# Properties

 Label 3920.2.a.bw.1.2 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +O(q^{10})$$ $$q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +5.82843 q^{11} -1.58579 q^{13} -2.41421 q^{15} +5.24264 q^{17} +6.00000 q^{19} -4.58579 q^{23} +1.00000 q^{25} -0.414214 q^{27} +2.65685 q^{29} +1.75736 q^{31} +14.0711 q^{33} -6.24264 q^{37} -3.82843 q^{39} -2.24264 q^{41} -2.00000 q^{43} -2.82843 q^{45} +1.24264 q^{47} +12.6569 q^{51} -4.24264 q^{53} -5.82843 q^{55} +14.4853 q^{57} +6.24264 q^{59} -2.82843 q^{61} +1.58579 q^{65} -0.242641 q^{67} -11.0711 q^{69} +8.82843 q^{71} -8.48528 q^{73} +2.41421 q^{75} +15.4853 q^{79} -9.48528 q^{81} -5.24264 q^{85} +6.41421 q^{87} +8.00000 q^{89} +4.24264 q^{93} -6.00000 q^{95} -4.75736 q^{97} +16.4853 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{11} - 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 12 q^{31} + 14 q^{33} - 4 q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} - 6 q^{47} + 14 q^{51} - 6 q^{55} + 12 q^{57} + 4 q^{59} + 6 q^{65} + 8 q^{67} - 8 q^{69} + 12 q^{71} + 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} + 10 q^{87} + 16 q^{89} - 12 q^{95} - 18 q^{97} + 16 q^{99} + O(q^{100})$$

## 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
$$3$$ 2.41421 1.39385 0.696923 0.717146i $$-0.254552\pi$$
0.696923 + 0.717146i $$0.254552\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.82843 0.942809
$$10$$ 0 0
$$11$$ 5.82843 1.75734 0.878668 0.477432i $$-0.158432\pi$$
0.878668 + 0.477432i $$0.158432\pi$$
$$12$$ 0 0
$$13$$ −1.58579 −0.439818 −0.219909 0.975520i $$-0.570576\pi$$
−0.219909 + 0.975520i $$0.570576\pi$$
$$14$$ 0 0
$$15$$ −2.41421 −0.623347
$$16$$ 0 0
$$17$$ 5.24264 1.27153 0.635764 0.771884i $$-0.280685\pi$$
0.635764 + 0.771884i $$0.280685\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.58579 −0.956203 −0.478101 0.878305i $$-0.658675\pi$$
−0.478101 + 0.878305i $$0.658675\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −0.414214 −0.0797154
$$28$$ 0 0
$$29$$ 2.65685 0.493365 0.246683 0.969096i $$-0.420659\pi$$
0.246683 + 0.969096i $$0.420659\pi$$
$$30$$ 0 0
$$31$$ 1.75736 0.315631 0.157816 0.987469i $$-0.449555\pi$$
0.157816 + 0.987469i $$0.449555\pi$$
$$32$$ 0 0
$$33$$ 14.0711 2.44946
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.24264 −1.02628 −0.513142 0.858304i $$-0.671519\pi$$
−0.513142 + 0.858304i $$0.671519\pi$$
$$38$$ 0 0
$$39$$ −3.82843 −0.613039
$$40$$ 0 0
$$41$$ −2.24264 −0.350242 −0.175121 0.984547i $$-0.556032\pi$$
−0.175121 + 0.984547i $$0.556032\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ −2.82843 −0.421637
$$46$$ 0 0
$$47$$ 1.24264 0.181258 0.0906289 0.995885i $$-0.471112\pi$$
0.0906289 + 0.995885i $$0.471112\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 12.6569 1.77231
$$52$$ 0 0
$$53$$ −4.24264 −0.582772 −0.291386 0.956606i $$-0.594116\pi$$
−0.291386 + 0.956606i $$0.594116\pi$$
$$54$$ 0 0
$$55$$ −5.82843 −0.785905
$$56$$ 0 0
$$57$$ 14.4853 1.91862
$$58$$ 0 0
$$59$$ 6.24264 0.812723 0.406361 0.913712i $$-0.366797\pi$$
0.406361 + 0.913712i $$0.366797\pi$$
$$60$$ 0 0
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.58579 0.196693
$$66$$ 0 0
$$67$$ −0.242641 −0.0296433 −0.0148216 0.999890i $$-0.504718\pi$$
−0.0148216 + 0.999890i $$0.504718\pi$$
$$68$$ 0 0
$$69$$ −11.0711 −1.33280
$$70$$ 0 0
$$71$$ 8.82843 1.04774 0.523871 0.851798i $$-0.324487\pi$$
0.523871 + 0.851798i $$0.324487\pi$$
$$72$$ 0 0
$$73$$ −8.48528 −0.993127 −0.496564 0.868000i $$-0.665405\pi$$
−0.496564 + 0.868000i $$0.665405\pi$$
$$74$$ 0 0
$$75$$ 2.41421 0.278769
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 15.4853 1.74223 0.871115 0.491079i $$-0.163397\pi$$
0.871115 + 0.491079i $$0.163397\pi$$
$$80$$ 0 0
$$81$$ −9.48528 −1.05392
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −5.24264 −0.568644
$$86$$ 0 0
$$87$$ 6.41421 0.687676
$$88$$ 0 0
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.24264 0.439941
