# Properties

 Label 1728.2.c.g.1727.2 Level $1728$ Weight $2$ Character 1728.1727 Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1727.2 Root $$0.258819 - 0.965926i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1727 Dual form 1728.2.c.g.1727.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.14626i q^{5} +3.44949i q^{7} +O(q^{10})$$ $$q-3.14626i q^{5} +3.44949i q^{7} -4.56048 q^{11} +6.89898 q^{13} -3.46410i q^{17} -4.89898i q^{19} -2.82843 q^{23} -4.89898 q^{25} +2.19275i q^{29} -2.55051i q^{31} +10.8530 q^{35} +4.89898 q^{37} -4.09978i q^{41} -2.89898i q^{43} +2.19275 q^{47} -4.89898 q^{49} -12.9029i q^{53} +14.3485i q^{55} +2.19275 q^{59} -4.00000 q^{61} -21.7060i q^{65} -14.8990i q^{67} +13.2207 q^{71} -7.89898 q^{73} -15.7313i q^{77} +2.00000i q^{79} -12.4101 q^{83} -10.8990 q^{85} +5.02118i q^{89} +23.7980i q^{91} -15.4135 q^{95} -5.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$5$$ − 3.14626i − 1.40705i −0.710669 0.703526i $$-0.751608\pi$$
0.710669 0.703526i $$-0.248392\pi$$
$$6$$ 0 0
$$7$$ 3.44949i 1.30378i 0.758312 + 0.651892i $$0.226025\pi$$
−0.758312 + 0.651892i $$0.773975\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.56048 −1.37504 −0.687518 0.726167i $$-0.741300\pi$$
−0.687518 + 0.726167i $$0.741300\pi$$
$$12$$ 0 0
$$13$$ 6.89898 1.91343 0.956716 0.291022i $$-0.0939953\pi$$
0.956716 + 0.291022i $$0.0939953\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.46410i − 0.840168i −0.907485 0.420084i $$-0.862001\pi$$
0.907485 0.420084i $$-0.137999\pi$$
$$18$$ 0 0
$$19$$ − 4.89898i − 1.12390i −0.827170 0.561951i $$-0.810051\pi$$
0.827170 0.561951i $$-0.189949\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ −4.89898 −0.979796
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.19275i 0.407184i 0.979056 + 0.203592i $$0.0652616\pi$$
−0.979056 + 0.203592i $$0.934738\pi$$
$$30$$ 0 0
$$31$$ − 2.55051i − 0.458085i −0.973416 0.229043i $$-0.926440\pi$$
0.973416 0.229043i $$-0.0735595\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 10.8530 1.83449
$$36$$ 0 0
$$37$$ 4.89898 0.805387 0.402694 0.915335i $$-0.368074\pi$$
0.402694 + 0.915335i $$0.368074\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 4.09978i − 0.640277i −0.947371 0.320139i $$-0.896270\pi$$
0.947371 0.320139i $$-0.103730\pi$$
$$42$$ 0 0
$$43$$ − 2.89898i − 0.442090i −0.975264 0.221045i $$-0.929053\pi$$
0.975264 0.221045i $$-0.0709468\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.19275 0.319846 0.159923 0.987130i $$-0.448875\pi$$
0.159923 + 0.987130i $$0.448875\pi$$
$$48$$ 0 0
$$49$$ −4.89898 −0.699854
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 12.9029i − 1.77235i −0.463352 0.886174i $$-0.653353\pi$$
0.463352 0.886174i $$-0.346647\pi$$
$$54$$ 0 0
$$55$$ 14.3485i 1.93475i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 2.19275 0.285472 0.142736 0.989761i $$-0.454410\pi$$
0.142736 + 0.989761i $$0.454410\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 21.7060i − 2.69230i
$$66$$ 0 0
$$67$$ − 14.8990i − 1.82020i −0.414389 0.910100i $$-0.636005\pi$$
0.414389 0.910100i $$-0.363995\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.2207 1.56901 0.784506 0.620121i $$-0.212917\pi$$
0.784506 + 0.620121i $$0.212917\pi$$
$$72$$ 0 0
$$73$$ −7.89898 −0.924506 −0.462253 0.886748i $$-0.652959\pi$$
−0.462253 + 0.886748i $$0.652959\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 15.7313i − 1.79275i
$$78$$ 0 0
$$79$$ 2.00000i 0.225018i 0.993651 + 0.112509i $$0.0358886\pi$$
−0.993651 + 0.112509i $$0.964111\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −12.4101 −1.36218 −0.681092 0.732198i $$-0.738495\pi$$
−0.681092 + 0.732198i $$0.738495\pi$$
$$84$$ 0 0
$$85$$ −10.8990 −1.18216
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 5.02118i 0.532244i 0.963939 + 0.266122i $$0.0857424\pi$$
−0.963939 + 0.266122i $$0.914258\pi$$
$$90$$ 0 0
$$91$$ 23.7980i 2.49470i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −15.4135 −1.58139
$$96$$ 0 0
