# Properties

 Label 4864.2.a.r.1.2 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 1.2 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 $$-0.234058\pi$$
0.741620 + 0.670820i $$0.234058\pi$$
$$6$$ 0 0
$$7$$ −3.31662 −1.25357 −0.626783 0.779194i $$-0.715629\pi$$
−0.626783 + 0.779194i $$0.715629\pi$$
$$8$$ 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 $$-0.743078\pi$$
−0.691564 + 0.722315i $$0.743078\pi$$
$$24$$ 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 $$-0.288800\pi$$
0.615882 + 0.787839i $$0.288800\pi$$
$$30$$ 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 $$-0.683565\pi$$
−0.545250 + 0.838274i $$0.683565\pi$$
$$38$$ 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 $$-0.241530\pi$$
0.725669 + 0.688044i $$0.241530\pi$$
$$48$$ 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 $$-0.864802\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 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 $$-0.280185\pi$$
0.636975 + 0.770884i $$0.280185\pi$$
$$62$$ 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 $$-0.371226\pi$$
0.393611 + 0.919277i $$0.371226\pi$$
$$72$$ 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 $$-0.768171\pi$$
−0.746299 + 0.665611i $$0.768171\pi$$
$$80$$ 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 $$-0.394031\pi$$
0.326797 + 0.945095i $$0.394031\pi$$
$$104$$ 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 $$-0.280856\pi$$
0.635350 + 0.772224i $$0.280856\pi$$
$$110$$ 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 $$-0.404913\pi$$
0.294303 + 0.955712i $$0.404913\pi$$
$$128$$ 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 $$-0.456622\pi$$
0.135854 + 0.990729i $$0.456622\pi$$
$$150$$ 0 0
$$151$$ −6.63325 −0.539806 −0.269903 0.962887i $$-0.586992\pi$$
−0.269903 + 0.962887i $$0.586992\pi$$
$$152$$ 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 $$-0.417382\pi$$
0.256648 + 0.966505i $$0.417382\pi$$
$$168$$ 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 $$-0.773080\pi$$
−0.756475 + 0.654023i $$0.773080\pi$$
$$174$$ 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 $$-0.235030\pi$$
0.739568 + 0.673081i $$0.235030\pi$$
$$182$$ 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 $$-0.704815\pi$$
−0.599956 + 0.800033i $$0.704815\pi$$
$$192$$ 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 $$-0.894132\pi$$
−0.945199 + 0.326495i $$0.894132\pi$$
$$198$$ 0 0
$$199$$ 16.5831 1.17555 0.587773 0.809026i $$-0.300005\pi$$
0.587773 + 0.809026i $$0.300005\pi$$
$$200$$ 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 $$-0.267880\pi$$
0.666293 + 0.745690i $$0.267880\pi$$
$$224$$ 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 $$-0.778296\pi$$
−0.767091 + 0.641539i $$0.778296\pi$$
$$230$$ 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 $$-0.465790\pi$$
0.107267 + 0.994230i $$0.465790\pi$$
$$240$$ 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 $$-0.467394\pi$$
0.102256 + 0.994758i $$0.467394\pi$$
$$264$$ 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 $$-0.632532\pi$$
−0.404436 + 0.914566i $$0.632532\pi$$
$$270$$ 0 0
$$271$$ 19.8997 1.20882 0.604412 0.796672i $$-0.293408\pi$$
0.604412 + 0.796672i $$0.293408\pi$$
$$272$$ 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 $$-0.531769\pi$$
−0.0996383 + 0.995024i $$0.531769\pi$$
$$278$$ 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 $$-0.217731\pi$$
0.775037 + 0.631916i $$0.217731\pi$$
$$294$$ 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 $$-0.344192\pi$$
0.470171 + 0.882575i $$0.344192\pi$$
$$312$$ 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 $$-0.440357\pi$$
0.186280 + 0.982497i $$0.440357\pi$$
$$318$$ 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 $$-0.414198\pi$$
0.266302 + 0.963890i $$0.414198\pi$$
$$350$$ 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 $$-0.709899\pi$$
−0.612657 + 0.790349i $$0.709899\pi$$
$$360$$ 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 $$-0.444613\pi$$
0.173126 + 0.984900i $$0.444613\pi$$
$$368$$ 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 $$-0.389931\pi$$
0.338943 + 0.940807i $$0.389931\pi$$
$$384$$ 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 $$-0.361890\pi$$
0.420399 + 0.907339i $$0.361890\pi$$
$$390$$ 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 $$-0.580327\pi$$
−0.249685 + 0.968327i $$0.580327\pi$$
$$398$$ 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 $$-0.395213\pi$$
0.323285 + 0.946302i $$0.395213\pi$$
$$422$$ 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 $$-0.720662\pi$$
−0.639025 + 0.769186i $$0.720662\pi$$
$$432$$ 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 $$-0.342491\pi$$
0.474882 + 0.880050i $$0.342491\pi$$