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −4.75736 −0.483037 −0.241518 0.970396i $$-0.577645\pi$$
−0.241518 + 0.970396i $$0.577645\pi$$
$$98$$ 0 0
$$99$$ 16.4853 1.65683
$$100$$ 0 0
$$101$$ 14.4853 1.44134 0.720670 0.693279i $$-0.243834\pi$$
0.720670 + 0.693279i $$0.243834\pi$$
$$102$$ 0 0
$$103$$ 10.7574 1.05995 0.529977 0.848012i $$-0.322201\pi$$
0.529977 + 0.848012i $$0.322201\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.4853 −1.40035 −0.700173 0.713974i $$-0.746894\pi$$
−0.700173 + 0.713974i $$0.746894\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ −15.0711 −1.43048
$$112$$ 0 0
$$113$$ 1.07107 0.100758 0.0503788 0.998730i $$-0.483957\pi$$
0.0503788 + 0.998730i $$0.483957\pi$$
$$114$$ 0 0
$$115$$ 4.58579 0.427627
$$116$$ 0 0
$$117$$ −4.48528 −0.414664
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 22.9706 2.08823
$$122$$ 0 0
$$123$$ −5.41421 −0.488183
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 0.242641 0.0215309 0.0107654 0.999942i $$-0.496573\pi$$
0.0107654 + 0.999942i $$0.496573\pi$$
$$128$$ 0 0
$$129$$ −4.82843 −0.425119
$$130$$ 0 0
$$131$$ 3.75736 0.328282 0.164141 0.986437i $$-0.447515\pi$$
0.164141 + 0.986437i $$0.447515\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0.414214 0.0356498
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 16.2426 1.37768 0.688841 0.724912i $$-0.258120\pi$$
0.688841 + 0.724912i $$0.258120\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 0 0
$$143$$ −9.24264 −0.772908
$$144$$ 0 0
$$145$$ −2.65685 −0.220640
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.8284 −1.21479 −0.607396 0.794399i $$-0.707786\pi$$
−0.607396 + 0.794399i $$0.707786\pi$$
$$150$$ 0 0
$$151$$ −9.48528 −0.771901 −0.385951 0.922519i $$-0.626126\pi$$
−0.385951 + 0.922519i $$0.626126\pi$$
$$152$$ 0 0
$$153$$ 14.8284 1.19881
$$154$$ 0 0
$$155$$ −1.75736 −0.141154
$$156$$ 0 0
$$157$$ 20.8284 1.66229 0.831145 0.556056i $$-0.187686\pi$$
0.831145 + 0.556056i $$0.187686\pi$$
$$158$$ 0 0
$$159$$ −10.2426 −0.812294
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1.75736 0.137647 0.0688235 0.997629i $$-0.478075\pi$$
0.0688235 + 0.997629i $$0.478075\pi$$
$$164$$ 0 0
$$165$$ −14.0711 −1.09543
$$166$$ 0 0
$$167$$ 9.24264 0.715217 0.357609 0.933872i $$-0.383592\pi$$
0.357609 + 0.933872i $$0.383592\pi$$
$$168$$ 0 0
$$169$$ −10.4853 −0.806560
$$170$$ 0 0
$$171$$ 16.9706 1.29777
$$172$$ 0 0
$$173$$ −1.24264 −0.0944762 −0.0472381 0.998884i $$-0.515042\pi$$
−0.0472381 + 0.998884i $$0.515042\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 15.0711 1.13281
$$178$$ 0 0
$$179$$ −2.48528 −0.185759 −0.0928793 0.995677i $$-0.529607\pi$$
−0.0928793 + 0.995677i $$0.529607\pi$$
$$180$$ 0 0
$$181$$ −6.72792 −0.500083 −0.250041 0.968235i $$-0.580444\pi$$
−0.250041 + 0.968235i $$0.580444\pi$$
$$182$$ 0 0
$$183$$ −6.82843 −0.504772
$$184$$ 0 0
$$185$$ 6.24264 0.458968
$$186$$ 0 0
$$187$$ 30.5563 2.23450
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.9706 1.44502 0.722510 0.691361i $$-0.242989\pi$$
0.722510 + 0.691361i $$0.242989\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 0 0
$$195$$ 3.82843 0.274159
$$196$$ 0 0
$$197$$ 10.5858 0.754206 0.377103 0.926171i $$-0.376920\pi$$
0.377103 + 0.926171i $$0.376920\pi$$
$$198$$ 0 0
$$199$$ −4.58579 −0.325078 −0.162539 0.986702i $$-0.551968\pi$$
−0.162539 + 0.986702i $$0.551968\pi$$
$$200$$ 0 0
$$201$$ −0.585786 −0.0413182
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.24264 0.156633
$$206$$ 0 0
$$207$$ −12.9706 −0.901516
$$208$$ 0 0
$$209$$ 34.9706 2.41896
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 0 0
$$213$$ 21.3137 1.46039
$$214$$ 0 0
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −20.4853 −1.38427
$$220$$ 0 0
$$221$$ −8.31371 −0.559241
$$222$$ 0 0
$$223$$ 18.2132 1.21965 0.609823 0.792538i $$-0.291240\pi$$
0.609823 + 0.792538i $$0.291240\pi$$
$$224$$ 0 0
$$225$$ 2.82843 0.188562
$$226$$ 0 0
$$227$$ −9.72792 −0.645665 −0.322832 0.946456i $$-0.604635\pi$$
