# Properties

 Label 16.22.a.a.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)

 $$f(q)$$ $$=$$ $$q-71604.0 q^{3} -2.86938e7 q^{5} +8.53202e8 q^{7} -5.33322e9 q^{9} +O(q^{10})$$ $$q-71604.0 q^{3} -2.86938e7 q^{5} +8.53202e8 q^{7} -5.33322e9 q^{9} -8.67312e10 q^{11} -8.95323e11 q^{13} +2.05459e12 q^{15} +3.25757e12 q^{17} -2.30325e13 q^{19} -6.10927e13 q^{21} -1.46496e14 q^{23} +3.46495e14 q^{25} +1.13088e15 q^{27} -7.34052e14 q^{29} +3.14666e15 q^{31} +6.21030e15 q^{33} -2.44816e16 q^{35} -1.29638e16 q^{37} +6.41087e16 q^{39} +4.57146e16 q^{41} +2.40736e16 q^{43} +1.53030e17 q^{45} +4.49992e17 q^{47} +1.69408e17 q^{49} -2.33255e17 q^{51} +2.06484e18 q^{53} +2.48864e18 q^{55} +1.64922e18 q^{57} +3.78050e18 q^{59} -7.61981e18 q^{61} -4.55032e18 q^{63} +2.56902e19 q^{65} +1.87912e19 q^{67} +1.04897e19 q^{69} +4.52649e18 q^{71} -2.55715e19 q^{73} -2.48104e19 q^{75} -7.39992e19 q^{77} -9.93364e19 q^{79} -2.51884e19 q^{81} -2.95818e18 q^{83} -9.34719e19 q^{85} +5.25610e19 q^{87} +1.18803e20 q^{89} -7.63892e20 q^{91} -2.25314e20 q^{93} +6.60888e20 q^{95} -5.69053e20 q^{97} +4.62556e20 q^{99} +O(q^{100})$$

## 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
$$3$$ −71604.0 −0.700106 −0.350053 0.936730i $$-0.613837\pi$$
−0.350053 + 0.936730i $$0.613837\pi$$
$$4$$ 0 0
$$5$$ −2.86938e7 −1.31402 −0.657011 0.753881i $$-0.728179\pi$$
−0.657011 + 0.753881i $$0.728179\pi$$
$$6$$ 0 0
$$7$$ 8.53202e8 1.14162 0.570811 0.821081i $$-0.306629\pi$$
0.570811 + 0.821081i $$0.306629\pi$$
$$8$$ 0 0
$$9$$ −5.33322e9 −0.509851
$$10$$ 0 0
$$11$$ −8.67312e10 −1.00821 −0.504106 0.863642i $$-0.668178\pi$$
−0.504106 + 0.863642i $$0.668178\pi$$
$$12$$ 0 0
$$13$$ −8.95323e11 −1.80125 −0.900627 0.434594i $$-0.856892\pi$$
−0.900627 + 0.434594i $$0.856892\pi$$
$$14$$ 0 0
$$15$$ 2.05459e12 0.919955
$$16$$ 0 0
$$17$$ 3.25757e12 0.391904 0.195952 0.980613i $$-0.437220\pi$$
0.195952 + 0.980613i $$0.437220\pi$$
$$18$$ 0 0
$$19$$ −2.30325e13 −0.861842 −0.430921 0.902390i $$-0.641811\pi$$
−0.430921 + 0.902390i $$0.641811\pi$$
$$20$$ 0 0
$$21$$ −6.10927e13 −0.799258
$$22$$ 0 0
$$23$$ −1.46496e14 −0.737365 −0.368683 0.929555i $$-0.620191\pi$$
−0.368683 + 0.929555i $$0.620191\pi$$
$$24$$ 0 0
$$25$$ 3.46495e14 0.726653
$$26$$ 0 0
$$27$$ 1.13088e15 1.05706
$$28$$ 0 0
$$29$$ −7.34052e14 −0.324002 −0.162001 0.986791i $$-0.551795\pi$$
−0.162001 + 0.986791i $$0.551795\pi$$
$$30$$ 0 0
$$31$$ 3.14666e15 0.689529 0.344765 0.938689i $$-0.387959\pi$$
0.344765 + 0.938689i $$0.387959\pi$$
$$32$$ 0 0
$$33$$ 6.21030e15 0.705856
$$34$$ 0 0
$$35$$ −2.44816e16 −1.50012
$$36$$ 0 0
$$37$$ −1.29638e16 −0.443215 −0.221608 0.975136i $$-0.571130\pi$$
−0.221608 + 0.975136i $$0.571130\pi$$
$$38$$ 0 0
$$39$$ 6.41087e16 1.26107
$$40$$ 0 0
$$41$$ 4.57146e16 0.531894 0.265947 0.963988i $$-0.414315\pi$$
0.265947 + 0.963988i $$0.414315\pi$$
$$42$$ 0 0
$$43$$ 2.40736e16 0.169872 0.0849361 0.996386i $$-0.472931\pi$$
0.0849361 + 0.996386i $$0.472931\pi$$
$$44$$ 0 0
$$45$$ 1.53030e17 0.669955
$$46$$ 0 0
$$47$$ 4.49992e17 1.24789 0.623946 0.781467i $$-0.285528\pi$$
0.623946 + 0.781467i $$0.285528\pi$$
$$48$$ 0 0
$$49$$ 1.69408e17 0.303303
$$50$$ 0 0
$$51$$ −2.33255e17 −0.274375
$$52$$ 0 0
$$53$$ 2.06484e18 1.62177 0.810885 0.585206i $$-0.198986\pi$$
0.810885 + 0.585206i $$0.198986\pi$$
$$54$$ 0 0
$$55$$ 2.48864e18 1.32481
$$56$$ 0 0
$$57$$ 1.64922e18 0.603381
$$58$$ 0 0
$$59$$ 3.78050e18 0.962948 0.481474 0.876460i $$-0.340102\pi$$
0.481474 + 0.876460i $$0.340102\pi$$
$$60$$ 0 0
$$61$$ −7.61981e18 −1.36767 −0.683835 0.729637i $$-0.739689\pi$$
−0.683835 + 0.729637i $$0.739689\pi$$
$$62$$ 0 0
$$63$$ −4.55032e18 −0.582057
$$64$$ 0 0
$$65$$ 2.56902e19 2.36689
$$66$$ 0 0
$$67$$ 1.87912e19 1.25941 0.629706 0.776833i $$-0.283175\pi$$
0.629706 + 0.776833i $$0.283175\pi$$
$$68$$ 0 0
$$69$$ 1.04897e19 0.516234
$$70$$ 0 0
$$71$$ 4.52649e18 0.165025 0.0825123 0.996590i $$-0.473706\pi$$
0.0825123 + 0.996590i $$0.473706\pi$$
$$72$$ 0 0
$$73$$ −2.55715e19 −0.696411 −0.348205 0.937418i $$-0.613209\pi$$
−0.348205 + 0.937418i $$0.613209\pi$$
$$74$$ 0 0
$$75$$ −2.48104e19 −0.508735
$$76$$ 0 0
$$77$$ −7.39992e19 −1.15100
$$78$$ 0 0
$$79$$ −9.93364e19 −1.18039 −0.590193 0.807262i $$-0.700948\pi$$
−0.590193 + 0.807262i $$0.700948\pi$$
$$80$$ 0 0
$$81$$ −2.51884e19 −0.230201
