# Properties

 Label 528.8.a.a.1.1 Level $528$ Weight $8$ Character 528.1 Self dual yes Analytic conductor $164.939$ 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] = [528,8,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$164.939293456$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 528.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +O(q^{10})$$ $$q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +1331.00 q^{11} +12958.0 q^{13} +11070.0 q^{15} +17062.0 q^{17} +54168.0 q^{19} -27756.0 q^{21} +11488.0 q^{23} +89975.0 q^{25} -19683.0 q^{27} -186654. q^{29} +188672. q^{31} -35937.0 q^{33} -421480. q^{35} +395886. q^{37} -349866. q^{39} -47546.0 q^{41} -602088. q^{43} -298890. q^{45} +647200. q^{47} +233241. q^{49} -460674. q^{51} -1.31272e6 q^{53} -545710. q^{55} -1.46254e6 q^{57} +2.68114e6 q^{59} +551190. q^{61} +749412. q^{63} -5.31278e6 q^{65} -459260. q^{67} -310176. q^{69} +18072.0 q^{71} -426062. q^{73} -2.42932e6 q^{75} +1.36827e6 q^{77} -297764. q^{79} +531441. q^{81} -5.68403e6 q^{83} -6.99542e6 q^{85} +5.03966e6 q^{87} -6.34297e6 q^{89} +1.33208e7 q^{91} -5.09414e6 q^{93} -2.22089e7 q^{95} +1.66516e7 q^{97} +970299. q^{99} +O(q^{100})$$

## 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
$$3$$ −27.0000 −0.577350
$$4$$ 0 0
$$5$$ −410.000 −1.46686 −0.733430 0.679765i $$-0.762082\pi$$
−0.733430 + 0.679765i $$0.762082\pi$$
$$6$$ 0 0
$$7$$ 1028.00 1.13279 0.566396 0.824133i $$-0.308337\pi$$
0.566396 + 0.824133i $$0.308337\pi$$
$$8$$ 0 0
$$9$$ 729.000 0.333333
$$10$$ 0 0
$$11$$ 1331.00 0.301511
$$12$$ 0 0
$$13$$ 12958.0 1.63582 0.817911 0.575344i $$-0.195132\pi$$
0.817911 + 0.575344i $$0.195132\pi$$
$$14$$ 0 0
$$15$$ 11070.0 0.846892
$$16$$ 0 0
$$17$$ 17062.0 0.842284 0.421142 0.906995i $$-0.361629\pi$$
0.421142 + 0.906995i $$0.361629\pi$$
$$18$$ 0 0
$$19$$ 54168.0 1.81178 0.905889 0.423514i $$-0.139204\pi$$
0.905889 + 0.423514i $$0.139204\pi$$
$$20$$ 0 0
$$21$$ −27756.0 −0.654017
$$22$$ 0 0
$$23$$ 11488.0 0.196878 0.0984390 0.995143i $$-0.468615\pi$$
0.0984390 + 0.995143i $$0.468615\pi$$
$$24$$ 0 0
$$25$$ 89975.0 1.15168
$$26$$ 0 0
$$27$$ −19683.0 −0.192450
$$28$$ 0 0
$$29$$ −186654. −1.42116 −0.710582 0.703614i $$-0.751568\pi$$
−0.710582 + 0.703614i $$0.751568\pi$$
$$30$$ 0 0
$$31$$ 188672. 1.13747 0.568737 0.822519i $$-0.307432\pi$$
0.568737 + 0.822519i $$0.307432\pi$$
$$32$$ 0 0
$$33$$ −35937.0 −0.174078
$$34$$ 0 0
$$35$$ −421480. −1.66165
$$36$$ 0 0
$$37$$ 395886. 1.28488 0.642442 0.766334i $$-0.277921\pi$$
0.642442 + 0.766334i $$0.277921\pi$$
$$38$$ 0 0
$$39$$ −349866. −0.944443
$$40$$ 0 0
$$41$$ −47546.0 −0.107738 −0.0538692 0.998548i $$-0.517155\pi$$
−0.0538692 + 0.998548i $$0.517155\pi$$
$$42$$ 0 0
$$43$$ −602088. −1.15484 −0.577418 0.816449i $$-0.695940\pi$$
−0.577418 + 0.816449i $$0.695940\pi$$
$$44$$ 0 0
$$45$$ −298890. −0.488954
$$46$$ 0 0
$$47$$ 647200. 0.909277 0.454638 0.890676i $$-0.349769\pi$$
0.454638 + 0.890676i $$0.349769\pi$$
$$48$$ 0 0
$$49$$ 233241. 0.283217
$$50$$ 0 0
$$51$$ −460674. −0.486293
$$52$$ 0 0
$$53$$ −1.31272e6 −1.21118 −0.605588 0.795778i $$-0.707062\pi$$
−0.605588 + 0.795778i $$0.707062\pi$$
$$54$$ 0 0
$$55$$ −545710. −0.442275
$$56$$ 0 0
$$57$$ −1.46254e6 −1.04603
$$58$$ 0 0
$$59$$ 2.68114e6 1.69956 0.849782 0.527135i $$-0.176734\pi$$
0.849782 + 0.527135i $$0.176734\pi$$
$$60$$ 0 0
$$61$$ 551190. 0.310919 0.155459 0.987842i $$-0.450314\pi$$
0.155459 + 0.987842i $$0.450314\pi$$
$$62$$ 0 0
$$63$$ 749412. 0.377597
$$64$$ 0 0
$$65$$ −5.31278e6 −2.39952
$$66$$ 0 0
$$67$$ −459260. −0.186551 −0.0932753 0.995640i $$-0.529734\pi$$
−0.0932753 + 0.995640i $$0.529734\pi$$
$$68$$ 0 0
$$69$$ −310176. −0.113668
$$70$$ 0 0
$$71$$ 18072.0 0.00599242 0.00299621 0.999996i $$-0.499046\pi$$
0.00299621 + 0.999996i $$0.499046\pi$$
$$72$$ 0 0
$$73$$ −426062. −0.128187 −0.0640933 0.997944i $$-0.520416\pi$$
−0.0640933 + 0.997944i $$0.520416\pi$$
$$74$$ 0 0
$$75$$ −2.42932e6 −0.664923
$$76$$ 0 0
$$77$$ 1.36827e6 0.341549
$$78$$ 0 0
$$79$$ −297764. −0.0679481 −0.0339741 0.999423i $$-0.510816\pi$$
−0.0339741 + 0.999423i $$0.510816\pi$$
$$80$$ 0 0
$$81$$ 531441. 0.111111
$$82$$ 0 0
$$83$$ −5.68403e6 −1.09115 −0.545573 0.838063i $$-0.683688\pi$$
−0.545573 + 0.838063i $$0.683688\pi$$
