# Properties

 Label 4864.2.a.r.1.1 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.31662$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -3.31662 q^{5} +3.31662 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -3.31662 q^{5} +3.31662 q^{7} +1.00000 q^{9} +5.00000 q^{11} +6.63325 q^{15} +5.00000 q^{17} +1.00000 q^{19} -6.63325 q^{21} +6.63325 q^{23} +6.00000 q^{25} +4.00000 q^{27} -6.63325 q^{29} -10.0000 q^{33} -11.0000 q^{35} +6.63325 q^{37} +6.00000 q^{41} -1.00000 q^{43} -3.31662 q^{45} -9.94987 q^{47} +4.00000 q^{49} -10.0000 q^{51} +13.2665 q^{53} -16.5831 q^{55} -2.00000 q^{57} -6.00000 q^{59} -9.94987 q^{61} +3.31662 q^{63} -8.00000 q^{67} -13.2665 q^{69} -6.63325 q^{71} +9.00000 q^{73} -12.0000 q^{75} +16.5831 q^{77} +13.2665 q^{79} -11.0000 q^{81} +4.00000 q^{83} -16.5831 q^{85} +13.2665 q^{87} +4.00000 q^{89} -3.31662 q^{95} -12.0000 q^{97} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} + 2q^{9} + 10q^{11} + 10q^{17} + 2q^{19} + 12q^{25} + 8q^{27} - 20q^{33} - 22q^{35} + 12q^{41} - 2q^{43} + 8q^{49} - 20q^{51} - 4q^{57} - 12q^{59} - 16q^{67} + 18q^{73} - 24q^{75} - 22q^{81} + 8q^{83} + 8q^{89} - 24q^{97} + 10q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ −3.31662 −1.48324 −0.741620 0.670820i $$-0.765942\pi$$
−0.741620 + 0.670820i $$0.765942\pi$$
$$6$$ 0 0
$$7$$ 3.31662 1.25357 0.626783 0.779194i $$-0.284371\pi$$
0.626783 + 0.779194i $$0.284371\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 6.63325 1.71270
$$16$$ 0 0
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −6.63325 −1.44749
$$22$$ 0 0
$$23$$ 6.63325 1.38313 0.691564 0.722315i $$-0.256922\pi$$
0.691564 + 0.722315i $$0.256922\pi$$
$$24$$ 0 0
$$25$$ 6.00000 1.20000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ −6.63325 −1.23176 −0.615882 0.787839i $$-0.711200\pi$$
−0.615882 + 0.787839i $$0.711200\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −10.0000 −1.74078
$$34$$ 0 0
$$35$$ −11.0000 −1.85934
$$36$$ 0 0
$$37$$ 6.63325 1.09050 0.545250 0.838274i $$-0.316435\pi$$
0.545250 + 0.838274i $$0.316435\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ −3.31662 −0.494413
$$46$$ 0 0
$$47$$ −9.94987 −1.45134 −0.725669 0.688044i $$-0.758470\pi$$
−0.725669 + 0.688044i $$0.758470\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ −10.0000 −1.40028
$$52$$ 0 0
$$53$$ 13.2665 1.82229 0.911147 0.412082i $$-0.135198\pi$$
0.911147 + 0.412082i $$0.135198\pi$$
$$54$$ 0 0
$$55$$ −16.5831 −2.23607
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −9.94987 −1.27395 −0.636975 0.770884i $$-0.719815\pi$$
−0.636975 + 0.770884i $$0.719815\pi$$
$$62$$ 0 0
$$63$$ 3.31662 0.417855
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ −13.2665 −1.59710
$$70$$ 0 0
$$71$$ −6.63325 −0.787222 −0.393611 0.919277i $$-0.628774\pi$$
−0.393611 + 0.919277i $$0.628774\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 0 0
$$75$$ −12.0000 −1.38564
$$76$$ 0 0
$$77$$ 16.5831 1.88982
$$78$$ 0 0
$$79$$ 13.2665 1.49260 0.746299 0.665611i $$-0.231829\pi$$
0.746299 + 0.665611i $$0.231829\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −16.5831 −1.79869
$$86$$ 0 0
$$87$$ 13.2665 1.42232
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.31662 −0.340279
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −6.63325 −0.653594 −0.326797 0.945095i $$-0.605969\pi$$