$$82$$ 0 0
$$83$$ −2.95818e18 −0.0209269 −0.0104634 0.999945i $$-0.503331\pi$$
−0.0104634 + 0.999945i $$0.503331\pi$$
$$84$$ 0 0
$$85$$ −9.34719e19 −0.514970
$$86$$ 0 0
$$87$$ 5.25610e19 0.226836
$$88$$ 0 0
$$89$$ 1.18803e20 0.403861 0.201931 0.979400i $$-0.435278\pi$$
0.201931 + 0.979400i $$0.435278\pi$$
$$90$$ 0 0
$$91$$ −7.63892e20 −2.05635
$$92$$ 0 0
$$93$$ −2.25314e20 −0.482744
$$94$$ 0 0
$$95$$ 6.60888e20 1.13248
$$96$$ 0 0
$$97$$ −5.69053e20 −0.783519 −0.391759 0.920068i $$-0.628133\pi$$
−0.391759 + 0.920068i $$0.628133\pi$$
$$98$$ 0 0
$$99$$ 4.62556e20 0.514038
$$100$$ 0 0
$$101$$ −2.65047e20 −0.238753 −0.119376 0.992849i $$-0.538090\pi$$
−0.119376 + 0.992849i $$0.538090\pi$$
$$102$$ 0 0
$$103$$ −1.51294e21 −1.10925 −0.554625 0.832100i $$-0.687138\pi$$
−0.554625 + 0.832100i $$0.687138\pi$$
$$104$$ 0 0
$$105$$ 1.75298e21 1.05024
$$106$$ 0 0
$$107$$ 1.78568e21 0.877552 0.438776 0.898596i $$-0.355412\pi$$
0.438776 + 0.898596i $$0.355412\pi$$
$$108$$ 0 0
$$109$$ 3.33894e21 1.35092 0.675462 0.737395i $$-0.263944\pi$$
0.675462 + 0.737395i $$0.263944\pi$$
$$110$$ 0 0
$$111$$ 9.28261e20 0.310298
$$112$$ 0 0
$$113$$ −3.04712e21 −0.844434 −0.422217 0.906495i $$-0.638748\pi$$
−0.422217 + 0.906495i $$0.638748\pi$$
$$114$$ 0 0
$$115$$ 4.20351e21 0.968914
$$116$$ 0 0
$$117$$ 4.77496e21 0.918371
$$118$$ 0 0
$$119$$ 2.77936e21 0.447407
$$120$$ 0 0
$$121$$ 1.22048e20 0.0164924
$$122$$ 0 0
$$123$$ −3.27335e21 −0.372382
$$124$$ 0 0
$$125$$ 3.74000e21 0.359184
$$126$$ 0 0
$$127$$ 7.04680e21 0.572866 0.286433 0.958100i $$-0.407530\pi$$
0.286433 + 0.958100i $$0.407530\pi$$
$$128$$ 0 0
$$129$$ −1.72377e21 −0.118929
$$130$$ 0 0
$$131$$ −9.42787e21 −0.553433 −0.276716 0.960952i $$-0.589246\pi$$
−0.276716 + 0.960952i $$0.589246\pi$$
$$132$$ 0 0
$$133$$ −1.96514e22 −0.983899
$$134$$ 0 0
$$135$$ −3.24493e22 −1.38900
$$136$$ 0 0
$$137$$ −2.01753e21 −0.0740039 −0.0370020 0.999315i $$-0.511781\pi$$
−0.0370020 + 0.999315i $$0.511781\pi$$
$$138$$ 0 0
$$139$$ −3.38653e22 −1.06684 −0.533420 0.845851i $$-0.679093\pi$$
−0.533420 + 0.845851i $$0.679093\pi$$
$$140$$ 0 0
$$141$$ −3.22212e22 −0.873658
$$142$$ 0 0
$$143$$ 7.76525e22 1.81605
$$144$$ 0 0
$$145$$ 2.10627e22 0.425746
$$146$$ 0 0
$$147$$ −1.21303e22 −0.212344
$$148$$ 0 0
$$149$$ −3.40800e22 −0.517658 −0.258829 0.965923i $$-0.583337\pi$$
−0.258829 + 0.965923i $$0.583337\pi$$
$$150$$ 0 0
$$151$$ −2.38856e22 −0.315413 −0.157706 0.987486i $$-0.550410\pi$$
−0.157706 + 0.987486i $$0.550410\pi$$
$$152$$ 0 0
$$153$$ −1.73733e22 −0.199813
$$154$$ 0 0
$$155$$ −9.02897e22 −0.906056
$$156$$ 0 0
$$157$$ 1.27315e23 1.11669 0.558345 0.829609i $$-0.311436\pi$$
0.558345 + 0.829609i $$0.311436\pi$$
$$158$$ 0 0
$$159$$ −1.47851e23 −1.13541
$$160$$ 0 0
$$161$$ −1.24990e23 −0.841793
$$162$$ 0 0
$$163$$ 2.36289e23 1.39789 0.698946 0.715175i $$-0.253653\pi$$
0.698946 + 0.715175i $$0.253653\pi$$
$$164$$ 0 0
$$165$$ −1.78197e23 −0.927510
$$166$$ 0 0
$$167$$ 3.15464e23 1.44686 0.723429 0.690398i $$-0.242565\pi$$
0.723429 + 0.690398i $$0.242565\pi$$
$$168$$ 0 0
$$169$$ 5.54540e23 2.24451
$$170$$ 0 0
$$171$$ 1.22837e23 0.439411
$$172$$ 0 0
$$173$$ 4.08353e23 1.29286 0.646431 0.762972i $$-0.276261\pi$$
0.646431 + 0.762972i $$0.276261\pi$$
$$174$$ 0 0
$$175$$ 2.95631e23 0.829564
$$176$$ 0 0
$$177$$ −2.70699e23 −0.674166
$$178$$ 0 0
$$179$$ −7.91545e23 −1.75194 −0.875970 0.482366i $$-0.839777\pi$$
−0.875970 + 0.482366i $$0.839777\pi$$
$$180$$ 0 0
$$181$$ −4.73055e23 −0.931722 −0.465861 0.884858i $$-0.654255\pi$$
−0.465861 + 0.884858i $$0.654255\pi$$
$$182$$ 0 0
$$183$$ 5.45609e23 0.957514
$$184$$ 0 0
$$185$$ 3.71981e23 0.582394
$$186$$ 0 0
$$187$$ −2.82533e23 −0.395122
$$188$$ 0 0
$$189$$ 9.64872e23 1.20676
$$190$$ 0 0
$$191$$ 4.27960e23 0.479239 0.239620 0.970867i $$-0.422977\pi$$
0.239620 + 0.970867i $$0.422977\pi$$
$$192$$ 0 0
$$193$$ −1.88476e24 −1.89193 −0.945964 0.324273i $$-0.894880\pi$$
−0.945964 + 0.324273i $$0.894880\pi$$
$$194$$ 0 0
$$195$$ −1.83952e24 −1.65707
$$196$$ 0 0
$$197$$ 8.32416e22 0.0673666 0.0336833 0.999433i $$-0.489276\pi$$
0.0336833 + 0.999433i $$0.489276\pi$$
$$198$$ 0 0
$$199$$ 8.81455e23 0.641568 0.320784 0.947152i $$-0.396054\pi$$
0.320784 + 0.947152i $$0.396054\pi$$
$$200$$ 0 0
$$201$$ −1.34552e24 −0.881723
$$202$$ 0 0
$$203$$ −6.26295e23 −0.369888
$$204$$ 0 0