$$84$$ 0 0
$$85$$ −6.99542e6 −1.23551
$$86$$ 0 0
$$87$$ 5.03966e6 0.820510
$$88$$ 0 0
$$89$$ −6.34297e6 −0.953734 −0.476867 0.878975i $$-0.658228\pi$$
−0.476867 + 0.878975i $$0.658228\pi$$
$$90$$ 0 0
$$91$$ 1.33208e7 1.85305
$$92$$ 0 0
$$93$$ −5.09414e6 −0.656721
$$94$$ 0 0
$$95$$ −2.22089e7 −2.65763
$$96$$ 0 0
$$97$$ 1.66516e7 1.85248 0.926242 0.376929i $$-0.123020\pi$$
0.926242 + 0.376929i $$0.123020\pi$$
$$98$$ 0 0
$$99$$ 970299. 0.100504
$$100$$ 0 0
$$101$$ −2.08327e6 −0.201197 −0.100598 0.994927i $$-0.532076\pi$$
−0.100598 + 0.994927i $$0.532076\pi$$
$$102$$ 0 0
$$103$$ 2.39046e6 0.215552 0.107776 0.994175i $$-0.465627\pi$$
0.107776 + 0.994175i $$0.465627\pi$$
$$104$$ 0 0
$$105$$ 1.13800e7 0.959352
$$106$$ 0 0
$$107$$ 1.40615e7 1.10965 0.554827 0.831966i $$-0.312785\pi$$
0.554827 + 0.831966i $$0.312785\pi$$
$$108$$ 0 0
$$109$$ −1.11321e7 −0.823347 −0.411674 0.911331i $$-0.635056\pi$$
−0.411674 + 0.911331i $$0.635056\pi$$
$$110$$ 0 0
$$111$$ −1.06889e7 −0.741828
$$112$$ 0 0
$$113$$ 5.66903e6 0.369602 0.184801 0.982776i $$-0.440836\pi$$
0.184801 + 0.982776i $$0.440836\pi$$
$$114$$ 0 0
$$115$$ −4.71008e6 −0.288792
$$116$$ 0 0
$$117$$ 9.44638e6 0.545274
$$118$$ 0 0
$$119$$ 1.75397e7 0.954132
$$120$$ 0 0
$$121$$ 1.77156e6 0.0909091
$$122$$ 0 0
$$123$$ 1.28374e6 0.0622028
$$124$$ 0 0
$$125$$ −4.85850e6 −0.222493
$$126$$ 0 0
$$127$$ 2.09170e7 0.906123 0.453061 0.891479i $$-0.350332\pi$$
0.453061 + 0.891479i $$0.350332\pi$$
$$128$$ 0 0
$$129$$ 1.62564e7 0.666745
$$130$$ 0 0
$$131$$ 1.12649e7 0.437802 0.218901 0.975747i $$-0.429753\pi$$
0.218901 + 0.975747i $$0.429753\pi$$
$$132$$ 0 0
$$133$$ 5.56847e7 2.05237
$$134$$ 0 0
$$135$$ 8.07003e6 0.282297
$$136$$ 0 0
$$137$$ 444290. 0.0147620 0.00738099 0.999973i $$-0.497651\pi$$
0.00738099 + 0.999973i $$0.497651\pi$$
$$138$$ 0 0
$$139$$ −3.42613e7 −1.08206 −0.541030 0.841003i $$-0.681966\pi$$
−0.541030 + 0.841003i $$0.681966\pi$$
$$140$$ 0 0
$$141$$ −1.74744e7 −0.524971
$$142$$ 0 0
$$143$$ 1.72471e7 0.493219
$$144$$ 0 0
$$145$$ 7.65281e7 2.08465
$$146$$ 0 0
$$147$$ −6.29751e6 −0.163515
$$148$$ 0 0
$$149$$ −4.82211e7 −1.19422 −0.597112 0.802158i $$-0.703685\pi$$
−0.597112 + 0.802158i $$0.703685\pi$$
$$150$$ 0 0
$$151$$ 4.48693e7 1.06055 0.530273 0.847827i $$-0.322089\pi$$
0.530273 + 0.847827i $$0.322089\pi$$
$$152$$ 0 0
$$153$$ 1.24382e7 0.280761
$$154$$ 0 0
$$155$$ −7.73555e7 −1.66852
$$156$$ 0 0
$$157$$ −5.38907e6 −0.111139 −0.0555693 0.998455i $$-0.517697\pi$$
−0.0555693 + 0.998455i $$0.517697\pi$$
$$158$$ 0 0
$$159$$ 3.54435e7 0.699273
$$160$$ 0 0
$$161$$ 1.18097e7 0.223022
$$162$$ 0 0
$$163$$ −9.81674e7 −1.77546 −0.887730 0.460365i $$-0.847719\pi$$
−0.887730 + 0.460365i $$0.847719\pi$$
$$164$$ 0 0
$$165$$ 1.47342e7 0.255348
$$166$$ 0 0
$$167$$ 4.40611e7 0.732062 0.366031 0.930603i $$-0.380716\pi$$
0.366031 + 0.930603i $$0.380716\pi$$
$$168$$ 0 0
$$169$$ 1.05161e8 1.67592
$$170$$ 0 0
$$171$$ 3.94885e7 0.603926
$$172$$ 0 0
$$173$$ 6.71087e7 0.985411 0.492706 0.870196i $$-0.336008\pi$$
0.492706 + 0.870196i $$0.336008\pi$$
$$174$$ 0 0
$$175$$ 9.24943e7 1.30461
$$176$$ 0 0
$$177$$ −7.23908e7 −0.981244
$$178$$ 0 0
$$179$$ −4.34929e6 −0.0566804 −0.0283402 0.999598i $$-0.509022\pi$$
−0.0283402 + 0.999598i $$0.509022\pi$$
$$180$$ 0 0
$$181$$ −1.20238e7 −0.150719 −0.0753593 0.997156i $$-0.524010\pi$$
−0.0753593 + 0.997156i $$0.524010\pi$$
$$182$$ 0 0
$$183$$ −1.48821e7 −0.179509
$$184$$ 0 0
$$185$$ −1.62313e8 −1.88475
$$186$$ 0 0
$$187$$ 2.27095e7 0.253958
$$188$$ 0 0
$$189$$ −2.02341e7 −0.218006
$$190$$ 0 0
$$191$$ −5.96399e7 −0.619327 −0.309664 0.950846i $$-0.600216\pi$$
−0.309664 + 0.950846i $$0.600216\pi$$
$$192$$ 0 0
$$193$$ −9.81036e7 −0.982278 −0.491139 0.871081i $$-0.663419\pi$$
−0.491139 + 0.871081i $$0.663419\pi$$
$$194$$ 0 0
$$195$$ 1.43445e8 1.38537
$$196$$ 0 0
$$197$$ −1.09317e8 −1.01872 −0.509361 0.860553i $$-0.670118\pi$$
−0.509361 + 0.860553i $$0.670118\pi$$
$$198$$ 0 0
$$199$$ 3.64317e7 0.327713 0.163857 0.986484i $$-0.447607\pi$$
0.163857 + 0.986484i $$0.447607\pi$$
$$200$$ 0 0
$$201$$ 1.24000e7 0.107705
$$202$$ 0 0
$$203$$ −1.91880e8 −1.60988
$$204$$ 0 0
$$205$$ 1.94939e7 0.158037
$$206$$ 0 0
$$207$$ 8.37475e6 0.0656260
$$208$$ 0 0
$$209$$ 7.20976e7 0.546272
