# Properties

 Label 432.9.e.g.161.1 Level $432$ Weight $9$ Character 432.161 Analytic conductor $175.988$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,9,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 432.e (of order $$2$$, degree $$1$$, not 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: $$175.987559546$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.1 Root $$-1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 432.161 Dual form 432.9.e.g.161.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-678.823i q^{5} +2065.00 q^{7} +O(q^{10})$$ $$q-678.823i q^{5} +2065.00 q^{7} +6652.46i q^{11} +8063.00 q^{13} +21586.6i q^{17} +226609. q^{19} -368329. i q^{23} -70175.0 q^{25} +937047. i q^{29} -826370. q^{31} -1.40177e6i q^{35} +1.34458e6 q^{37} +5.19191e6i q^{41} +6.14774e6 q^{43} +5.91078e6i q^{47} -1.50058e6 q^{49} +768156. i q^{53} +4.51584e6 q^{55} +473954. i q^{59} -1.49857e7 q^{61} -5.47335e6i q^{65} +1.00237e7 q^{67} +4.54849e7i q^{71} -2.32616e7 q^{73} +1.37373e7i q^{77} -1.42672e7 q^{79} +3.61918e7i q^{83} +1.46534e7 q^{85} -1.15088e8i q^{89} +1.66501e7 q^{91} -1.53827e8i q^{95} -4.05716e7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4130 q^{7}+O(q^{10})$$ 2 * q + 4130 * q^7 $$2 q + 4130 q^{7} + 16126 q^{13} + 453218 q^{19} - 140350 q^{25} - 1652740 q^{31} + 2689150 q^{37} + 12295484 q^{43} - 3001152 q^{49} + 9031680 q^{55} - 29971394 q^{61} + 20047394 q^{67} - 46523138 q^{73} - 28534366 q^{79} + 29306880 q^{85} + 33300190 q^{91} - 81143234 q^{97}+O(q^{100})$$ 2 * q + 4130 * q^7 + 16126 * q^13 + 453218 * q^19 - 140350 * q^25 - 1652740 * q^31 + 2689150 * q^37 + 12295484 * q^43 - 3001152 * q^49 + 9031680 * q^55 - 29971394 * q^61 + 20047394 * q^67 - 46523138 * q^73 - 28534366 * q^79 + 29306880 * q^85 + 33300190 * q^91 - 81143234 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 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 $$-0.358503\pi$$
0.430029 + 0.902815i $$0.358503\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6652.46i 0.454372i 0.973851 + 0.227186i $$0.0729525\pi$$
−0.973851 + 0.227186i $$0.927047\pi$$
$$12$$ 0 0
$$13$$ 8063.00 0.282308 0.141154 0.989988i $$-0.454919\pi$$
0.141154 + 0.989988i $$0.454919\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 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 $$-0.164489\pi$$
0.869426 + 0.494063i $$0.164489\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 368329.i − 1.31621i −0.752927 0.658104i $$-0.771359\pi$$
0.752927 0.658104i $$-0.228641\pi$$
$$24$$ 0 0
$$25$$ −70175.0 −0.179648
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 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 $$-0.647651\pi$$
−0.447402 + 0.894333i $$0.647651\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 1.40177e6i − 0.934123i
$$36$$ 0 0
$$37$$ 1.34458e6 0.717428 0.358714 0.933448i $$-0.383215\pi$$
0.358714 + 0.933448i $$0.383215\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 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 $$-0.144216\pi$$
0.899108 + 0.437727i $$0.144216\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.91078e6i 1.21130i 0.795729 + 0.605652i $$0.207088\pi$$
−0.795729 + 0.605652i $$0.792912\pi$$
$$48$$ 0 0
$$49$$ −1.50058e6 −0.260300
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 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
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 473954.i 0.0391136i 0.999809 + 0.0195568i $$0.00622552\pi$$
−0.999809 + 0.0195568i $$0.993774\pi$$
$$60$$ 0 0
$$61$$ −1.49857e7 −1.08232 −0.541162 0.840918i $$-0.682016\pi$$
−0.541162 + 0.840918i $$0.682016\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 5.47335e6i − 0.306619i
$$66$$ 0 0
