# Properties

 Label 54.9.b.a.53.1 Level $54$ Weight $9$ Character 54.53 Analytic conductor $21.998$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,9,Mod(53,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("54.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 54.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$21.9984449433$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 53.1 Root $$-1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 54.53 Dual form 54.9.b.a.53.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-11.3137i q^{2} -128.000 q^{4} -678.823i q^{5} -2065.00 q^{7} +1448.15i q^{8} +O(q^{10})$$ $$q-11.3137i q^{2} -128.000 q^{4} -678.823i q^{5} -2065.00 q^{7} +1448.15i q^{8} -7680.00 q^{10} -6652.46i q^{11} +8063.00 q^{13} +23362.8i q^{14} +16384.0 q^{16} +21586.6i q^{17} -226609. q^{19} +86889.3i q^{20} -75264.0 q^{22} +368329. i q^{23} -70175.0 q^{25} -91222.4i q^{26} +264320. q^{28} +937047. i q^{29} +826370. q^{31} -185364. i q^{32} +244224. q^{34} +1.40177e6i q^{35} +1.34458e6 q^{37} +2.56379e6i q^{38} +983040. q^{40} +5.19191e6i q^{41} -6.14774e6 q^{43} +851515. i q^{44} +4.16717e6 q^{46} -5.91078e6i q^{47} -1.50058e6 q^{49} +793939. i q^{50} -1.03206e6 q^{52} +768156. i q^{53} -4.51584e6 q^{55} -2.99044e6i q^{56} +1.06015e7 q^{58} -473954. i q^{59} -1.49857e7 q^{61} -9.34931e6i q^{62} -2.09715e6 q^{64} -5.47335e6i q^{65} -1.00237e7 q^{67} -2.76308e6i q^{68} +1.58592e7 q^{70} -4.54849e7i q^{71} -2.32616e7 q^{73} -1.52121e7i q^{74} +2.90060e7 q^{76} +1.37373e7i q^{77} +1.42672e7 q^{79} -1.11218e7i q^{80} +5.87397e7 q^{82} -3.61918e7i q^{83} +1.46534e7 q^{85} +6.95538e7i q^{86} +9.63379e6 q^{88} -1.15088e8i q^{89} -1.66501e7 q^{91} -4.71461e7i q^{92} -6.68728e7 q^{94} +1.53827e8i q^{95} -4.05716e7 q^{97} +1.69771e7i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 256 q^{4} - 4130 q^{7}+O(q^{10})$$ 2 * q - 256 * q^4 - 4130 * q^7 $$2 q - 256 q^{4} - 4130 q^{7} - 15360 q^{10} + 16126 q^{13} + 32768 q^{16} - 453218 q^{19} - 150528 q^{22} - 140350 q^{25} + 528640 q^{28} + 1652740 q^{31} + 488448 q^{34} + 2689150 q^{37} + 1966080 q^{40} - 12295484 q^{43} + 8334336 q^{46} - 3001152 q^{49} - 2064128 q^{52} - 9031680 q^{55} + 21202944 q^{58} - 29971394 q^{61} - 4194304 q^{64} - 20047394 q^{67} + 31718400 q^{70} - 46523138 q^{73} + 58011904 q^{76} + 28534366 q^{79} + 117479424 q^{82} + 29306880 q^{85} + 19267584 q^{88} - 33300190 q^{91} - 133745664 q^{94} - 81143234 q^{97}+O(q^{100})$$ 2 * q - 256 * q^4 - 4130 * q^7 - 15360 * q^10 + 16126 * q^13 + 32768 * q^16 - 453218 * q^19 - 150528 * q^22 - 140350 * q^25 + 528640 * q^28 + 1652740 * q^31 + 488448 * q^34 + 2689150 * q^37 + 1966080 * q^40 - 12295484 * q^43 + 8334336 * q^46 - 3001152 * q^49 - 2064128 * q^52 - 9031680 * q^55 + 21202944 * q^58 - 29971394 * q^61 - 4194304 * q^64 - 20047394 * q^67 + 31718400 * q^70 - 46523138 * q^73 + 58011904 * q^76 + 28534366 * q^79 + 117479424 * q^82 + 29306880 * q^85 + 19267584 * q^88 - 33300190 * q^91 - 133745664 * q^94 - 81143234 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 11.3137i − 0.707107i
$$3$$ 0 0
$$4$$ −128.000 −0.500000
$$5$$ − 678.823i − 1.08612i −0.839695 0.543058i $$-0.817266\pi$$
0.839695 0.543058i $$-0.182734\pi$$
$$6$$ 0 0
$$7$$ −2065.00 −0.860058 −0.430029 0.902815i $$-0.641497\pi$$
−0.430029 + 0.902815i $$0.641497\pi$$
$$8$$ 1448.15i 0.353553i
$$9$$ 0 0
$$10$$ −7680.00 −0.768000
$$11$$ − 6652.46i − 0.454372i −0.973851 0.227186i $$-0.927047\pi$$
0.973851 0.227186i $$-0.0729525\pi$$
$$12$$ 0 0
$$13$$ 8063.00 0.282308 0.141154 0.989988i $$-0.454919\pi$$
0.141154 + 0.989988i $$0.454919\pi$$
$$14$$ 23362.8i 0.608153i
$$15$$ 0 0
$$16$$ 16384.0 0.250000
$$17$$ 21586.6i 0.258457i 0.991615 + 0.129228i $$0.0412500\pi$$
−0.991615 + 0.129228i $$0.958750\pi$$
$$18$$ 0 0
$$19$$ −226609. −1.73885 −0.869426 0.494063i $$-0.835511\pi$$
−0.869426 + 0.494063i $$0.835511\pi$$
$$20$$ 86889.3i 0.543058i
$$21$$ 0 0
$$22$$ −75264.0 −0.321290
$$23$$ 368329.i 1.31621i 0.752927 + 0.658104i $$0.228641\pi$$
−0.752927 + 0.658104i $$0.771359\pi$$
$$24$$ 0 0
$$25$$ −70175.0 −0.179648
$$26$$ − 91222.4i − 0.199622i
$$27$$ 0 0
$$28$$ 264320. 0.430029
$$29$$ 937047.i 1.32486i 0.749125 + 0.662429i $$0.230474\pi$$
−0.749125 + 0.662429i $$0.769526\pi$$
$$30$$ 0 0
$$31$$ 826370. 0.894804 0.447402 0.894333i $$-0.352349\pi$$
0.447402 + 0.894333i $$0.352349\pi$$
$$32$$ − 185364.i − 0.176777i
$$33$$ 0 0
$$34$$ 244224. 0.182756
$$35$$ 1.40177e6i 0.934123i
$$36$$ 0 0
$$37$$ 1.34458e6 0.717428 0.358714 0.933448i $$-0.383215\pi$$
0.358714 + 0.933448i $$0.383215\pi$$
$$38$$ 2.56379e6i 1.22955i
$$39$$ 0 0
$$40$$ 983040. 0.384000
$$41$$ 5.19191e6i 1.83735i 0.395017 + 0.918674i $$0.370739\pi$$
−0.395017 + 0.918674i $$0.629261\pi$$
$$42$$ 0 0
$$43$$ −6.14774e6 −1.79822 −0.899108 0.437727i $$-0.855784\pi$$
−0.899108 + 0.437727i $$0.855784\pi$$
$$44$$ 851515.i 0.227186i
$$45$$ 0 0
$$46$$ 4.16717e6 0.930700
$$47$$ − 5.91078e6i − 1.21130i −0.795729 0.605652i $$-0.792912\pi$$