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 1.58919i − 0.158130i −0.996869 0.0790650i $$-0.974807\pi$$
0.996869 0.0790650i $$-0.0251935\pi$$
$$102$$ 0 0
$$103$$ − 10.0000i − 0.985329i −0.870219 0.492665i $$-0.836023\pi$$
0.870219 0.492665i $$-0.163977\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.66025 −0.837218 −0.418609 0.908166i $$-0.637482\pi$$
−0.418609 + 0.908166i $$0.637482\pi$$
$$108$$ 0 0
$$109$$ −4.89898 −0.469237 −0.234619 0.972088i $$-0.575384\pi$$
−0.234619 + 0.972088i $$0.575384\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.1421i 1.33038i 0.746674 + 0.665190i $$0.231650\pi$$
−0.746674 + 0.665190i $$0.768350\pi$$
$$114$$ 0 0
$$115$$ 8.89898i 0.829834i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 11.9494 1.09540
$$120$$ 0 0
$$121$$ 9.79796 0.890724
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 0.317837i − 0.0284282i
$$126$$ 0 0
$$127$$ − 9.24745i − 0.820578i −0.911955 0.410289i $$-0.865428\pi$$
0.911955 0.410289i $$-0.134572\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.3171 1.25089 0.625446 0.780268i $$-0.284917\pi$$
0.625446 + 0.780268i $$0.284917\pi$$
$$132$$ 0 0
$$133$$ 16.8990 1.46533
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 17.6062i − 1.50420i −0.659048 0.752101i $$-0.729040\pi$$
0.659048 0.752101i $$-0.270960\pi$$
$$138$$ 0 0
$$139$$ − 20.0000i − 1.69638i −0.529694 0.848189i $$-0.677693\pi$$
0.529694 0.848189i $$-0.322307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −31.4626 −2.63104
$$144$$ 0 0
$$145$$ 6.89898 0.572929
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.61037i 0.541542i 0.962644 + 0.270771i $$0.0872787\pi$$
−0.962644 + 0.270771i $$0.912721\pi$$
$$150$$ 0 0
$$151$$ − 7.24745i − 0.589789i −0.955530 0.294895i $$-0.904715\pi$$
0.955530 0.294895i $$-0.0952845\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.02458 −0.644550
$$156$$ 0 0
$$157$$ −5.79796 −0.462728 −0.231364 0.972867i $$-0.574319\pi$$
−0.231364 + 0.972867i $$0.574319\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 9.75663i − 0.768930i
$$162$$ 0 0
$$163$$ 6.00000i 0.469956i 0.972001 + 0.234978i $$0.0755019\pi$$
−0.972001 + 0.234978i $$0.924498\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 11.6637 0.902561 0.451280 0.892382i $$-0.350967\pi$$
0.451280 + 0.892382i $$0.350967\pi$$
$$168$$ 0 0
$$169$$ 34.5959 2.66122
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.80312i 0.669289i 0.942344 + 0.334644i $$0.108616\pi$$
−0.942344 + 0.334644i $$0.891384\pi$$
$$174$$ 0 0
$$175$$ − 16.8990i − 1.27744i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.66025 0.647298 0.323649 0.946177i $$-0.395090\pi$$
0.323649 + 0.946177i $$0.395090\pi$$
$$180$$ 0 0
$$181$$ −9.79796 −0.728277 −0.364138 0.931345i $$-0.618636\pi$$
−0.364138 + 0.931345i $$0.618636\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 15.4135i − 1.13322i
$$186$$ 0 0
$$187$$ 15.7980i 1.15526i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.6780 0.772635 0.386318 0.922366i $$-0.373747\pi$$
0.386318 + 0.922366i $$0.373747\pi$$
$$192$$ 0 0
$$193$$ 11.8990 0.856507 0.428254 0.903659i $$-0.359129\pi$$
0.428254 + 0.903659i $$0.359129\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 1.87492i − 0.133582i −0.997767 0.0667911i $$-0.978724\pi$$
0.997767 0.0667911i $$-0.0212761\pi$$
$$198$$ 0 0
$$199$$ 5.65153i 0.400626i 0.979732 + 0.200313i $$0.0641960\pi$$
−0.979732 + 0.200313i $$0.935804\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −7.56388 −0.530880
$$204$$ 0 0
$$205$$ −12.8990 −0.900904
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 22.3417i 1.54541i
$$210$$ 0 0
$$211$$ 20.4949i 1.41093i 0.708746 + 0.705463i $$0.249261\pi$$
−0.708746 + 0.705463i $$0.750739\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −9.12096 −0.622044
$$216$$ 0 0
$$217$$ 8.79796 0.597244
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 23.8988i − 1.60760i
$$222$$ 0 0
$$223$$ 19.7980i 1.32577i 0.748721 + 0.662885i $$0.230668\pi$$