−0.322832 + 0.946456i $$0.604635\pi$$
$$228$$ 0 0
$$229$$ 30.0416 1.98521 0.992603 0.121402i $$-0.0387390\pi$$
0.992603 + 0.121402i $$0.0387390\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.8284 −0.971443 −0.485721 0.874114i $$-0.661443\pi$$
−0.485721 + 0.874114i $$0.661443\pi$$
$$234$$ 0 0
$$235$$ −1.24264 −0.0810609
$$236$$ 0 0
$$237$$ 37.3848 2.42840
$$238$$ 0 0
$$239$$ 0.514719 0.0332944 0.0166472 0.999861i $$-0.494701\pi$$
0.0166472 + 0.999861i $$0.494701\pi$$
$$240$$ 0 0
$$241$$ −24.7279 −1.59287 −0.796433 0.604727i $$-0.793282\pi$$
−0.796433 + 0.604727i $$0.793282\pi$$
$$242$$ 0 0
$$243$$ −21.6569 −1.38929
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.51472 −0.605407
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −25.2132 −1.59144 −0.795722 0.605663i $$-0.792908\pi$$
−0.795722 + 0.605663i $$0.792908\pi$$
$$252$$ 0 0
$$253$$ −26.7279 −1.68037
$$254$$ 0 0
$$255$$ −12.6569 −0.792603
$$256$$ 0 0
$$257$$ 26.4853 1.65211 0.826053 0.563592i $$-0.190581\pi$$
0.826053 + 0.563592i $$0.190581\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 7.51472 0.465149
$$262$$ 0 0
$$263$$ −28.6274 −1.76524 −0.882621 0.470085i $$-0.844223\pi$$
−0.882621 + 0.470085i $$0.844223\pi$$
$$264$$ 0 0
$$265$$ 4.24264 0.260623
$$266$$ 0 0
$$267$$ 19.3137 1.18198
$$268$$ 0 0
$$269$$ 7.75736 0.472975 0.236487 0.971635i $$-0.424004\pi$$
0.236487 + 0.971635i $$0.424004\pi$$
$$270$$ 0 0
$$271$$ −23.3137 −1.41621 −0.708103 0.706109i $$-0.750449\pi$$
−0.708103 + 0.706109i $$0.750449\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.82843 0.351467
$$276$$ 0 0
$$277$$ −27.2132 −1.63508 −0.817541 0.575870i $$-0.804664\pi$$
−0.817541 + 0.575870i $$0.804664\pi$$
$$278$$ 0 0
$$279$$ 4.97056 0.297580
$$280$$ 0 0
$$281$$ −20.3137 −1.21181 −0.605907 0.795535i $$-0.707190\pi$$
−0.605907 + 0.795535i $$0.707190\pi$$
$$282$$ 0 0
$$283$$ −6.55635 −0.389735 −0.194867 0.980830i $$-0.562428\pi$$
−0.194867 + 0.980830i $$0.562428\pi$$
$$284$$ 0 0
$$285$$ −14.4853 −0.858034
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 10.4853 0.616781
$$290$$ 0 0
$$291$$ −11.4853 −0.673279
$$292$$ 0 0
$$293$$ −0.272078 −0.0158950 −0.00794748 0.999968i $$-0.502530\pi$$
−0.00794748 + 0.999968i $$0.502530\pi$$
$$294$$ 0 0
$$295$$ −6.24264 −0.363461
$$296$$ 0 0
$$297$$ −2.41421 −0.140087
$$298$$ 0 0
$$299$$ 7.27208 0.420555
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 34.9706 2.00901
$$304$$ 0 0
$$305$$ 2.82843 0.161955
$$306$$ 0 0
$$307$$ 11.1005 0.633539 0.316770 0.948503i $$-0.397402\pi$$
0.316770 + 0.948503i $$0.397402\pi$$
$$308$$ 0 0
$$309$$ 25.9706 1.47741
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ 0 0
$$313$$ −24.2132 −1.36861 −0.684306 0.729195i $$-0.739895\pi$$
−0.684306 + 0.729195i $$0.739895\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −11.6569 −0.654714 −0.327357 0.944901i $$-0.606158\pi$$
−0.327357 + 0.944901i $$0.606158\pi$$
$$318$$ 0 0
$$319$$ 15.4853 0.867009
$$320$$ 0 0
$$321$$ −34.9706 −1.95187
$$322$$ 0 0
$$323$$ 31.4558 1.75025
$$324$$ 0 0
$$325$$ −1.58579 −0.0879636
$$326$$ 0 0
$$327$$ 12.0711 0.667532
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −23.4558 −1.28925 −0.644625 0.764499i $$-0.722986\pi$$
−0.644625 + 0.764499i $$0.722986\pi$$
$$332$$ 0 0
$$333$$ −17.6569 −0.967590
$$334$$ 0 0
$$335$$ 0.242641 0.0132569
$$336$$ 0 0
$$337$$ 13.7574 0.749411 0.374706 0.927144i $$-0.377744\pi$$
0.374706 + 0.927144i $$0.377744\pi$$
$$338$$ 0 0
$$339$$ 2.58579 0.140441
$$340$$ 0 0
$$341$$ 10.2426 0.554670
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 11.0711 0.596046
$$346$$ 0 0
$$347$$ 1.07107 0.0574979 0.0287490 0.999587i $$-0.490848\pi$$
0.0287490 + 0.999587i $$0.490848\pi$$
$$348$$ 0 0
$$349$$ −22.9706 −1.22959 −0.614793 0.788688i $$-0.710760\pi$$
−0.614793 + 0.788688i $$0.710760\pi$$
$$350$$ 0 0
$$351$$ 0.656854 0.0350603
$$352$$ 0 0
$$353$$ −36.2132 −1.92743 −0.963717 0.266925i $$-0.913992\pi$$