−0.326797 + 0.945095i $$0.605969\pi$$
$$104$$ 0 0
$$105$$ 22.0000 2.14698
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ −13.2665 −1.27070 −0.635350 0.772224i $$-0.719144\pi$$
−0.635350 + 0.772224i $$0.719144\pi$$
$$110$$ 0 0
$$111$$ −13.2665 −1.25920
$$112$$ 0 0
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ −22.0000 −2.05151
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 16.5831 1.52017
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −12.0000 −1.08200
$$124$$ 0 0
$$125$$ −3.31662 −0.296648
$$126$$ 0 0
$$127$$ −6.63325 −0.588606 −0.294303 0.955712i $$-0.595087\pi$$
−0.294303 + 0.955712i $$0.595087\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 3.31662 0.287588
$$134$$ 0 0
$$135$$ −13.2665 −1.14180
$$136$$ 0 0
$$137$$ −5.00000 −0.427179 −0.213589 0.976924i $$-0.568515\pi$$
−0.213589 + 0.976924i $$0.568515\pi$$
$$138$$ 0 0
$$139$$ 11.0000 0.933008 0.466504 0.884519i $$-0.345513\pi$$
0.466504 + 0.884519i $$0.345513\pi$$
$$140$$ 0 0
$$141$$ 19.8997 1.67586
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 22.0000 1.82700
$$146$$ 0 0
$$147$$ −8.00000 −0.659829
$$148$$ 0 0
$$149$$ −3.31662 −0.271708 −0.135854 0.990729i $$-0.543378\pi$$
−0.135854 + 0.990729i $$0.543378\pi$$
$$150$$ 0 0
$$151$$ 6.63325 0.539806 0.269903 0.962887i $$-0.413008\pi$$
0.269903 + 0.962887i $$0.413008\pi$$
$$152$$ 0 0
$$153$$ 5.00000 0.404226
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ −26.5330 −2.10420
$$160$$ 0 0
$$161$$ 22.0000 1.73384
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ 33.1662 2.58199
$$166$$ 0 0
$$167$$ −6.63325 −0.513296 −0.256648 0.966505i $$-0.582618\pi$$
−0.256648 + 0.966505i $$0.582618\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 19.8997 1.51295 0.756475 0.654023i $$-0.226920\pi$$
0.756475 + 0.654023i $$0.226920\pi$$
$$174$$ 0 0
$$175$$ 19.8997 1.50428
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −19.8997 −1.47914 −0.739568 0.673081i $$-0.764970\pi$$
−0.739568 + 0.673081i $$0.764970\pi$$
$$182$$ 0 0
$$183$$ 19.8997 1.47103
$$184$$ 0 0
$$185$$ −22.0000 −1.61747
$$186$$ 0 0
$$187$$ 25.0000 1.82818
$$188$$ 0 0
$$189$$ 13.2665 0.964996
$$190$$ 0 0
$$191$$ 16.5831 1.19991 0.599956 0.800033i $$-0.295185\pi$$
0.599956 + 0.800033i $$0.295185\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.5330 1.89040 0.945199 0.326495i $$-0.105868\pi$$
0.945199 + 0.326495i $$0.105868\pi$$
$$198$$ 0 0
$$199$$ −16.5831 −1.17555 −0.587773 0.809026i $$-0.699995\pi$$
−0.587773 + 0.809026i $$0.699995\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ −22.0000 −1.54410
$$204$$ 0 0
$$205$$ −19.8997 −1.38986
$$206$$ 0 0
$$207$$ 6.63325 0.461043
$$208$$ 0 0
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 0 0
$$213$$ 13.2665 0.909006
$$214$$ 0 0
$$215$$ 3.31662 0.226192
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −18.0000 −1.21633
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −19.8997 −1.33259 −0.666293 0.745690i $$-0.732120\pi$$
−0.666293 + 0.745690i $$0.732120\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ 23.2164 1.53418 0.767091 0.641539i $$-0.221704\pi$$
0.767091 + 0.641539i $$0.221704\pi$$
$$230$$ 0 0
$$231$$ −33.1662 −2.18218
$$232$$ 0 0
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 0 0
$$235$$ 33.0000 2.15268
$$236$$ 0 0
$$237$$ −26.5330 −1.72350
$$238$$ 0 0