0.795729 0.605652i $$-0.207088\pi$$
$$48$$ 0 0
$$49$$ −1.50058e6 −0.260300
$$50$$ 793939.i 0.127030i
$$51$$ 0 0
$$52$$ −1.03206e6 −0.141154
$$53$$ 768156.i 0.0973522i 0.998815 + 0.0486761i $$0.0155002\pi$$
−0.998815 + 0.0486761i $$0.984500\pi$$
$$54$$ 0 0
$$55$$ −4.51584e6 −0.493501
$$56$$ − 2.99044e6i − 0.304077i
$$57$$ 0 0
$$58$$ 1.06015e7 0.936816
$$59$$ − 473954.i − 0.0391136i −0.999809 0.0195568i $$-0.993774\pi$$
0.999809 0.0195568i $$-0.00622552\pi$$
$$60$$ 0 0
$$61$$ −1.49857e7 −1.08232 −0.541162 0.840918i $$-0.682016\pi$$
−0.541162 + 0.840918i $$0.682016\pi$$
$$62$$ − 9.34931e6i − 0.632722i
$$63$$ 0 0
$$64$$ −2.09715e6 −0.125000
$$65$$ − 5.47335e6i − 0.306619i
$$66$$ 0 0
$$67$$ −1.00237e7 −0.497426 −0.248713 0.968577i $$-0.580008\pi$$
−0.248713 + 0.968577i $$0.580008\pi$$
$$68$$ − 2.76308e6i − 0.129228i
$$69$$ 0 0
$$70$$ 1.58592e7 0.660525
$$71$$ − 4.54849e7i − 1.78992i −0.446145 0.894961i $$-0.647203\pi$$
0.446145 0.894961i $$-0.352797\pi$$
$$72$$ 0 0
$$73$$ −2.32616e7 −0.819120 −0.409560 0.912283i $$-0.634318\pi$$
−0.409560 + 0.912283i $$0.634318\pi$$
$$74$$ − 1.52121e7i − 0.507298i
$$75$$ 0 0
$$76$$ 2.90060e7 0.869426
$$77$$ 1.37373e7i 0.390786i
$$78$$ 0 0
$$79$$ 1.42672e7 0.366294 0.183147 0.983086i $$-0.441372\pi$$
0.183147 + 0.983086i $$0.441372\pi$$
$$80$$ − 1.11218e7i − 0.271529i
$$81$$ 0 0
$$82$$ 5.87397e7 1.29920
$$83$$ − 3.61918e7i − 0.762602i −0.924451 0.381301i $$-0.875476\pi$$
0.924451 0.381301i $$-0.124524\pi$$
$$84$$ 0 0
$$85$$ 1.46534e7 0.280714
$$86$$ 6.95538e7i 1.27153i
$$87$$ 0 0
$$88$$ 9.63379e6 0.160645
$$89$$ − 1.15088e8i − 1.83429i −0.398549 0.917147i $$-0.630486\pi$$
0.398549 0.917147i $$-0.369514\pi$$
$$90$$ 0 0
$$91$$ −1.66501e7 −0.242801
$$92$$ − 4.71461e7i − 0.658104i
$$93$$ 0 0
$$94$$ −6.68728e7 −0.856522
$$95$$ 1.53827e8i 1.88860i
$$96$$ 0 0
$$97$$ −4.05716e7 −0.458285 −0.229142 0.973393i $$-0.573592\pi$$
−0.229142 + 0.973393i $$0.573592\pi$$
$$98$$ 1.69771e7i 0.184060i
$$99$$ 0 0
$$100$$ 8.98240e6 0.0898240
$$101$$ 6.49465e7i 0.624123i 0.950062 + 0.312061i $$0.101019\pi$$
−0.950062 + 0.312061i $$0.898981\pi$$
$$102$$ 0 0
$$103$$ −1.37263e8 −1.21956 −0.609782 0.792569i $$-0.708743\pi$$
−0.609782 + 0.792569i $$0.708743\pi$$
$$104$$ 1.16765e7i 0.0998110i
$$105$$ 0 0
$$106$$ 8.69069e6 0.0688384
$$107$$ 1.39108e8i 1.06125i 0.847607 + 0.530625i $$0.178043\pi$$
−0.847607 + 0.530625i $$0.821957\pi$$
$$108$$ 0 0
$$109$$ −4.29417e7 −0.304210 −0.152105 0.988364i $$-0.548605\pi$$
−0.152105 + 0.988364i $$0.548605\pi$$
$$110$$ 5.10909e7i 0.348958i
$$111$$ 0 0
$$112$$ −3.38330e7 −0.215015
$$113$$ 2.91649e8i 1.78874i 0.447329 + 0.894369i $$0.352375\pi$$
−0.447329 + 0.894369i $$0.647625\pi$$
$$114$$ 0 0
$$115$$ 2.50030e8 1.42956
$$116$$ − 1.19942e8i − 0.662429i
$$117$$ 0 0
$$118$$ −5.36218e6 −0.0276575
$$119$$ − 4.45762e7i − 0.222288i
$$120$$ 0 0
$$121$$ 1.70104e8 0.793546
$$122$$ 1.69544e8i 0.765319i
$$123$$ 0 0
$$124$$ −1.05775e8 −0.447402
$$125$$ − 2.17529e8i − 0.890997i
$$126$$ 0 0
$$127$$ −3.39515e8 −1.30510 −0.652551 0.757745i $$-0.726301\pi$$
−0.652551 + 0.757745i $$0.726301\pi$$
$$128$$ 2.37266e7i 0.0883883i
$$129$$ 0 0
$$130$$ −6.19238e7 −0.216813
$$131$$ 2.24537e8i 0.762436i 0.924485 + 0.381218i $$0.124495\pi$$
−0.924485 + 0.381218i $$0.875505\pi$$
$$132$$ 0 0
$$133$$ 4.67948e8 1.49551
$$134$$ 1.13405e8i 0.351733i
$$135$$ 0 0
$$136$$ −3.12607e7 −0.0913782
$$137$$ − 4.01263e8i − 1.13906i −0.821970 0.569531i $$-0.807125\pi$$
0.821970 0.569531i $$-0.192875\pi$$
$$138$$ 0 0
$$139$$ 2.69764e8 0.722645 0.361322 0.932441i $$-0.382325\pi$$
0.361322 + 0.932441i $$0.382325\pi$$
$$140$$ − 1.79426e8i − 0.467062i
$$141$$ 0 0
$$142$$ −5.14603e8 −1.26567
$$143$$ − 5.36388e7i − 0.128273i
$$144$$ 0 0
$$145$$ 6.36088e8 1.43895
$$146$$ 2.63175e8i 0.579205i
$$147$$ 0 0
$$148$$ −1.72106e8 −0.358714
$$149$$ 1.94198e8i 0.394003i 0.980403 + 0.197002i $$0.0631204\pi$$
−0.980403 + 0.197002i $$0.936880\pi$$
$$150$$ 0 0
$$151$$ −8.75100e7 −0.168325 −0.0841627 0.996452i $$-0.526822\pi$$
−0.0841627 + 0.996452i $$0.526822\pi$$
$$152$$ − 3.28165e8i − 0.614777i
$$153$$ 0 0
$$154$$ 1.55420e8 0.276328
$$155$$ − 5.60959e8i − 0.971861i
$$156$$ 0 0
$$157$$ 2.84655e8 0.468512 0.234256 0.972175i $$-0.424735\pi$$
0.234256 + 0.972175i $$0.424735\pi$$
$$158$$ − 1.61415e8i − 0.259009i
$$159$$ 0 0
$$160$$ −1.25829e8 −0.192000
$$161$$ − 7.60600e8i − 1.13202i
$$162$$ 0 0
$$163$$ −2.63153e8 −0.372785 −0.186393 0.982475i $$-0.559680\pi$$
−0.186393 + 0.982475i $$0.559680\pi$$
$$164$$ − 6.64564e8i − 0.918674i
$$165$$ 0 0
$$166$$ −4.09464e8 −0.539241
$$167$$ − 1.16384e9i − 1.49633i −0.663511 0.748167i $$-0.730934\pi$$
0.663511 0.748167i $$-0.269066\pi$$
$$168$$ 0 0
$$169$$ −7.50719e8 −0.920302
$$170$$ − 1.65785e8i − 0.198495i
$$171$$ 0 0
$$172$$ 7.86911e8 0.899108
$$173$$ 1.32417e9i 1.47829i 0.673547 + 0.739145i $$0.264770\pi$$
−0.673547 + 0.739145i $$0.735230\pi$$
$$174$$ 0 0
$$175$$ 1.44911e8 0.154508
$$176$$ − 1.08994e8i − 0.113593i
$$177$$ 0 0
$$178$$ −1.30207e9 −1.29704