$$67$$ 1.00237e7 0.497426 0.248713 0.968577i $$-0.419992\pi$$
0.248713 + 0.968577i $$0.419992\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.54849e7i 1.78992i 0.446145 + 0.894961i $$0.352797\pi$$
−0.446145 + 0.894961i $$0.647203\pi$$
$$72$$ 0 0
$$73$$ −2.32616e7 −0.819120 −0.409560 0.912283i $$-0.634318\pi$$
−0.409560 + 0.912283i $$0.634318\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.37373e7i 0.390786i
$$78$$ 0 0
$$79$$ −1.42672e7 −0.366294 −0.183147 0.983086i $$-0.558628\pi$$
−0.183147 + 0.983086i $$0.558628\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.61918e7i 0.762602i 0.924451 + 0.381301i $$0.124524\pi$$
−0.924451 + 0.381301i $$0.875476\pi$$
$$84$$ 0 0
$$85$$ 1.46534e7 0.280714
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 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
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 1.53827e8i − 1.88860i
$$96$$ 0 0
$$97$$ −4.05716e7 −0.458285 −0.229142 0.973393i $$-0.573592\pi$$
−0.229142 + 0.973393i $$0.573592\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 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 $$-0.291257\pi$$
0.609782 + 0.792569i $$0.291257\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1.39108e8i − 1.06125i −0.847607 0.530625i $$-0.821957\pi$$
0.847607 0.530625i $$-0.178043\pi$$
$$108$$ 0 0
$$109$$ −4.29417e7 −0.304210 −0.152105 0.988364i $$-0.548605\pi$$
−0.152105 + 0.988364i $$0.548605\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 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
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.45762e7i 0.222288i
$$120$$ 0 0
$$121$$ 1.70104e8 0.793546
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 2.17529e8i − 0.890997i
$$126$$ 0 0
$$127$$ 3.39515e8 1.30510 0.652551 0.757745i $$-0.273699\pi$$
0.652551 + 0.757745i $$0.273699\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 2.24537e8i − 0.762436i −0.924485 0.381218i $$-0.875505\pi$$
0.924485 0.381218i $$-0.124495\pi$$
$$132$$ 0 0
$$133$$ 4.67948e8 1.49551
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 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 $$-0.617675\pi$$
−0.361322 + 0.932441i $$0.617675\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 5.36388e7i 0.128273i
$$144$$ 0 0
$$145$$ 6.36088e8 1.43895
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 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 $$-0.473178\pi$$
0.0841627 + 0.996452i $$0.473178\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 5.60959e8i 0.971861i
$$156$$ 0 0
$$157$$ 2.84655e8 0.468512 0.234256 0.972175i $$-0.424735\pi$$
0.234256 + 0.972175i $$0.424735\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 7.60600e8i − 1.13202i
$$162$$ 0 0
$$163$$ 2.63153e8 0.372785 0.186393 0.982475i $$-0.440320\pi$$
0.186393 + 0.982475i $$0.440320\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.16384e9i 1.49633i 0.663511 + 0.748167i $$0.269066\pi$$
−0.663511 + 0.748167i $$0.730934\pi$$
$$168$$ 0 0
$$169$$ −7.50719e8 −0.920302
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 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
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 4.06696e8i − 0.396148i −0.980187 0.198074i $$-0.936531\pi$$
0.980187 0.198074i $$-0.0634686\pi$$
$$180$$ 0 0
$$181$$ −1.29071e9 −1.20258 −0.601289 0.799032i $$-0.705346\pi$$
−0.601289 + 0.799032i $$0.705346\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 9.12728e8i − 0.779210i
$$186$$ 0 0
$$187$$ −1.43604e8 −0.117435
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 2.97714e8i − 0.223700i −0.993725 0.111850i $$-0.964322\pi$$
0.993725 0.111850i $$-0.0356776\pi$$
$$192$$ 0 0
$$193$$ 2.22004e9 1.60004 0.800021 0.599972i $$-0.204822\pi$$