$$239$$ −3.31662 −0.214535 −0.107267 0.994230i $$-0.534210\pi$$
−0.107267 + 0.994230i $$0.534210\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ −13.2665 −0.847566
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 5.00000 0.315597 0.157799 0.987471i $$-0.449560\pi$$
0.157799 + 0.987471i $$0.449560\pi$$
$$252$$ 0 0
$$253$$ 33.1662 2.08514
$$254$$ 0 0
$$255$$ 33.1662 2.07695
$$256$$ 0 0
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ 22.0000 1.36701
$$260$$ 0 0
$$261$$ −6.63325 −0.410588
$$262$$ 0 0
$$263$$ −3.31662 −0.204512 −0.102256 0.994758i $$-0.532606\pi$$
−0.102256 + 0.994758i $$0.532606\pi$$
$$264$$ 0 0
$$265$$ −44.0000 −2.70290
$$266$$ 0 0
$$267$$ −8.00000 −0.489592
$$268$$ 0 0
$$269$$ 13.2665 0.808873 0.404436 0.914566i $$-0.367468\pi$$
0.404436 + 0.914566i $$0.367468\pi$$
$$270$$ 0 0
$$271$$ −19.8997 −1.20882 −0.604412 0.796672i $$-0.706592\pi$$
−0.604412 + 0.796672i $$0.706592\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 30.0000 1.80907
$$276$$ 0 0
$$277$$ 3.31662 0.199277 0.0996383 0.995024i $$-0.468231\pi$$
0.0996383 + 0.995024i $$0.468231\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ 0 0
$$285$$ 6.63325 0.392920
$$286$$ 0 0
$$287$$ 19.8997 1.17465
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 24.0000 1.40690
$$292$$ 0 0
$$293$$ −26.5330 −1.55007 −0.775037 0.631916i $$-0.782269\pi$$
−0.775037 + 0.631916i $$0.782269\pi$$
$$294$$ 0 0
$$295$$ 19.8997 1.15861
$$296$$ 0 0
$$297$$ 20.0000 1.16052
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −3.31662 −0.191167
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 33.0000 1.88957
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 13.2665 0.754705
$$310$$ 0 0
$$311$$ −16.5831 −0.940343 −0.470171 0.882575i $$-0.655808\pi$$
−0.470171 + 0.882575i $$0.655808\pi$$
$$312$$ 0 0
$$313$$ 34.0000 1.92179 0.960897 0.276907i $$-0.0893093\pi$$
0.960897 + 0.276907i $$0.0893093\pi$$
$$314$$ 0 0
$$315$$ −11.0000 −0.619780
$$316$$ 0 0
$$317$$ −6.63325 −0.372560 −0.186280 0.982497i $$-0.559643\pi$$
−0.186280 + 0.982497i $$0.559643\pi$$
$$318$$ 0 0
$$319$$ −33.1662 −1.85695
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 5.00000 0.278207
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 26.5330 1.46728
$$328$$ 0 0
$$329$$ −33.0000 −1.81935
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 6.63325 0.363500
$$334$$ 0 0
$$335$$ 26.5330 1.44965
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ −36.0000 −1.95525
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −9.94987 −0.537243
$$344$$ 0 0
$$345$$ 44.0000 2.36888
$$346$$ 0 0
$$347$$ 27.0000 1.44944 0.724718 0.689046i $$-0.241970\pi$$
0.724718 + 0.689046i $$0.241970\pi$$
$$348$$ 0 0
$$349$$ −9.94987 −0.532605 −0.266302 0.963890i $$-0.585802\pi$$
−0.266302 + 0.963890i $$0.585802\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 22.0000 1.16764
$$356$$ 0 0
$$357$$ −33.1662 −1.75534
$$358$$ 0 0
$$359$$ 23.2164 1.22531 0.612657 0.790349i $$-0.290101\pi$$
0.612657 + 0.790349i $$0.290101\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −28.0000 −1.46962
$$364$$ 0 0
$$365$$ −29.8496 −1.56240
$$366$$ 0 0
$$367$$ −6.63325 −0.346253 −0.173126 0.984900i $$-0.555387\pi$$
−0.173126 + 0.984900i $$0.555387\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 44.0000 2.28437
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 6.63325 0.342540