$$179$$ 4.06696e8i 0.396148i 0.980187 + 0.198074i $$0.0634686\pi$$
−0.980187 + 0.198074i $$0.936531\pi$$
$$180$$ 0 0
$$181$$ −1.29071e9 −1.20258 −0.601289 0.799032i $$-0.705346\pi$$
−0.601289 + 0.799032i $$0.705346\pi$$
$$182$$ 1.88374e8i 0.171687i
$$183$$ 0 0
$$184$$ −5.33398e8 −0.465350
$$185$$ − 9.12728e8i − 0.779210i
$$186$$ 0 0
$$187$$ 1.43604e8 0.117435
$$188$$ 7.56580e8i 0.605652i
$$189$$ 0 0
$$190$$ 1.74036e9 1.33544
$$191$$ 2.97714e8i 0.223700i 0.993725 + 0.111850i $$0.0356776\pi$$
−0.993725 + 0.111850i $$0.964322\pi$$
$$192$$ 0 0
$$193$$ 2.22004e9 1.60004 0.800021 0.599972i $$-0.204822\pi$$
0.800021 + 0.599972i $$0.204822\pi$$
$$194$$ 4.59015e8i 0.324056i
$$195$$ 0 0
$$196$$ 1.92074e8 0.130150
$$197$$ 1.91580e9i 1.27199i 0.771693 + 0.635996i $$0.219410\pi$$
−0.771693 + 0.635996i $$0.780590\pi$$
$$198$$ 0 0
$$199$$ −1.75472e9 −1.11891 −0.559457 0.828859i $$-0.688990\pi$$
−0.559457 + 0.828859i $$0.688990\pi$$
$$200$$ − 1.01624e8i − 0.0635152i
$$201$$ 0 0
$$202$$ 7.34786e8 0.441322
$$203$$ − 1.93500e9i − 1.13945i
$$204$$ 0 0
$$205$$ 3.52438e9 1.99557
$$206$$ 1.55295e9i 0.862362i
$$207$$ 0 0
$$208$$ 1.32104e8 0.0705770
$$209$$ 1.50751e9i 0.790086i
$$210$$ 0 0
$$211$$ −2.15389e9 −1.08666 −0.543331 0.839519i $$-0.682837\pi$$
−0.543331 + 0.839519i $$0.682837\pi$$
$$212$$ − 9.83239e7i − 0.0486761i
$$213$$ 0 0
$$214$$ 1.57383e9 0.750417
$$215$$ 4.17323e9i 1.95307i
$$216$$ 0 0
$$217$$ −1.70645e9 −0.769583
$$218$$ 4.85829e8i 0.215109i
$$219$$ 0 0
$$220$$ 5.78028e8 0.246750
$$221$$ 1.74052e8i 0.0729644i
$$222$$ 0 0
$$223$$ −2.39374e9 −0.967959 −0.483980 0.875079i $$-0.660809\pi$$
−0.483980 + 0.875079i $$0.660809\pi$$
$$224$$ 3.82776e8i 0.152038i
$$225$$ 0 0
$$226$$ 3.29963e9 1.26483
$$227$$ 9.92657e8i 0.373849i 0.982374 + 0.186924i $$0.0598519\pi$$
−0.982374 + 0.186924i $$0.940148\pi$$
$$228$$ 0 0
$$229$$ −2.22150e9 −0.807800 −0.403900 0.914803i $$-0.632346\pi$$
−0.403900 + 0.914803i $$0.632346\pi$$
$$230$$ − 2.82877e9i − 1.01085i
$$231$$ 0 0
$$232$$ −1.35699e9 −0.468408
$$233$$ − 1.24042e9i − 0.420867i −0.977608 0.210433i $$-0.932512\pi$$
0.977608 0.210433i $$-0.0674875\pi$$
$$234$$ 0 0
$$235$$ −4.01237e9 −1.31562
$$236$$ 6.06661e7i 0.0195568i
$$237$$ 0 0
$$238$$ −5.04323e8 −0.157181
$$239$$ 7.59290e8i 0.232711i 0.993208 + 0.116355i $$0.0371211\pi$$
−0.993208 + 0.116355i $$0.962879\pi$$
$$240$$ 0 0
$$241$$ 4.47467e9 1.32646 0.663228 0.748417i $$-0.269186\pi$$
0.663228 + 0.748417i $$0.269186\pi$$
$$242$$ − 1.92450e9i − 0.561122i
$$243$$ 0 0
$$244$$ 1.91817e9 0.541162
$$245$$ 1.01862e9i 0.282716i
$$246$$ 0 0
$$247$$ −1.82715e9 −0.490892
$$248$$ 1.19671e9i 0.316361i
$$249$$ 0 0
$$250$$ −2.46106e9 −0.630030
$$251$$ 1.07356e9i 0.270477i 0.990813 + 0.135239i $$0.0431801\pi$$
−0.990813 + 0.135239i $$0.956820\pi$$
$$252$$ 0 0
$$253$$ 2.45029e9 0.598048
$$254$$ 3.84118e9i 0.922846i
$$255$$ 0 0
$$256$$ 2.68435e8 0.0625000
$$257$$ − 7.58666e9i − 1.73907i −0.493867 0.869537i $$-0.664417\pi$$
0.493867 0.869537i $$-0.335583\pi$$
$$258$$ 0 0
$$259$$ −2.77655e9 −0.617030
$$260$$ 7.00588e8i 0.153310i
$$261$$ 0 0
$$262$$ 2.54035e9 0.539124
$$263$$ − 2.81146e9i − 0.587637i −0.955861 0.293819i $$-0.905074\pi$$
0.955861 0.293819i $$-0.0949262\pi$$
$$264$$ 0 0
$$265$$ 5.21441e8 0.105736
$$266$$ − 5.29422e9i − 1.05749i
$$267$$ 0 0
$$268$$ 1.28303e9 0.248713
$$269$$ − 2.96285e9i − 0.565849i −0.959142 0.282924i $$-0.908695\pi$$
0.959142 0.282924i $$-0.0913045\pi$$
$$270$$ 0 0
$$271$$ −8.40415e9 −1.55818 −0.779089 0.626914i $$-0.784318\pi$$
−0.779089 + 0.626914i $$0.784318\pi$$
$$272$$ 3.53674e8i 0.0646142i
$$273$$ 0 0
$$274$$ −4.53978e9 −0.805438
$$275$$ 4.66836e8i 0.0816270i
$$276$$ 0 0
$$277$$ 5.98162e9 1.01601 0.508007 0.861353i $$-0.330382\pi$$
0.508007 + 0.861353i $$0.330382\pi$$
$$278$$ − 3.05203e9i − 0.510987i
$$279$$ 0 0
$$280$$ −2.02998e9 −0.330262
$$281$$ − 1.08341e10i − 1.73767i −0.495105 0.868833i $$-0.664870\pi$$
0.495105 0.868833i $$-0.335130\pi$$
$$282$$ 0 0
$$283$$ 3.78670e9 0.590357 0.295179 0.955442i $$-0.404621\pi$$
0.295179 + 0.955442i $$0.404621\pi$$
$$284$$ 5.82207e9i 0.894961i
$$285$$ 0 0
$$286$$ −6.06854e8 −0.0907026
$$287$$ − 1.07213e10i − 1.58023i
$$288$$ 0 0
$$289$$ 6.50978e9 0.933200
$$290$$ − 7.19652e9i − 1.01749i
$$291$$ 0 0
$$292$$ 2.97748e9 0.409560
$$293$$ 7.32039e9i 0.993262i 0.867962 + 0.496631i $$0.165430\pi$$
−0.867962 + 0.496631i $$0.834570\pi$$
$$294$$ 0 0
$$295$$ −3.21731e8 −0.0424819
$$296$$ 1.94715e9i 0.253649i
$$297$$ 0 0
$$298$$ 2.19710e9 0.278602
$$299$$ 2.96984e9i 0.371576i
$$300$$ 0 0
$$301$$ 1.26951e10 1.54657
$$302$$ 9.90063e8i 0.119024i
$$303$$ 0 0
$$304$$ −3.71276e9 −0.434713
$$305$$ 1.01726e10i 1.17553i
$$306$$ 0 0
$$307$$ −2.21986e8 −0.0249903 −0.0124951 0.999922i $$-0.503977\pi$$
−0.0124951 + 0.999922i $$0.503977\pi$$
$$308$$ − 1.75838e9i − 0.195393i
$$309$$ 0 0
$$310$$ −6.34652e9 −0.687209
$$311$$ 7.19555e9i 0.769171i 0.923089 + 0.384585i $$0.125656\pi$$
−0.923089 + 0.384585i $$0.874344\pi$$
$$312$$ 0 0
$$313$$ 4.23341e9 0.441076 0.220538 0.975378i $$-0.429219\pi$$