$$205$$ −1.31173e24 −0.698920
$$206$$ 0 0
$$207$$ 7.81294e23 0.375946
$$208$$ 0 0
$$209$$ 1.99763e24 0.868920
$$210$$ 0 0
$$211$$ −3.79755e24 −1.49464 −0.747322 0.664462i $$-0.768661\pi$$
−0.747322 + 0.664462i $$0.768661\pi$$
$$212$$ 0 0
$$213$$ −3.24115e23 −0.115535
$$214$$ 0 0
$$215$$ −6.90763e23 −0.223216
$$216$$ 0 0
$$217$$ 2.68474e24 0.787182
$$218$$ 0 0
$$219$$ 1.83102e24 0.487562
$$220$$ 0 0
$$221$$ −2.91658e24 −0.705918
$$222$$ 0 0
$$223$$ 3.92011e24 0.863171 0.431586 0.902072i $$-0.357954\pi$$
0.431586 + 0.902072i $$0.357954\pi$$
$$224$$ 0 0
$$225$$ −1.84794e24 −0.370485
$$226$$ 0 0
$$227$$ 1.12173e24 0.204936 0.102468 0.994736i $$-0.467326\pi$$
0.102468 + 0.994736i $$0.467326\pi$$
$$228$$ 0 0
$$229$$ 3.79307e23 0.0632002 0.0316001 0.999501i $$-0.489940\pi$$
0.0316001 + 0.999501i $$0.489940\pi$$
$$230$$ 0 0
$$231$$ 5.29864e24 0.805821
$$232$$ 0 0
$$233$$ 4.03014e24 0.559865 0.279933 0.960020i $$-0.409688\pi$$
0.279933 + 0.960020i $$0.409688\pi$$
$$234$$ 0 0
$$235$$ −1.29120e25 −1.63976
$$236$$ 0 0
$$237$$ 7.11289e24 0.826396
$$238$$ 0 0
$$239$$ 1.75220e25 1.86383 0.931916 0.362673i $$-0.118136\pi$$
0.931916 + 0.362673i $$0.118136\pi$$
$$240$$ 0 0
$$241$$ 9.55213e24 0.930939 0.465470 0.885064i $$-0.345885\pi$$
0.465470 + 0.885064i $$0.345885\pi$$
$$242$$ 0 0
$$243$$ −1.00258e25 −0.895891
$$244$$ 0 0
$$245$$ −4.86097e24 −0.398546
$$246$$ 0 0
$$247$$ 2.06215e25 1.55240
$$248$$ 0 0
$$249$$ 2.11818e23 0.0146511
$$250$$ 0 0
$$251$$ −6.29655e24 −0.400432 −0.200216 0.979752i $$-0.564164\pi$$
−0.200216 + 0.979752i $$0.564164\pi$$
$$252$$ 0 0
$$253$$ 1.27057e25 0.743421
$$254$$ 0 0
$$255$$ 6.69296e24 0.360534
$$256$$ 0 0
$$257$$ −1.45248e25 −0.720794 −0.360397 0.932799i $$-0.617359\pi$$
−0.360397 + 0.932799i $$0.617359\pi$$
$$258$$ 0 0
$$259$$ −1.10608e25 −0.505985
$$260$$ 0 0
$$261$$ 3.91486e24 0.165193
$$262$$ 0 0
$$263$$ −4.13430e25 −1.61015 −0.805075 0.593173i $$-0.797875\pi$$
−0.805075 + 0.593173i $$0.797875\pi$$
$$264$$ 0 0
$$265$$ −5.92480e25 −2.13104
$$266$$ 0 0
$$267$$ −8.50677e24 −0.282746
$$268$$ 0 0
$$269$$ −5.18786e25 −1.59437 −0.797185 0.603735i $$-0.793678\pi$$
−0.797185 + 0.603735i $$0.793678\pi$$
$$270$$ 0 0
$$271$$ 2.35538e25 0.669706 0.334853 0.942270i $$-0.391313\pi$$
0.334853 + 0.942270i $$0.391313\pi$$
$$272$$ 0 0
$$273$$ 5.46977e25 1.43967
$$274$$ 0 0
$$275$$ −3.00519e25 −0.732621
$$276$$ 0 0
$$277$$ 5.29041e25 1.19523 0.597615 0.801783i $$-0.296115\pi$$
0.597615 + 0.801783i $$0.296115\pi$$
$$278$$ 0 0
$$279$$ −1.67819e25 −0.351557
$$280$$ 0 0
$$281$$ 2.17922e25 0.423530 0.211765 0.977321i $$-0.432079\pi$$
0.211765 + 0.977321i $$0.432079\pi$$
$$282$$ 0 0
$$283$$ 5.39243e24 0.0972808 0.0486404 0.998816i $$-0.484511\pi$$
0.0486404 + 0.998816i $$0.484511\pi$$
$$284$$ 0 0
$$285$$ −4.73222e25 −0.792856
$$286$$ 0 0
$$287$$ 3.90038e25 0.607222
$$288$$ 0 0
$$289$$ −5.84802e25 −0.846411
$$290$$ 0 0
$$291$$ 4.07465e25 0.548547
$$292$$ 0 0
$$293$$ 3.92947e25 0.492293 0.246147 0.969233i $$-0.420836\pi$$
0.246147 + 0.969233i $$0.420836\pi$$
$$294$$ 0 0
$$295$$ −1.08477e26 −1.26533
$$296$$ 0 0
$$297$$ −9.80828e25 −1.06574
$$298$$ 0 0
$$299$$ 1.31161e26 1.32818
$$300$$ 0 0
$$301$$ 2.05397e25 0.193930
$$302$$ 0 0
$$303$$ 1.89784e25 0.167152
$$304$$ 0 0
$$305$$ 2.18641e26 1.79715
$$306$$ 0 0
$$307$$ −1.23827e26 −0.950306 −0.475153 0.879903i $$-0.657607\pi$$
−0.475153 + 0.879903i $$0.657607\pi$$
$$308$$ 0 0
$$309$$ 1.08332e26 0.776593
$$310$$ 0 0
$$311$$ 1.58511e26 1.06188 0.530941 0.847409i $$-0.321839\pi$$
0.530941 + 0.847409i $$0.321839\pi$$
$$312$$ 0 0
$$313$$ 1.15882e26 0.725770 0.362885 0.931834i $$-0.381792\pi$$
0.362885 + 0.931834i $$0.381792\pi$$
$$314$$ 0 0
$$315$$ 1.30566e26 0.764836
$$316$$ 0 0
$$317$$ −5.09758e25 −0.279410 −0.139705 0.990193i $$-0.544615\pi$$
−0.139705 + 0.990193i $$0.544615\pi$$
$$318$$ 0 0
$$319$$ 6.36652e25 0.326663
$$320$$ 0 0
$$321$$ −1.27862e26 −0.614380
$$322$$ 0 0
$$323$$ −7.50298e25 −0.337759
$$324$$ 0 0
$$325$$ −3.10225e26 −1.30889
$$326$$ 0 0
$$327$$ −2.39082e26 −0.945790
$$328$$ 0 0
$$329$$ 3.83934e26 1.42462
$$330$$ 0 0
$$331$$ 4.64134e26 1.61603 0.808015 0.589162i $$-0.200542\pi$$
0.808015 + 0.589162i $$0.200542\pi$$
$$332$$ 0 0
$$333$$ 6.91389e25 0.225974
$$334$$ 0 0
$$335$$ −5.39189e26 −1.65490
$$336$$ 0 0