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 6.00000 0.308199 0.154100 0.988055i $$-0.450752\pi$$
0.154100 + 0.988055i $$0.450752\pi$$
$$380$$ 0 0
$$381$$ 13.2665 0.679663
$$382$$ 0 0
$$383$$ −13.2665 −0.677886 −0.338943 0.940807i $$-0.610069\pi$$
−0.338943 + 0.940807i $$0.610069\pi$$
$$384$$ 0 0
$$385$$ −55.0000 −2.80306
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 0 0
$$389$$ −16.5831 −0.840798 −0.420399 0.907339i $$-0.638110\pi$$
−0.420399 + 0.907339i $$0.638110\pi$$
$$390$$ 0 0
$$391$$ 33.1662 1.67729
$$392$$ 0 0
$$393$$ −30.0000 −1.51330
$$394$$ 0 0
$$395$$ −44.0000 −2.21388
$$396$$ 0 0
$$397$$ 9.94987 0.499370 0.249685 0.968327i $$-0.419673\pi$$
0.249685 + 0.968327i $$0.419673\pi$$
$$398$$ 0 0
$$399$$ −6.63325 −0.332078
$$400$$ 0 0
$$401$$ 16.0000 0.799002 0.399501 0.916733i $$-0.369183\pi$$
0.399501 + 0.916733i $$0.369183\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 36.4829 1.81285
$$406$$ 0 0
$$407$$ 33.1662 1.64399
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 0 0
$$413$$ −19.8997 −0.979203
$$414$$ 0 0
$$415$$ −13.2665 −0.651227
$$416$$ 0 0
$$417$$ −22.0000 −1.07734
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −13.2665 −0.646570 −0.323285 0.946302i $$-0.604787\pi$$
−0.323285 + 0.946302i $$0.604787\pi$$
$$422$$ 0 0
$$423$$ −9.94987 −0.483779
$$424$$ 0 0
$$425$$ 30.0000 1.45521
$$426$$ 0 0
$$427$$ −33.0000 −1.59698
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 26.5330 1.27805 0.639025 0.769186i $$-0.279338\pi$$
0.639025 + 0.769186i $$0.279338\pi$$
$$432$$ 0 0
$$433$$ −38.0000 −1.82616 −0.913082 0.407777i $$-0.866304\pi$$
−0.913082 + 0.407777i $$0.866304\pi$$
$$434$$ 0 0
$$435$$ −44.0000 −2.10964
$$436$$ 0 0
$$437$$ 6.63325 0.317311
$$438$$ 0 0
$$439$$ −19.8997 −0.949763 −0.474882 0.880050i $$-0.657509\pi$$
−0.474882 + 0.880050i $$0.657509\pi$$
$$440$$ 0 0
$$441$$ 4.00000 0.190476
$$442$$ 0 0
$$443$$ −13.0000 −0.617649 −0.308824 0.951119i $$-0.599936\pi$$
−0.308824 + 0.951119i $$0.599936\pi$$
$$444$$ 0 0
$$445$$ −13.2665 −0.628892
$$446$$ 0 0
$$447$$ 6.63325 0.313742
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 30.0000 1.41264
$$452$$ 0 0
$$453$$ −13.2665 −0.623315
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 0 0
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ 16.5831 0.772353 0.386177 0.922425i $$-0.373796\pi$$
0.386177 + 0.922425i $$0.373796\pi$$
$$462$$ 0 0
$$463$$ 23.2164 1.07896 0.539478 0.842000i $$-0.318622\pi$$
0.539478 + 0.842000i $$0.318622\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 0 0
$$469$$ −26.5330 −1.22518
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.00000 −0.229900
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ 13.2665 0.607431
$$478$$ 0 0
$$479$$ −6.63325 −0.303081 −0.151540 0.988451i $$-0.548423\pi$$
−0.151540 + 0.988451i $$0.548423\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −44.0000 −2.00207
$$484$$ 0 0
$$485$$ 39.7995 1.80720
$$486$$ 0 0
$$487$$ −33.1662 −1.50291 −0.751453 0.659787i $$-0.770647\pi$$
−0.751453 + 0.659787i $$0.770647\pi$$
$$488$$ 0 0
$$489$$ −40.0000 −1.80886
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ −33.1662 −1.49373
$$494$$ 0 0
$$495$$ −16.5831 −0.745356
$$496$$ 0 0
$$497$$ −22.0000 −0.986835
$$498$$ 0 0
$$499$$ 3.00000 0.134298 0.0671492 0.997743i $$-0.478610\pi$$