−0.963717 + 0.266925i $$0.913992\pi$$
$$354$$ 0 0
$$355$$ −8.82843 −0.468564
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11.3137 −0.597115 −0.298557 0.954392i $$-0.596505\pi$$
−0.298557 + 0.954392i $$0.596505\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 55.4558 2.91068
$$364$$ 0 0
$$365$$ 8.48528 0.444140
$$366$$ 0 0
$$367$$ 29.8701 1.55920 0.779602 0.626275i $$-0.215421\pi$$
0.779602 + 0.626275i $$0.215421\pi$$
$$368$$ 0 0
$$369$$ −6.34315 −0.330211
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0.485281 0.0251269 0.0125635 0.999921i $$-0.496001\pi$$
0.0125635 + 0.999921i $$0.496001\pi$$
$$374$$ 0 0
$$375$$ −2.41421 −0.124669
$$376$$ 0 0
$$377$$ −4.21320 −0.216991
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ 0.585786 0.0300107
$$382$$ 0 0
$$383$$ −12.4853 −0.637968 −0.318984 0.947760i $$-0.603342\pi$$
−0.318984 + 0.947760i $$0.603342\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −5.65685 −0.287554
$$388$$ 0 0
$$389$$ −35.1421 −1.78178 −0.890889 0.454222i $$-0.849917\pi$$
−0.890889 + 0.454222i $$0.849917\pi$$
$$390$$ 0 0
$$391$$ −24.0416 −1.21584
$$392$$ 0 0
$$393$$ 9.07107 0.457575
$$394$$ 0 0
$$395$$ −15.4853 −0.779149
$$396$$ 0 0
$$397$$ −4.41421 −0.221543 −0.110772 0.993846i $$-0.535332\pi$$
−0.110772 + 0.993846i $$0.535332\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.17157 0.308194 0.154097 0.988056i $$-0.450753\pi$$
0.154097 + 0.988056i $$0.450753\pi$$
$$402$$ 0 0
$$403$$ −2.78680 −0.138820
$$404$$ 0 0
$$405$$ 9.48528 0.471327
$$406$$ 0 0
$$407$$ −36.3848 −1.80353
$$408$$ 0 0
$$409$$ 14.4853 0.716251 0.358126 0.933673i $$-0.383416\pi$$
0.358126 + 0.933673i $$0.383416\pi$$
$$410$$ 0 0
$$411$$ 28.9706 1.42901
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 39.2132 1.92028
$$418$$ 0 0
$$419$$ −6.72792 −0.328681 −0.164340 0.986404i $$-0.552550\pi$$
−0.164340 + 0.986404i $$0.552550\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 0 0
$$423$$ 3.51472 0.170891
$$424$$ 0 0
$$425$$ 5.24264 0.254305
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −22.3137 −1.07732
$$430$$ 0 0
$$431$$ −10.7990 −0.520169 −0.260085 0.965586i $$-0.583750\pi$$
−0.260085 + 0.965586i $$0.583750\pi$$
$$432$$ 0 0
$$433$$ 22.9706 1.10389 0.551947 0.833879i $$-0.313885\pi$$
0.551947 + 0.833879i $$0.313885\pi$$
$$434$$ 0 0
$$435$$ −6.41421 −0.307538
$$436$$ 0 0
$$437$$ −27.5147 −1.31621
$$438$$ 0 0
$$439$$ 6.38478 0.304729 0.152364 0.988324i $$-0.451311\pi$$
0.152364 + 0.988324i $$0.451311\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −20.8284 −0.989588 −0.494794 0.869010i $$-0.664757\pi$$
−0.494794 + 0.869010i $$0.664757\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ 0 0
$$447$$ −35.7990 −1.69323
$$448$$ 0 0
$$449$$ −29.8284 −1.40769 −0.703845 0.710353i $$-0.748535\pi$$
−0.703845 + 0.710353i $$0.748535\pi$$
$$450$$ 0 0
$$451$$ −13.0711 −0.615493
$$452$$ 0 0
$$453$$ −22.8995 −1.07591
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.2426 −0.946911 −0.473455 0.880818i $$-0.656993\pi$$
−0.473455 + 0.880818i $$0.656993\pi$$
$$458$$ 0 0
$$459$$ −2.17157 −0.101360
$$460$$ 0 0
$$461$$ −36.9706 −1.72189 −0.860945 0.508697i $$-0.830127\pi$$
−0.860945 + 0.508697i $$0.830127\pi$$
$$462$$ 0 0
$$463$$ −29.4558 −1.36893 −0.684465 0.729046i $$-0.739964\pi$$
−0.684465 + 0.729046i $$0.739964\pi$$
$$464$$ 0 0
$$465$$ −4.24264 −0.196748
$$466$$ 0 0
$$467$$ 19.7279 0.912899 0.456450 0.889749i $$-0.349121\pi$$
0.456450 + 0.889749i $$0.349121\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 50.2843 2.31698
$$472$$ 0 0
$$473$$ −11.6569 −0.535983
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ 20.2426 0.924910 0.462455 0.886643i $$-0.346969\pi$$
0.462455 + 0.886643i $$0.346969\pi$$
$$480$$ 0 0
$$481$$ 9.89949 0.451378
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.75736 0.216021
$$486$$ 0 0
$$487$$ 31.6985 1.43640 0.718198 0.695839i $$-0.244967\pi$$
0.718198 + 0.695839i $$0.244967\pi$$
$$488$$ 0 0
$$489$$ 4.24264 0.191859