0.220538 + 0.975378i $$0.429219\pi$$
$$314$$ − 3.22051e9i − 0.331288i
$$315$$ 0 0
$$316$$ −1.82620e9 −0.183147
$$317$$ 5.40044e9i 0.534801i 0.963586 + 0.267400i $$0.0861646\pi$$
−0.963586 + 0.267400i $$0.913835\pi$$
$$318$$ 0 0
$$319$$ 6.23367e9 0.601978
$$320$$ 1.42359e9i 0.135765i
$$321$$ 0 0
$$322$$ −8.60520e9 −0.800456
$$323$$ − 4.89171e9i − 0.449418i
$$324$$ 0 0
$$325$$ −5.65821e8 −0.0507161
$$326$$ 2.97724e9i 0.263599i
$$327$$ 0 0
$$328$$ −7.51868e9 −0.649601
$$329$$ 1.22058e10i 1.04179i
$$330$$ 0 0
$$331$$ −1.35941e10 −1.13250 −0.566252 0.824232i $$-0.691607\pi$$
−0.566252 + 0.824232i $$0.691607\pi$$
$$332$$ 4.63255e9i 0.381301i
$$333$$ 0 0
$$334$$ −1.31674e10 −1.05807
$$335$$ 6.80431e9i 0.540263i
$$336$$ 0 0
$$337$$ −2.35433e9 −0.182536 −0.0912680 0.995826i $$-0.529092\pi$$
−0.0912680 + 0.995826i $$0.529092\pi$$
$$338$$ 8.49341e9i 0.650752i
$$339$$ 0 0
$$340$$ −1.87564e9 −0.140357
$$341$$ − 5.49739e9i − 0.406574i
$$342$$ 0 0
$$343$$ 1.50030e10 1.08393
$$344$$ − 8.90288e9i − 0.635765i
$$345$$ 0 0
$$346$$ 1.49813e10 1.04531
$$347$$ 8.33815e9i 0.575111i 0.957764 + 0.287555i $$0.0928426\pi$$
−0.957764 + 0.287555i $$0.907157\pi$$
$$348$$ 0 0
$$349$$ −1.26463e10 −0.852434 −0.426217 0.904621i $$-0.640154\pi$$
−0.426217 + 0.904621i $$0.640154\pi$$
$$350$$ − 1.63949e9i − 0.109253i
$$351$$ 0 0
$$352$$ −1.23313e9 −0.0803224
$$353$$ 3.76139e9i 0.242242i 0.992638 + 0.121121i $$0.0386490\pi$$
−0.992638 + 0.121121i $$0.961351\pi$$
$$354$$ 0 0
$$355$$ −3.08762e10 −1.94406
$$356$$ 1.47312e10i 0.917147i
$$357$$ 0 0
$$358$$ 4.60124e9 0.280119
$$359$$ − 1.64726e10i − 0.991707i −0.868406 0.495853i $$-0.834855\pi$$
0.868406 0.495853i $$-0.165145\pi$$
$$360$$ 0 0
$$361$$ 3.43681e10 2.02361
$$362$$ 1.46027e10i 0.850351i
$$363$$ 0 0
$$364$$ 2.13121e9 0.121401
$$365$$ 1.57905e10i 0.889659i
$$366$$ 0 0
$$367$$ 7.38723e9 0.407209 0.203605 0.979053i $$-0.434734\pi$$
0.203605 + 0.979053i $$0.434734\pi$$
$$368$$ 6.03470e9i 0.329052i
$$369$$ 0 0
$$370$$ −1.03263e10 −0.550984
$$371$$ − 1.58624e9i − 0.0837286i
$$372$$ 0 0
$$373$$ 2.30025e10 1.18834 0.594168 0.804341i $$-0.297482\pi$$
0.594168 + 0.804341i $$0.297482\pi$$
$$374$$ − 1.62469e9i − 0.0830394i
$$375$$ 0 0
$$376$$ 8.55972e9 0.428261
$$377$$ 7.55541e9i 0.374018i
$$378$$ 0 0
$$379$$ −2.05891e10 −0.997883 −0.498942 0.866636i $$-0.666278\pi$$
−0.498942 + 0.866636i $$0.666278\pi$$
$$380$$ − 1.96899e10i − 0.944298i
$$381$$ 0 0
$$382$$ 3.36825e9 0.158180
$$383$$ 1.43645e10i 0.667569i 0.942649 + 0.333785i $$0.108326\pi$$
−0.942649 + 0.333785i $$0.891674\pi$$
$$384$$ 0 0
$$385$$ 9.32521e9 0.424439
$$386$$ − 2.51169e10i − 1.13140i
$$387$$ 0 0
$$388$$ 5.19317e9 0.229142
$$389$$ − 3.33296e10i − 1.45557i −0.685808 0.727783i $$-0.740551\pi$$
0.685808 0.727783i $$-0.259449\pi$$
$$390$$ 0 0
$$391$$ −7.95096e9 −0.340183
$$392$$ − 2.17307e9i − 0.0920298i
$$393$$ 0 0
$$394$$ 2.16747e10 0.899434
$$395$$ − 9.68488e9i − 0.397838i
$$396$$ 0 0
$$397$$ −4.32631e9 −0.174163 −0.0870814 0.996201i $$-0.527754\pi$$
−0.0870814 + 0.996201i $$0.527754\pi$$
$$398$$ 1.98524e10i 0.791192i
$$399$$ 0 0
$$400$$ −1.14975e9 −0.0449120
$$401$$ − 1.20770e10i − 0.467069i −0.972348 0.233535i $$-0.924971\pi$$
0.972348 0.233535i $$-0.0750293\pi$$
$$402$$ 0 0
$$403$$ 6.66302e9 0.252610
$$404$$ − 8.31315e9i − 0.312061i
$$405$$ 0 0
$$406$$ −2.18920e10 −0.805716
$$407$$ − 8.94473e9i − 0.325979i
$$408$$ 0 0
$$409$$ −3.81221e10 −1.36234 −0.681168 0.732127i $$-0.738528\pi$$
−0.681168 + 0.732127i $$0.738528\pi$$
$$410$$ − 3.98738e10i − 1.41108i
$$411$$ 0 0
$$412$$ 1.75697e10 0.609782
$$413$$ 9.78715e8i 0.0336400i
$$414$$ 0 0
$$415$$ −2.45678e10 −0.828275
$$416$$ − 1.49459e9i − 0.0499055i
$$417$$ 0 0
$$418$$ 1.70555e10 0.558675
$$419$$ − 2.52616e10i − 0.819605i −0.912174 0.409803i $$-0.865598\pi$$
0.912174 0.409803i $$-0.134402\pi$$
$$420$$ 0 0
$$421$$ −3.33965e9 −0.106310 −0.0531548 0.998586i $$-0.516928\pi$$
−0.0531548 + 0.998586i $$0.516928\pi$$
$$422$$ 2.43685e10i 0.768386i
$$423$$ 0 0
$$424$$ −1.11241e9 −0.0344192
$$425$$ − 1.51484e9i − 0.0464312i
$$426$$ 0 0
$$427$$ 3.09455e10 0.930862
$$428$$ − 1.78059e10i − 0.530625i
$$429$$ 0 0
$$430$$ 4.72147e10 1.38103
$$431$$ 5.55244e9i 0.160907i 0.996758 + 0.0804534i $$0.0256368\pi$$
−0.996758 + 0.0804534i $$0.974363\pi$$
$$432$$ 0 0
$$433$$ 1.14713e10 0.326334 0.163167 0.986598i $$-0.447829\pi$$
0.163167 + 0.986598i $$0.447829\pi$$
$$434$$ 1.93063e10i 0.544178i
$$435$$ 0 0
$$436$$ 5.49653e9 0.152105
$$437$$ − 8.34667e10i − 2.28869i
$$438$$ 0 0
$$439$$ −5.98486e10 −1.61137 −0.805686 0.592343i $$-0.798203\pi$$
−0.805686 + 0.592343i $$0.798203\pi$$
$$440$$ − 6.53963e9i − 0.174479i
$$441$$ 0 0
$$442$$ 1.96918e9 0.0515936
$$443$$ 3.61609e9i 0.0938910i 0.998897 + 0.0469455i $$0.0149487\pi$$
−0.998897 + 0.0469455i $$0.985051\pi$$
$$444$$ 0 0
$$445$$ −7.81241e10 −1.99226
$$446$$ 2.70821e10i 0.684451i
$$447$$ 0 0
$$448$$ 4.33062e9 0.107507
$$449$$ − 2.39980e9i − 0.0590459i −0.999564 0.0295230i $$-0.990601\pi$$
0.999564 0.0295230i $$-0.00939882\pi$$
$$450$$ 0 0