$$337$$ −2.39093e26 −0.689371 −0.344686 0.938718i $$-0.612014\pi$$
−0.344686 + 0.938718i $$0.612014\pi$$
$$338$$ 0 0
$$339$$ 2.18186e26 0.591194
$$340$$ 0 0
$$341$$ −2.72914e26 −0.695192
$$342$$ 0 0
$$343$$ −3.32013e26 −0.795366
$$344$$ 0 0
$$345$$ −3.00988e26 −0.678343
$$346$$ 0 0
$$347$$ 3.01925e26 0.640383 0.320192 0.947353i $$-0.396253\pi$$
0.320192 + 0.947353i $$0.396253\pi$$
$$348$$ 0 0
$$349$$ 4.92021e26 0.982464 0.491232 0.871029i $$-0.336547\pi$$
0.491232 + 0.871029i $$0.336547\pi$$
$$350$$ 0 0
$$351$$ −1.01251e27 −1.90403
$$352$$ 0 0
$$353$$ −6.69643e26 −1.18634 −0.593169 0.805078i $$-0.702124\pi$$
−0.593169 + 0.805078i $$0.702124\pi$$
$$354$$ 0 0
$$355$$ −1.29882e26 −0.216846
$$356$$ 0 0
$$357$$ −1.99014e26 −0.313232
$$358$$ 0 0
$$359$$ −5.89124e26 −0.874410 −0.437205 0.899362i $$-0.644032\pi$$
−0.437205 + 0.899362i $$0.644032\pi$$
$$360$$ 0 0
$$361$$ −1.83715e26 −0.257228
$$362$$ 0 0
$$363$$ −8.73909e24 −0.0115464
$$364$$ 0 0
$$365$$ 7.33741e26 0.915099
$$366$$ 0 0
$$367$$ 1.61827e27 1.90571 0.952854 0.303428i $$-0.0981313\pi$$
0.952854 + 0.303428i $$0.0981313\pi$$
$$368$$ 0 0
$$369$$ −2.43806e26 −0.271187
$$370$$ 0 0
$$371$$ 1.76172e27 1.85145
$$372$$ 0 0
$$373$$ 1.12579e26 0.111819 0.0559095 0.998436i $$-0.482194\pi$$
0.0559095 + 0.998436i $$0.482194\pi$$
$$374$$ 0 0
$$375$$ −2.67799e26 −0.251467
$$376$$ 0 0
$$377$$ 6.57214e26 0.583610
$$378$$ 0 0
$$379$$ −1.61687e27 −1.35820 −0.679101 0.734045i $$-0.737630\pi$$
−0.679101 + 0.734045i $$0.737630\pi$$
$$380$$ 0 0
$$381$$ −5.04579e26 −0.401067
$$382$$ 0 0
$$383$$ 7.30139e25 0.0549311 0.0274656 0.999623i $$-0.491256\pi$$
0.0274656 + 0.999623i $$0.491256\pi$$
$$384$$ 0 0
$$385$$ 2.12332e27 1.51244
$$386$$ 0 0
$$387$$ −1.28390e26 −0.0866095
$$388$$ 0 0
$$389$$ 4.38998e26 0.280538 0.140269 0.990113i $$-0.455203\pi$$
0.140269 + 0.990113i $$0.455203\pi$$
$$390$$ 0 0
$$391$$ −4.77220e26 −0.288976
$$392$$ 0 0
$$393$$ 6.75073e26 0.387462
$$394$$ 0 0
$$395$$ 2.85034e27 1.55105
$$396$$ 0 0
$$397$$ −1.33852e27 −0.690756 −0.345378 0.938464i $$-0.612249\pi$$
−0.345378 + 0.938464i $$0.612249\pi$$
$$398$$ 0 0
$$399$$ 1.40712e27 0.688834
$$400$$ 0 0
$$401$$ −2.07292e27 −0.962870 −0.481435 0.876482i $$-0.659884\pi$$
−0.481435 + 0.876482i $$0.659884\pi$$
$$402$$ 0 0
$$403$$ −2.81728e27 −1.24202
$$404$$ 0 0
$$405$$ 7.22750e26 0.302489
$$406$$ 0 0
$$407$$ 1.12437e27 0.446855
$$408$$ 0 0
$$409$$ −4.12924e26 −0.155874 −0.0779372 0.996958i $$-0.524833\pi$$
−0.0779372 + 0.996958i $$0.524833\pi$$
$$410$$ 0 0
$$411$$ 1.44463e26 0.0518106
$$412$$ 0 0
$$413$$ 3.22553e27 1.09932
$$414$$ 0 0
$$415$$ 8.48813e25 0.0274984
$$416$$ 0 0
$$417$$ 2.42489e27 0.746901
$$418$$ 0 0
$$419$$ 3.55921e27 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$420$$ 0 0
$$421$$ −3.99663e27 −1.11361 −0.556804 0.830644i $$-0.687972\pi$$
−0.556804 + 0.830644i $$0.687972\pi$$
$$422$$ 0 0
$$423$$ −2.39991e27 −0.636239
$$424$$ 0 0
$$425$$ 1.12873e27 0.284778
$$426$$ 0 0
$$427$$ −6.50124e27 −1.56136
$$428$$ 0 0
$$429$$ −5.56023e27 −1.27143
$$430$$ 0 0
$$431$$ −5.74296e27 −1.25062 −0.625308 0.780378i $$-0.715027\pi$$
−0.625308 + 0.780378i $$0.715027\pi$$
$$432$$ 0 0
$$433$$ −6.99214e27 −1.45040 −0.725199 0.688540i $$-0.758252\pi$$
−0.725199 + 0.688540i $$0.758252\pi$$
$$434$$ 0 0
$$435$$ −1.50817e27 −0.298067
$$436$$ 0 0
$$437$$ 3.37416e27 0.635492
$$438$$ 0 0
$$439$$ 8.74047e27 1.56912 0.784562 0.620050i $$-0.212888\pi$$
0.784562 + 0.620050i $$0.212888\pi$$
$$440$$ 0 0
$$441$$ −9.03493e26 −0.154639
$$442$$ 0 0
$$443$$ −1.15967e28 −1.89276 −0.946380 0.323055i $$-0.895290\pi$$
−0.946380 + 0.323055i $$0.895290\pi$$
$$444$$ 0 0
$$445$$ −3.40891e27 −0.530683
$$446$$ 0 0
$$447$$ 2.44026e27 0.362416
$$448$$ 0 0
$$449$$ 6.44084e27 0.912758 0.456379 0.889785i $$-0.349146\pi$$
0.456379 + 0.889785i $$0.349146\pi$$
$$450$$ 0 0
$$451$$ −3.96489e27 −0.536262
$$452$$ 0 0
$$453$$ 1.71031e27 0.220822
$$454$$ 0 0
$$455$$ 2.19189e28 2.70209
$$456$$ 0 0
$$457$$ −8.27692e27 −0.974425 −0.487212 0.873283i $$-0.661986\pi$$
−0.487212 + 0.873283i $$0.661986\pi$$
$$458$$ 0 0
$$459$$ 3.68393e27 0.414265
$$460$$ 0 0
$$461$$ 4.93315e27 0.529986 0.264993 0.964250i $$-0.414630\pi$$
0.264993 + 0.964250i $$0.414630\pi$$
$$462$$ 0 0
$$463$$ 4.15139e26 0.0426181 0.0213090 0.999773i $$-0.493217\pi$$
0.0213090 + 0.999773i $$0.493217\pi$$