−0.748721 + 0.662885i $$0.769332\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 22.9774 1.52506 0.762531 0.646952i $$-0.223957\pi$$
0.762531 + 0.646952i $$0.223957\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 7.56388i 0.495526i 0.968821 + 0.247763i $$0.0796954\pi$$
−0.968821 + 0.247763i $$0.920305\pi$$
$$234$$ 0 0
$$235$$ − 6.89898i − 0.450040i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 7.21393 0.466630 0.233315 0.972401i $$-0.425043\pi$$
0.233315 + 0.972401i $$0.425043\pi$$
$$240$$ 0 0
$$241$$ −0.202041 −0.0130146 −0.00650730 0.999979i $$-0.502071\pi$$
−0.00650730 + 0.999979i $$0.502071\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 15.4135i 0.984731i
$$246$$ 0 0
$$247$$ − 33.7980i − 2.15051i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −17.3205 −1.09326 −0.546630 0.837374i $$-0.684090\pi$$
−0.546630 + 0.837374i $$0.684090\pi$$
$$252$$ 0 0
$$253$$ 12.8990 0.810952
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.9774i 1.43329i 0.697439 + 0.716644i $$0.254323\pi$$
−0.697439 + 0.716644i $$0.745677\pi$$
$$258$$ 0 0
$$259$$ 16.8990i 1.05005i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 30.5412 1.88325 0.941627 0.336659i $$-0.109297\pi$$
0.941627 + 0.336659i $$0.109297\pi$$
$$264$$ 0 0
$$265$$ −40.5959 −2.49379
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.3923i 0.633630i 0.948487 + 0.316815i $$0.102613\pi$$
−0.948487 + 0.316815i $$0.897387\pi$$
$$270$$ 0 0
$$271$$ 17.4495i 1.05998i 0.848004 + 0.529991i $$0.177805\pi$$
−0.848004 + 0.529991i $$0.822195\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 22.3417 1.34725
$$276$$ 0 0
$$277$$ −5.79796 −0.348366 −0.174183 0.984713i $$-0.555728\pi$$
−0.174183 + 0.984713i $$0.555728\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 21.7060i 1.29487i 0.762120 + 0.647436i $$0.224159\pi$$
−0.762120 + 0.647436i $$0.775841\pi$$
$$282$$ 0 0
$$283$$ − 12.8990i − 0.766765i −0.923590 0.383382i $$-0.874759\pi$$
0.923590 0.383382i $$-0.125241\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 14.1421 0.834784
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 28.6342i 1.67283i 0.548098 + 0.836414i $$0.315352\pi$$
−0.548098 + 0.836414i $$0.684648\pi$$
$$294$$ 0 0
$$295$$ − 6.89898i − 0.401674i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −19.5133 −1.12848
$$300$$ 0 0
$$301$$ 10.0000 0.576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12.5851i 0.720618i
$$306$$ 0 0
$$307$$ 18.0000i 1.02731i 0.857996 + 0.513657i $$0.171710\pi$$
−0.857996 + 0.513657i $$0.828290\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.75663 −0.553248 −0.276624 0.960978i $$-0.589216\pi$$
−0.276624 + 0.960978i $$0.589216\pi$$
$$312$$ 0 0
$$313$$ −19.4949 −1.10192 −0.550958 0.834533i $$-0.685738\pi$$
−0.550958 + 0.834533i $$0.685738\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.4456i 0.867511i 0.901031 + 0.433755i $$0.142812\pi$$
−0.901031 + 0.433755i $$0.857188\pi$$
$$318$$ 0 0
$$319$$ − 10.0000i − 0.559893i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −16.9706 −0.944267
$$324$$ 0 0
$$325$$ −33.7980 −1.87477
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 7.56388i 0.417010i
$$330$$ 0 0
$$331$$ − 28.6969i − 1.57733i −0.614825 0.788663i $$-0.710774\pi$$
0.614825 0.788663i $$-0.289226\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −46.8761 −2.56112
$$336$$ 0 0
$$337$$ 23.7980 1.29636 0.648179 0.761488i $$-0.275531\pi$$
0.648179 + 0.761488i $$0.275531\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 11.6315i 0.629884i
$$342$$ 0 0
$$343$$ 7.24745i 0.391325i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −22.8024 −1.22410 −0.612048 0.790820i $$-0.709654\pi$$
−0.612048 + 0.790820i $$0.709654\pi$$
$$348$$ 0 0
$$349$$ −15.1010 −0.808339 −0.404170 0.914684i $$-0.632439\pi$$
−0.404170 + 0.914684i $$0.632439\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 14.1421i − 0.752710i −0.926476 0.376355i $$-0.877177\pi$$
0.926476 0.376355i $$-0.122823\pi$$
$$354$$ 0 0