$$451$$ 3.45390e10 0.834839
$$452$$ − 3.73311e10i − 0.894369i
$$453$$ 0 0
$$454$$ 1.12306e10 0.264351
$$455$$ 1.13025e10i 0.263710i
$$456$$ 0 0
$$457$$ −4.11731e10 −0.943948 −0.471974 0.881612i $$-0.656458\pi$$
−0.471974 + 0.881612i $$0.656458\pi$$
$$458$$ 2.51334e10i 0.571201i
$$459$$ 0 0
$$460$$ −3.20039e10 −0.714778
$$461$$ 2.07400e10i 0.459203i 0.973285 + 0.229601i $$0.0737422\pi$$
−0.973285 + 0.229601i $$0.926258\pi$$
$$462$$ 0 0
$$463$$ −7.35380e9 −0.160025 −0.0800125 0.996794i $$-0.525496\pi$$
−0.0800125 + 0.996794i $$0.525496\pi$$
$$464$$ 1.53526e10i 0.331214i
$$465$$ 0 0
$$466$$ −1.40337e10 −0.297598
$$467$$ 9.09281e10i 1.91175i 0.293776 + 0.955874i $$0.405088\pi$$
−0.293776 + 0.955874i $$0.594912\pi$$
$$468$$ 0 0
$$469$$ 2.06989e10 0.427816
$$470$$ 4.53948e10i 0.930282i
$$471$$ 0 0
$$472$$ 6.86359e8 0.0138288
$$473$$ 4.08976e10i 0.817059i
$$474$$ 0 0
$$475$$ 1.59023e10 0.312381
$$476$$ 5.70576e9i 0.111144i
$$477$$ 0 0
$$478$$ 8.59039e9 0.164551
$$479$$ 1.91197e10i 0.363194i 0.983373 + 0.181597i $$0.0581266\pi$$
−0.983373 + 0.181597i $$0.941873\pi$$
$$480$$ 0 0
$$481$$ 1.08413e10 0.202536
$$482$$ − 5.06251e10i − 0.937946i
$$483$$ 0 0
$$484$$ −2.17733e10 −0.396773
$$485$$ 2.75409e10i 0.497750i
$$486$$ 0 0
$$487$$ 5.28737e10 0.939992 0.469996 0.882669i $$-0.344255\pi$$
0.469996 + 0.882669i $$0.344255\pi$$
$$488$$ − 2.17016e10i − 0.382660i
$$489$$ 0 0
$$490$$ 1.15244e10 0.199910
$$491$$ − 9.78032e10i − 1.68278i −0.540429 0.841389i $$-0.681738\pi$$
0.540429 0.841389i $$-0.318262\pi$$
$$492$$ 0 0
$$493$$ −2.02276e10 −0.342418
$$494$$ 2.06718e10i 0.347113i
$$495$$ 0 0
$$496$$ 1.35392e10 0.223701
$$497$$ 9.39263e10i 1.53944i
$$498$$ 0 0
$$499$$ 8.89351e10 1.43440 0.717201 0.696866i $$-0.245423\pi$$
0.717201 + 0.696866i $$0.245423\pi$$
$$500$$ 2.78437e10i 0.445499i
$$501$$ 0 0
$$502$$ 1.21459e10 0.191256
$$503$$ 2.97791e10i 0.465200i 0.972572 + 0.232600i $$0.0747233\pi$$
−0.972572 + 0.232600i $$0.925277\pi$$
$$504$$ 0 0
$$505$$ 4.40871e10 0.677870
$$506$$ − 2.77219e10i − 0.422884i
$$507$$ 0 0
$$508$$ 4.34579e10 0.652551
$$509$$ − 2.67186e10i − 0.398054i −0.979994 0.199027i $$-0.936222\pi$$
0.979994 0.199027i $$-0.0637781\pi$$
$$510$$ 0 0
$$511$$ 4.80351e10 0.704491
$$512$$ − 3.03700e9i − 0.0441942i
$$513$$ 0 0
$$514$$ −8.58333e10 −1.22971
$$515$$ 9.31772e10i 1.32459i
$$516$$ 0 0
$$517$$ −3.93212e10 −0.550383
$$518$$ 3.14130e10i 0.436306i
$$519$$ 0 0
$$520$$ 7.92625e9 0.108406
$$521$$ − 1.06779e10i − 0.144922i −0.997371 0.0724610i $$-0.976915\pi$$
0.997371 0.0724610i $$-0.0230853\pi$$
$$522$$ 0 0
$$523$$ 4.63928e10 0.620075 0.310037 0.950724i $$-0.399658\pi$$
0.310037 + 0.950724i $$0.399658\pi$$
$$524$$ − 2.87408e10i − 0.381218i
$$525$$ 0 0
$$526$$ −3.18081e10 −0.415522
$$527$$ 1.78385e10i 0.231268i
$$528$$ 0 0
$$529$$ −5.73553e10 −0.732405
$$530$$ − 5.89943e9i − 0.0747665i
$$531$$ 0 0
$$532$$ −5.98973e10 −0.747757
$$533$$ 4.18623e10i 0.518698i
$$534$$ 0 0
$$535$$ 9.44298e10 1.15264
$$536$$ − 1.45159e10i − 0.175867i
$$537$$ 0 0
$$538$$ −3.35208e10 −0.400115
$$539$$ 9.98252e9i 0.118273i
$$540$$ 0 0
$$541$$ −1.22420e11 −1.42911 −0.714553 0.699581i $$-0.753370\pi$$
−0.714553 + 0.699581i $$0.753370\pi$$
$$542$$ 9.50822e10i 1.10180i
$$543$$ 0 0
$$544$$ 4.00137e9 0.0456891
$$545$$ 2.91498e10i 0.330407i
$$546$$ 0 0
$$547$$ −5.53975e10 −0.618786 −0.309393 0.950934i $$-0.600126\pi$$
−0.309393 + 0.950934i $$0.600126\pi$$
$$548$$ 5.13617e10i 0.569531i
$$549$$ 0 0
$$550$$ 5.28165e9 0.0577190
$$551$$ − 2.12343e11i − 2.30373i
$$552$$ 0 0
$$553$$ −2.94617e10 −0.315034
$$554$$ − 6.76743e10i − 0.718431i
$$555$$ 0 0
$$556$$ −3.45298e10 −0.361322
$$557$$ − 1.17293e11i − 1.21857i −0.792951 0.609285i $$-0.791456\pi$$
0.792951 0.609285i $$-0.208544\pi$$
$$558$$ 0 0
$$559$$ −4.95692e10 −0.507651
$$560$$ 2.29666e10i 0.233531i
$$561$$ 0 0
$$562$$ −1.22574e11 −1.22872
$$563$$ − 1.03752e11i − 1.03267i −0.856385 0.516337i $$-0.827295\pi$$
0.856385 0.516337i $$-0.172705\pi$$
$$564$$ 0 0
$$565$$ 1.97978e11 1.94278
$$566$$ − 4.28416e10i − 0.417445i
$$567$$ 0 0
$$568$$ 6.58692e10 0.632833
$$569$$ − 2.67515e10i − 0.255211i −0.991825 0.127605i $$-0.959271\pi$$
0.991825 0.127605i $$-0.0407291\pi$$
$$570$$ 0 0
$$571$$ 1.14161e11 1.07393 0.536963 0.843606i $$-0.319572\pi$$
0.536963 + 0.843606i $$0.319572\pi$$
$$572$$ 6.86577e9i 0.0641364i
$$573$$ 0 0
$$574$$ −1.21298e11 −1.11739
$$575$$ − 2.58475e10i − 0.236454i
$$576$$ 0 0
$$577$$ 1.43414e11 1.29386 0.646930 0.762549i $$-0.276052\pi$$
0.646930 + 0.762549i $$0.276052\pi$$
$$578$$ − 7.36497e10i − 0.659872i
$$579$$ 0 0
$$580$$ −8.14193e10 −0.719475
$$581$$ 7.47361e10i 0.655883i
$$582$$ 0 0
$$583$$ 5.11012e9 0.0442341
$$584$$ − 3.36864e10i − 0.289603i
$$585$$ 0 0
$$586$$ 8.28207e10 0.702342
$$587$$ − 1.93256e11i − 1.62773i −0.581057 0.813863i $$-0.697361\pi$$
0.581057 0.813863i $$-0.302639\pi$$
$$588$$ 0 0
$$589$$ −1.87263e11 −1.55593
$$590$$ 3.63997e9i 0.0300393i
$$591$$ 0 0
$$592$$ 2.20295e10 0.179357
$$593$$ 2.21385e10i 0.179031i 0.995985 + 0.0895156i $$0.0285319\pi$$