$$490$$ 0 0
$$491$$ −19.2843 −0.870287 −0.435143 0.900361i $$-0.643302\pi$$
−0.435143 + 0.900361i $$0.643302\pi$$
$$492$$ 0 0
$$493$$ 13.9289 0.627328
$$494$$ 0 0
$$495$$ −16.4853 −0.740958
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3.00000 −0.134298 −0.0671492 0.997743i $$-0.521390\pi$$
−0.0671492 + 0.997743i $$0.521390\pi$$
$$500$$ 0 0
$$501$$ 22.3137 0.996903
$$502$$ 0 0
$$503$$ −32.7574 −1.46058 −0.730289 0.683138i $$-0.760615\pi$$
−0.730289 + 0.683138i $$0.760615\pi$$
$$504$$ 0 0
$$505$$ −14.4853 −0.644587
$$506$$ 0 0
$$507$$ −25.3137 −1.12422
$$508$$ 0 0
$$509$$ −17.2132 −0.762962 −0.381481 0.924377i $$-0.624586\pi$$
−0.381481 + 0.924377i $$0.624586\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.48528 −0.109728
$$514$$ 0 0
$$515$$ −10.7574 −0.474026
$$516$$ 0 0
$$517$$ 7.24264 0.318531
$$518$$ 0 0
$$519$$ −3.00000 −0.131685
$$520$$ 0 0
$$521$$ 18.9706 0.831115 0.415558 0.909567i $$-0.363586\pi$$
0.415558 + 0.909567i $$0.363586\pi$$
$$522$$ 0 0
$$523$$ 15.5147 0.678411 0.339206 0.940712i $$-0.389842\pi$$
0.339206 + 0.940712i $$0.389842\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.21320 0.401333
$$528$$ 0 0
$$529$$ −1.97056 −0.0856766
$$530$$ 0 0
$$531$$ 17.6569 0.766242
$$532$$ 0 0
$$533$$ 3.55635 0.154043
$$534$$ 0 0
$$535$$ 14.4853 0.626253
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11.9706 0.514655 0.257327 0.966324i $$-0.417158\pi$$
0.257327 + 0.966324i $$0.417158\pi$$
$$542$$ 0 0
$$543$$ −16.2426 −0.697038
$$544$$ 0 0
$$545$$ −5.00000 −0.214176
$$546$$ 0 0
$$547$$ 7.51472 0.321306 0.160653 0.987011i $$-0.448640\pi$$
0.160653 + 0.987011i $$0.448640\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 15.9411 0.679115
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 15.0711 0.639731
$$556$$ 0 0
$$557$$ 31.7990 1.34737 0.673683 0.739020i $$-0.264711\pi$$
0.673683 + 0.739020i $$0.264711\pi$$
$$558$$ 0 0
$$559$$ 3.17157 0.134143
$$560$$ 0 0
$$561$$ 73.7696 3.11455
$$562$$ 0 0
$$563$$ 35.9411 1.51474 0.757369 0.652987i $$-0.226484\pi$$
0.757369 + 0.652987i $$0.226484\pi$$
$$564$$ 0 0
$$565$$ −1.07107 −0.0450602
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −2.14214 −0.0898030 −0.0449015 0.998991i $$-0.514297\pi$$
−0.0449015 + 0.998991i $$0.514297\pi$$
$$570$$ 0 0
$$571$$ 34.4853 1.44316 0.721582 0.692329i $$-0.243415\pi$$
0.721582 + 0.692329i $$0.243415\pi$$
$$572$$ 0 0
$$573$$ 48.2132 2.01414
$$574$$ 0 0
$$575$$ −4.58579 −0.191241
$$576$$ 0 0
$$577$$ 9.72792 0.404979 0.202489 0.979284i $$-0.435097\pi$$
0.202489 + 0.979284i $$0.435097\pi$$
$$578$$ 0 0
$$579$$ −38.6274 −1.60530
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −24.7279 −1.02413
$$584$$ 0 0
$$585$$ 4.48528 0.185444
$$586$$ 0 0
$$587$$ 13.4558 0.555382 0.277691 0.960670i $$-0.410431\pi$$
0.277691 + 0.960670i $$0.410431\pi$$
$$588$$ 0 0
$$589$$ 10.5442 0.434464
$$590$$ 0 0
$$591$$ 25.5563 1.05125
$$592$$ 0 0
$$593$$ −10.7574 −0.441752 −0.220876 0.975302i $$-0.570892\pi$$
−0.220876 + 0.975302i $$0.570892\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −11.0711 −0.453109
$$598$$ 0 0
$$599$$ 12.1716 0.497317 0.248658 0.968591i $$-0.420010\pi$$
0.248658 + 0.968591i $$0.420010\pi$$
$$600$$ 0 0
$$601$$ −22.9706 −0.936989 −0.468494 0.883466i $$-0.655203\pi$$
−0.468494 + 0.883466i $$0.655203\pi$$
$$602$$ 0 0
$$603$$ −0.686292 −0.0279480
$$604$$ 0 0
$$605$$ −22.9706 −0.933886
$$606$$ 0 0
$$607$$ −24.8995 −1.01064 −0.505320 0.862932i $$-0.668625\pi$$
−0.505320 + 0.862932i $$0.668625\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.97056 −0.0797204
$$612$$ 0 0
$$613$$ 43.9411 1.77477 0.887383 0.461034i $$-0.152521\pi$$
0.887383 + 0.461034i $$0.152521\pi$$
$$614$$ 0 0
$$615$$ 5.41421 0.218322
$$616$$ 0 0
$$617$$ 7.41421 0.298485 0.149242 0.988801i $$-0.452316\pi$$
0.149242 + 0.988801i $$0.452316\pi$$
$$618$$ 0 0
$$619$$ 31.0711 1.24885 0.624426 0.781084i $$-0.285333\pi$$
0.624426 + 0.781084i $$0.285333\pi$$
$$620$$ 0 0
$$621$$ 1.89949 0.0762241