0.0671492 + 0.997743i $$0.478610\pi$$
$$500$$ 0 0
$$501$$ 13.2665 0.592703
$$502$$ 0 0
$$503$$ −19.8997 −0.887286 −0.443643 0.896204i $$-0.646314\pi$$
−0.443643 + 0.896204i $$0.646314\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 26.0000 1.15470
$$508$$ 0 0
$$509$$ 26.5330 1.17605 0.588027 0.808841i $$-0.299905\pi$$
0.588027 + 0.808841i $$0.299905\pi$$
$$510$$ 0 0
$$511$$ 29.8496 1.32047
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 22.0000 0.969436
$$516$$ 0 0
$$517$$ −49.7494 −2.18797
$$518$$ 0 0
$$519$$ −39.7995 −1.74700
$$520$$ 0 0
$$521$$ −16.0000 −0.700973 −0.350486 0.936568i $$-0.613984\pi$$
−0.350486 + 0.936568i $$0.613984\pi$$
$$522$$ 0 0
$$523$$ 6.00000 0.262362 0.131181 0.991358i $$-0.458123\pi$$
0.131181 + 0.991358i $$0.458123\pi$$
$$524$$ 0 0
$$525$$ −39.7995 −1.73699
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 21.0000 0.913043
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 19.8997 0.860341
$$536$$ 0 0
$$537$$ 36.0000 1.55351
$$538$$ 0 0
$$539$$ 20.0000 0.861461
$$540$$ 0 0
$$541$$ −9.94987 −0.427779 −0.213889 0.976858i $$-0.568613\pi$$
−0.213889 + 0.976858i $$0.568613\pi$$
$$542$$ 0 0
$$543$$ 39.7995 1.70796
$$544$$ 0 0
$$545$$ 44.0000 1.88475
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 0 0
$$549$$ −9.94987 −0.424650
$$550$$ 0 0
$$551$$ −6.63325 −0.282586
$$552$$ 0 0
$$553$$ 44.0000 1.87107
$$554$$ 0 0
$$555$$ 44.0000 1.86770
$$556$$ 0 0
$$557$$ −3.31662 −0.140530 −0.0702650 0.997528i $$-0.522384\pi$$
−0.0702650 + 0.997528i $$0.522384\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −50.0000 −2.11100
$$562$$ 0 0
$$563$$ −14.0000 −0.590030 −0.295015 0.955493i $$-0.595325\pi$$
−0.295015 + 0.955493i $$0.595325\pi$$
$$564$$ 0 0
$$565$$ −59.6992 −2.51157
$$566$$ 0 0
$$567$$ −36.4829 −1.53214
$$568$$ 0 0
$$569$$ −20.0000 −0.838444 −0.419222 0.907884i $$-0.637697\pi$$
−0.419222 + 0.907884i $$0.637697\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ −33.1662 −1.38554
$$574$$ 0 0
$$575$$ 39.7995 1.65975
$$576$$ 0 0
$$577$$ −43.0000 −1.79011 −0.895057 0.445952i $$-0.852865\pi$$
−0.895057 + 0.445952i $$0.852865\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13.2665 0.550387
$$582$$ 0 0
$$583$$ 66.3325 2.74721
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.00000 0.206372 0.103186 0.994662i $$-0.467096\pi$$
0.103186 + 0.994662i $$0.467096\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −53.0660 −2.18284
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ −55.0000 −2.25478
$$596$$ 0 0
$$597$$ 33.1662 1.35740
$$598$$ 0 0
$$599$$ −13.2665 −0.542054 −0.271027 0.962572i $$-0.587363\pi$$
−0.271027 + 0.962572i $$0.587363\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ −46.4327 −1.88776
$$606$$ 0 0
$$607$$ −26.5330 −1.07694 −0.538471 0.842644i $$-0.680998\pi$$
−0.538471 + 0.842644i $$0.680998\pi$$
$$608$$ 0 0
$$609$$ 44.0000 1.78297
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 3.31662 0.133957 0.0669786 0.997754i $$-0.478664\pi$$
0.0669786 + 0.997754i $$0.478664\pi$$
$$614$$ 0 0
$$615$$ 39.7995 1.60487
$$616$$ 0 0
$$617$$ 41.0000 1.65060 0.825299 0.564696i $$-0.191007\pi$$
0.825299 + 0.564696i $$0.191007\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 26.5330 1.06473
$$622$$ 0 0
$$623$$ 13.2665 0.531511
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ −10.0000 −0.399362
$$628$$ 0 0