$$355$$ − 41.5959i − 2.20768i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0.349945 0.0184694 0.00923470 0.999957i $$-0.497060\pi$$
0.00923470 + 0.999957i $$0.497060\pi$$
$$360$$ 0 0
$$361$$ −5.00000 −0.263158
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 24.8523i 1.30083i
$$366$$ 0 0
$$367$$ 13.4495i 0.702058i 0.936365 + 0.351029i $$0.114168\pi$$
−0.936365 + 0.351029i $$0.885832\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 44.5084 2.31076
$$372$$ 0 0
$$373$$ 14.4949 0.750517 0.375259 0.926920i $$-0.377554\pi$$
0.375259 + 0.926920i $$0.377554\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 15.1278i 0.779119i
$$378$$ 0 0
$$379$$ 8.00000i 0.410932i 0.978664 + 0.205466i $$0.0658711\pi$$
−0.978664 + 0.205466i $$0.934129\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −24.5344 −1.25365 −0.626826 0.779160i $$-0.715646\pi$$
−0.626826 + 0.779160i $$0.715646\pi$$
$$384$$ 0 0
$$385$$ −49.4949 −2.52249
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 21.6739i − 1.09891i −0.835523 0.549455i $$-0.814835\pi$$
0.835523 0.549455i $$-0.185165\pi$$
$$390$$ 0 0
$$391$$ 9.79796i 0.495504i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 6.29253 0.316611
$$396$$ 0 0
$$397$$ 20.6969 1.03875 0.519375 0.854547i $$-0.326165\pi$$
0.519375 + 0.854547i $$0.326165\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 22.6274i − 1.12996i −0.825105 0.564980i $$-0.808884\pi$$
0.825105 0.564980i $$-0.191116\pi$$
$$402$$ 0 0
$$403$$ − 17.5959i − 0.876515i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −22.3417 −1.10744
$$408$$ 0 0
$$409$$ −7.00000 −0.346128 −0.173064 0.984911i $$-0.555367\pi$$
−0.173064 + 0.984911i $$0.555367\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 7.56388i 0.372194i
$$414$$ 0 0
$$415$$ 39.0454i 1.91666i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −19.1633 −0.936189 −0.468095 0.883678i $$-0.655059\pi$$
−0.468095 + 0.883678i $$0.655059\pi$$
$$420$$ 0 0
$$421$$ 10.2020 0.497217 0.248609 0.968604i $$-0.420027\pi$$
0.248609 + 0.968604i $$0.420027\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 16.9706i 0.823193i
$$426$$ 0 0
$$427$$ − 13.7980i − 0.667730i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.57826 0.316864 0.158432 0.987370i $$-0.449356\pi$$
0.158432 + 0.987370i $$0.449356\pi$$
$$432$$ 0 0
$$433$$ −26.3939 −1.26841 −0.634204 0.773165i $$-0.718672\pi$$
−0.634204 + 0.773165i $$0.718672\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 13.8564i 0.662842i
$$438$$ 0 0
$$439$$ 5.24745i 0.250447i 0.992129 + 0.125224i $$0.0399648\pi$$
−0.992129 + 0.125224i $$0.960035\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6.00680 −0.285392 −0.142696 0.989767i $$-0.545577\pi$$
−0.142696 + 0.989767i $$0.545577\pi$$
$$444$$ 0 0
$$445$$ 15.7980 0.748895
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.37113i 0.253479i 0.991936 + 0.126740i $$0.0404512\pi$$
−0.991936 + 0.126740i $$0.959549\pi$$
$$450$$ 0 0
$$451$$ 18.6969i 0.880404i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 74.8747 3.51018
$$456$$ 0 0
$$457$$ 25.0000 1.16945 0.584725 0.811231i $$-0.301202\pi$$
0.584725 + 0.811231i $$0.301202\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 12.2672i − 0.571341i −0.958328 0.285671i $$-0.907784\pi$$
0.958328 0.285671i $$-0.0922164\pi$$
$$462$$ 0 0
$$463$$ 17.0454i 0.792167i 0.918214 + 0.396084i $$0.129631\pi$$
−0.918214 + 0.396084i $$0.870369\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17.1455 0.793401 0.396700 0.917948i $$-0.370155\pi$$
0.396700 + 0.917948i $$0.370155\pi$$
$$468$$ 0 0
$$469$$ 51.3939 2.37315
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 13.2207i 0.607890i
$$474$$ 0 0
$$475$$ 24.0000i 1.10120i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −19.1633 −0.875594 −0.437797 0.899074i $$-0.644241\pi$$
−0.437797 + 0.899074i $$0.644241\pi$$
$$480$$ 0 0
$$481$$ 33.7980 1.54105
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 15.7313i 0.714323i
$$486$$ 0 0