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 84.4264 3.37167
$$628$$ 0 0
$$629$$ −32.7279 −1.30495
$$630$$ 0 0
$$631$$ −8.45584 −0.336622 −0.168311 0.985734i $$-0.553831\pi$$
−0.168311 + 0.985734i $$0.553831\pi$$
$$632$$ 0 0
$$633$$ −21.7279 −0.863607
$$634$$ 0 0
$$635$$ −0.242641 −0.00962890
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 24.9706 0.987820
$$640$$ 0 0
$$641$$ −0.686292 −0.0271069 −0.0135534 0.999908i $$-0.504314\pi$$
−0.0135534 + 0.999908i $$0.504314\pi$$
$$642$$ 0 0
$$643$$ 27.7279 1.09348 0.546741 0.837302i $$-0.315868\pi$$
0.546741 + 0.837302i $$0.315868\pi$$
$$644$$ 0 0
$$645$$ 4.82843 0.190119
$$646$$ 0 0
$$647$$ −28.4853 −1.11987 −0.559936 0.828536i $$-0.689174\pi$$
−0.559936 + 0.828536i $$0.689174\pi$$
$$648$$ 0 0
$$649$$ 36.3848 1.42823
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.02944 0.0402850 0.0201425 0.999797i $$-0.493588\pi$$
0.0201425 + 0.999797i $$0.493588\pi$$
$$654$$ 0 0
$$655$$ −3.75736 −0.146812
$$656$$ 0 0
$$657$$ −24.0000 −0.936329
$$658$$ 0 0
$$659$$ 13.9706 0.544216 0.272108 0.962267i $$-0.412279\pi$$
0.272108 + 0.962267i $$0.412279\pi$$
$$660$$ 0 0
$$661$$ 37.4558 1.45686 0.728432 0.685118i $$-0.240250\pi$$
0.728432 + 0.685118i $$0.240250\pi$$
$$662$$ 0 0
$$663$$ −20.0711 −0.779496
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.1838 −0.471757
$$668$$ 0 0
$$669$$ 43.9706 1.70000
$$670$$ 0 0
$$671$$ −16.4853 −0.636407
$$672$$ 0 0
$$673$$ 20.4853 0.789650 0.394825 0.918756i $$-0.370805\pi$$
0.394825 + 0.918756i $$0.370805\pi$$
$$674$$ 0 0
$$675$$ −0.414214 −0.0159431
$$676$$ 0 0
$$677$$ −1.78680 −0.0686722 −0.0343361 0.999410i $$-0.510932\pi$$
−0.0343361 + 0.999410i $$0.510932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −23.4853 −0.899958
$$682$$ 0 0
$$683$$ 7.79899 0.298420 0.149210 0.988806i $$-0.452327\pi$$
0.149210 + 0.988806i $$0.452327\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ 72.5269 2.76707
$$688$$ 0 0
$$689$$ 6.72792 0.256313
$$690$$ 0 0
$$691$$ 9.17157 0.348903 0.174452 0.984666i $$-0.444185\pi$$
0.174452 + 0.984666i $$0.444185\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −16.2426 −0.616118
$$696$$ 0 0
$$697$$ −11.7574 −0.445342
$$698$$ 0 0
$$699$$ −35.7990 −1.35404
$$700$$ 0 0
$$701$$ −4.45584 −0.168295 −0.0841475 0.996453i $$-0.526817\pi$$
−0.0841475 + 0.996453i $$0.526817\pi$$
$$702$$ 0 0
$$703$$ −37.4558 −1.41267
$$704$$ 0 0
$$705$$ −3.00000 −0.112987
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 17.0000 0.638448 0.319224 0.947679i $$-0.396578\pi$$
0.319224 + 0.947679i $$0.396578\pi$$
$$710$$ 0 0
$$711$$ 43.7990 1.64259
$$712$$ 0 0
$$713$$ −8.05887 −0.301807
$$714$$ 0 0
$$715$$ 9.24264 0.345655
$$716$$ 0 0
$$717$$ 1.24264 0.0464073
$$718$$ 0 0
$$719$$ 17.2132 0.641944 0.320972 0.947089i $$-0.395990\pi$$
0.320972 + 0.947089i $$0.395990\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −59.6985 −2.22021
$$724$$ 0 0
$$725$$ 2.65685 0.0986731
$$726$$ 0 0
$$727$$ −29.3137 −1.08719 −0.543593 0.839349i $$-0.682936\pi$$
−0.543593 + 0.839349i $$0.682936\pi$$
$$728$$ 0 0
$$729$$ −23.8284 −0.882534
$$730$$ 0 0
$$731$$ −10.4853 −0.387812
$$732$$ 0 0
$$733$$ 44.6985 1.65098 0.825488 0.564420i $$-0.190900\pi$$
0.825488 + 0.564420i $$0.190900\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.41421 −0.0520932
$$738$$ 0 0
$$739$$ 3.97056 0.146060 0.0730298 0.997330i $$-0.476733\pi$$
0.0730298 + 0.997330i $$0.476733\pi$$
$$740$$ 0 0
$$741$$ −22.9706 −0.843845
$$742$$ 0 0
$$743$$ −36.7279 −1.34742 −0.673708 0.738997i $$-0.735300\pi$$
−0.673708 + 0.738997i $$0.735300\pi$$
$$744$$ 0 0
$$745$$ 14.8284 0.543272
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.5147 −0.456669 −0.228334 0.973583i $$-0.573328\pi$$
−0.228334 + 0.973583i $$0.573328\pi$$
$$752$$ 0 0
$$753$$ −60.8701 −2.21823
$$754$$ 0 0
$$755$$ 9.48528 0.345205
$$756$$ 0 0
$$757$$ −16.4853 −0.599168 −0.299584 0.954070i $$-0.596848\pi$$
−0.299584 + 0.954070i $$0.596848\pi$$