$$210$$ 0 0
$$211$$ −1.38637e7 −0.101599 −0.0507997 0.998709i $$-0.516177\pi$$
−0.0507997 + 0.998709i $$0.516177\pi$$
$$212$$ 0 0
$$213$$ −487944. −0.00345972
$$214$$ 0 0
$$215$$ 2.46856e8 1.69398
$$216$$ 0 0
$$217$$ 1.93955e8 1.28852
$$218$$ 0 0
$$219$$ 1.15037e7 0.0740086
$$220$$ 0 0
$$221$$ 2.21089e8 1.37783
$$222$$ 0 0
$$223$$ −1.35935e8 −0.820850 −0.410425 0.911894i $$-0.634620\pi$$
−0.410425 + 0.911894i $$0.634620\pi$$
$$224$$ 0 0
$$225$$ 6.55918e7 0.383893
$$226$$ 0 0
$$227$$ −2.82203e7 −0.160129 −0.0800646 0.996790i $$-0.525513\pi$$
−0.0800646 + 0.996790i $$0.525513\pi$$
$$228$$ 0 0
$$229$$ −5.31215e7 −0.292312 −0.146156 0.989262i $$-0.546690\pi$$
−0.146156 + 0.989262i $$0.546690\pi$$
$$230$$ 0 0
$$231$$ −3.69432e7 −0.197194
$$232$$ 0 0
$$233$$ 1.54589e8 0.800631 0.400316 0.916377i $$-0.368901\pi$$
0.400316 + 0.916377i $$0.368901\pi$$
$$234$$ 0 0
$$235$$ −2.65352e8 −1.33378
$$236$$ 0 0
$$237$$ 8.03963e6 0.0392299
$$238$$ 0 0
$$239$$ 1.86143e8 0.881972 0.440986 0.897514i $$-0.354629\pi$$
0.440986 + 0.897514i $$0.354629\pi$$
$$240$$ 0 0
$$241$$ 2.62107e8 1.20620 0.603100 0.797666i $$-0.293932\pi$$
0.603100 + 0.797666i $$0.293932\pi$$
$$242$$ 0 0
$$243$$ −1.43489e7 −0.0641500
$$244$$ 0 0
$$245$$ −9.56288e7 −0.415439
$$246$$ 0 0
$$247$$ 7.01909e8 2.96375
$$248$$ 0 0
$$249$$ 1.53469e8 0.629973
$$250$$ 0 0
$$251$$ 2.75827e8 1.10098 0.550489 0.834842i $$-0.314441\pi$$
0.550489 + 0.834842i $$0.314441\pi$$
$$252$$ 0 0
$$253$$ 1.52905e7 0.0593609
$$254$$ 0 0
$$255$$ 1.88876e8 0.713324
$$256$$ 0 0
$$257$$ 1.06856e6 0.00392675 0.00196338 0.999998i $$-0.499375\pi$$
0.00196338 + 0.999998i $$0.499375\pi$$
$$258$$ 0 0
$$259$$ 4.06971e8 1.45551
$$260$$ 0 0
$$261$$ −1.36071e8 −0.473721
$$262$$ 0 0
$$263$$ 7.92924e7 0.268774 0.134387 0.990929i $$-0.457094\pi$$
0.134387 + 0.990929i $$0.457094\pi$$
$$264$$ 0 0
$$265$$ 5.38216e8 1.77663
$$266$$ 0 0
$$267$$ 1.71260e8 0.550639
$$268$$ 0 0
$$269$$ 2.10170e8 0.658321 0.329160 0.944274i $$-0.393234\pi$$
0.329160 + 0.944274i $$0.393234\pi$$
$$270$$ 0 0
$$271$$ 2.65510e8 0.810378 0.405189 0.914233i $$-0.367206\pi$$
0.405189 + 0.914233i $$0.367206\pi$$
$$272$$ 0 0
$$273$$ −3.59662e8 −1.06986
$$274$$ 0 0
$$275$$ 1.19757e8 0.347245
$$276$$ 0 0
$$277$$ −6.23529e8 −1.76270 −0.881349 0.472466i $$-0.843364\pi$$
−0.881349 + 0.472466i $$0.843364\pi$$
$$278$$ 0 0
$$279$$ 1.37542e8 0.379158
$$280$$ 0 0
$$281$$ 1.30611e8 0.351162 0.175581 0.984465i $$-0.443820\pi$$
0.175581 + 0.984465i $$0.443820\pi$$
$$282$$ 0 0
$$283$$ 2.20874e7 0.0579283 0.0289642 0.999580i $$-0.490779\pi$$
0.0289642 + 0.999580i $$0.490779\pi$$
$$284$$ 0 0
$$285$$ 5.99640e8 1.53438
$$286$$ 0 0
$$287$$ −4.88773e7 −0.122045
$$288$$ 0 0
$$289$$ −1.19227e8 −0.290557
$$290$$ 0 0
$$291$$ −4.49593e8 −1.06953
$$292$$ 0 0
$$293$$ 2.00188e8 0.464944 0.232472 0.972603i $$-0.425319\pi$$
0.232472 + 0.972603i $$0.425319\pi$$
$$294$$ 0 0
$$295$$ −1.09927e9 −2.49302
$$296$$ 0 0
$$297$$ −2.61981e7 −0.0580259
$$298$$ 0 0
$$299$$ 1.48862e8 0.322057
$$300$$ 0 0
$$301$$ −6.18946e8 −1.30819
$$302$$ 0 0
$$303$$ 5.62483e7 0.116161
$$304$$ 0 0
$$305$$ −2.25988e8 −0.456074
$$306$$ 0 0
$$307$$ 4.79736e8 0.946276 0.473138 0.880988i $$-0.343121\pi$$
0.473138 + 0.880988i $$0.343121\pi$$
$$308$$ 0 0
$$309$$ −6.45425e7 −0.124449
$$310$$ 0 0
$$311$$ 5.19734e8 0.979761 0.489880 0.871790i $$-0.337040\pi$$
0.489880 + 0.871790i $$0.337040\pi$$
$$312$$ 0 0
$$313$$ 9.69759e8 1.78755 0.893776 0.448514i $$-0.148047\pi$$
0.893776 + 0.448514i $$0.148047\pi$$
$$314$$ 0 0
$$315$$ −3.07259e8 −0.553882
$$316$$ 0 0
$$317$$ 7.56875e8 1.33450 0.667248 0.744836i $$-0.267472\pi$$
0.667248 + 0.744836i $$0.267472\pi$$
$$318$$ 0 0
$$319$$ −2.48436e8 −0.428497
$$320$$ 0 0
$$321$$ −3.79660e8 −0.640659
$$322$$ 0 0
$$323$$ 9.24214e8 1.52603
$$324$$ 0 0
$$325$$ 1.16590e9 1.88394
$$326$$ 0 0
$$327$$ 3.00566e8 0.475360
$$328$$ 0 0
$$329$$ 6.65322e8 1.03002
$$330$$ 0 0
$$331$$ 1.79867e8 0.272618 0.136309 0.990666i $$-0.456476\pi$$
0.136309 + 0.990666i $$0.456476\pi$$
$$332$$ 0 0
$$333$$ 2.88601e8 0.428295
$$334$$ 0 0
$$335$$ 1.88297e8 0.273644
$$336$$ 0 0
$$337$$ −1.38092e9 −1.96546 −0.982728 0.185054i $$-0.940754\pi$$
−0.982728 + 0.185054i $$0.940754\pi$$
$$338$$ 0 0
$$339$$ −1.53064e8 −0.213390