$$487$$ − 15.7980i − 0.715874i −0.933746 0.357937i $$-0.883480\pi$$
0.933746 0.357937i $$-0.116520\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.00340 −0.135542 −0.0677708 0.997701i $$-0.521589\pi$$
−0.0677708 + 0.997701i $$0.521589\pi$$
$$492$$ 0 0
$$493$$ 7.59592 0.342103
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 45.6048i 2.04565i
$$498$$ 0 0
$$499$$ 5.10102i 0.228353i 0.993460 + 0.114177i $$0.0364229\pi$$
−0.993460 + 0.114177i $$0.963577\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 31.1127 1.38725 0.693623 0.720338i $$-0.256013\pi$$
0.693623 + 0.720338i $$0.256013\pi$$
$$504$$ 0 0
$$505$$ −5.00000 −0.222497
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 44.0156i 1.95096i 0.220095 + 0.975478i $$0.429363\pi$$
−0.220095 + 0.975478i $$0.570637\pi$$
$$510$$ 0 0
$$511$$ − 27.2474i − 1.20536i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −31.4626 −1.38641
$$516$$ 0 0
$$517$$ −10.0000 −0.439799
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0.921404i 0.0403674i 0.999796 + 0.0201837i $$0.00642511\pi$$
−0.999796 + 0.0201837i $$0.993575\pi$$
$$522$$ 0 0
$$523$$ − 31.3939i − 1.37276i −0.727244 0.686379i $$-0.759199\pi$$
0.727244 0.686379i $$-0.240801\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.83523 −0.384869
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 28.2843i − 1.22513i
$$534$$ 0 0
$$535$$ 27.2474i 1.17801i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 22.3417 0.962325
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 15.4135i 0.660241i
$$546$$ 0 0
$$547$$ − 4.20204i − 0.179666i −0.995957 0.0898332i $$-0.971367\pi$$
0.995957 0.0898332i $$-0.0286334\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 10.7423 0.457635
$$552$$ 0 0
$$553$$ −6.89898 −0.293374
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 29.8735i 1.26578i 0.774242 + 0.632890i $$0.218131\pi$$
−0.774242 + 0.632890i $$0.781869\pi$$
$$558$$ 0 0
$$559$$ − 20.0000i − 0.845910i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −5.48188 −0.231034 −0.115517 0.993306i $$-0.536852\pi$$
−0.115517 + 0.993306i $$0.536852\pi$$
$$564$$ 0 0
$$565$$ 44.4949 1.87191
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 19.5133i 0.818038i 0.912526 + 0.409019i $$0.134129\pi$$
−0.912526 + 0.409019i $$0.865871\pi$$
$$570$$ 0 0
$$571$$ − 0.202041i − 0.00845515i −0.999991 0.00422758i $$-0.998654\pi$$
0.999991 0.00422758i $$-0.00134568\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 13.8564 0.577852
$$576$$ 0 0
$$577$$ 15.7980 0.657678 0.328839 0.944386i $$-0.393343\pi$$
0.328839 + 0.944386i $$0.393343\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 42.8085i − 1.77599i
$$582$$ 0 0
$$583$$ 58.8434i 2.43704i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 33.4804 1.38188 0.690942 0.722910i $$-0.257196\pi$$
0.690942 + 0.722910i $$0.257196\pi$$
$$588$$ 0 0
$$589$$ −12.4949 −0.514843
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 42.7764i − 1.75661i −0.478097 0.878307i $$-0.658673\pi$$
0.478097 0.878307i $$-0.341327\pi$$
$$594$$ 0 0
$$595$$ − 37.5959i − 1.54128i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.8338 −1.50499 −0.752493 0.658600i $$-0.771149\pi$$
−0.752493 + 0.658600i $$0.771149\pi$$
$$600$$ 0 0
$$601$$ 39.6969 1.61927 0.809636 0.586932i $$-0.199665\pi$$
0.809636 + 0.586932i $$0.199665\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 30.8270i − 1.25329i
$$606$$ 0 0
$$607$$ 4.20204i 0.170556i 0.996357 + 0.0852778i $$0.0271778\pi$$
−0.996357 + 0.0852778i $$0.972822\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.1278 0.612003
$$612$$ 0 0
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 0.285729i − 0.0115030i −0.999983 0.00575151i $$-0.998169\pi$$
0.999983 0.00575151i $$-0.00183077\pi$$
$$618$$ 0 0
$$619$$ 17.5959i 0.707240i 0.935389 + 0.353620i $$0.115049\pi$$
−0.935389 + 0.353620i $$0.884951\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −17.3205 −0.693932
$$624$$ 0 0
$$625$$ −25.4949 −1.01980