$$464$$ 0 0
$$465$$ 6.46510e27 0.634336
$$466$$ 0 0
$$467$$ −7.30716e27 −0.685364 −0.342682 0.939452i $$-0.611335\pi$$
−0.342682 + 0.939452i $$0.611335\pi$$
$$468$$ 0 0
$$469$$ 1.60327e28 1.43777
$$470$$ 0 0
$$471$$ −9.11626e27 −0.781802
$$472$$ 0 0
$$473$$ −2.08793e27 −0.171267
$$474$$ 0 0
$$475$$ −7.98064e27 −0.626260
$$476$$ 0 0
$$477$$ −1.10122e28 −0.826861
$$478$$ 0 0
$$479$$ 1.30805e28 0.939946 0.469973 0.882681i $$-0.344264\pi$$
0.469973 + 0.882681i $$0.344264\pi$$
$$480$$ 0 0
$$481$$ 1.16068e28 0.798343
$$482$$ 0 0
$$483$$ 8.94982e27 0.589345
$$484$$ 0 0
$$485$$ 1.63283e28 1.02956
$$486$$ 0 0
$$487$$ 1.68131e28 1.01530 0.507650 0.861564i $$-0.330514\pi$$
0.507650 + 0.861564i $$0.330514\pi$$
$$488$$ 0 0
$$489$$ −1.69192e28 −0.978673
$$490$$ 0 0
$$491$$ 2.67382e28 1.48175 0.740877 0.671641i $$-0.234410\pi$$
0.740877 + 0.671641i $$0.234410\pi$$
$$492$$ 0 0
$$493$$ −2.39122e27 −0.126978
$$494$$ 0 0
$$495$$ −1.32725e28 −0.675457
$$496$$ 0 0
$$497$$ 3.86201e27 0.188396
$$498$$ 0 0
$$499$$ −1.92819e28 −0.901765 −0.450882 0.892583i $$-0.648891\pi$$
−0.450882 + 0.892583i $$0.648891\pi$$
$$500$$ 0 0
$$501$$ −2.25884e28 −1.01296
$$502$$ 0 0
$$503$$ −2.46340e27 −0.105943 −0.0529714 0.998596i $$-0.516869\pi$$
−0.0529714 + 0.998596i $$0.516869\pi$$
$$504$$ 0 0
$$505$$ 7.60520e27 0.313726
$$506$$ 0 0
$$507$$ −3.97072e28 −1.57140
$$508$$ 0 0
$$509$$ 2.34782e28 0.891513 0.445757 0.895154i $$-0.352935\pi$$
0.445757 + 0.895154i $$0.352935\pi$$
$$510$$ 0 0
$$511$$ −2.18176e28 −0.795038
$$512$$ 0 0
$$513$$ −2.60470e28 −0.911016
$$514$$ 0 0
$$515$$ 4.34118e28 1.45758
$$516$$ 0 0
$$517$$ −3.90283e28 −1.25814
$$518$$ 0 0
$$519$$ −2.92397e28 −0.905141
$$520$$ 0 0
$$521$$ 5.65820e28 1.68222 0.841109 0.540865i $$-0.181903\pi$$
0.841109 + 0.540865i $$0.181903\pi$$
$$522$$ 0 0
$$523$$ 2.74538e28 0.784033 0.392016 0.919958i $$-0.371778\pi$$
0.392016 + 0.919958i $$0.371778\pi$$
$$524$$ 0 0
$$525$$ −2.11683e28 −0.580783
$$526$$ 0 0
$$527$$ 1.02505e28 0.270229
$$528$$ 0 0
$$529$$ −1.80106e28 −0.456293
$$530$$ 0 0
$$531$$ −2.01622e28 −0.490960
$$532$$ 0 0
$$533$$ −4.09294e28 −0.958075
$$534$$ 0 0
$$535$$ −5.12378e28 −1.15312
$$536$$ 0 0
$$537$$ 5.66778e28 1.22654
$$538$$ 0 0
$$539$$ −1.46930e28 −0.305794
$$540$$ 0 0
$$541$$ −1.54302e28 −0.308888 −0.154444 0.988002i $$-0.549359\pi$$
−0.154444 + 0.988002i $$0.549359\pi$$
$$542$$ 0 0
$$543$$ 3.38726e28 0.652305
$$544$$ 0 0
$$545$$ −9.58069e28 −1.77514
$$546$$ 0 0
$$547$$ −8.66492e27 −0.154489 −0.0772445 0.997012i $$-0.524612\pi$$
−0.0772445 + 0.997012i $$0.524612\pi$$
$$548$$ 0 0
$$549$$ 4.06381e28 0.697307
$$550$$ 0 0
$$551$$ 1.69070e28 0.279238
$$552$$ 0 0
$$553$$ −8.47541e28 −1.34756
$$554$$ 0 0
$$555$$ −2.66353e28 −0.407738
$$556$$ 0 0
$$557$$ 1.12653e29 1.66060 0.830300 0.557317i $$-0.188169\pi$$
0.830300 + 0.557317i $$0.188169\pi$$
$$558$$ 0 0
$$559$$ −2.15537e28 −0.305983
$$560$$ 0 0
$$561$$ 2.02305e28 0.276628
$$562$$ 0 0
$$563$$ 1.30345e29 1.71695 0.858475 0.512856i $$-0.171412\pi$$
0.858475 + 0.512856i $$0.171412\pi$$
$$564$$ 0 0
$$565$$ 8.74333e28 1.10960
$$566$$ 0 0
$$567$$ −2.14908e28 −0.262803
$$568$$ 0 0
$$569$$ 1.64552e29 1.93921 0.969606 0.244671i $$-0.0786800\pi$$
0.969606 + 0.244671i $$0.0786800\pi$$
$$570$$ 0 0
$$571$$ 9.69589e28 1.10131 0.550654 0.834734i $$-0.314378\pi$$
0.550654 + 0.834734i $$0.314378\pi$$
$$572$$ 0 0
$$573$$ −3.06436e28 −0.335519
$$574$$ 0 0
$$575$$ −5.07601e28 −0.535809
$$576$$ 0 0
$$577$$ 1.25423e28 0.127653 0.0638263 0.997961i $$-0.479670\pi$$
0.0638263 + 0.997961i $$0.479670\pi$$
$$578$$ 0 0
$$579$$ 1.34956e29 1.32455
$$580$$ 0 0
$$581$$ −2.52393e27 −0.0238906
$$582$$ 0 0
$$583$$ −1.79086e29 −1.63509
$$584$$ 0 0
$$585$$ −1.37012e29 −1.20676
$$586$$ 0 0
$$587$$ 2.13236e29 1.81201 0.906003 0.423271i $$-0.139118\pi$$
0.906003 + 0.423271i $$0.139118\pi$$
$$588$$ 0 0
$$589$$ −7.24754e28 −0.594265
$$590$$ 0 0
$$591$$ −5.96043e27 −0.0471638
$$592$$ 0 0
$$593$$ 5.74182e28 0.438506 0.219253 0.975668i $$-0.429638\pi$$
0.219253 + 0.975668i $$0.429638\pi$$
$$594$$ 0 0
$$595$$ −7.97504e28 −0.587902
$$596$$ 0 0
$$597$$ −6.31157e28 −0.449166
$$598$$ 0 0
$$599$$ 1.18461e28 0.0813941 0.0406970 0.999172i $$-0.487042\pi$$
0.0406970 + 0.999172i $$0.487042\pi$$
$$600$$ 0 0
$$601$$ 1.60567e28 0.106531 0.0532653 0.998580i $$-0.483037\pi$$