$$340$$ 0 0
$$341$$ 2.51122e8 0.342961
$$342$$ 0 0
$$343$$ −6.06830e8 −0.811966
$$344$$ 0 0
$$345$$ 1.27172e8 0.166734
$$346$$ 0 0
$$347$$ −7.66253e8 −0.984507 −0.492254 0.870452i $$-0.663827\pi$$
−0.492254 + 0.870452i $$0.663827\pi$$
$$348$$ 0 0
$$349$$ −2.68852e8 −0.338552 −0.169276 0.985569i $$-0.554143\pi$$
−0.169276 + 0.985569i $$0.554143\pi$$
$$350$$ 0 0
$$351$$ −2.55052e8 −0.314814
$$352$$ 0 0
$$353$$ −3.95002e8 −0.477956 −0.238978 0.971025i $$-0.576812\pi$$
−0.238978 + 0.971025i $$0.576812\pi$$
$$354$$ 0 0
$$355$$ −7.40952e6 −0.00879004
$$356$$ 0 0
$$357$$ −4.73573e8 −0.550869
$$358$$ 0 0
$$359$$ 4.25768e7 0.0485671 0.0242836 0.999705i $$-0.492270\pi$$
0.0242836 + 0.999705i $$0.492270\pi$$
$$360$$ 0 0
$$361$$ 2.04030e9 2.28254
$$362$$ 0 0
$$363$$ −4.78321e7 −0.0524864
$$364$$ 0 0
$$365$$ 1.74685e8 0.188032
$$366$$ 0 0
$$367$$ −1.85295e9 −1.95673 −0.978366 0.206882i $$-0.933668\pi$$
−0.978366 + 0.206882i $$0.933668\pi$$
$$368$$ 0 0
$$369$$ −3.46610e7 −0.0359128
$$370$$ 0 0
$$371$$ −1.34948e9 −1.37201
$$372$$ 0 0
$$373$$ −4.83602e7 −0.0482511 −0.0241256 0.999709i $$-0.507680\pi$$
−0.0241256 + 0.999709i $$0.507680\pi$$
$$374$$ 0 0
$$375$$ 1.31180e8 0.128457
$$376$$ 0 0
$$377$$ −2.41866e9 −2.32477
$$378$$ 0 0
$$379$$ 2.26078e8 0.213315 0.106658 0.994296i $$-0.465985\pi$$
0.106658 + 0.994296i $$0.465985\pi$$
$$380$$ 0 0
$$381$$ −5.64760e8 −0.523150
$$382$$ 0 0
$$383$$ 1.35198e9 1.22963 0.614815 0.788671i $$-0.289231\pi$$
0.614815 + 0.788671i $$0.289231\pi$$
$$384$$ 0 0
$$385$$ −5.60990e8 −0.501005
$$386$$ 0 0
$$387$$ −4.38922e8 −0.384945
$$388$$ 0 0
$$389$$ −1.09107e9 −0.939789 −0.469894 0.882723i $$-0.655708\pi$$
−0.469894 + 0.882723i $$0.655708\pi$$
$$390$$ 0 0
$$391$$ 1.96008e8 0.165827
$$392$$ 0 0
$$393$$ −3.04152e8 −0.252765
$$394$$ 0 0
$$395$$ 1.22083e8 0.0996704
$$396$$ 0 0
$$397$$ −6.97868e8 −0.559766 −0.279883 0.960034i $$-0.590296\pi$$
−0.279883 + 0.960034i $$0.590296\pi$$
$$398$$ 0 0
$$399$$ −1.50349e9 −1.18494
$$400$$ 0 0
$$401$$ 1.74689e9 1.35288 0.676441 0.736497i $$-0.263521\pi$$
0.676441 + 0.736497i $$0.263521\pi$$
$$402$$ 0 0
$$403$$ 2.44481e9 1.86071
$$404$$ 0 0
$$405$$ −2.17891e8 −0.162985
$$406$$ 0 0
$$407$$ 5.26924e8 0.387407
$$408$$ 0 0
$$409$$ −1.30304e9 −0.941729 −0.470865 0.882205i $$-0.656058\pi$$
−0.470865 + 0.882205i $$0.656058\pi$$
$$410$$ 0 0
$$411$$ −1.19958e7 −0.00852283
$$412$$ 0 0
$$413$$ 2.75621e9 1.92525
$$414$$ 0 0
$$415$$ 2.33045e9 1.60056
$$416$$ 0 0
$$417$$ 9.25054e8 0.624728
$$418$$ 0 0
$$419$$ −2.87139e9 −1.90697 −0.953484 0.301443i $$-0.902532\pi$$
−0.953484 + 0.301443i $$0.902532\pi$$
$$420$$ 0 0
$$421$$ 1.15946e9 0.757299 0.378650 0.925540i $$-0.376389\pi$$
0.378650 + 0.925540i $$0.376389\pi$$
$$422$$ 0 0
$$423$$ 4.71809e8 0.303092
$$424$$ 0 0
$$425$$ 1.53515e9 0.970042
$$426$$ 0 0
$$427$$ 5.66623e8 0.352206
$$428$$ 0 0
$$429$$ −4.65672e8 −0.284760
$$430$$ 0 0
$$431$$ 1.66703e9 1.00294 0.501468 0.865176i $$-0.332793\pi$$
0.501468 + 0.865176i $$0.332793\pi$$
$$432$$ 0 0
$$433$$ 6.34094e8 0.375358 0.187679 0.982230i $$-0.439903\pi$$
0.187679 + 0.982230i $$0.439903\pi$$
$$434$$ 0 0
$$435$$ −2.06626e9 −1.20357
$$436$$ 0 0
$$437$$ 6.22282e8 0.356699
$$438$$ 0 0
$$439$$ 1.22368e9 0.690307 0.345154 0.938546i $$-0.387827\pi$$
0.345154 + 0.938546i $$0.387827\pi$$
$$440$$ 0 0
$$441$$ 1.70033e8 0.0944055
$$442$$ 0 0
$$443$$ 1.23213e9 0.673355 0.336677 0.941620i $$-0.390697\pi$$
0.336677 + 0.941620i $$0.390697\pi$$
$$444$$ 0 0
$$445$$ 2.60062e9 1.39900
$$446$$ 0 0
$$447$$ 1.30197e9 0.689485
$$448$$ 0 0
$$449$$ −3.07511e9 −1.60324 −0.801621 0.597833i $$-0.796029\pi$$
−0.801621 + 0.597833i $$0.796029\pi$$
$$450$$ 0 0
$$451$$ −6.32837e7 −0.0324843
$$452$$ 0 0
$$453$$ −1.21147e9 −0.612307
$$454$$ 0 0
$$455$$ −5.46154e9 −2.71816
$$456$$ 0 0
$$457$$ −2.44730e9 −1.19945 −0.599723 0.800207i $$-0.704723\pi$$
−0.599723 + 0.800207i $$0.704723\pi$$
$$458$$ 0 0
$$459$$ −3.35831e8 −0.162098
$$460$$ 0 0
$$461$$ 9.52419e8 0.452767 0.226383 0.974038i $$-0.427310\pi$$
0.226383 + 0.974038i $$0.427310\pi$$
$$462$$ 0 0
$$463$$ 6.05200e8 0.283378 0.141689 0.989911i $$-0.454747\pi$$
0.141689 + 0.989911i $$0.454747\pi$$
$$464$$ 0 0
$$465$$ 2.08860e9 0.963318
$$466$$ 0 0