0.800021 + 0.599972i $$0.204822\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 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 $$-0.311010\pi$$
0.559457 + 0.828859i $$0.311010\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.93500e9i 1.13945i
$$204$$ 0 0
$$205$$ 3.52438e9 1.99557
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.50751e9i 0.790086i
$$210$$ 0 0
$$211$$ 2.15389e9 1.08666 0.543331 0.839519i $$-0.317163\pi$$
0.543331 + 0.839519i $$0.317163\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 4.17323e9i − 1.95307i
$$216$$ 0 0
$$217$$ −1.70645e9 −0.769583
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.74052e8i 0.0729644i
$$222$$ 0 0
$$223$$ 2.39374e9 0.967959 0.483980 0.875079i $$-0.339191\pi$$
0.483980 + 0.875079i $$0.339191\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 9.92657e8i − 0.373849i −0.982374 0.186924i $$-0.940148\pi$$
0.982374 0.186924i $$-0.0598519\pi$$
$$228$$ 0 0
$$229$$ −2.22150e9 −0.807800 −0.403900 0.914803i $$-0.632346\pi$$
−0.403900 + 0.914803i $$0.632346\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 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
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 7.59290e8i − 0.232711i −0.993208 0.116355i $$-0.962879\pi$$
0.993208 0.116355i $$-0.0371211\pi$$
$$240$$ 0 0
$$241$$ 4.47467e9 1.32646 0.663228 0.748417i $$-0.269186\pi$$
0.663228 + 0.748417i $$0.269186\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.01862e9i 0.282716i
$$246$$ 0 0
$$247$$ 1.82715e9 0.490892
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 1.07356e9i − 0.270477i −0.990813 0.135239i $$-0.956820\pi$$
0.990813 0.135239i $$-0.0431801\pi$$
$$252$$ 0 0
$$253$$ 2.45029e9 0.598048
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 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
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 2.81146e9i 0.587637i 0.955861 + 0.293819i $$0.0949262\pi$$
−0.955861 + 0.293819i $$0.905074\pi$$
$$264$$ 0 0
$$265$$ 5.21441e8 0.105736
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 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 $$-0.215682\pi$$
0.779089 + 0.626914i $$0.215682\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 4.66836e8i − 0.0816270i
$$276$$ 0 0
$$277$$ 5.98162e9 1.01601 0.508007 0.861353i $$-0.330382\pi$$
0.508007 + 0.861353i $$0.330382\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 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 $$-0.595379\pi$$
−0.295179 + 0.955442i $$0.595379\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.07213e10i 1.58023i
$$288$$ 0 0
$$289$$ 6.50978e9 0.933200
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 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
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 2.96984e9i − 0.371576i
$$300$$ 0 0
$$301$$ 1.26951e10 1.54657
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1.01726e10i 1.17553i
$$306$$ 0 0
$$307$$ 2.21986e8 0.0249903 0.0124951 0.999922i $$-0.496023\pi$$
0.0124951 + 0.999922i $$0.496023\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 7.19555e9i − 0.769171i −0.923089 0.384585i $$-0.874344\pi$$
0.923089 0.384585i $$-0.125656\pi$$
$$312$$ 0 0
$$313$$ 4.23341e9 0.441076 0.220538 0.975378i $$-0.429219\pi$$
0.220538 + 0.975378i $$0.429219\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 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
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.89171e9i 0.449418i
$$324$$ 0 0
$$325$$ −5.65821e8 −0.0507161
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1.22058e10i 1.04179i
$$330$$ 0 0
$$331$$ 1.35941e10 1.13250 0.566252 0.824232i $$-0.308393\pi$$