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 16.9706i − 0.676661i
$$630$$ 0 0
$$631$$ − 27.7423i − 1.10441i −0.833710 0.552203i $$-0.813787\pi$$
0.833710 0.552203i $$-0.186213\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −29.0949 −1.15460
$$636$$ 0 0
$$637$$ −33.7980 −1.33912
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 20.4989i − 0.809657i −0.914393 0.404829i $$-0.867331\pi$$
0.914393 0.404829i $$-0.132669\pi$$
$$642$$ 0 0
$$643$$ − 21.3939i − 0.843692i −0.906667 0.421846i $$-0.861382\pi$$
0.906667 0.421846i $$-0.138618\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 20.7846 0.817127 0.408564 0.912730i $$-0.366030\pi$$
0.408564 + 0.912730i $$0.366030\pi$$
$$648$$ 0 0
$$649$$ −10.0000 −0.392534
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 1.23924i − 0.0484952i −0.999706 0.0242476i $$-0.992281\pi$$
0.999706 0.0242476i $$-0.00771901\pi$$
$$654$$ 0 0
$$655$$ − 45.0454i − 1.76007i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 17.4313 0.679026 0.339513 0.940601i $$-0.389738\pi$$
0.339513 + 0.940601i $$0.389738\pi$$
$$660$$ 0 0
$$661$$ −44.4949 −1.73065 −0.865325 0.501210i $$-0.832888\pi$$
−0.865325 + 0.501210i $$0.832888\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 53.1687i − 2.06179i
$$666$$ 0 0
$$667$$ − 6.20204i − 0.240144i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 18.2419 0.704221
$$672$$ 0 0
$$673$$ 2.59592 0.100065 0.0500326 0.998748i $$-0.484067\pi$$
0.0500326 + 0.998748i $$0.484067\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 45.6048i 1.75273i 0.481644 + 0.876367i $$0.340040\pi$$
−0.481644 + 0.876367i $$0.659960\pi$$
$$678$$ 0 0
$$679$$ − 17.2474i − 0.661896i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −10.3923 −0.397650 −0.198825 0.980035i $$-0.563713\pi$$
−0.198825 + 0.980035i $$0.563713\pi$$
$$684$$ 0 0
$$685$$ −55.3939 −2.11649
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 89.0168i − 3.39127i
$$690$$ 0 0
$$691$$ − 4.00000i − 0.152167i −0.997101 0.0760836i $$-0.975758\pi$$
0.997101 0.0760836i $$-0.0242416\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −62.9253 −2.38689
$$696$$ 0 0
$$697$$ −14.2020 −0.537941
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 13.2528i − 0.500553i −0.968174 0.250276i $$-0.919478\pi$$
0.968174 0.250276i $$-0.0805215\pi$$
$$702$$ 0 0
$$703$$ − 24.0000i − 0.905177i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 5.48188 0.206167
$$708$$ 0 0
$$709$$ 40.0000 1.50223 0.751116 0.660171i $$-0.229516\pi$$
0.751116 + 0.660171i $$0.229516\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 7.21393i 0.270164i
$$714$$ 0 0
$$715$$ 98.9898i 3.70201i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6.92820 −0.258378 −0.129189 0.991620i $$-0.541237\pi$$
−0.129189 + 0.991620i $$0.541237\pi$$
$$720$$ 0 0
$$721$$ 34.4949 1.28466
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 10.7423i − 0.398957i
$$726$$ 0 0
$$727$$ 17.2474i 0.639672i 0.947473 + 0.319836i $$0.103628\pi$$
−0.947473 + 0.319836i $$0.896372\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −10.0424 −0.371430
$$732$$ 0 0
$$733$$ 44.4949 1.64346 0.821728 0.569880i $$-0.193010\pi$$
0.821728 + 0.569880i $$0.193010\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 67.9465i 2.50284i
$$738$$ 0 0
$$739$$ 39.3939i 1.44913i 0.689208 + 0.724564i $$0.257959\pi$$
−0.689208 + 0.724564i $$0.742041\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 3.17837 0.116603 0.0583016 0.998299i $$-0.481431\pi$$
0.0583016 + 0.998299i $$0.481431\pi$$
$$744$$ 0 0
$$745$$ 20.7980 0.761978
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 29.8735i − 1.09155i
$$750$$ 0 0
$$751$$ − 25.7423i − 0.939352i −0.882839 0.469676i $$-0.844371\pi$$
0.882839 0.469676i $$-0.155629\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −22.8024 −0.829864
$$756$$ 0 0
$$757$$ −19.3939 −0.704882 −0.352441 0.935834i $$-0.614648\pi$$
−0.352441 + 0.935834i $$0.614648\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 30.2555i − 1.09676i −0.836229 0.548381i $$-0.815244\pi$$