$$467$$ 1.37708e9 0.625676 0.312838 0.949806i $$-0.398720\pi$$
0.312838 + 0.949806i $$0.398720\pi$$
$$468$$ 0 0
$$469$$ −4.72119e8 −0.211323
$$470$$ 0 0
$$471$$ 1.45505e8 0.0641659
$$472$$ 0 0
$$473$$ −8.01379e8 −0.348196
$$474$$ 0 0
$$475$$ 4.87377e9 2.08659
$$476$$ 0 0
$$477$$ −9.56974e8 −0.403725
$$478$$ 0 0
$$479$$ 4.00222e9 1.66390 0.831949 0.554851i $$-0.187225\pi$$
0.831949 + 0.554851i $$0.187225\pi$$
$$480$$ 0 0
$$481$$ 5.12989e9 2.10184
$$482$$ 0 0
$$483$$ −3.18861e8 −0.128762
$$484$$ 0 0
$$485$$ −6.82715e9 −2.71734
$$486$$ 0 0
$$487$$ 2.88677e9 1.13256 0.566279 0.824214i $$-0.308383\pi$$
0.566279 + 0.824214i $$0.308383\pi$$
$$488$$ 0 0
$$489$$ 2.65052e9 1.02506
$$490$$ 0 0
$$491$$ −1.19743e8 −0.0456525 −0.0228262 0.999739i $$-0.507266\pi$$
−0.0228262 + 0.999739i $$0.507266\pi$$
$$492$$ 0 0
$$493$$ −3.18469e9 −1.19702
$$494$$ 0 0
$$495$$ −3.97823e8 −0.147425
$$496$$ 0 0
$$497$$ 1.85780e7 0.00678816
$$498$$ 0 0
$$499$$ 4.78950e9 1.72559 0.862796 0.505552i $$-0.168711\pi$$
0.862796 + 0.505552i $$0.168711\pi$$
$$500$$ 0 0
$$501$$ −1.18965e9 −0.422656
$$502$$ 0 0
$$503$$ −3.83047e9 −1.34203 −0.671017 0.741442i $$-0.734142\pi$$
−0.671017 + 0.741442i $$0.734142\pi$$
$$504$$ 0 0
$$505$$ 8.54141e8 0.295127
$$506$$ 0 0
$$507$$ −2.83935e9 −0.967591
$$508$$ 0 0
$$509$$ −2.34385e9 −0.787803 −0.393902 0.919153i $$-0.628875\pi$$
−0.393902 + 0.919153i $$0.628875\pi$$
$$510$$ 0 0
$$511$$ −4.37992e8 −0.145209
$$512$$ 0 0
$$513$$ −1.06619e9 −0.348677
$$514$$ 0 0
$$515$$ −9.80090e8 −0.316185
$$516$$ 0 0
$$517$$ 8.61423e8 0.274157
$$518$$ 0 0
$$519$$ −1.81194e9 −0.568928
$$520$$ 0 0
$$521$$ 5.77085e9 1.78775 0.893877 0.448313i $$-0.147975\pi$$
0.893877 + 0.448313i $$0.147975\pi$$
$$522$$ 0 0
$$523$$ 3.49411e8 0.106802 0.0534012 0.998573i $$-0.482994\pi$$
0.0534012 + 0.998573i $$0.482994\pi$$
$$524$$ 0 0
$$525$$ −2.49735e9 −0.753219
$$526$$ 0 0
$$527$$ 3.21912e9 0.958077
$$528$$ 0 0
$$529$$ −3.27285e9 −0.961239
$$530$$ 0 0
$$531$$ 1.95455e9 0.566521
$$532$$ 0 0
$$533$$ −6.16101e8 −0.176241
$$534$$ 0 0
$$535$$ −5.76520e9 −1.62771
$$536$$ 0 0
$$537$$ 1.17431e8 0.0327244
$$538$$ 0 0
$$539$$ 3.10444e8 0.0853930
$$540$$ 0 0
$$541$$ −5.10025e9 −1.38484 −0.692422 0.721493i $$-0.743456\pi$$
−0.692422 + 0.721493i $$0.743456\pi$$
$$542$$ 0 0
$$543$$ 3.24643e8 0.0870175
$$544$$ 0 0
$$545$$ 4.56415e9 1.20774
$$546$$ 0 0
$$547$$ −4.96217e9 −1.29633 −0.648166 0.761499i $$-0.724464\pi$$
−0.648166 + 0.761499i $$0.724464\pi$$
$$548$$ 0 0
$$549$$ 4.01818e8 0.103640
$$550$$ 0 0
$$551$$ −1.01107e10 −2.57484
$$552$$ 0 0
$$553$$ −3.06101e8 −0.0769710
$$554$$ 0 0
$$555$$ 4.38246e9 1.08816
$$556$$ 0 0
$$557$$ 1.42590e9 0.349620 0.174810 0.984602i $$-0.444069\pi$$
0.174810 + 0.984602i $$0.444069\pi$$
$$558$$ 0 0
$$559$$ −7.80186e9 −1.88911
$$560$$ 0 0
$$561$$ −6.13157e8 −0.146623
$$562$$ 0 0
$$563$$ −5.96929e9 −1.40975 −0.704876 0.709330i $$-0.748998\pi$$
−0.704876 + 0.709330i $$0.748998\pi$$
$$564$$ 0 0
$$565$$ −2.32430e9 −0.542155
$$566$$ 0 0
$$567$$ 5.46321e8 0.125866
$$568$$ 0 0
$$569$$ −3.51616e9 −0.800158 −0.400079 0.916481i $$-0.631017\pi$$
−0.400079 + 0.916481i $$0.631017\pi$$
$$570$$ 0 0
$$571$$ −6.44706e8 −0.144922 −0.0724611 0.997371i $$-0.523085\pi$$
−0.0724611 + 0.997371i $$0.523085\pi$$
$$572$$ 0 0
$$573$$ 1.61028e9 0.357569
$$574$$ 0 0
$$575$$ 1.03363e9 0.226740
$$576$$ 0 0
$$577$$ −2.63322e9 −0.570652 −0.285326 0.958430i $$-0.592102\pi$$
−0.285326 + 0.958430i $$0.592102\pi$$
$$578$$ 0 0
$$579$$ 2.64880e9 0.567118
$$580$$ 0 0
$$581$$ −5.84318e9 −1.23604
$$582$$ 0 0
$$583$$ −1.74723e9 −0.365183
$$584$$ 0 0
$$585$$ −3.87302e9 −0.799841
$$586$$ 0 0
$$587$$ −6.76347e9 −1.38018 −0.690090 0.723723i $$-0.742429\pi$$
−0.690090 + 0.723723i $$0.742429\pi$$
$$588$$ 0 0
$$589$$ 1.02200e10 2.06085
$$590$$ 0 0
$$591$$ 2.95156e9 0.588159
$$592$$ 0 0
$$593$$ −4.22718e9 −0.832452 −0.416226 0.909261i $$-0.636648\pi$$
−0.416226 + 0.909261i $$0.636648\pi$$
$$594$$ 0 0
$$595$$ −7.19129e9 −1.39958
$$596$$ 0 0
$$597$$ −9.83657e8 −0.189205
$$598$$ 0 0
$$599$$ −4.00299e9 −0.761010 −0.380505 0.924779i $$-0.624250\pi$$
−0.380505 + 0.924779i $$0.624250\pi$$
$$600$$ 0 0
$$601$$ 6.67554e9 1.25437 0.627185 0.778870i $$-0.284207\pi$$
0.627185 + 0.778870i $$0.284207\pi$$
$$602$$ 0 0
$$603$$ −3.34801e8 −0.0621836