−0.995985 + 0.0895156i $$0.971468\pi$$
$$594$$ 0 0
$$595$$ −3.02594e10 −0.241430
$$596$$ − 2.48574e10i − 0.197002i
$$597$$ 0 0
$$598$$ 3.35999e10 0.262744
$$599$$ 1.32315e11i 1.02778i 0.857856 + 0.513891i $$0.171796\pi$$
−0.857856 + 0.513891i $$0.828204\pi$$
$$600$$ 0 0
$$601$$ −2.29562e10 −0.175955 −0.0879775 0.996122i $$-0.528040\pi$$
−0.0879775 + 0.996122i $$0.528040\pi$$
$$602$$ − 1.43629e11i − 1.09359i
$$603$$ 0 0
$$604$$ 1.12013e10 0.0841627
$$605$$ − 1.15470e11i − 0.861883i
$$606$$ 0 0
$$607$$ 1.56978e11 1.15634 0.578169 0.815917i $$-0.303767\pi$$
0.578169 + 0.815917i $$0.303767\pi$$
$$608$$ 4.20051e10i 0.307389i
$$609$$ 0 0
$$610$$ 1.15090e11 0.831225
$$611$$ − 4.76586e10i − 0.341961i
$$612$$ 0 0
$$613$$ −1.72931e11 −1.22470 −0.612352 0.790586i $$-0.709776\pi$$
−0.612352 + 0.790586i $$0.709776\pi$$
$$614$$ 2.51148e9i 0.0176708i
$$615$$ 0 0
$$616$$ −1.98938e10 −0.138164
$$617$$ − 1.42425e10i − 0.0982758i −0.998792 0.0491379i $$-0.984353\pi$$
0.998792 0.0491379i $$-0.0156474\pi$$
$$618$$ 0 0
$$619$$ 1.29983e11 0.885367 0.442684 0.896678i $$-0.354027\pi$$
0.442684 + 0.896678i $$0.354027\pi$$
$$620$$ 7.18027e10i 0.485930i
$$621$$ 0 0
$$622$$ 8.14084e10 0.543886
$$623$$ 2.37656e11i 1.57760i
$$624$$ 0 0
$$625$$ −1.75075e11 −1.14737
$$626$$ − 4.78956e10i − 0.311888i
$$627$$ 0 0
$$628$$ −3.64359e10 −0.234256
$$629$$ 2.90247e10i 0.185424i
$$630$$ 0 0
$$631$$ −5.79110e9 −0.0365295 −0.0182648 0.999833i $$-0.505814\pi$$
−0.0182648 + 0.999833i $$0.505814\pi$$
$$632$$ 2.06611e10i 0.129505i
$$633$$ 0 0
$$634$$ 6.10990e10 0.378161
$$635$$ 2.30471e11i 1.41749i
$$636$$ 0 0
$$637$$ −1.20991e10 −0.0734847
$$638$$ − 7.05259e10i − 0.425663i
$$639$$ 0 0
$$640$$ 1.61061e10 0.0960000
$$641$$ − 2.18083e11i − 1.29178i −0.763429 0.645892i $$-0.776486\pi$$
0.763429 0.645892i $$-0.223514\pi$$
$$642$$ 0 0
$$643$$ 1.74278e11 1.01953 0.509764 0.860314i $$-0.329733\pi$$
0.509764 + 0.860314i $$0.329733\pi$$
$$644$$ 9.73567e10i 0.566008i
$$645$$ 0 0
$$646$$ −5.53434e10 −0.317786
$$647$$ − 2.22876e11i − 1.27188i −0.771738 0.635941i $$-0.780612\pi$$
0.771738 0.635941i $$-0.219388\pi$$
$$648$$ 0 0
$$649$$ −3.15296e9 −0.0177721
$$650$$ 6.40153e9i 0.0358617i
$$651$$ 0 0
$$652$$ 3.36836e10 0.186393
$$653$$ 2.56934e11i 1.41308i 0.707671 + 0.706542i $$0.249746\pi$$
−0.707671 + 0.706542i $$0.750254\pi$$
$$654$$ 0 0
$$655$$ 1.52421e11 0.828094
$$656$$ 8.50642e10i 0.459337i
$$657$$ 0 0
$$658$$ 1.38092e11 0.736658
$$659$$ 2.69559e11i 1.42926i 0.699500 + 0.714632i $$0.253406\pi$$
−0.699500 + 0.714632i $$0.746594\pi$$
$$660$$ 0 0
$$661$$ −2.56807e11 −1.34525 −0.672623 0.739986i $$-0.734832\pi$$
−0.672623 + 0.739986i $$0.734832\pi$$
$$662$$ 1.53800e11i 0.800802i
$$663$$ 0 0
$$664$$ 5.24114e10 0.269621
$$665$$ − 3.17653e11i − 1.62430i
$$666$$ 0 0
$$667$$ −3.45142e11 −1.74379
$$668$$ 1.48972e11i 0.748167i
$$669$$ 0 0
$$670$$ 7.69820e10 0.382023
$$671$$ 9.96918e10i 0.491778i
$$672$$ 0 0
$$673$$ 1.67895e11 0.818421 0.409210 0.912440i $$-0.365804\pi$$
0.409210 + 0.912440i $$0.365804\pi$$
$$674$$ 2.66363e10i 0.129072i
$$675$$ 0 0
$$676$$ 9.60920e10 0.460151
$$677$$ 1.70201e11i 0.810230i 0.914266 + 0.405115i $$0.132769\pi$$
−0.914266 + 0.405115i $$0.867231\pi$$
$$678$$ 0 0
$$679$$ 8.37804e10 0.394152
$$680$$ 2.12204e10i 0.0992473i
$$681$$ 0 0
$$682$$ −6.21959e10 −0.287491
$$683$$ 3.04016e11i 1.39706i 0.715582 + 0.698529i $$0.246162\pi$$
−0.715582 + 0.698529i $$0.753838\pi$$
$$684$$ 0 0
$$685$$ −2.72387e11 −1.23715
$$686$$ − 1.69740e11i − 0.766455i
$$687$$ 0 0
$$688$$ −1.00725e11 −0.449554
$$689$$ 6.19364e9i 0.0274833i
$$690$$ 0 0
$$691$$ −4.17969e10 −0.183329 −0.0916646 0.995790i $$-0.529219\pi$$
−0.0916646 + 0.995790i $$0.529219\pi$$
$$692$$ − 1.69494e11i − 0.739145i
$$693$$ 0 0
$$694$$ 9.43354e10 0.406665
$$695$$ − 1.83122e11i − 0.784876i
$$696$$ 0 0
$$697$$ −1.12075e11 −0.474875
$$698$$ 1.43076e11i 0.602762i
$$699$$ 0 0
$$700$$ −1.85487e10 −0.0772539
$$701$$ 2.07821e11i 0.860631i 0.902678 + 0.430316i $$0.141598\pi$$
−0.902678 + 0.430316i $$0.858402\pi$$
$$702$$ 0 0
$$703$$ −3.04693e11 −1.24750
$$704$$ 1.39512e10i 0.0567965i
$$705$$ 0 0
$$706$$ 4.25553e10 0.171291
$$707$$ − 1.34114e11i − 0.536782i
$$708$$ 0 0
$$709$$ −3.47189e11 −1.37398 −0.686990 0.726667i $$-0.741068\pi$$
−0.686990 + 0.726667i $$0.741068\pi$$
$$710$$ 3.49324e11i 1.37466i
$$711$$ 0 0
$$712$$ 1.66665e11 0.648521
$$713$$ 3.04376e11i 1.17775i
$$714$$ 0 0
$$715$$ −3.64112e10 −0.139319
$$716$$ − 5.20570e10i − 0.198074i
$$717$$ 0 0
$$718$$ −1.86366e11 −0.701242
$$719$$ − 1.56177e11i − 0.584387i −0.956359 0.292194i $$-0.905615\pi$$
0.956359 0.292194i $$-0.0943851\pi$$
$$720$$ 0 0
$$721$$ 2.83448e11 1.04890
$$722$$ − 3.88830e11i − 1.43091i
$$723$$ 0 0
$$724$$ 1.65210e11 0.601289
$$725$$ − 6.57572e10i − 0.238008i
$$726$$ 0 0
$$727$$ −2.70116e11 −0.966969 −0.483485 0.875353i $$-0.660629\pi$$
−0.483485 + 0.875353i $$0.660629\pi$$
$$728$$ − 2.41119e10i − 0.0858433i
$$729$$ 0 0
$$730$$ 1.78649e11 0.629084
$$731$$ − 1.32709e11i − 0.464761i
$$732$$ 0 0