0.0532653 + 0.998580i $$0.483037\pi$$
$$602$$ 0 0
$$603$$ −1.00217e29 −0.642113
$$604$$ 0 0
$$605$$ −3.50200e27 −0.0216713
$$606$$ 0 0
$$607$$ −9.90701e28 −0.592190 −0.296095 0.955158i $$-0.595685\pi$$
−0.296095 + 0.955158i $$0.595685\pi$$
$$608$$ 0 0
$$609$$ 4.48452e28 0.258961
$$610$$ 0 0
$$611$$ −4.02888e29 −2.24777
$$612$$ 0 0
$$613$$ 1.45877e29 0.786416 0.393208 0.919450i $$-0.371365\pi$$
0.393208 + 0.919450i $$0.371365\pi$$
$$614$$ 0 0
$$615$$ 9.39248e28 0.489318
$$616$$ 0 0
$$617$$ −9.52438e28 −0.479560 −0.239780 0.970827i $$-0.577075\pi$$
−0.239780 + 0.970827i $$0.577075\pi$$
$$618$$ 0 0
$$619$$ 3.36526e29 1.63782 0.818911 0.573921i $$-0.194578\pi$$
0.818911 + 0.573921i $$0.194578\pi$$
$$620$$ 0 0
$$621$$ −1.65670e29 −0.779437
$$622$$ 0 0
$$623$$ 1.01363e29 0.461057
$$624$$ 0 0
$$625$$ −2.72537e29 −1.19863
$$626$$ 0 0
$$627$$ −1.43039e29 −0.608336
$$628$$ 0 0
$$629$$ −4.22305e28 −0.173698
$$630$$ 0 0
$$631$$ −8.77108e27 −0.0348935 −0.0174468 0.999848i $$-0.505554\pi$$
−0.0174468 + 0.999848i $$0.505554\pi$$
$$632$$ 0 0
$$633$$ 2.71920e29 1.04641
$$634$$ 0 0
$$635$$ −2.02199e29 −0.752758
$$636$$ 0 0
$$637$$ −1.51675e29 −0.546325
$$638$$ 0 0
$$639$$ −2.41408e28 −0.0841379
$$640$$ 0 0
$$641$$ 4.29903e28 0.144998 0.0724988 0.997368i $$-0.476903\pi$$
0.0724988 + 0.997368i $$0.476903\pi$$
$$642$$ 0 0
$$643$$ −4.61999e28 −0.150809 −0.0754043 0.997153i $$-0.524025\pi$$
−0.0754043 + 0.997153i $$0.524025\pi$$
$$644$$ 0 0
$$645$$ 4.94614e28 0.156275
$$646$$ 0 0
$$647$$ 4.54118e28 0.138891 0.0694455 0.997586i $$-0.477877\pi$$
0.0694455 + 0.997586i $$0.477877\pi$$
$$648$$ 0 0
$$649$$ −3.27887e29 −0.970856
$$650$$ 0 0
$$651$$ −1.92238e29 −0.551111
$$652$$ 0 0
$$653$$ −1.64556e29 −0.456799 −0.228400 0.973567i $$-0.573349\pi$$
−0.228400 + 0.973567i $$0.573349\pi$$
$$654$$ 0 0
$$655$$ 2.70521e29 0.727223
$$656$$ 0 0
$$657$$ 1.36378e29 0.355066
$$658$$ 0 0
$$659$$ −2.21390e29 −0.558291 −0.279146 0.960249i $$-0.590051\pi$$
−0.279146 + 0.960249i $$0.590051\pi$$
$$660$$ 0 0
$$661$$ 2.54395e29 0.621431 0.310716 0.950503i $$-0.399431\pi$$
0.310716 + 0.950503i $$0.399431\pi$$
$$662$$ 0 0
$$663$$ 2.08839e29 0.494218
$$664$$ 0 0
$$665$$ 5.63872e29 1.29286
$$666$$ 0 0
$$667$$ 1.07535e29 0.238908
$$668$$ 0 0
$$669$$ −2.80695e29 −0.604312
$$670$$ 0 0
$$671$$ 6.60875e29 1.37890
$$672$$ 0 0
$$673$$ 7.99394e29 1.61660 0.808300 0.588770i $$-0.200388\pi$$
0.808300 + 0.588770i $$0.200388\pi$$
$$674$$ 0 0
$$675$$ 3.91846e29 0.768114
$$676$$ 0 0
$$677$$ 4.61947e29 0.877831 0.438916 0.898528i $$-0.355363\pi$$
0.438916 + 0.898528i $$0.355363\pi$$
$$678$$ 0 0
$$679$$ −4.85517e29 −0.894483
$$680$$ 0 0
$$681$$ −8.03207e28 −0.143477
$$682$$ 0 0
$$683$$ −3.35857e29 −0.581751 −0.290876 0.956761i $$-0.593947\pi$$
−0.290876 + 0.956761i $$0.593947\pi$$
$$684$$ 0 0
$$685$$ 5.78906e28 0.0972428
$$686$$ 0 0
$$687$$ −2.71599e28 −0.0442469
$$688$$ 0 0
$$689$$ −1.84870e30 −2.92122
$$690$$ 0 0
$$691$$ 4.41273e29 0.676376 0.338188 0.941079i $$-0.390186\pi$$
0.338188 + 0.941079i $$0.390186\pi$$
$$692$$ 0 0
$$693$$ 3.94654e29 0.586838
$$694$$ 0 0
$$695$$ 9.71722e29 1.40185
$$696$$ 0 0
$$697$$ 1.48919e29 0.208451
$$698$$ 0 0
$$699$$ −2.88574e29 −0.391965
$$700$$ 0 0
$$701$$ 1.23969e29 0.163408 0.0817040 0.996657i $$-0.473964\pi$$
0.0817040 + 0.996657i $$0.473964\pi$$
$$702$$ 0 0
$$703$$ 2.98589e29 0.381982
$$704$$ 0 0
$$705$$ 9.24548e29 1.14801
$$706$$ 0 0
$$707$$ −2.26139e29 −0.272566
$$708$$ 0 0
$$709$$ 1.27607e30 1.49311 0.746553 0.665326i $$-0.231708\pi$$
0.746553 + 0.665326i $$0.231708\pi$$
$$710$$ 0 0
$$711$$ 5.29783e29 0.601821
$$712$$ 0 0
$$713$$ −4.60973e29 −0.508435
$$714$$ 0 0
$$715$$ −2.22814e30 −2.38632
$$716$$ 0 0
$$717$$ −1.25465e30 −1.30488
$$718$$ 0 0
$$719$$ −1.07039e30 −1.08116 −0.540580 0.841293i $$-0.681795\pi$$
−0.540580 + 0.841293i $$0.681795\pi$$
$$720$$ 0 0
$$721$$ −1.29084e30 −1.26635
$$722$$ 0 0
$$723$$ −6.83971e29 −0.651757
$$724$$ 0 0
$$725$$ −2.54345e29 −0.235437
$$726$$ 0 0
$$727$$ 7.64279e29 0.687291 0.343645 0.939100i $$-0.388338\pi$$
0.343645 + 0.939100i $$0.388338\pi$$
$$728$$ 0 0
$$729$$ 9.81370e29 0.857420
$$730$$ 0 0
$$731$$ 7.84214e28 0.0665736
$$732$$ 0 0
$$733$$ −1.18755e30 −0.979629 −0.489814 0.871827i $$-0.662935\pi$$
−0.489814 + 0.871827i $$0.662935\pi$$
$$734$$ 0 0
$$735$$ 3.48065e29 0.279025