$$629$$ 33.1662 1.32242
$$630$$ 0 0
$$631$$ 9.94987 0.396098 0.198049 0.980192i $$-0.436539\pi$$
0.198049 + 0.980192i $$0.436539\pi$$
$$632$$ 0 0
$$633$$ −28.0000 −1.11290
$$634$$ 0 0
$$635$$ 22.0000 0.873043
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.63325 −0.262407
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ 0 0
$$645$$ −6.63325 −0.261184
$$646$$ 0 0
$$647$$ −16.5831 −0.651950 −0.325975 0.945378i $$-0.605693\pi$$
−0.325975 + 0.945378i $$0.605693\pi$$
$$648$$ 0 0
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 36.4829 1.42769 0.713843 0.700306i $$-0.246953\pi$$
0.713843 + 0.700306i $$0.246953\pi$$
$$654$$ 0 0
$$655$$ −49.7494 −1.94387
$$656$$ 0 0
$$657$$ 9.00000 0.351123
$$658$$ 0 0
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 0 0
$$661$$ 39.7995 1.54802 0.774011 0.633173i $$-0.218248\pi$$
0.774011 + 0.633173i $$0.218248\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −11.0000 −0.426562
$$666$$ 0 0
$$667$$ −44.0000 −1.70369
$$668$$ 0 0
$$669$$ 39.7995 1.53874
$$670$$ 0 0
$$671$$ −49.7494 −1.92055
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 24.0000 0.923760
$$676$$ 0 0
$$677$$ −19.8997 −0.764809 −0.382405 0.923995i $$-0.624904\pi$$
−0.382405 + 0.923995i $$0.624904\pi$$
$$678$$ 0 0
$$679$$ −39.7995 −1.52736
$$680$$ 0 0
$$681$$ −16.0000 −0.613121
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 16.5831 0.633609
$$686$$ 0 0
$$687$$ −46.4327 −1.77152
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −31.0000 −1.17930 −0.589648 0.807661i $$-0.700733\pi$$
−0.589648 + 0.807661i $$0.700733\pi$$
$$692$$ 0 0
$$693$$ 16.5831 0.629941
$$694$$ 0 0
$$695$$ −36.4829 −1.38387
$$696$$ 0 0
$$697$$ 30.0000 1.13633
$$698$$ 0 0
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 6.63325 0.250178
$$704$$ 0 0
$$705$$ −66.0000 −2.48570
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −13.2665 −0.498234 −0.249117 0.968473i $$-0.580140\pi$$
−0.249117 + 0.968473i $$0.580140\pi$$
$$710$$ 0 0
$$711$$ 13.2665 0.497533
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.63325 0.247723
$$718$$ 0 0
$$719$$ 3.31662 0.123689 0.0618446 0.998086i $$-0.480302\pi$$
0.0618446 + 0.998086i $$0.480302\pi$$
$$720$$ 0 0
$$721$$ −22.0000 −0.819323
$$722$$ 0 0
$$723$$ −52.0000 −1.93390
$$724$$ 0 0
$$725$$ −39.7995 −1.47812
$$726$$ 0 0
$$727$$ −16.5831 −0.615034 −0.307517 0.951543i $$-0.599498\pi$$
−0.307517 + 0.951543i $$0.599498\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −5.00000 −0.184932
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 26.5330 0.978684
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ −45.0000 −1.65535 −0.827676 0.561206i $$-0.810337\pi$$
−0.827676 + 0.561206i $$0.810337\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 53.0660 1.94680 0.973401 0.229107i $$-0.0735805\pi$$
0.973401 + 0.229107i $$0.0735805\pi$$
$$744$$ 0 0
$$745$$ 11.0000 0.403009
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ −19.8997 −0.727121
$$750$$ 0 0
$$751$$ 39.7995 1.45230 0.726152 0.687534i $$-0.241307\pi$$
0.726152 + 0.687534i $$0.241307\pi$$
$$752$$ 0 0
$$753$$ −10.0000 −0.364420
$$754$$ 0 0
$$755$$ −22.0000 −0.800662
$$756$$ 0 0
$$757$$ −9.94987 −0.361634 −0.180817 0.983517i $$-0.557874\pi$$
−0.180817 + 0.983517i $$0.557874\pi$$
$$758$$ 0 0
$$759$$ −66.3325 −2.40772
$$760$$ 0 0