0.566252 + 0.824232i $$0.308393\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 6.80431e9i − 0.540263i
$$336$$ 0 0
$$337$$ −2.35433e9 −0.182536 −0.0912680 0.995826i $$-0.529092\pi$$
−0.0912680 + 0.995826i $$0.529092\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 5.49739e9i − 0.406574i
$$342$$ 0 0
$$343$$ −1.50030e10 −1.08393
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 8.33815e9i − 0.575111i −0.957764 0.287555i $$-0.907157\pi$$
0.957764 0.287555i $$-0.0928426\pi$$
$$348$$ 0 0
$$349$$ −1.26463e10 −0.852434 −0.426217 0.904621i $$-0.640154\pi$$
−0.426217 + 0.904621i $$0.640154\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 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
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 1.64726e10i 0.991707i 0.868406 + 0.495853i $$0.165145\pi$$
−0.868406 + 0.495853i $$0.834855\pi$$
$$360$$ 0 0
$$361$$ 3.43681e10 2.02361
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.57905e10i 0.889659i
$$366$$ 0 0
$$367$$ −7.38723e9 −0.407209 −0.203605 0.979053i $$-0.565266\pi$$
−0.203605 + 0.979053i $$0.565266\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.58624e9i 0.0837286i
$$372$$ 0 0
$$373$$ 2.30025e10 1.18834 0.594168 0.804341i $$-0.297482\pi$$
0.594168 + 0.804341i $$0.297482\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.55541e9i 0.374018i
$$378$$ 0 0
$$379$$ 2.05891e10 0.997883 0.498942 0.866636i $$-0.333722\pi$$
0.498942 + 0.866636i $$0.333722\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 1.43645e10i − 0.667569i −0.942649 0.333785i $$-0.891674\pi$$
0.942649 0.333785i $$-0.108326\pi$$
$$384$$ 0 0
$$385$$ 9.32521e9 0.424439
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 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
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 9.68488e9i 0.397838i
$$396$$ 0 0
$$397$$ −4.32631e9 −0.174163 −0.0870814 0.996201i $$-0.527754\pi$$
−0.0870814 + 0.996201i $$0.527754\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 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
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.94473e9i 0.325979i
$$408$$ 0 0
$$409$$ −3.81221e10 −1.36234 −0.681168 0.732127i $$-0.738528\pi$$
−0.681168 + 0.732127i $$0.738528\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 9.78715e8i 0.0336400i
$$414$$ 0 0
$$415$$ 2.45678e10 0.828275
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2.52616e10i 0.819605i 0.912174 + 0.409803i $$0.134402\pi$$
−0.912174 + 0.409803i $$0.865598\pi$$
$$420$$ 0 0
$$421$$ −3.33965e9 −0.106310 −0.0531548 0.998586i $$-0.516928\pi$$
−0.0531548 + 0.998586i $$0.516928\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 1.51484e9i − 0.0464312i
$$426$$ 0 0
$$427$$ −3.09455e10 −0.930862
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 5.55244e9i − 0.160907i −0.996758 0.0804534i $$-0.974363\pi$$
0.996758 0.0804534i $$-0.0256368\pi$$
$$432$$ 0 0
$$433$$ 1.14713e10 0.326334 0.163167 0.986598i $$-0.447829\pi$$
0.163167 + 0.986598i $$0.447829\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 8.34667e10i − 2.28869i
$$438$$ 0 0
$$439$$ 5.98486e10 1.61137 0.805686 0.592343i $$-0.201797\pi$$
0.805686 + 0.592343i $$0.201797\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 3.61609e9i − 0.0938910i −0.998897 0.0469455i $$-0.985051\pi$$
0.998897 0.0469455i $$-0.0149487\pi$$
$$444$$ 0 0
$$445$$ −7.81241e10 −1.99226
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 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
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 1.13025e10i − 0.263710i
$$456$$ 0 0
$$457$$ −4.11731e10 −0.943948 −0.471974 0.881612i $$-0.656458\pi$$
−0.471974 + 0.881612i $$0.656458\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 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 $$-0.474504\pi$$