$$736$$ 0 0
$$737$$ −1.62978e30 −1.26976
$$738$$ 0 0
$$739$$ −1.62225e29 −0.122843 −0.0614217 0.998112i $$-0.519563\pi$$
−0.0614217 + 0.998112i $$0.519563\pi$$
$$740$$ 0 0
$$741$$ −1.47658e30 −1.08684
$$742$$ 0 0
$$743$$ −1.09499e30 −0.783478 −0.391739 0.920076i $$-0.628126\pi$$
−0.391739 + 0.920076i $$0.628126\pi$$
$$744$$ 0 0
$$745$$ 9.77883e29 0.680214
$$746$$ 0 0
$$747$$ 1.57766e28 0.0106696
$$748$$ 0 0
$$749$$ 1.52354e30 1.00183
$$750$$ 0 0
$$751$$ 1.43876e30 0.919962 0.459981 0.887929i $$-0.347856\pi$$
0.459981 + 0.887929i $$0.347856\pi$$
$$752$$ 0 0
$$753$$ 4.50858e29 0.280345
$$754$$ 0 0
$$755$$ 6.85369e29 0.414459
$$756$$ 0 0
$$757$$ 2.13374e29 0.125498 0.0627488 0.998029i $$-0.480013\pi$$
0.0627488 + 0.998029i $$0.480013\pi$$
$$758$$ 0 0
$$759$$ −9.09782e29 −0.520474
$$760$$ 0 0
$$761$$ 3.15854e30 1.75771 0.878857 0.477086i $$-0.158307\pi$$
0.878857 + 0.477086i $$0.158307\pi$$
$$762$$ 0 0
$$763$$ 2.84879e30 1.54225
$$764$$ 0 0
$$765$$ 4.98506e29 0.262558
$$766$$ 0 0
$$767$$ −3.38477e30 −1.73451
$$768$$ 0 0
$$769$$ −1.20266e30 −0.599675 −0.299838 0.953990i $$-0.596933\pi$$
−0.299838 + 0.953990i $$0.596933\pi$$
$$770$$ 0 0
$$771$$ 1.04003e30 0.504633
$$772$$ 0 0
$$773$$ −2.33922e30 −1.10455 −0.552277 0.833661i $$-0.686241\pi$$
−0.552277 + 0.833661i $$0.686241\pi$$
$$774$$ 0 0
$$775$$ 1.09030e30 0.501049
$$776$$ 0 0
$$777$$ 7.91994e29 0.354243
$$778$$ 0 0
$$779$$ −1.05292e30 −0.458408
$$780$$ 0 0
$$781$$ −3.92588e29 −0.166380
$$782$$ 0 0
$$783$$ −8.30127e29 −0.342488
$$784$$ 0 0
$$785$$ −3.65314e30 −1.46736
$$786$$ 0 0
$$787$$ 1.29344e30 0.505838 0.252919 0.967487i $$-0.418609\pi$$
0.252919 + 0.967487i $$0.418609\pi$$
$$788$$ 0 0
$$789$$ 2.96032e30 1.12728
$$790$$ 0 0
$$791$$ −2.59981e30 −0.964025
$$792$$ 0 0
$$793$$ 6.82220e30 2.46352
$$794$$ 0 0
$$795$$ 4.24239e30 1.49196
$$796$$ 0 0
$$797$$ −4.15582e29 −0.142346 −0.0711729 0.997464i $$-0.522674\pi$$
−0.0711729 + 0.997464i $$0.522674\pi$$
$$798$$ 0 0
$$799$$ 1.46588e30 0.489054
$$800$$ 0 0
$$801$$ −6.33602e29 −0.205909
$$802$$ 0 0
$$803$$ 2.21784e30 0.702130
$$804$$ 0 0
$$805$$ 3.58645e30 1.10613
$$806$$ 0 0
$$807$$ 3.71471e30 1.11623
$$808$$ 0 0
$$809$$ −6.34707e30 −1.85829 −0.929146 0.369713i $$-0.879456\pi$$
−0.929146 + 0.369713i $$0.879456\pi$$
$$810$$ 0 0
$$811$$ 1.03289e30 0.294669 0.147335 0.989087i $$-0.452931\pi$$
0.147335 + 0.989087i $$0.452931\pi$$
$$812$$ 0 0
$$813$$ −1.68655e30 −0.468865
$$814$$ 0 0
$$815$$ −6.78002e30 −1.83686
$$816$$ 0 0
$$817$$ −5.54475e29 −0.146403
$$818$$ 0 0
$$819$$ 4.07400e30 1.04843
$$820$$ 0 0
$$821$$ 5.48361e30 1.37551 0.687755 0.725943i $$-0.258596\pi$$
0.687755 + 0.725943i $$0.258596\pi$$
$$822$$ 0 0
$$823$$ −1.50751e30 −0.368605 −0.184302 0.982870i $$-0.559003\pi$$
−0.184302 + 0.982870i $$0.559003\pi$$
$$824$$ 0 0
$$825$$ 2.15184e30 0.512913
$$826$$ 0 0
$$827$$ −3.24916e30 −0.755027 −0.377514 0.926004i $$-0.623221\pi$$
−0.377514 + 0.926004i $$0.623221\pi$$
$$828$$ 0 0
$$829$$ −3.88679e30 −0.880579 −0.440290 0.897856i $$-0.645124\pi$$
−0.440290 + 0.897856i $$0.645124\pi$$
$$830$$ 0 0
$$831$$ −3.78815e30 −0.836789
$$832$$ 0 0
$$833$$ 5.51859e29 0.118866
$$834$$ 0 0
$$835$$ −9.05184e30 −1.90120
$$836$$ 0 0
$$837$$ 3.55851e30 0.728871
$$838$$ 0 0
$$839$$ 4.76500e30 0.951836 0.475918 0.879490i $$-0.342116\pi$$
0.475918 + 0.879490i $$0.342116\pi$$
$$840$$ 0 0
$$841$$ −4.59401e30 −0.895023
$$842$$ 0 0
$$843$$ −1.56041e30 −0.296516
$$844$$ 0 0
$$845$$ −1.59118e31 −2.94934
$$846$$ 0 0
$$847$$ 1.04131e29 0.0188281
$$848$$ 0 0
$$849$$ −3.86120e29 −0.0681069
$$850$$ 0 0
$$851$$ 1.89914e30 0.326811
$$852$$ 0 0
$$853$$ 2.09705e30 0.352083 0.176041 0.984383i $$-0.443671\pi$$
0.176041 + 0.984383i $$0.443671\pi$$
$$854$$ 0 0
$$855$$ −3.52466e30 −0.577396
$$856$$ 0 0
$$857$$ 5.93597e30 0.948840 0.474420 0.880299i $$-0.342658\pi$$
0.474420 + 0.880299i $$0.342658\pi$$
$$858$$ 0 0
$$859$$ 2.89046e30 0.450856 0.225428 0.974260i $$-0.427622\pi$$
0.225428 + 0.974260i $$0.427622\pi$$
$$860$$ 0 0
$$861$$ −2.79283e30 −0.425120
$$862$$ 0 0
$$863$$ 5.57344e30 0.827961 0.413981 0.910286i $$-0.364138\pi$$
0.413981 + 0.910286i $$0.364138\pi$$
$$864$$ 0 0
$$865$$ −1.17172e31 −1.69885
$$866$$ 0 0
$$867$$ 4.18742e30 0.592578
$$868$$ 0 0
$$869$$ 8.61557e30 1.19008
$$870$$ 0 0
$$871$$ −1.68242e31 −2.26852