$$761$$ −1.00000 −0.0362500 −0.0181250 0.999836i $$-0.505770\pi$$
−0.0181250 + 0.999836i $$0.505770\pi$$
$$762$$ 0 0
$$763$$ −44.0000 −1.59291
$$764$$ 0 0
$$765$$ −16.5831 −0.599564
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 33.0000 1.19001 0.595005 0.803722i $$-0.297150\pi$$
0.595005 + 0.803722i $$0.297150\pi$$
$$770$$ 0 0
$$771$$ 16.0000 0.576226
$$772$$ 0 0
$$773$$ −6.63325 −0.238581 −0.119291 0.992859i $$-0.538062\pi$$
−0.119291 + 0.992859i $$0.538062\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −44.0000 −1.57849
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −33.1662 −1.18678
$$782$$ 0 0
$$783$$ −26.5330 −0.948212
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 0 0
$$789$$ 6.63325 0.236150
$$790$$ 0 0
$$791$$ 59.6992 2.12266
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 88.0000 3.12104
$$796$$ 0 0
$$797$$ 26.5330 0.939847 0.469924 0.882707i $$-0.344281\pi$$
0.469924 + 0.882707i $$0.344281\pi$$
$$798$$ 0 0
$$799$$ −49.7494 −1.76001
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ 45.0000 1.58802
$$804$$ 0 0
$$805$$ −72.9657 −2.57170
$$806$$ 0 0
$$807$$ −26.5330 −0.934006
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 39.7995 1.39583
$$814$$ 0 0
$$815$$ −66.3325 −2.32353
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9.94987 0.347253 0.173627 0.984812i $$-0.444451\pi$$
0.173627 + 0.984812i $$0.444451\pi$$
$$822$$ 0 0
$$823$$ −29.8496 −1.04049 −0.520246 0.854016i $$-0.674160\pi$$
−0.520246 + 0.854016i $$0.674160\pi$$
$$824$$ 0 0
$$825$$ −60.0000 −2.08893
$$826$$ 0 0
$$827$$ −32.0000 −1.11275 −0.556375 0.830932i $$-0.687808\pi$$
−0.556375 + 0.830932i $$0.687808\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ −6.63325 −0.230105
$$832$$ 0 0
$$833$$ 20.0000 0.692959
$$834$$ 0 0
$$835$$ 22.0000 0.761341
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 46.4327 1.60304 0.801518 0.597970i $$-0.204026\pi$$
0.801518 + 0.597970i $$0.204026\pi$$
$$840$$ 0 0
$$841$$ 15.0000 0.517241
$$842$$ 0 0
$$843$$ −52.0000 −1.79098
$$844$$ 0 0
$$845$$ 43.1161 1.48324
$$846$$ 0 0
$$847$$ 46.4327 1.59545
$$848$$ 0 0
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 44.0000 1.50830
$$852$$ 0 0
$$853$$ 53.0660 1.81695 0.908473 0.417945i $$-0.137249\pi$$
0.908473 + 0.417945i $$0.137249\pi$$
$$854$$ 0 0
$$855$$ −3.31662 −0.113426
$$856$$ 0 0
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ −41.0000 −1.39890 −0.699451 0.714681i $$-0.746572\pi$$
−0.699451 + 0.714681i $$0.746572\pi$$
$$860$$ 0 0
$$861$$ −39.7995 −1.35636
$$862$$ 0 0
$$863$$ 33.1662 1.12899 0.564496 0.825436i $$-0.309071\pi$$
0.564496 + 0.825436i $$0.309071\pi$$
$$864$$ 0 0
$$865$$ −66.0000 −2.24407
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 66.3325 2.25018
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −12.0000 −0.406138
$$874$$ 0 0
$$875$$ −11.0000 −0.371868
$$876$$ 0 0
$$877$$ −6.63325 −0.223989 −0.111994 0.993709i $$-0.535724\pi$$
−0.111994 + 0.993709i $$0.535724\pi$$
$$878$$ 0 0
$$879$$ 53.0660 1.78987
$$880$$ 0 0
$$881$$ 51.0000 1.71823 0.859117 0.511780i $$-0.171014\pi$$
0.859117 + 0.511780i $$0.171014\pi$$
$$882$$ 0 0
$$883$$ 31.0000 1.04323 0.521617 0.853180i $$-0.325329\pi$$
0.521617 + 0.853180i $$0.325329\pi$$
$$884$$ 0 0
$$885$$ −39.7995 −1.33785
$$886$$ 0 0
$$887$$ 19.8997 0.668168 0.334084 0.942543i $$-0.391573\pi$$
0.334084 + 0.942543i $$0.391573\pi$$