0.0800125 + 0.996794i $$0.474504\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 9.09281e10i − 1.91175i −0.293776 0.955874i $$-0.594912\pi$$
0.293776 0.955874i $$-0.405088\pi$$
$$468$$ 0 0
$$469$$ 2.06989e10 0.427816
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4.08976e10i 0.817059i
$$474$$ 0 0
$$475$$ −1.59023e10 −0.312381
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 1.91197e10i − 0.363194i −0.983373 0.181597i $$-0.941873\pi$$
0.983373 0.181597i $$-0.0581266\pi$$
$$480$$ 0 0
$$481$$ 1.08413e10 0.202536
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.75409e10i 0.497750i
$$486$$ 0 0
$$487$$ −5.28737e10 −0.939992 −0.469996 0.882669i $$-0.655745\pi$$
−0.469996 + 0.882669i $$0.655745\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 9.78032e10i 1.68278i 0.540429 + 0.841389i $$0.318262\pi$$
−0.540429 + 0.841389i $$0.681738\pi$$
$$492$$ 0 0
$$493$$ −2.02276e10 −0.342418
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 9.39263e10i 1.53944i
$$498$$ 0 0
$$499$$ −8.89351e10 −1.43440 −0.717201 0.696866i $$-0.754577\pi$$
−0.717201 + 0.696866i $$0.754577\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 2.97791e10i − 0.465200i −0.972572 0.232600i $$-0.925277\pi$$
0.972572 0.232600i $$-0.0747233\pi$$
$$504$$ 0 0
$$505$$ 4.40871e10 0.677870
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 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
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 9.31772e10i − 1.32459i
$$516$$ 0 0
$$517$$ −3.93212e10 −0.550383
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 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 $$-0.600342\pi$$
−0.310037 + 0.950724i $$0.600342\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1.78385e10i − 0.231268i
$$528$$ 0 0
$$529$$ −5.73553e10 −0.732405
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.18623e10i 0.518698i
$$534$$ 0 0
$$535$$ −9.44298e10 −1.15264
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 9.98252e9i − 0.118273i
$$540$$ 0 0
$$541$$ −1.22420e11 −1.42911 −0.714553 0.699581i $$-0.753370\pi$$
−0.714553 + 0.699581i $$0.753370\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.91498e10i 0.330407i
$$546$$ 0 0
$$547$$ 5.53975e10 0.618786 0.309393 0.950934i $$-0.399874\pi$$
0.309393 + 0.950934i $$0.399874\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.12343e11i 2.30373i
$$552$$ 0 0
$$553$$ −2.94617e10 −0.315034
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 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
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.03752e11i 1.03267i 0.856385 + 0.516337i $$0.172705\pi$$
−0.856385 + 0.516337i $$0.827295\pi$$
$$564$$ 0 0
$$565$$ 1.97978e11 1.94278
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 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 $$-0.680428\pi$$
−0.536963 + 0.843606i $$0.680428\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.58475e10i 0.236454i
$$576$$ 0 0
$$577$$ 1.43414e11 1.29386 0.646930 0.762549i $$-0.276052\pi$$
0.646930 + 0.762549i $$0.276052\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.47361e10i 0.655883i
$$582$$ 0 0
$$583$$ −5.11012e9 −0.0442341
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.93256e11i 1.62773i 0.581057 + 0.813863i $$0.302639\pi$$
−0.581057 + 0.813863i $$0.697361\pi$$
$$588$$ 0 0
$$589$$ −1.87263e11 −1.55593
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 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
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 1.32315e11i − 1.02778i −0.857856 0.513891i $$-0.828204\pi$$
0.857856 0.513891i $$-0.171796\pi$$
$$600$$ 0 0
$$601$$ −2.29562e10 −0.175955 −0.0879775 0.996122i $$-0.528040\pi$$