0.836229 0.548381i $$-0.184756\pi$$
$$762$$ 0 0
$$763$$ − 16.8990i − 0.611784i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 15.1278 0.546232
$$768$$ 0 0
$$769$$ −37.4949 −1.35210 −0.676050 0.736855i $$-0.736310\pi$$
−0.676050 + 0.736855i $$0.736310\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 0.349945i − 0.0125867i −0.999980 0.00629333i $$-0.997997\pi$$
0.999980 0.00629333i $$-0.00200324\pi$$
$$774$$ 0 0
$$775$$ 12.4949i 0.448830i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −20.0847 −0.719610
$$780$$ 0 0
$$781$$ −60.2929 −2.15745
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 18.2419i 0.651082i
$$786$$ 0 0
$$787$$ 43.1918i 1.53962i 0.638271 + 0.769811i $$0.279650\pi$$
−0.638271 + 0.769811i $$0.720350\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −48.7832 −1.73453
$$792$$ 0 0
$$793$$ −27.5959 −0.979960
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.0745i 0.356856i 0.983953 + 0.178428i $$0.0571012\pi$$
−0.983953 + 0.178428i $$0.942899\pi$$
$$798$$ 0 0
$$799$$ − 7.59592i − 0.268724i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 36.0231 1.27123
$$804$$ 0 0
$$805$$ −30.6969 −1.08192
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 5.02118i 0.176535i 0.996097 + 0.0882676i $$0.0281331\pi$$
−0.996097 + 0.0882676i $$0.971867\pi$$
$$810$$ 0 0
$$811$$ 24.4949i 0.860132i 0.902797 + 0.430066i $$0.141510\pi$$
−0.902797 + 0.430066i $$0.858490\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 18.8776 0.661253
$$816$$ 0 0
$$817$$ −14.2020 −0.496867
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 11.6637i − 0.407064i −0.979068 0.203532i $$-0.934758\pi$$
0.979068 0.203532i $$-0.0652421\pi$$
$$822$$ 0 0
$$823$$ 44.3485i 1.54589i 0.634473 + 0.772945i $$0.281217\pi$$
−0.634473 + 0.772945i $$0.718783\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −51.2616 −1.78254 −0.891271 0.453471i $$-0.850185\pi$$
−0.891271 + 0.453471i $$0.850185\pi$$
$$828$$ 0 0
$$829$$ −1.59592 −0.0554285 −0.0277143 0.999616i $$-0.508823\pi$$
−0.0277143 + 0.999616i $$0.508823\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 16.9706i 0.587995i
$$834$$ 0 0
$$835$$ − 36.6969i − 1.26995i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 31.4626 1.08621 0.543106 0.839664i $$-0.317248\pi$$
0.543106 + 0.839664i $$0.317248\pi$$
$$840$$ 0 0
$$841$$ 24.1918 0.834201
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 108.848i − 3.74448i
$$846$$ 0 0
$$847$$ 33.7980i 1.16131i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −13.8564 −0.474991
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 22.9774i − 0.784892i −0.919775 0.392446i $$-0.871629\pi$$
0.919775 0.392446i $$-0.128371\pi$$
$$858$$ 0 0
$$859$$ 0.404082i 0.0137871i 0.999976 + 0.00689355i $$0.00219430\pi$$
−0.999976 + 0.00689355i $$0.997806\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 6.92820 0.235839 0.117919 0.993023i $$-0.462378\pi$$
0.117919 + 0.993023i $$0.462378\pi$$
$$864$$ 0 0
$$865$$ 27.6969 0.941724
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 9.12096i − 0.309407i
$$870$$ 0 0
$$871$$ − 102.788i − 3.48283i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1.09638 0.0370643
$$876$$ 0 0
$$877$$ 35.7980 1.20881 0.604406 0.796677i $$-0.293411\pi$$
0.604406 + 0.796677i $$0.293411\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 41.2193i − 1.38871i −0.719631 0.694356i $$-0.755689\pi$$
0.719631 0.694356i $$-0.244311\pi$$
$$882$$ 0 0
$$883$$ − 43.1010i − 1.45046i −0.688504 0.725232i $$-0.741732\pi$$
0.688504 0.725232i $$-0.258268\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −17.0348 −0.571972 −0.285986 0.958234i $$-0.592321\pi$$
−0.285986 + 0.958234i $$0.592321\pi$$
$$888$$ 0 0
$$889$$ 31.8990 1.06986
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 10.7423i − 0.359476i
$$894$$ 0 0
$$895$$ − 27.2474i − 0.910782i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 5.59264 0.186525
$$900$$ 0 0
$$901$$ −44.6969 −1.48907