$$872$$ 0 0
$$873$$ 3.03489e30 0.399478
$$874$$ 0 0
$$875$$ 3.19098e30 0.410052
$$876$$ 0 0
$$877$$ 3.23623e30 0.406016 0.203008 0.979177i $$-0.434928\pi$$
0.203008 + 0.979177i $$0.434928\pi$$
$$878$$ 0 0
$$879$$ −2.81366e30 −0.344658
$$880$$ 0 0
$$881$$ 3.38000e30 0.404268 0.202134 0.979358i $$-0.435212\pi$$
0.202134 + 0.979358i $$0.435212\pi$$
$$882$$ 0 0
$$883$$ 4.75979e30 0.555904 0.277952 0.960595i $$-0.410344\pi$$
0.277952 + 0.960595i $$0.410344\pi$$
$$884$$ 0 0
$$885$$ 7.76737e30 0.885869
$$886$$ 0 0
$$887$$ −4.28065e30 −0.476773 −0.238387 0.971170i $$-0.576619\pi$$
−0.238387 + 0.971170i $$0.576619\pi$$
$$888$$ 0 0
$$889$$ 6.01235e30 0.653997
$$890$$ 0 0
$$891$$ 2.18462e30 0.232092
$$892$$ 0 0
$$893$$ −1.03644e31 −1.07549
$$894$$ 0 0
$$895$$ 2.27124e31 2.30209
$$896$$ 0 0
$$897$$ −9.39166e30 −0.929868
$$898$$ 0 0
$$899$$ −2.30981e30 −0.223409
$$900$$ 0 0
$$901$$ 6.72635e30 0.635578
$$902$$ 0 0
$$903$$ −1.47072e30 −0.135772
$$904$$ 0 0
$$905$$ 1.35737e31 1.22430
$$906$$ 0 0
$$907$$ −4.03498e30 −0.355602 −0.177801 0.984066i $$-0.556898\pi$$
−0.177801 + 0.984066i $$0.556898\pi$$
$$908$$ 0 0
$$909$$ 1.41355e30 0.121728
$$910$$ 0 0
$$911$$ 1.00965e31 0.849623 0.424812 0.905282i $$-0.360340\pi$$
0.424812 + 0.905282i $$0.360340\pi$$
$$912$$ 0 0
$$913$$ 2.56566e29 0.0210987
$$914$$ 0 0
$$915$$ −1.56556e31 −1.25819
$$916$$ 0 0
$$917$$ −8.04388e30 −0.631811
$$918$$ 0 0
$$919$$ 2.04869e31 1.57277 0.786383 0.617739i $$-0.211951\pi$$
0.786383 + 0.617739i $$0.211951\pi$$
$$920$$ 0 0
$$921$$ 8.86653e30 0.665315
$$922$$ 0 0
$$923$$ −4.05267e30 −0.297251
$$924$$ 0 0
$$925$$ −4.49190e30 −0.322064
$$926$$ 0 0
$$927$$ 8.06882e30 0.565552
$$928$$ 0 0
$$929$$ 1.15155e31 0.789072 0.394536 0.918880i $$-0.370905\pi$$
0.394536 + 0.918880i $$0.370905\pi$$
$$930$$ 0 0
$$931$$ −3.90189e30 −0.261399
$$932$$ 0 0
$$933$$ −1.13500e31 −0.743431
$$934$$ 0 0
$$935$$ 8.10693e30 0.519200
$$936$$ 0 0
$$937$$ 1.40172e31 0.877801 0.438900 0.898536i $$-0.355368\pi$$
0.438900 + 0.898536i $$0.355368\pi$$
$$938$$ 0 0
$$939$$ −8.29758e30 −0.508116
$$940$$ 0 0
$$941$$ −9.63512e30 −0.576987 −0.288493 0.957482i $$-0.593154\pi$$
−0.288493 + 0.957482i $$0.593154\pi$$
$$942$$ 0 0
$$943$$ −6.69700e30 −0.392200
$$944$$ 0 0
$$945$$ −2.76858e31 −1.58571
$$946$$ 0 0
$$947$$ −1.70818e31 −0.956880 −0.478440 0.878120i $$-0.658798\pi$$
−0.478440 + 0.878120i $$0.658798\pi$$
$$948$$ 0 0
$$949$$ 2.28947e31 1.25441
$$950$$ 0 0
$$951$$ 3.65007e30 0.195617
$$952$$ 0 0
$$953$$ −1.43239e31 −0.750904 −0.375452 0.926842i $$-0.622513\pi$$
−0.375452 + 0.926842i $$0.622513\pi$$
$$954$$ 0 0
$$955$$ −1.22798e31 −0.629731
$$956$$ 0 0
$$957$$ −4.55868e30 −0.228699
$$958$$ 0 0
$$959$$ −1.72136e30 −0.0844846
$$960$$ 0 0
$$961$$ −1.09240e31 −0.524550
$$962$$ 0 0
$$963$$ −9.52340e30 −0.447421
$$964$$ 0 0
$$965$$ 5.40809e31 2.48603
$$966$$ 0 0
$$967$$ −1.43951e31 −0.647493 −0.323747 0.946144i $$-0.604943\pi$$
−0.323747 + 0.946144i $$0.604943\pi$$
$$968$$ 0 0
$$969$$ 5.37243e30 0.236467
$$970$$ 0 0
$$971$$ 2.67107e31 1.15049 0.575245 0.817981i $$-0.304907\pi$$
0.575245 + 0.817981i $$0.304907\pi$$
$$972$$ 0 0
$$973$$ −2.88939e31 −1.21793
$$974$$ 0 0
$$975$$ 2.22134e31 0.916360
$$976$$ 0 0
$$977$$ −3.63714e31 −1.46848 −0.734238 0.678892i $$-0.762460\pi$$
−0.734238 + 0.678892i $$0.762460\pi$$
$$978$$ 0 0
$$979$$ −1.03039e31 −0.407178
$$980$$ 0 0
$$981$$ −1.78073e31 −0.688770
$$982$$ 0 0
$$983$$ 2.81078e31 1.06418 0.532089 0.846688i $$-0.321407\pi$$
0.532089 + 0.846688i $$0.321407\pi$$
$$984$$ 0 0
$$985$$ −2.38851e30 −0.0885212
$$986$$ 0 0
$$987$$ −2.74912e31 −0.997388
$$988$$ 0 0
$$989$$ −3.52668e30 −0.125258
$$990$$ 0 0
$$991$$ −3.20038e31 −1.11283 −0.556413 0.830906i $$-0.687823\pi$$
−0.556413 + 0.830906i $$0.687823\pi$$
$$992$$ 0 0
$$993$$ −3.32339e31 −1.13139
$$994$$ 0 0
$$995$$ −2.52923e31 −0.843035
$$996$$ 0 0
$$997$$ −4.21028e31 −1.37408 −0.687040 0.726620i $$-0.741090\pi$$
−0.687040 + 0.726620i $$0.741090\pi$$
$$998$$ 0 0
$$999$$ −1.46606e31 −0.468504
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.a.1.1 1
4.3 odd 2 2.22.a.a.1.1 1
8.3 odd 2 64.22.a.b.1.1 1
8.5 even 2 64.22.a.f.1.1 1
12.11 even 2 18.22.a.e.1.1 1
20.3 even 4 50.22.b.a.49.2 2
20.7 even 4 50.22.b.a.49.1 2
20.19 odd 2 50.22.a.c.1.1 1

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