−0.0879775 + 0.996122i $$0.528040\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 1.15470e11i − 0.861883i
$$606$$ 0 0
$$607$$ −1.56978e11 −1.15634 −0.578169 0.815917i $$-0.696233\pi$$
−0.578169 + 0.815917i $$0.696233\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.76586e10i 0.341961i
$$612$$ 0 0
$$613$$ −1.72931e11 −1.22470 −0.612352 0.790586i $$-0.709776\pi$$
−0.612352 + 0.790586i $$0.709776\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 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 $$-0.645973\pi$$
−0.442684 + 0.896678i $$0.645973\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 2.37656e11i − 1.57760i
$$624$$ 0 0
$$625$$ −1.75075e11 −1.14737
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2.90247e10i 0.185424i
$$630$$ 0 0
$$631$$ 5.79110e9 0.0365295 0.0182648 0.999833i $$-0.494186\pi$$
0.0182648 + 0.999833i $$0.494186\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 2.30471e11i − 1.41749i
$$636$$ 0 0
$$637$$ −1.20991e10 −0.0734847
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 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 $$-0.670267\pi$$
−0.509764 + 0.860314i $$0.670267\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.22876e11i 1.27188i 0.771738 + 0.635941i $$0.219388\pi$$
−0.771738 + 0.635941i $$0.780612\pi$$
$$648$$ 0 0
$$649$$ −3.15296e9 −0.0177721
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 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
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 2.69559e11i − 1.42926i −0.699500 0.714632i $$-0.746594\pi$$
0.699500 0.714632i $$-0.253406\pi$$
$$660$$ 0 0
$$661$$ −2.56807e11 −1.34525 −0.672623 0.739986i $$-0.734832\pi$$
−0.672623 + 0.739986i $$0.734832\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 3.17653e11i − 1.62430i
$$666$$ 0 0
$$667$$ 3.45142e11 1.74379
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 9.96918e10i − 0.491778i
$$672$$ 0 0
$$673$$ 1.67895e11 0.818421 0.409210 0.912440i $$-0.365804\pi$$
0.409210 + 0.912440i $$0.365804\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 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
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 3.04016e11i − 1.39706i −0.715582 0.698529i $$-0.753838\pi$$
0.715582 0.698529i $$-0.246162\pi$$
$$684$$ 0 0
$$685$$ −2.72387e11 −1.23715
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 6.19364e9i 0.0274833i
$$690$$ 0 0
$$691$$ 4.17969e10 0.183329 0.0916646 0.995790i $$-0.470781\pi$$
0.0916646 + 0.995790i $$0.470781\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.83122e11i 0.784876i
$$696$$ 0 0
$$697$$ −1.12075e11 −0.474875
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 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
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.34114e11i 0.536782i
$$708$$ 0 0
$$709$$ −3.47189e11 −1.37398 −0.686990 0.726667i $$-0.741068\pi$$
−0.686990 + 0.726667i $$0.741068\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.04376e11i 1.17775i
$$714$$ 0 0
$$715$$ 3.64112e10 0.139319
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1.56177e11i 0.584387i 0.956359 + 0.292194i $$0.0943851\pi$$
−0.956359 + 0.292194i $$0.905615\pi$$
$$720$$ 0 0
$$721$$ 2.83448e11 1.04890
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 6.57572e10i − 0.238008i
$$726$$ 0 0
$$727$$ 2.70116e11 0.966969 0.483485 0.875353i $$-0.339371\pi$$
0.483485 + 0.875353i $$0.339371\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.32709e11i 0.464761i
$$732$$ 0 0
$$733$$ 1.53459e11 0.531589 0.265795 0.964030i $$-0.414366\pi$$
0.265795 + 0.964030i $$0.414366\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.66822e10i 0.226017i
$$738$$ 0 0