$$440$$ 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 $$-0.626204\pi$$
−0.386177 + 0.922425i $$0.626204\pi$$
$$462$$ 0 0
$$463$$ −23.2164 −1.07896 −0.539478 0.842000i $$-0.681378\pi$$
−0.539478 + 0.842000i $$0.681378\pi$$
$$464$$ 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 $$-0.451577\pi$$
0.151540 + 0.988451i $$0.451577\pi$$
$$480$$ 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 $$-0.229353\pi$$
0.751453 + 0.659787i $$0.229353\pi$$
$$488$$ 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 $$-0.353686\pi$$
0.443643 + 0.896204i $$0.353686\pi$$
$$504$$ 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 $$-0.700095\pi$$
−0.588027 + 0.808841i $$0.700095\pi$$
$$510$$ 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 $$-0.431387\pi$$
0.213889 + 0.976858i $$0.431387\pi$$
$$542$$ 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 $$-0.477616\pi$$
0.0702650 + 0.997528i $$0.477616\pi$$
$$558$$ 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 $$-0.412637\pi$$
0.271027 + 0.962572i $$0.412637\pi$$
$$600$$ 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 $$-0.319002\pi$$
0.538471 + 0.842644i $$0.319002\pi$$
$$608$$ 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 $$-0.521336\pi$$
−0.0669786 + 0.997754i $$0.521336\pi$$
$$614$$ 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 $$-0.563461\pi$$
−0.198049 + 0.980192i $$0.563461\pi$$
$$632$$ 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 $$-0.394307\pi$$
0.325975 + 0.945378i $$0.394307\pi$$
$$648$$ 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 $$-0.753047\pi$$
−0.713843 + 0.700306i $$0.753047\pi$$
$$654$$ 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 $$-0.781752\pi$$
−0.774011 + 0.633173i $$0.781752\pi$$
$$662$$ 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 $$-0.375096\pi$$
0.382405 + 0.923995i $$0.375096\pi$$
$$678$$ 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 $$-0.419860\pi$$
0.249117 + 0.968473i $$0.419860\pi$$
$$710$$ 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 $$-0.519698\pi$$
−0.0618446 + 0.998086i $$0.519698\pi$$
$$720$$ 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 $$-0.400502\pi$$
0.307517 + 0.951543i $$0.400502\pi$$
$$728$$ 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 $$-0.926420\pi$$
−0.973401 + 0.229107i $$0.926420\pi$$
$$744$$ 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 $$-0.758693\pi$$
−0.726152 + 0.687534i $$0.758693\pi$$
$$752$$ 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 $$-0.442126\pi$$
0.180817 + 0.983517i $$0.442126\pi$$
$$758$$ 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 $$-0.461938\pi$$
0.119291 + 0.992859i $$0.461938\pi$$
$$774$$ 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 $$-0.655719\pi$$
−0.469924 + 0.882707i $$0.655719\pi$$
$$798$$ 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 $$-0.555549\pi$$
−0.173627 + 0.984812i $$0.555549\pi$$
$$822$$ 0 0
$$823$$ 29.8496 1.04049 0.520246 0.854016i $$-0.325840\pi$$
0.520246 + 0.854016i $$0.325840\pi$$
$$824$$ 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 $$-0.795974\pi$$
−0.801518 + 0.597970i $$0.795974\pi$$
$$840$$ 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 $$-0.862751\pi$$
−0.908473 + 0.417945i $$0.862751\pi$$
$$854$$ 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 $$-0.690929\pi$$
−0.564496 + 0.825436i $$0.690929\pi$$
$$864$$ 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 $$-0.464276\pi$$
0.111994 + 0.993709i $$0.464276\pi$$
$$878$$ 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 $$-0.608427\pi$$
−0.334084 + 0.942543i $$0.608427\pi$$
$$888$$ 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 $$-0.606932\pi$$
−0.329654 + 0.944102i $$0.606932\pi$$
$$912$$ 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 $$-0.315760\pi$$
0.547027 + 0.837115i $$0.315760\pi$$
$$920$$ 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 $$-0.773254\pi$$
−0.756832 + 0.653609i $$0.773254\pi$$
$$942$$ 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 $$-0.679040\pi$$
−0.533277 + 0.845940i $$0.679040\pi$$
$$968$$ 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 $$-0.432143\pi$$
0.211568 + 0.977363i $$0.432143\pi$$
$$984$$ 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 $$-0.676601\pi$$
−0.526780 + 0.850001i $$0.676601\pi$$
$$992$$ 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 $$-0.739216\pi$$
−0.682751 + 0.730651i $$0.739216\pi$$
$$998$$ 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.1.2 2
4.3 odd 2 4864.2.a.y.1.2 2
8.3 odd 2 inner 4864.2.a.r.1.1 2
8.5 even 2 4864.2.a.y.1.1 2
16.3 odd 4 1216.2.c.f.609.3 yes 4
16.5 even 4 1216.2.c.f.609.4 yes 4
16.11 odd 4 1216.2.c.f.609.2 yes 4
16.13 even 4 1216.2.c.f.609.1 4

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