$$733$$ 1.53459e11 0.531589 0.265795 0.964030i $$-0.414366\pi$$
0.265795 + 0.964030i $$0.414366\pi$$
$$734$$ − 8.35770e10i − 0.287940i
$$735$$ 0 0
$$736$$ 6.82749e10 0.232675
$$737$$ 6.66822e10i 0.226017i
$$738$$ 0 0
$$739$$ 7.60266e10 0.254911 0.127455 0.991844i $$-0.459319\pi$$
0.127455 + 0.991844i $$0.459319\pi$$
$$740$$ 1.16829e11i 0.389605i
$$741$$ 0 0
$$742$$ −1.79463e10 −0.0592050
$$743$$ 4.18589e11i 1.37351i 0.726888 + 0.686756i $$0.240966\pi$$
−0.726888 + 0.686756i $$0.759034\pi$$
$$744$$ 0 0
$$745$$ 1.31826e11 0.427933
$$746$$ − 2.60243e11i − 0.840280i
$$747$$ 0 0
$$748$$ −1.83813e10 −0.0587177
$$749$$ − 2.87259e11i − 0.912737i
$$750$$ 0 0
$$751$$ 3.73788e11 1.17507 0.587537 0.809197i $$-0.300098\pi$$
0.587537 + 0.809197i $$0.300098\pi$$
$$752$$ − 9.68422e10i − 0.302826i
$$753$$ 0 0
$$754$$ 8.54797e10 0.264471
$$755$$ 5.94038e10i 0.182821i
$$756$$ 0 0
$$757$$ 4.74806e11 1.44588 0.722941 0.690910i $$-0.242790\pi$$
0.722941 + 0.690910i $$0.242790\pi$$
$$758$$ 2.32939e11i 0.705610i
$$759$$ 0 0
$$760$$ −2.22766e11 −0.667719
$$761$$ − 5.91673e11i − 1.76418i −0.471079 0.882091i $$-0.656135\pi$$
0.471079 0.882091i $$-0.343865\pi$$
$$762$$ 0 0
$$763$$ 8.86745e10 0.261638
$$764$$ − 3.81074e10i − 0.111850i
$$765$$ 0 0
$$766$$ 1.62516e11 0.472043
$$767$$ − 3.82149e9i − 0.0110421i
$$768$$ 0 0
$$769$$ 4.29184e11 1.22727 0.613633 0.789592i $$-0.289708\pi$$
0.613633 + 0.789592i $$0.289708\pi$$
$$770$$ − 1.05503e11i − 0.300124i
$$771$$ 0 0
$$772$$ −2.84165e11 −0.800021
$$773$$ 1.95537e11i 0.547660i 0.961778 + 0.273830i $$0.0882906\pi$$
−0.961778 + 0.273830i $$0.911709\pi$$
$$774$$ 0 0
$$775$$ −5.79905e10 −0.160750
$$776$$ − 5.87540e10i − 0.162028i
$$777$$ 0 0
$$778$$ −3.77082e11 −1.02924
$$779$$ − 1.17653e12i − 3.19488i
$$780$$ 0 0
$$781$$ −3.02587e11 −0.813290
$$782$$ 8.99548e10i 0.240546i
$$783$$ 0 0
$$784$$ −2.45854e10 −0.0650749
$$785$$ − 1.93230e11i − 0.508858i
$$786$$ 0 0
$$787$$ −1.30204e10 −0.0339409 −0.0169705 0.999856i $$-0.505402\pi$$
−0.0169705 + 0.999856i $$0.505402\pi$$
$$788$$ − 2.45222e11i − 0.635996i
$$789$$ 0 0
$$790$$ −1.09572e11 −0.281314
$$791$$ − 6.02256e11i − 1.53842i
$$792$$ 0 0
$$793$$ −1.20830e11 −0.305549
$$794$$ 4.89466e10i 0.123152i
$$795$$ 0 0
$$796$$ 2.24605e11 0.559457
$$797$$ − 1.33234e11i − 0.330204i −0.986277 0.165102i $$-0.947205\pi$$
0.986277 0.165102i $$-0.0527953\pi$$
$$798$$ 0 0
$$799$$ 1.27593e11 0.313070
$$800$$ 1.30079e10i 0.0317576i
$$801$$ 0 0
$$802$$ −1.36636e11 −0.330268
$$803$$ 1.54747e11i 0.372185i
$$804$$ 0 0
$$805$$ −5.16312e11 −1.22950
$$806$$ − 7.53835e10i − 0.178622i
$$807$$ 0 0
$$808$$ −9.40525e10 −0.220661
$$809$$ 7.75707e11i 1.81094i 0.424413 + 0.905469i $$0.360481\pi$$
−0.424413 + 0.905469i $$0.639519\pi$$
$$810$$ 0 0
$$811$$ −7.76807e11 −1.79568 −0.897842 0.440318i $$-0.854866\pi$$
−0.897842 + 0.440318i $$0.854866\pi$$
$$812$$ 2.47680e11i 0.569727i
$$813$$ 0 0
$$814$$ −1.01198e11 −0.230502
$$815$$ 1.78634e11i 0.404888i
$$816$$ 0 0
$$817$$ 1.39313e12 3.12683
$$818$$ 4.31303e11i 0.963317i
$$819$$ 0 0
$$820$$ −4.51121e11 −0.997786
$$821$$ 2.75649e10i 0.0606713i 0.999540 + 0.0303356i $$0.00965761\pi$$
−0.999540 + 0.0303356i $$0.990342\pi$$
$$822$$ 0 0
$$823$$ 2.84608e11 0.620366 0.310183 0.950677i $$-0.399610\pi$$
0.310183 + 0.950677i $$0.399610\pi$$
$$824$$ − 1.98778e11i − 0.431181i
$$825$$ 0 0
$$826$$ 1.10729e10 0.0237871
$$827$$ − 2.82378e11i − 0.603683i −0.953358 0.301842i $$-0.902399\pi$$
0.953358 0.301842i $$-0.0976014\pi$$
$$828$$ 0 0
$$829$$ 8.40257e11 1.77907 0.889537 0.456863i $$-0.151027\pi$$
0.889537 + 0.456863i $$0.151027\pi$$
$$830$$ 2.77953e11i 0.585679i
$$831$$ 0 0
$$832$$ −1.69093e10 −0.0352885
$$833$$ − 3.23923e10i − 0.0672762i
$$834$$ 0 0
$$835$$ −7.90043e11 −1.62519
$$836$$ − 1.92961e11i − 0.395043i
$$837$$ 0 0
$$838$$ −2.85802e11 −0.579548
$$839$$ 3.34946e11i 0.675968i 0.941152 + 0.337984i $$0.109745\pi$$
−0.941152 + 0.337984i $$0.890255\pi$$
$$840$$ 0 0
$$841$$ −3.77810e11 −0.755248
$$842$$ 3.77838e10i 0.0751722i
$$843$$ 0 0
$$844$$ 2.75698e11 0.543331
$$845$$ 5.09605e11i 0.999555i
$$846$$ 0 0
$$847$$ −3.51264e11 −0.682496
$$848$$ 1.25855e10i 0.0243380i
$$849$$ 0 0
$$850$$ −1.71384e10 −0.0328318
$$851$$ 4.95246e11i 0.944284i
$$852$$ 0 0
$$853$$ 6.09757e10 0.115176 0.0575878 0.998340i $$-0.481659\pi$$
0.0575878 + 0.998340i $$0.481659\pi$$
$$854$$ − 3.50108e11i − 0.658219i
$$855$$ 0 0
$$856$$ −2.01450e11 −0.375209
$$857$$ − 4.12820e11i − 0.765311i −0.923891 0.382655i $$-0.875010\pi$$
0.923891 0.382655i $$-0.124990\pi$$
$$858$$ 0 0
$$859$$ 4.28958e11 0.787847 0.393924 0.919143i $$-0.371117\pi$$
0.393924 + 0.919143i $$0.371117\pi$$
$$860$$ − 5.34173e11i − 0.976535i
$$861$$ 0 0
$$862$$ 6.28186e10 0.113778
$$863$$ 9.65828e11i 1.74123i 0.491963 + 0.870616i $$0.336280\pi$$
−0.491963 + 0.870616i $$0.663720\pi$$
$$864$$ 0 0
$$865$$ 8.98876e11 1.60559
$$866$$ − 1.29783e11i − 0.230753i
$$867$$ 0 0
$$868$$ 2.18426e11 0.384792
$$869$$ − 9.49119e10i − 0.166434i
$$870$$ 0 0
$$871$$ −8.08211e10 −0.140427
$$872$$ − 6.21862e10i − 0.107554i
$$873$$ 0 0