$$739$$ −7.60266e10 −0.254911 −0.127455 0.991844i $$-0.540681\pi$$
−0.127455 + 0.991844i $$0.540681\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 4.18589e11i − 1.37351i −0.726888 0.686756i $$-0.759034\pi$$
0.726888 0.686756i $$-0.240966\pi$$
$$744$$ 0 0
$$745$$ 1.31826e11 0.427933
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 2.87259e11i − 0.912737i
$$750$$ 0 0
$$751$$ −3.73788e11 −1.17507 −0.587537 0.809197i $$-0.699902\pi$$
−0.587537 + 0.809197i $$0.699902\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 5.94038e10i − 0.182821i
$$756$$ 0 0
$$757$$ 4.74806e11 1.44588 0.722941 0.690910i $$-0.242790\pi$$
0.722941 + 0.690910i $$0.242790\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 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
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.82149e9i 0.0110421i
$$768$$ 0 0
$$769$$ 4.29184e11 1.22727 0.613633 0.789592i $$-0.289708\pi$$
0.613633 + 0.789592i $$0.289708\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 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
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.17653e12i 3.19488i
$$780$$ 0 0
$$781$$ −3.02587e11 −0.813290
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 1.93230e11i − 0.508858i
$$786$$ 0 0
$$787$$ 1.30204e10 0.0339409 0.0169705 0.999856i $$-0.494598\pi$$
0.0169705 + 0.999856i $$0.494598\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.02256e11i 1.53842i
$$792$$ 0 0
$$793$$ −1.20830e11 −0.305549
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 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
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 1.54747e11i − 0.372185i
$$804$$ 0 0
$$805$$ −5.16312e11 −1.22950
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 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 $$-0.145134\pi$$
0.897842 + 0.440318i $$0.145134\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1.78634e11i − 0.404888i
$$816$$ 0 0
$$817$$ 1.39313e12 3.12683
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 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 $$-0.600390\pi$$
−0.310183 + 0.950677i $$0.600390\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.82378e11i 0.603683i 0.953358 + 0.301842i $$0.0976014\pi$$
−0.953358 + 0.301842i $$0.902399\pi$$
$$828$$ 0 0
$$829$$ 8.40257e11 1.77907 0.889537 0.456863i $$-0.151027\pi$$
0.889537 + 0.456863i $$0.151027\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 3.23923e10i − 0.0672762i
$$834$$ 0 0
$$835$$ 7.90043e11 1.62519
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 3.34946e11i − 0.675968i −0.941152 0.337984i $$-0.890255\pi$$
0.941152 0.337984i $$-0.109745\pi$$
$$840$$ 0 0
$$841$$ −3.77810e11 −0.755248
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 5.09605e11i 0.999555i
$$846$$ 0 0
$$847$$ 3.51264e11 0.682496
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 4.95246e11i − 0.944284i
$$852$$ 0 0
$$853$$ 6.09757e10 0.115176 0.0575878 0.998340i $$-0.481659\pi$$
0.0575878 + 0.998340i $$0.481659\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 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 $$-0.628883\pi$$
−0.393924 + 0.919143i $$0.628883\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 9.65828e11i − 1.74123i −0.491963 0.870616i $$-0.663720\pi$$
0.491963 0.870616i $$-0.336280\pi$$
$$864$$ 0 0
$$865$$ 8.98876e11 1.60559
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 9.49119e10i − 0.166434i
$$870$$ 0 0
$$871$$ 8.08211e10 0.140427
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 4.49197e11i − 0.766310i
$$876$$ 0 0
$$877$$ −1.01992e12 −1.72412 −0.862059 0.506808i $$-0.830825\pi$$
−0.862059 + 0.506808i $$0.830825\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 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 $$-0.535492\pi$$