$$604$$ 0 0
$$605$$ −7.26340e8 −0.133351
$$606$$ 0 0
$$607$$ 5.30634e9 0.963018 0.481509 0.876441i $$-0.340089\pi$$
0.481509 + 0.876441i $$0.340089\pi$$
$$608$$ 0 0
$$609$$ 5.18077e9 0.929466
$$610$$ 0 0
$$611$$ 8.38642e9 1.48742
$$612$$ 0 0
$$613$$ −8.65802e9 −1.51812 −0.759061 0.651019i $$-0.774342\pi$$
−0.759061 + 0.651019i $$0.774342\pi$$
$$614$$ 0 0
$$615$$ −5.26334e8 −0.0912428
$$616$$ 0 0
$$617$$ −7.38891e9 −1.26643 −0.633217 0.773974i $$-0.718266\pi$$
−0.633217 + 0.773974i $$0.718266\pi$$
$$618$$ 0 0
$$619$$ −9.99141e9 −1.69321 −0.846603 0.532225i $$-0.821356\pi$$
−0.846603 + 0.532225i $$0.821356\pi$$
$$620$$ 0 0
$$621$$ −2.26118e8 −0.0378892
$$622$$ 0 0
$$623$$ −6.52057e9 −1.08038
$$624$$ 0 0
$$625$$ −5.03731e9 −0.825313
$$626$$ 0 0
$$627$$ −1.94664e9 −0.315390
$$628$$ 0 0
$$629$$ 6.75461e9 1.08224
$$630$$ 0 0
$$631$$ 3.29834e9 0.522628 0.261314 0.965254i $$-0.415844\pi$$
0.261314 + 0.965254i $$0.415844\pi$$
$$632$$ 0 0
$$633$$ 3.74320e8 0.0586584
$$634$$ 0 0
$$635$$ −8.57598e9 −1.32916
$$636$$ 0 0
$$637$$ 3.02234e9 0.463292
$$638$$ 0 0
$$639$$ 1.31745e7 0.00199747
$$640$$ 0 0
$$641$$ −9.76971e9 −1.46514 −0.732569 0.680692i $$-0.761679\pi$$
−0.732569 + 0.680692i $$0.761679\pi$$
$$642$$ 0 0
$$643$$ 4.18444e9 0.620724 0.310362 0.950618i $$-0.399550\pi$$
0.310362 + 0.950618i $$0.399550\pi$$
$$644$$ 0 0
$$645$$ −6.66511e9 −0.978022
$$646$$ 0 0
$$647$$ 6.96085e8 0.101041 0.0505204 0.998723i $$-0.483912\pi$$
0.0505204 + 0.998723i $$0.483912\pi$$
$$648$$ 0 0
$$649$$ 3.56860e9 0.512438
$$650$$ 0 0
$$651$$ −5.23678e9 −0.743928
$$652$$ 0 0
$$653$$ 6.20046e9 0.871420 0.435710 0.900087i $$-0.356497\pi$$
0.435710 + 0.900087i $$0.356497\pi$$
$$654$$ 0 0
$$655$$ −4.61861e9 −0.642194
$$656$$ 0 0
$$657$$ −3.10599e8 −0.0427289
$$658$$ 0 0
$$659$$ 1.11404e10 1.51636 0.758178 0.652047i $$-0.226090\pi$$
0.758178 + 0.652047i $$0.226090\pi$$
$$660$$ 0 0
$$661$$ 4.56096e9 0.614258 0.307129 0.951668i $$-0.400632\pi$$
0.307129 + 0.951668i $$0.400632\pi$$
$$662$$ 0 0
$$663$$ −5.96941e9 −0.795489
$$664$$ 0 0
$$665$$ −2.28307e10 −3.01054
$$666$$ 0 0
$$667$$ −2.14428e9 −0.279796
$$668$$ 0 0
$$669$$ 3.67024e9 0.473918
$$670$$ 0 0
$$671$$ 7.33634e8 0.0937455
$$672$$ 0 0
$$673$$ 5.82879e9 0.737099 0.368550 0.929608i $$-0.379854\pi$$
0.368550 + 0.929608i $$0.379854\pi$$
$$674$$ 0 0
$$675$$ −1.77098e9 −0.221641
$$676$$ 0 0
$$677$$ 4.99624e9 0.618846 0.309423 0.950924i $$-0.399864\pi$$
0.309423 + 0.950924i $$0.399864\pi$$
$$678$$ 0 0
$$679$$ 1.71178e10 2.09848
$$680$$ 0 0
$$681$$ 7.61947e8 0.0924507
$$682$$ 0 0
$$683$$ −1.21371e10 −1.45762 −0.728808 0.684718i $$-0.759925\pi$$
−0.728808 + 0.684718i $$0.759925\pi$$
$$684$$ 0 0
$$685$$ −1.82159e8 −0.0216538
$$686$$ 0 0
$$687$$ 1.43428e9 0.168766
$$688$$ 0 0
$$689$$ −1.70103e10 −1.98127
$$690$$ 0 0
$$691$$ 9.23403e9 1.06468 0.532339 0.846531i $$-0.321313\pi$$
0.532339 + 0.846531i $$0.321313\pi$$
$$692$$ 0 0
$$693$$ 9.97467e8 0.113850
$$694$$ 0 0
$$695$$ 1.40471e10 1.58723
$$696$$ 0 0
$$697$$ −8.11230e8 −0.0907464
$$698$$ 0 0
$$699$$ −4.17390e9 −0.462245
$$700$$ 0 0
$$701$$ 4.74530e9 0.520296 0.260148 0.965569i $$-0.416229\pi$$
0.260148 + 0.965569i $$0.416229\pi$$
$$702$$ 0 0
$$703$$ 2.14444e10 2.32793
$$704$$ 0 0
$$705$$ 7.16450e9 0.770059
$$706$$ 0 0
$$707$$ −2.14160e9 −0.227914
$$708$$ 0 0
$$709$$ 1.34547e10 1.41779 0.708894 0.705315i $$-0.249195\pi$$
0.708894 + 0.705315i $$0.249195\pi$$
$$710$$ 0 0
$$711$$ −2.17070e8 −0.0226494
$$712$$ 0 0
$$713$$ 2.16746e9 0.223944
$$714$$ 0 0
$$715$$ −7.07131e9 −0.723484
$$716$$ 0 0
$$717$$ −5.02587e9 −0.509207
$$718$$ 0 0
$$719$$ −2.63976e9 −0.264858 −0.132429 0.991192i $$-0.542278\pi$$
−0.132429 + 0.991192i $$0.542278\pi$$
$$720$$ 0 0
$$721$$ 2.45740e9 0.244175
$$722$$ 0 0
$$723$$ −7.07689e9 −0.696400
$$724$$ 0 0
$$725$$ −1.67942e10 −1.63673
$$726$$ 0 0
$$727$$ 3.52707e9 0.340442 0.170221 0.985406i $$-0.445552\pi$$
0.170221 + 0.985406i $$0.445552\pi$$
$$728$$ 0 0
$$729$$ 3.87420e8 0.0370370
$$730$$ 0 0
$$731$$ −1.02728e10 −0.972700
$$732$$ 0 0
$$733$$ −1.03828e10 −0.973760 −0.486880 0.873469i $$-0.661865\pi$$
−0.486880 + 0.873469i $$0.661865\pi$$
$$734$$ 0 0
$$735$$ 2.58198e9 0.239854
$$736$$ 0 0
$$737$$ −6.11275e8 −0.0562471
$$738$$ 0 0