$$874$$ −9.44318e11 −1.61835
$$875$$ 4.49197e11i 0.766310i
$$876$$ 0 0
$$877$$ −1.01992e12 −1.72412 −0.862059 0.506808i $$-0.830825\pi$$
−0.862059 + 0.506808i $$0.830825\pi$$
$$878$$ 6.77110e11i 1.13941i
$$879$$ 0 0
$$880$$ −7.39875e10 −0.123375
$$881$$ − 7.44050e11i − 1.23509i −0.786535 0.617545i $$-0.788127\pi$$
0.786535 0.617545i $$-0.211873\pi$$
$$882$$ 0 0
$$883$$ 1.35286e11 0.222540 0.111270 0.993790i $$-0.464508\pi$$
0.111270 + 0.993790i $$0.464508\pi$$
$$884$$ − 2.22787e10i − 0.0364822i
$$885$$ 0 0
$$886$$ 4.09113e10 0.0663909
$$887$$ − 5.56566e11i − 0.899129i −0.893248 0.449565i $$-0.851579\pi$$
0.893248 0.449565i $$-0.148421\pi$$
$$888$$ 0 0
$$889$$ 7.01099e11 1.12246
$$890$$ 8.83874e11i 1.40874i
$$891$$ 0 0
$$892$$ 3.06398e11 0.483980
$$893$$ 1.33944e12i 2.10628i
$$894$$ 0 0
$$895$$ 2.76074e11 0.430263
$$896$$ − 4.89954e10i − 0.0760191i
$$897$$ 0 0
$$898$$ −2.71507e10 −0.0417518
$$899$$ 7.74347e11i 1.18549i
$$900$$ 0 0
$$901$$ −1.65818e10 −0.0251613
$$902$$ − 3.90764e11i − 0.590321i
$$903$$ 0 0
$$904$$ −4.22353e11 −0.632415
$$905$$ 8.76161e11i 1.30614i
$$906$$ 0 0
$$907$$ 3.75761e10 0.0555243 0.0277621 0.999615i $$-0.491162\pi$$
0.0277621 + 0.999615i $$0.491162\pi$$
$$908$$ − 1.27060e11i − 0.186924i
$$909$$ 0 0
$$910$$ 1.27873e11 0.186471
$$911$$ − 5.43769e11i − 0.789479i −0.918793 0.394740i $$-0.870835\pi$$
0.918793 0.394740i $$-0.129165\pi$$
$$912$$ 0 0
$$913$$ −2.40765e11 −0.346505
$$914$$ 4.65820e11i 0.667472i
$$915$$ 0 0
$$916$$ 2.84352e11 0.403900
$$917$$ − 4.63670e11i − 0.655740i
$$918$$ 0 0
$$919$$ 9.08317e10 0.127343 0.0636715 0.997971i $$-0.479719\pi$$
0.0636715 + 0.997971i $$0.479719\pi$$
$$920$$ 3.62082e11i 0.505424i
$$921$$ 0 0
$$922$$ 2.34646e11 0.324705
$$923$$ − 3.66745e11i − 0.505309i
$$924$$ 0 0
$$925$$ −9.43556e10 −0.128884
$$926$$ 8.31987e10i 0.113155i
$$927$$ 0 0
$$928$$ 1.73695e11 0.234204
$$929$$ 1.38421e11i 0.185841i 0.995674 + 0.0929203i $$0.0296202\pi$$
−0.995674 + 0.0929203i $$0.970380\pi$$
$$930$$ 0 0
$$931$$ 3.40044e11 0.452623
$$932$$ 1.58774e11i 0.210433i
$$933$$ 0 0
$$934$$ 1.02873e12 1.35181
$$935$$ − 9.74814e10i − 0.127549i
$$936$$ 0 0
$$937$$ 1.28993e12 1.67343 0.836714 0.547640i $$-0.184474\pi$$
0.836714 + 0.547640i $$0.184474\pi$$
$$938$$ − 2.34182e11i − 0.302511i
$$939$$ 0 0
$$940$$ 5.13583e11 0.657809
$$941$$ − 1.44651e12i − 1.84486i −0.386163 0.922431i $$-0.626200\pi$$
0.386163 0.922431i $$-0.373800\pi$$
$$942$$ 0 0
$$943$$ −1.91233e12 −2.41833
$$944$$ − 7.76526e9i − 0.00977841i
$$945$$ 0 0
$$946$$ 4.62704e11 0.577748
$$947$$ 6.92009e11i 0.860423i 0.902728 + 0.430212i $$0.141561\pi$$
−0.902728 + 0.430212i $$0.858439\pi$$
$$948$$ 0 0
$$949$$ −1.87558e11 −0.231244
$$950$$ − 1.79914e11i − 0.220887i
$$951$$ 0 0
$$952$$ 6.45533e10 0.0785906
$$953$$ 3.33259e11i 0.404027i 0.979383 + 0.202013i $$0.0647484\pi$$
−0.979383 + 0.202013i $$0.935252\pi$$
$$954$$ 0 0
$$955$$ 2.02095e11 0.242964
$$956$$ − 9.71891e10i − 0.116355i
$$957$$ 0 0
$$958$$ 2.16314e11 0.256817
$$959$$ 8.28609e11i 0.979659i
$$960$$ 0 0
$$961$$ −1.70004e11 −0.199326
$$962$$ − 1.22655e11i − 0.143214i
$$963$$ 0 0
$$964$$ −5.72758e11 −0.663228
$$965$$ − 1.50701e12i − 1.73783i
$$966$$ 0 0
$$967$$ −1.25085e12 −1.43054 −0.715270 0.698848i $$-0.753696\pi$$
−0.715270 + 0.698848i $$0.753696\pi$$
$$968$$ 2.46336e11i 0.280561i
$$969$$ 0 0
$$970$$ 3.11590e11 0.351963
$$971$$ − 7.48637e11i − 0.842160i −0.907024 0.421080i $$-0.861651\pi$$
0.907024 0.421080i $$-0.138349\pi$$
$$972$$ 0 0
$$973$$ −5.57063e11 −0.621516
$$974$$ − 5.98198e11i − 0.664675i
$$975$$ 0 0
$$976$$ −2.45526e11 −0.270581
$$977$$ 6.85254e11i 0.752096i 0.926600 + 0.376048i $$0.122717\pi$$
−0.926600 + 0.376048i $$0.877283\pi$$
$$978$$ 0 0
$$979$$ −7.65616e11 −0.833452
$$980$$ − 1.30384e11i − 0.141358i
$$981$$ 0 0
$$982$$ −1.10652e12 −1.18990
$$983$$ − 1.15973e12i − 1.24206i −0.783785 0.621032i $$-0.786714\pi$$
0.783785 0.621032i $$-0.213286\pi$$
$$984$$ 0 0
$$985$$ 1.30048e12 1.38153
$$986$$ 2.28849e11i 0.242126i
$$987$$ 0 0
$$988$$ 2.33875e11 0.245446
$$989$$ − 2.26439e12i − 2.36683i
$$990$$ 0 0
$$991$$ −1.55112e12 −1.60824 −0.804121 0.594465i $$-0.797364\pi$$
−0.804121 + 0.594465i $$0.797364\pi$$
$$992$$ − 1.53179e11i − 0.158180i
$$993$$ 0 0
$$994$$ 1.06266e12 1.08855
$$995$$ 1.19115e12i 1.21527i
$$996$$ 0 0
$$997$$ −3.61247e11 −0.365615 −0.182807 0.983149i $$-0.558518\pi$$
−0.182807 + 0.983149i $$0.558518\pi$$
$$998$$ − 1.00619e12i − 1.01428i
$$999$$ 0 0
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 54.9.b.a.53.1 2
3.2 odd 2 inner 54.9.b.a.53.2 yes 2
4.3 odd 2 432.9.e.g.161.1 2
9.2 odd 6 162.9.d.c.53.2 4
9.4 even 3 162.9.d.c.107.2 4
9.5 odd 6 162.9.d.c.107.1 4
9.7 even 3 162.9.d.c.53.1 4
12.11 even 2 432.9.e.g.161.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
54.9.b.a.53.1 2 1.1 even 1 trivial
54.9.b.a.53.2 yes 2 3.2 odd 2 inner
162.9.d.c.53.1 4 9.7 even 3
162.9.d.c.53.2 4 9.2 odd 6
162.9.d.c.107.1 4 9.5 odd 6
162.9.d.c.107.2 4 9.4 even 3
432.9.e.g.161.1 2 4.3 odd 2
432.9.e.g.161.2 2 12.11 even 2