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 30.8270i 1.02472i
$$906$$ 0 0
$$907$$ 44.4949i 1.47743i 0.674019 + 0.738714i $$0.264567\pi$$
−0.674019 + 0.738714i $$0.735433\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 21.7060 0.719152 0.359576 0.933116i $$-0.382921\pi$$
0.359576 + 0.933116i $$0.382921\pi$$
$$912$$ 0 0
$$913$$ 56.5959 1.87305
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 49.3867i 1.63089i
$$918$$ 0 0
$$919$$ − 27.2474i − 0.898810i −0.893328 0.449405i $$-0.851636\pi$$
0.893328 0.449405i $$-0.148364\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 91.2096 3.00220
$$924$$ 0 0
$$925$$ −24.0000 −0.789115
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 26.4415i − 0.867516i −0.901029 0.433758i $$-0.857187\pi$$
0.901029 0.433758i $$-0.142813\pi$$
$$930$$ 0 0
$$931$$ 24.0000i 0.786568i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 49.7046 1.62551
$$936$$ 0 0
$$937$$ −15.0000 −0.490029 −0.245014 0.969519i $$-0.578793\pi$$
−0.245014 + 0.969519i $$0.578793\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 10.4244i − 0.339826i −0.985459 0.169913i $$-0.945651\pi$$
0.985459 0.169913i $$-0.0543487\pi$$
$$942$$ 0 0
$$943$$ 11.5959i 0.377615i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −33.4804 −1.08797 −0.543984 0.839096i $$-0.683085\pi$$
−0.543984 + 0.839096i $$0.683085\pi$$
$$948$$ 0 0
$$949$$ −54.4949 −1.76898
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 44.6834i − 1.44744i −0.690096 0.723718i $$-0.742432\pi$$
0.690096 0.723718i $$-0.257568\pi$$
$$954$$ 0 0
$$955$$ − 33.5959i − 1.08714i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 60.7325 1.96116
$$960$$ 0 0
$$961$$ 24.4949 0.790158
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 37.4373i − 1.20515i
$$966$$ 0 0
$$967$$ 24.7526i 0.795988i 0.917388 + 0.397994i $$0.130294\pi$$
−0.917388 + 0.397994i $$0.869706\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −8.94598 −0.287090 −0.143545 0.989644i $$-0.545850\pi$$
−0.143545 + 0.989644i $$0.545850\pi$$
$$972$$ 0 0
$$973$$ 68.9898 2.21171
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 3.17837i − 0.101685i −0.998707 0.0508426i $$-0.983809\pi$$
0.998707 0.0508426i $$-0.0161907\pi$$
$$978$$ 0 0
$$979$$ − 22.8990i − 0.731855i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 4.03556 0.128714 0.0643572 0.997927i $$-0.479500\pi$$
0.0643572 + 0.997927i $$0.479500\pi$$
$$984$$ 0 0
$$985$$ −5.89898 −0.187957
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.19955i 0.260731i
$$990$$ 0 0
$$991$$ − 33.2474i − 1.05614i −0.849201 0.528070i $$-0.822916\pi$$
0.849201 0.528070i $$-0.177084\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 17.7812 0.563702
$$996$$ 0 0
$$997$$ 31.3031 0.991378 0.495689 0.868500i $$-0.334916\pi$$
0.495689 + 0.868500i $$0.334916\pi$$
$$998$$ 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 1728.2.c.g.1727.2 8
3.2 odd 2 inner 1728.2.c.g.1727.8 8
4.3 odd 2 inner 1728.2.c.g.1727.1 8
8.3 odd 2 864.2.c.a.863.7 yes 8
8.5 even 2 864.2.c.a.863.8 yes 8
12.11 even 2 inner 1728.2.c.g.1727.7 8
24.5 odd 2 864.2.c.a.863.2 yes 8
24.11 even 2 864.2.c.a.863.1 8
72.5 odd 6 2592.2.s.a.1727.1 8
72.11 even 6 2592.2.s.h.863.3 8
72.13 even 6 2592.2.s.h.1727.3 8
72.29 odd 6 2592.2.s.h.863.4 8
72.43 odd 6 2592.2.s.a.863.1 8
72.59 even 6 2592.2.s.a.1727.2 8
72.61 even 6 2592.2.s.a.863.2 8
72.67 odd 6 2592.2.s.h.1727.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.a.863.1 8 24.11 even 2
864.2.c.a.863.2 yes 8 24.5 odd 2
864.2.c.a.863.7 yes 8 8.3 odd 2
864.2.c.a.863.8 yes 8 8.5 even 2
1728.2.c.g.1727.1 8 4.3 odd 2 inner
1728.2.c.g.1727.2 8 1.1 even 1 trivial
1728.2.c.g.1727.7 8 12.11 even 2 inner
1728.2.c.g.1727.8 8 3.2 odd 2 inner
2592.2.s.a.863.1 8 72.43 odd 6
2592.2.s.a.863.2 8 72.61 even 6
2592.2.s.a.1727.1 8 72.5 odd 6
2592.2.s.a.1727.2 8 72.59 even 6
2592.2.s.h.863.3 8 72.11 even 6
2592.2.s.h.863.4 8 72.29 odd 6
2592.2.s.h.1727.3 8 72.13 even 6
2592.2.s.h.1727.4 8 72.67 odd 6