$$758$$ 0 0
$$759$$ −64.5269 −2.34218
$$760$$ 0 0
$$761$$ −12.7279 −0.461387 −0.230693 0.973026i $$-0.574099\pi$$
−0.230693 + 0.973026i $$0.574099\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −14.8284 −0.536123
$$766$$ 0 0
$$767$$ −9.89949 −0.357450
$$768$$ 0 0
$$769$$ 3.17157 0.114370 0.0571849 0.998364i $$-0.481788\pi$$
0.0571849 + 0.998364i $$0.481788\pi$$
$$770$$ 0 0
$$771$$ 63.9411 2.30278
$$772$$ 0 0
$$773$$ −36.2132 −1.30250 −0.651249 0.758864i $$-0.725755\pi$$
−0.651249 + 0.758864i $$0.725755\pi$$
$$774$$ 0 0
$$775$$ 1.75736 0.0631262
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −13.4558 −0.482106
$$780$$ 0 0
$$781$$ 51.4558 1.84123
$$782$$ 0 0
$$783$$ −1.10051 −0.0393288
$$784$$ 0 0
$$785$$ −20.8284 −0.743398
$$786$$ 0 0
$$787$$ 45.7279 1.63002 0.815012 0.579444i $$-0.196730\pi$$
0.815012 + 0.579444i $$0.196730\pi$$
$$788$$ 0 0
$$789$$ −69.1127 −2.46048
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4.48528 0.159277
$$794$$ 0 0
$$795$$ 10.2426 0.363269
$$796$$ 0 0
$$797$$ −55.1838 −1.95471 −0.977355 0.211608i $$-0.932130\pi$$
−0.977355 + 0.211608i $$0.932130\pi$$
$$798$$ 0 0
$$799$$ 6.51472 0.230474
$$800$$ 0 0
$$801$$ 22.6274 0.799500
$$802$$ 0 0
$$803$$ −49.4558 −1.74526
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.7279 0.659254
$$808$$ 0 0
$$809$$ 30.5980 1.07577 0.537884 0.843019i $$-0.319224\pi$$
0.537884 + 0.843019i $$0.319224\pi$$
$$810$$ 0 0
$$811$$ −23.3553 −0.820117 −0.410058 0.912059i $$-0.634492\pi$$
−0.410058 + 0.912059i $$0.634492\pi$$
$$812$$ 0 0
$$813$$ −56.2843 −1.97398
$$814$$ 0 0
$$815$$ −1.75736 −0.0615576
$$816$$ 0 0
$$817$$ −12.0000 −0.419827
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.51472 −0.227365 −0.113683 0.993517i $$-0.536265\pi$$
−0.113683 + 0.993517i $$0.536265\pi$$
$$822$$ 0 0
$$823$$ 8.72792 0.304236 0.152118 0.988362i $$-0.451391\pi$$
0.152118 + 0.988362i $$0.451391\pi$$
$$824$$ 0 0
$$825$$ 14.0711 0.489892
$$826$$ 0 0
$$827$$ 6.04163 0.210088 0.105044 0.994468i $$-0.466502\pi$$
0.105044 + 0.994468i $$0.466502\pi$$
$$828$$ 0 0
$$829$$ −54.0416 −1.87694 −0.938472 0.345356i $$-0.887758\pi$$
−0.938472 + 0.345356i $$0.887758\pi$$
$$830$$ 0 0
$$831$$ −65.6985 −2.27906
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −9.24264 −0.319855
$$836$$ 0 0
$$837$$ −0.727922 −0.0251607
$$838$$ 0 0
$$839$$ −48.7279 −1.68227 −0.841137 0.540822i $$-0.818113\pi$$
−0.841137 + 0.540822i $$0.818113\pi$$
$$840$$ 0 0
$$841$$ −21.9411 −0.756591
$$842$$ 0 0
$$843$$ −49.0416 −1.68908
$$844$$ 0 0
$$845$$ 10.4853 0.360705
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −15.8284 −0.543230
$$850$$ 0 0
$$851$$ 28.6274 0.981335
$$852$$ 0 0
$$853$$ 22.9706 0.786497 0.393249 0.919432i $$-0.371351\pi$$
0.393249 + 0.919432i $$0.371351\pi$$
$$854$$ 0 0
$$855$$ −16.9706 −0.580381
$$856$$ 0 0
$$857$$ 5.51472 0.188379 0.0941896 0.995554i $$-0.469974\pi$$
0.0941896 + 0.995554i $$0.469974\pi$$
$$858$$ 0 0
$$859$$ 48.7696 1.66400 0.831998 0.554779i $$-0.187197\pi$$
0.831998 + 0.554779i $$0.187197\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 18.3848 0.625825 0.312913 0.949782i $$-0.398695\pi$$
0.312913 + 0.949782i $$0.398695\pi$$
$$864$$ 0 0
$$865$$ 1.24264 0.0422511
$$866$$ 0 0
$$867$$ 25.3137 0.859699
$$868$$ 0 0
$$869$$ 90.2548 3.06169
$$870$$ 0 0
$$871$$ 0.384776 0.0130376
$$872$$ 0 0
$$873$$ −13.4558 −0.455411
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.9706 1.58608 0.793042 0.609167i $$-0.208496\pi$$
0.793042 + 0.609167i $$0.208496\pi$$
$$878$$ 0 0
$$879$$ −0.656854 −0.0221551
$$880$$ 0 0
$$881$$ 19.0294 0.641118 0.320559 0.947229i $$-0.396129\pi$$
0.320559 + 0.947229i $$0.396129\pi$$
$$882$$ 0 0
$$883$$ −48.4853 −1.63166 −0.815830 0.578292i $$-0.803719\pi$$
−0.815830 + 0.578292i $$0.803719\pi$$
$$884$$ 0 0
$$885$$ −15.0711 −0.506608
$$886$$ 0 0
$$887$$ −30.9706 −1.03989 −0.519945 0.854200i $$-0.674048\pi$$
−0.519945 + 0.854200i $$0.674048\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −55.2843 −1.85209
