ −1.06740e31 −0.800942
$$922$$ 0 0
$$923$$ 1.32868e30 0.0974550
$$924$$ 0 0
$$925$$ −1.40733e31 −1.00904
$$926$$ 0 0
$$927$$ −4.82961e30 −0.338513
$$928$$ 0 0
$$929$$ 2.59042e30 0.177503 0.0887513 0.996054i $$-0.471712\pi$$
0.0887513 + 0.996054i $$0.471712\pi$$
$$930$$ 0 0
$$931$$ −1.63046e31 −1.09229
$$932$$ 0 0
$$933$$ 1.48154e31 0.970414
$$934$$ 0 0
$$935$$ −5.95164e30 −0.381167
$$936$$ 0 0
$$937$$ 5.29691e30 0.331709 0.165854 0.986150i $$-0.446962\pi$$
0.165854 + 0.986150i $$0.446962\pi$$
$$938$$ 0 0
$$939$$ −1.27408e31 −0.780207
$$940$$ 0 0
$$941$$ 1.38757e31 0.830930 0.415465 0.909609i $$-0.363619\pi$$
0.415465 + 0.909609i $$0.363619\pi$$
$$942$$ 0 0
$$943$$ 3.18744e30 0.186668
$$944$$ 0 0
$$945$$ −7.33088e30 −0.419877
$$946$$ 0 0
$$947$$ 8.16878e30 0.457596 0.228798 0.973474i $$-0.426520\pi$$
0.228798 + 0.973474i $$0.426520\pi$$
$$948$$ 0 0
$$949$$ −5.50819e30 −0.301796
$$950$$ 0 0
$$951$$ −1.81774e31 −0.974172
$$952$$ 0 0
$$953$$ 2.02516e31 1.06166 0.530828 0.847479i $$-0.321881\pi$$
0.530828 + 0.847479i $$0.321881\pi$$
$$954$$ 0 0
$$955$$ 4.79959e30 0.246132
$$956$$ 0 0
$$957$$ −1.50877e31 −0.756914
$$958$$ 0 0
$$959$$ 5.10480e31 2.50544
$$960$$ 0 0
$$961$$ −2.00537e31 −0.962937
$$962$$ 0 0
$$963$$ −1.65329e31 −0.776734
$$964$$ 0 0
$$965$$ −1.69755e30 −0.0780343
$$966$$ 0 0
$$967$$ −1.34251e31 −0.603864 −0.301932 0.953329i $$-0.597632\pi$$
−0.301932 + 0.953329i $$0.597632\pi$$
$$968$$ 0 0
$$969$$ −7.32632e30 −0.322468
$$970$$ 0 0
$$971$$ −4.11059e31 −1.77053 −0.885263 0.465090i $$-0.846022\pi$$
−0.885263 + 0.465090i $$0.846022\pi$$
$$972$$ 0 0
$$973$$ −3.00374e31 −1.26613
$$974$$ 0 0
$$975$$ −4.02640e30 −0.166100
$$976$$ 0 0
$$977$$ 3.60772e30 0.145660 0.0728299 0.997344i $$-0.476797\pi$$
0.0728299 + 0.997344i $$0.476797\pi$$
$$978$$ 0 0
$$979$$ −4.38005e30 −0.173086
$$980$$ 0 0
$$981$$ −1.40167e31 −0.542154
$$982$$ 0 0
$$983$$ −3.62646e31 −1.37300 −0.686500 0.727130i $$-0.740854\pi$$
−0.686500 + 0.727130i $$0.740854\pi$$
$$984$$ 0 0
$$985$$ −2.49062e30 −0.0923053
$$986$$ 0 0
$$987$$ 1.63009e31 0.591399
$$988$$ 0 0
$$989$$ −1.72739e31 −0.613520
$$990$$ 0 0
$$991$$ 2.38846e31 0.830508 0.415254 0.909706i $$-0.363693\pi$$
0.415254 + 0.909706i $$0.363693\pi$$
$$992$$ 0 0
$$993$$ −6.22256e30 −0.211837
$$994$$ 0 0
$$995$$ 2.53681e30 0.0845561
$$996$$ 0 0
$$997$$ 5.34032e31 1.74288 0.871442 0.490499i $$-0.163186\pi$$
0.871442 + 0.490499i $$0.163186\pi$$
$$998$$ 0 0
$$999$$ 3.21335e31 1.02688
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 16.22.a.b.1.1 1
4.3 odd 2 2.22.a.b.1.1 1
8.3 odd 2 64.22.a.c.1.1 1
8.5 even 2 64.22.a.e.1.1 1
12.11 even 2 18.22.a.b.1.1 1
20.3 even 4 50.22.b.c.49.1 2
20.7 even 4 50.22.b.c.49.2 2
20.19 odd 2 50.22.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.b.1.1 1 4.3 odd 2
16.22.a.b.1.1 1 1.1 even 1 trivial
18.22.a.b.1.1 1 12.11 even 2
50.22.a.a.1.1 1 20.19 odd 2
50.22.b.c.49.1 2 20.3 even 4
50.22.b.c.49.2 2 20.7 even 4
64.22.a.c.1.1 1 8.3 odd 2
64.22.a.e.1.1 1 8.5 even 2