$$739$$ −2.05418e9 −0.187233 −0.0936164 0.995608i $$-0.529843\pi$$
−0.0936164 + 0.995608i $$0.529843\pi$$
$$740$$ 0 0
$$741$$ −1.89515e10 −1.71112
$$742$$ 0 0
$$743$$ −4.87476e9 −0.436006 −0.218003 0.975948i $$-0.569954\pi$$
−0.218003 + 0.975948i $$0.569954\pi$$
$$744$$ 0 0
$$745$$ 1.97707e10 1.75176
$$746$$ 0 0
$$747$$ −4.14366e9 −0.363715
$$748$$ 0 0
$$749$$ 1.44552e10 1.25701
$$750$$ 0 0
$$751$$ −1.15809e10 −0.997705 −0.498853 0.866687i $$-0.666245\pi$$
−0.498853 + 0.866687i $$0.666245\pi$$
$$752$$ 0 0
$$753$$ −7.44733e9 −0.635650
$$754$$ 0 0
$$755$$ −1.83964e10 −1.55567
$$756$$ 0 0
$$757$$ 3.46735e9 0.290511 0.145255 0.989394i $$-0.453600\pi$$
0.145255 + 0.989394i $$0.453600\pi$$
$$758$$ 0 0
$$759$$ −4.12844e8 −0.0342720
$$760$$ 0 0
$$761$$ −1.14023e10 −0.937877 −0.468938 0.883231i $$-0.655363\pi$$
−0.468938 + 0.883231i $$0.655363\pi$$
$$762$$ 0 0
$$763$$ −1.14438e10 −0.932681
$$764$$ 0 0
$$765$$ −5.09966e9 −0.411838
$$766$$ 0 0
$$767$$ 3.47422e10 2.78018
$$768$$ 0 0
$$769$$ 2.30715e10 1.82951 0.914754 0.404012i $$-0.132384\pi$$
0.914754 + 0.404012i $$0.132384\pi$$
$$770$$ 0 0
$$771$$ −2.88512e7 −0.00226711
$$772$$ 0 0
$$773$$ 2.15091e10 1.67492 0.837461 0.546497i $$-0.184039\pi$$
0.837461 + 0.546497i $$0.184039\pi$$
$$774$$ 0 0
$$775$$ 1.69758e10 1.31001
$$776$$ 0 0
$$777$$ −1.09882e10 −0.840337
$$778$$ 0 0
$$779$$ −2.57547e9 −0.195198
$$780$$ 0 0
$$781$$ 2.40538e7 0.00180678
$$782$$ 0 0
$$783$$ 3.67391e9 0.273503
$$784$$ 0 0
$$785$$ 2.20952e9 0.163025
$$786$$ 0 0
$$787$$ 4.46678e9 0.326651 0.163325 0.986572i $$-0.447778\pi$$
0.163325 + 0.986572i $$0.447778\pi$$
$$788$$ 0 0
$$789$$ −2.14090e9 −0.155176
$$790$$ 0 0
$$791$$ 5.82777e9 0.418682
$$792$$ 0 0
$$793$$ 7.14232e9 0.508608
$$794$$ 0 0
$$795$$ −1.45318e10 −1.02574
$$796$$ 0 0
$$797$$ 2.43899e10 1.70650 0.853248 0.521505i $$-0.174629\pi$$
0.853248 + 0.521505i $$0.174629\pi$$
$$798$$ 0 0
$$799$$ 1.10425e10 0.765869
$$800$$ 0 0
$$801$$ −4.62402e9 −0.317911
$$802$$ 0 0
$$803$$ −5.67089e8 −0.0386497
$$804$$ 0 0
$$805$$ −4.84196e9 −0.327142
$$806$$ 0 0
$$807$$ −5.67459e9 −0.380082
$$808$$ 0 0
$$809$$ 9.88857e9 0.656620 0.328310 0.944570i $$-0.393521\pi$$
0.328310 + 0.944570i $$0.393521\pi$$
$$810$$ 0 0
$$811$$ 1.15204e10 0.758395 0.379198 0.925316i $$-0.376200\pi$$
0.379198 + 0.925316i $$0.376200\pi$$
$$812$$ 0 0
$$813$$ −7.16876e9 −0.467872
$$814$$ 0 0
$$815$$ 4.02487e10 2.60435
$$816$$ 0 0
$$817$$ −3.26139e10 −2.09231
$$818$$ 0 0
$$819$$ 9.71088e9 0.617682
$$820$$ 0 0
$$821$$ 2.63516e9 0.166191 0.0830953 0.996542i $$-0.473519\pi$$
0.0830953 + 0.996542i $$0.473519\pi$$
$$822$$ 0 0
$$823$$ 1.27039e10 0.794400 0.397200 0.917732i $$-0.369982\pi$$
0.397200 + 0.917732i $$0.369982\pi$$
$$824$$ 0 0
$$825$$ −3.23343e9 −0.200482
$$826$$ 0 0
$$827$$ 1.11339e10 0.684504 0.342252 0.939608i $$-0.388810\pi$$
0.342252 + 0.939608i $$0.388810\pi$$
$$828$$ 0 0
$$829$$ 2.79852e10 1.70604 0.853018 0.521881i $$-0.174770\pi$$
0.853018 + 0.521881i $$0.174770\pi$$
$$830$$ 0 0
$$831$$ 1.68353e10 1.01769
$$832$$ 0 0
$$833$$ 3.97956e9 0.238549
$$834$$ 0 0
$$835$$ −1.80651e10 −1.07383
$$836$$ 0 0
$$837$$ −3.71363e9 −0.218907
$$838$$ 0 0
$$839$$ −2.71170e9 −0.158517 −0.0792583 0.996854i $$-0.525255\pi$$
−0.0792583 + 0.996854i $$0.525255\pi$$
$$840$$ 0 0
$$841$$ 1.75898e10 1.01971
$$842$$ 0 0
$$843$$ −3.52650e9 −0.202744
$$844$$ 0 0
$$845$$ −4.31161e10 −2.45834
$$846$$ 0 0
$$847$$ 1.82116e9 0.102981
$$848$$ 0 0
$$849$$ −5.96359e8 −0.0334449
$$850$$ 0 0
$$851$$ 4.54794e9 0.252965
$$852$$ 0 0
$$853$$ 1.97175e10 1.08775 0.543877 0.839165i $$-0.316956\pi$$
0.543877 + 0.839165i $$0.316956\pi$$
$$854$$ 0 0
$$855$$ −1.61903e10 −0.885876
$$856$$ 0 0
$$857$$ 1.89411e10 1.02795 0.513976 0.857804i $$-0.328172\pi$$
0.513976 + 0.857804i $$0.328172\pi$$
$$858$$ 0 0
$$859$$ 6.77637e9 0.364772 0.182386 0.983227i $$-0.441618\pi$$
0.182386 + 0.983227i $$0.441618\pi$$
$$860$$ 0 0
$$861$$ 1.31969e9 0.0704628
$$862$$ 0 0
$$863$$ −2.80635e10 −1.48629 −0.743146 0.669129i $$-0.766667\pi$$
−0.743146 + 0.669129i $$0.766667\pi$$
$$864$$ 0 0
$$865$$ −2.75146e10 −1.44546
$$866$$ 0 0
$$867$$ 3.21912e9 0.167753
$$868$$ 0 0
$$869$$ −3.96324e8 −0.0204871
$$870$$ 0 0
$$871$$ −5.95109e9 −0.305164
$$872$$ 0 0
$$873$$ 1.21390e10 0.617495