$$888$$ 0 0
$$889$$ −22.0000 −0.737856
$$890$$ 0 0
$$891$$ −55.0000 −1.84257
$$892$$ 0 0
$$893$$ −9.94987 −0.332960
$$894$$ 0 0
$$895$$ 59.6992 1.99553
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 66.3325 2.20986
$$902$$ 0 0
$$903$$ 6.63325 0.220741
$$904$$ 0 0
$$905$$ 66.0000 2.19391
$$906$$ 0 0
$$907$$ 32.0000 1.06254 0.531271 0.847202i $$-0.321714\pi$$
0.531271 + 0.847202i $$0.321714\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 19.8997 0.659308 0.329654 0.944102i $$-0.393068\pi$$
0.329654 + 0.944102i $$0.393068\pi$$
$$912$$ 0 0
$$913$$ 20.0000 0.661903
$$914$$ 0 0
$$915$$ −66.0000 −2.18189
$$916$$ 0 0
$$917$$ 49.7494 1.64287
$$918$$ 0 0
$$919$$ −33.1662 −1.09405 −0.547027 0.837115i $$-0.684240\pi$$
−0.547027 + 0.837115i $$0.684240\pi$$
$$920$$ 0 0
$$921$$ 24.0000 0.790827
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 39.7995 1.30860
$$926$$ 0 0
$$927$$ −6.63325 −0.217865
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 0 0
$$933$$ 33.1662 1.08581
$$934$$ 0 0
$$935$$ −82.9156 −2.71163
$$936$$ 0 0
$$937$$ −1.00000 −0.0326686 −0.0163343 0.999867i $$-0.505200\pi$$
−0.0163343 + 0.999867i $$0.505200\pi$$
$$938$$ 0 0
$$939$$ −68.0000 −2.21910
$$940$$ 0 0
$$941$$ 46.4327 1.51366 0.756832 0.653609i $$-0.226746\pi$$
0.756832 + 0.653609i $$0.226746\pi$$
$$942$$ 0 0
$$943$$ 39.7995 1.29605
$$944$$ 0 0
$$945$$ −44.0000 −1.43132
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 13.2665 0.430196
$$952$$ 0 0
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ 0 0
$$955$$ −55.0000 −1.77976
$$956$$ 0 0
$$957$$ 66.3325 2.14423
$$958$$ 0 0
$$959$$ −16.5831 −0.535497
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 33.1662 1.06655 0.533277 0.845940i $$-0.320960\pi$$
0.533277 + 0.845940i $$0.320960\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ −32.0000 −1.02693 −0.513464 0.858111i $$-0.671638\pi$$
−0.513464 + 0.858111i $$0.671638\pi$$
$$972$$ 0 0
$$973$$ 36.4829 1.16959
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16.0000 0.511885 0.255943 0.966692i $$-0.417614\pi$$
0.255943 + 0.966692i $$0.417614\pi$$
$$978$$ 0 0
$$979$$ 20.0000 0.639203
$$980$$ 0 0
$$981$$ −13.2665 −0.423567
$$982$$ 0 0
$$983$$ −13.2665 −0.423136 −0.211568 0.977363i $$-0.567857\pi$$
−0.211568 + 0.977363i $$0.567857\pi$$
$$984$$ 0 0
$$985$$ −88.0000 −2.80391
$$986$$ 0 0
$$987$$ 66.0000 2.10080
$$988$$ 0 0
$$989$$ −6.63325 −0.210925
$$990$$ 0 0
$$991$$ 33.1662 1.05356 0.526780 0.850001i $$-0.323399\pi$$
0.526780 + 0.850001i $$0.323399\pi$$
$$992$$ 0 0
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ 55.0000 1.74362
$$996$$ 0 0
$$997$$ 43.1161 1.36550 0.682751 0.730651i $$-0.260784\pi$$
0.682751 + 0.730651i $$0.260784\pi$$
$$998$$ 0 0
$$999$$ 26.5330 0.839467
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 4864.2.a.r.1.1 2
4.3 odd 2 4864.2.a.y.1.1 2
8.3 odd 2 inner 4864.2.a.r.1.2 2
8.5 even 2 4864.2.a.y.1.2 2
16.3 odd 4 1216.2.c.f.609.4 yes 4
16.5 even 4 1216.2.c.f.609.3 yes 4
16.11 odd 4 1216.2.c.f.609.1 4
16.13 even 4 1216.2.c.f.609.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.f.609.1 4 16.11 odd 4
1216.2.c.f.609.2 yes 4 16.13 even 4
1216.2.c.f.609.3 yes 4 16.5 even 4
1216.2.c.f.609.4 yes 4 16.3 odd 4
4864.2.a.r.1.1 2 1.1 even 1 trivial
4864.2.a.r.1.2 2 8.3 odd 2 inner
4864.2.a.y.1.1 2 4.3 odd 2
4864.2.a.y.1.2 2 8.5 even 2