−0.111270 + 0.993790i $$0.535492\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 5.56566e11i 0.899129i 0.893248 + 0.449565i $$0.148421\pi$$
−0.893248 + 0.449565i $$0.851579\pi$$
$$888$$ 0 0
$$889$$ 7.01099e11 1.12246
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.33944e12i 2.10628i
$$894$$ 0 0
$$895$$ −2.76074e11 −0.430263
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 7.74347e11i − 1.18549i
$$900$$ 0 0
$$901$$ −1.65818e10 −0.0251613
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 8.76161e11i 1.30614i
$$906$$ 0 0
$$907$$ −3.75761e10 −0.0555243 −0.0277621 0.999615i $$-0.508838\pi$$
−0.0277621 + 0.999615i $$0.508838\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 5.43769e11i 0.789479i 0.918793 + 0.394740i $$0.129165\pi$$
−0.918793 + 0.394740i $$0.870835\pi$$
$$912$$ 0 0
$$913$$ −2.40765e11 −0.346505
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 4.63670e11i − 0.655740i
$$918$$ 0 0
$$919$$ −9.08317e10 −0.127343 −0.0636715 0.997971i $$-0.520281\pi$$
−0.0636715 + 0.997971i $$0.520281\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 3.66745e11i 0.505309i
$$924$$ 0 0
$$925$$ −9.43556e10 −0.128884
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 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
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 9.74814e10i 0.127549i
$$936$$ 0 0
$$937$$ 1.28993e12 1.67343 0.836714 0.547640i $$-0.184474\pi$$
0.836714 + 0.547640i $$0.184474\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 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
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 6.92009e11i − 0.860423i −0.902728 0.430212i $$-0.858439\pi$$
0.902728 0.430212i $$-0.141561\pi$$
$$948$$ 0 0
$$949$$ −1.87558e11 −0.231244
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 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
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 8.28609e11i − 0.979659i
$$960$$ 0 0
$$961$$ −1.70004e11 −0.199326
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 1.50701e12i − 1.73783i
$$966$$ 0 0
$$967$$ 1.25085e12 1.43054 0.715270 0.698848i $$-0.246304\pi$$
0.715270 + 0.698848i $$0.246304\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 7.48637e11i 0.842160i 0.907024 + 0.421080i $$0.138349\pi$$
−0.907024 + 0.421080i $$0.861651\pi$$
$$972$$ 0 0
$$973$$ −5.57063e11 −0.621516
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 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
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.15973e12i 1.24206i 0.783785 + 0.621032i $$0.213286\pi$$
−0.783785 + 0.621032i $$0.786714\pi$$
$$984$$ 0 0
$$985$$ 1.30048e12 1.38153
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 2.26439e12i − 2.36683i
$$990$$ 0 0
$$991$$ 1.55112e12 1.60824 0.804121 0.594465i $$-0.202636\pi$$
0.804121 + 0.594465i $$0.202636\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 1.19115e12i − 1.21527i
$$996$$ 0 0
$$997$$ −3.61247e11 −0.365615 −0.182807 0.983149i $$-0.558518\pi$$
−0.182807 + 0.983149i $$0.558518\pi$$
$$998$$ 0 0
$$999$$ 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 432.9.e.g.161.1 2
3.2 odd 2 inner 432.9.e.g.161.2 2
4.3 odd 2 54.9.b.a.53.1 2
12.11 even 2 54.9.b.a.53.2 yes 2
36.7 odd 6 162.9.d.c.53.1 4
36.11 even 6 162.9.d.c.53.2 4
36.23 even 6 162.9.d.c.107.1 4
36.31 odd 6 162.9.d.c.107.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
54.9.b.a.53.1 2 4.3 odd 2
54.9.b.a.53.2 yes 2 12.11 even 2
162.9.d.c.53.1 4 36.7 odd 6
162.9.d.c.53.2 4 36.11 even 6
162.9.d.c.107.1 4 36.23 even 6
162.9.d.c.107.2 4 36.31 odd 6
432.9.e.g.161.1 2 1.1 even 1 trivial
432.9.e.g.161.2 2 3.2 odd 2 inner