$$874$$ 0 0
$$875$$ −4.99454e9 −0.252039
$$876$$ 0 0
$$877$$ −1.01559e10 −0.508418 −0.254209 0.967149i $$-0.581815\pi$$
−0.254209 + 0.967149i $$0.581815\pi$$
$$878$$ 0 0
$$879$$ −5.40507e9 −0.268436
$$880$$ 0 0
$$881$$ −2.78023e10 −1.36982 −0.684912 0.728626i $$-0.740159\pi$$
−0.684912 + 0.728626i $$0.740159\pi$$
$$882$$ 0 0
$$883$$ −2.15199e10 −1.05191 −0.525954 0.850513i $$-0.676291\pi$$
−0.525954 + 0.850513i $$0.676291\pi$$
$$884$$ 0 0
$$885$$ 2.96802e10 1.43935
$$886$$ 0 0
$$887$$ 3.13376e10 1.50776 0.753881 0.657011i $$-0.228180\pi$$
0.753881 + 0.657011i $$0.228180\pi$$
$$888$$ 0 0
$$889$$ 2.15027e10 1.02645
$$890$$ 0 0
$$891$$ 7.07348e8 0.0335013
$$892$$ 0 0
$$893$$ 3.50575e10 1.64741
$$894$$ 0 0
$$895$$ 1.78321e9 0.0831423
$$896$$ 0 0
$$897$$ −4.01926e9 −0.185940
$$898$$ 0 0
$$899$$ −3.52164e10 −1.61654
$$900$$ 0 0
$$901$$ −2.23977e10 −1.02015
$$902$$ 0 0
$$903$$ 1.67116e10 0.755283
$$904$$ 0 0
$$905$$ 4.92976e9 0.221083
$$906$$ 0 0
$$907$$ 1.62459e10 0.722966 0.361483 0.932379i $$-0.382270\pi$$
0.361483 + 0.932379i $$0.382270\pi$$
$$908$$ 0 0
$$909$$ −1.51870e9 −0.0670656
$$910$$ 0 0
$$911$$ 3.15726e9 0.138356 0.0691778 0.997604i $$-0.477962\pi$$
0.0691778 + 0.997604i $$0.477962\pi$$
$$912$$ 0 0
$$913$$ −7.56544e9 −0.328993
$$914$$ 0 0
$$915$$ 6.10167e9 0.263315
$$916$$ 0 0
$$917$$ 1.15803e10 0.495938
$$918$$ 0 0
$$919$$ −2.53655e9 −0.107805 −0.0539026 0.998546i $$-0.517166\pi$$
−0.0539026 + 0.998546i $$0.517166\pi$$
$$920$$ 0 0
$$921$$ −1.29529e10 −0.546333
$$922$$ 0 0
$$923$$ 2.34177e8 0.00980253
$$924$$ 0 0
$$925$$ 3.56198e10 1.47978
$$926$$ 0 0
$$927$$ 1.74265e9 0.0718506
$$928$$ 0 0
$$929$$ −2.10282e10 −0.860494 −0.430247 0.902711i $$-0.641574\pi$$
−0.430247 + 0.902711i $$0.641574\pi$$
$$930$$ 0 0
$$931$$ 1.26342e10 0.513126
$$932$$ 0 0
$$933$$ −1.40328e10 −0.565665
$$934$$ 0 0
$$935$$ −9.31090e9 −0.372521
$$936$$ 0 0
$$937$$ 3.12893e10 1.24253 0.621265 0.783601i $$-0.286619\pi$$
0.621265 + 0.783601i $$0.286619\pi$$
$$938$$ 0 0
$$939$$ −2.61835e10 −1.03204
$$940$$ 0 0
$$941$$ 7.91706e9 0.309742 0.154871 0.987935i $$-0.450504\pi$$
0.154871 + 0.987935i $$0.450504\pi$$
$$942$$ 0 0
$$943$$ −5.46208e8 −0.0212113
$$944$$ 0 0
$$945$$ 8.29599e9 0.319784
$$946$$ 0 0
$$947$$ −8.55849e9 −0.327471 −0.163735 0.986504i $$-0.552354\pi$$
−0.163735 + 0.986504i $$0.552354\pi$$
$$948$$ 0 0
$$949$$ −5.52091e9 −0.209691
$$950$$ 0 0
$$951$$ −2.04356e10 −0.770471
$$952$$ 0 0
$$953$$ −8.49661e9 −0.317995 −0.158998 0.987279i $$-0.550826\pi$$
−0.158998 + 0.987279i $$0.550826\pi$$
$$954$$ 0 0
$$955$$ 2.44524e10 0.908466
$$956$$ 0 0
$$957$$ 6.70778e9 0.247393
$$958$$ 0 0
$$959$$ 4.56730e8 0.0167222
$$960$$ 0 0
$$961$$ 8.08451e9 0.293847
$$962$$ 0 0
$$963$$ 1.02508e10 0.369885
$$964$$ 0 0
$$965$$ 4.02225e10 1.44086
$$966$$ 0 0
$$967$$ 1.47988e10 0.526300 0.263150 0.964755i $$-0.415239\pi$$
0.263150 + 0.964755i $$0.415239\pi$$
$$968$$ 0 0
$$969$$ −2.49538e10 −0.881056
$$970$$ 0 0
$$971$$ 2.86157e10 1.00308 0.501542 0.865133i $$-0.332766\pi$$
0.501542 + 0.865133i $$0.332766\pi$$
$$972$$ 0 0
$$973$$ −3.52206e10 −1.22575
$$974$$ 0 0
$$975$$ −3.14792e10 −1.08770
$$976$$ 0 0
$$977$$ 3.37991e10 1.15951 0.579755 0.814791i $$-0.303148\pi$$
0.579755 + 0.814791i $$0.303148\pi$$
$$978$$ 0 0
$$979$$ −8.44249e9 −0.287562
$$980$$ 0 0
$$981$$ −8.11528e9 −0.274449
$$982$$ 0 0
$$983$$ −1.03134e9 −0.0346308 −0.0173154 0.999850i $$-0.505512\pi$$
−0.0173154 + 0.999850i $$0.505512\pi$$
$$984$$ 0 0
$$985$$ 4.48199e10 1.49432
$$986$$ 0 0
$$987$$ −1.79637e10 −0.594683
$$988$$ 0 0
$$989$$ −6.91679e9 −0.227362
$$990$$ 0 0
$$991$$ 5.63139e10 1.83805 0.919027 0.394195i $$-0.128977\pi$$
0.919027 + 0.394195i $$0.128977\pi$$
$$992$$ 0 0
$$993$$ −4.85642e9 −0.157396
$$994$$ 0 0
$$995$$ −1.49370e10 −0.480710
$$996$$ 0 0
$$997$$ 2.55531e10 0.816603 0.408301 0.912847i $$-0.366121\pi$$
0.408301 + 0.912847i $$0.366121\pi$$
$$998$$ 0 0
$$999$$ −7.79222e9 −0.247276
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 528.8.a.a.1.1 1
4.3 odd 2 33.8.a.a.1.1 1
12.11 even 2 99.8.a.a.1.1 1
44.43 even 2 363.8.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.8.a.a.1.1 1 4.3 odd 2
99.8.a.a.1.1 1 12.11 even 2
363.8.a.a.1.1 1 44.43 even 2
528.8.a.a.1.1 1 1.1 even 1 trivial