$$892$$ 0 0
$$893$$ 7.45584 0.249500
$$894$$ 0 0
$$895$$ 2.48528 0.0830738
$$896$$ 0 0
$$897$$ 17.5563 0.586189
$$898$$ 0 0
$$899$$ 4.66905 0.155721
$$900$$ 0 0
$$901$$ −22.2426 −0.741010
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.72792 0.223644
$$906$$ 0 0
$$907$$ −32.1838 −1.06864 −0.534322 0.845281i $$-0.679433\pi$$
−0.534322 + 0.845281i $$0.679433\pi$$
$$908$$ 0 0
$$909$$ 40.9706 1.35891
$$910$$ 0 0
$$911$$ 5.65685 0.187420 0.0937100 0.995600i $$-0.470127\pi$$
0.0937100 + 0.995600i $$0.470127\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 6.82843 0.225741
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38.4558 1.26854 0.634271 0.773111i $$-0.281301\pi$$
0.634271 + 0.773111i $$0.281301\pi$$
$$920$$ 0 0
$$921$$ 26.7990 0.883057
$$922$$ 0 0
$$923$$ −14.0000 −0.460816
$$924$$ 0 0
$$925$$ −6.24264 −0.205257
$$926$$ 0 0
$$927$$ 30.4264 0.999334
$$928$$ 0 0
$$929$$ −54.7279 −1.79556 −0.897782 0.440439i $$-0.854823\pi$$
−0.897782 + 0.440439i $$0.854823\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −24.1421 −0.790378
$$934$$ 0 0
$$935$$ −30.5563 −0.999299
$$936$$ 0 0
$$937$$ 17.4437 0.569859 0.284930 0.958548i $$-0.408030\pi$$
0.284930 + 0.958548i $$0.408030\pi$$
$$938$$ 0 0
$$939$$ −58.4558 −1.90763
$$940$$ 0 0
$$941$$ 4.97056 0.162036 0.0810179 0.996713i $$-0.474183\pi$$
0.0810179 + 0.996713i $$0.474183\pi$$
$$942$$ 0 0
$$943$$ 10.2843 0.334902
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 43.7574 1.42192 0.710962 0.703231i $$-0.248260\pi$$
0.710962 + 0.703231i $$0.248260\pi$$
$$948$$ 0 0
$$949$$ 13.4558 0.436795
$$950$$ 0 0
$$951$$ −28.1421 −0.912571
$$952$$ 0 0
$$953$$ −29.0122 −0.939797 −0.469899 0.882720i $$-0.655710\pi$$
−0.469899 + 0.882720i $$0.655710\pi$$
$$954$$ 0 0
$$955$$ −19.9706 −0.646232
$$956$$ 0 0
$$957$$ 37.3848 1.20848
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.9117 −0.900377
$$962$$ 0 0
$$963$$ −40.9706 −1.32026
$$964$$ 0 0
$$965$$ 16.0000 0.515058
$$966$$ 0 0
$$967$$ −24.4264 −0.785500 −0.392750 0.919645i $$-0.628476\pi$$
−0.392750 + 0.919645i $$0.628476\pi$$
$$968$$ 0 0
$$969$$ 75.9411 2.43958
$$970$$ 0 0
$$971$$ −42.7279 −1.37120 −0.685602 0.727976i $$-0.740461\pi$$
−0.685602 + 0.727976i $$0.740461\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −3.82843 −0.122608
$$976$$ 0 0
$$977$$ −30.7696 −0.984405 −0.492203 0.870481i $$-0.663808\pi$$
−0.492203 + 0.870481i $$0.663808\pi$$
$$978$$ 0 0
$$979$$ 46.6274 1.49022
$$980$$ 0 0
$$981$$ 14.1421 0.451524
$$982$$ 0 0
$$983$$ −42.2132 −1.34639 −0.673196 0.739464i $$-0.735079\pi$$
−0.673196 + 0.739464i $$0.735079\pi$$
$$984$$ 0 0
$$985$$ −10.5858 −0.337291
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.17157 0.291639
$$990$$ 0 0
$$991$$ 47.9411 1.52290 0.761450 0.648224i $$-0.224488\pi$$
0.761450 + 0.648224i $$0.224488\pi$$
$$992$$ 0 0
$$993$$ −56.6274 −1.79702
$$994$$ 0 0
$$995$$ 4.58579 0.145379
$$996$$ 0 0
$$997$$ −3.72792 −0.118064 −0.0590322 0.998256i $$-0.518801\pi$$
−0.0590322 + 0.998256i $$0.518801\pi$$
$$998$$ 0 0
$$999$$ 2.58579 0.0818107
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 3920.2.a.bw.1.2 2
4.3 odd 2 245.2.a.e.1.2 2
7.6 odd 2 3920.2.a.br.1.1 2
12.11 even 2 2205.2.a.v.1.1 2
20.3 even 4 1225.2.b.i.99.1 4
20.7 even 4 1225.2.b.i.99.4 4
20.19 odd 2 1225.2.a.r.1.1 2
28.3 even 6 245.2.e.f.226.1 4
28.11 odd 6 245.2.e.g.226.1 4
28.19 even 6 245.2.e.f.116.1 4
28.23 odd 6 245.2.e.g.116.1 4
28.27 even 2 245.2.a.f.1.2 yes 2
84.83 odd 2 2205.2.a.t.1.1 2
140.27 odd 4 1225.2.b.j.99.3 4
140.83 odd 4 1225.2.b.j.99.2 4
140.139 even 2 1225.2.a.p.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.2 2 4.3 odd 2
245.2.a.f.1.2 yes 2 28.27 even 2
245.2.e.f.116.1 4 28.19 even 6
245.2.e.f.226.1 4 28.3 even 6
245.2.e.g.116.1 4 28.23 odd 6
245.2.e.g.226.1 4 28.11 odd 6
1225.2.a.p.1.1 2 140.139 even 2
1225.2.a.r.1.1 2 20.19 odd 2
1225.2.b.i.99.1 4 20.3 even 4
1225.2.b.i.99.4 4 20.7 even 4
1225.2.b.j.99.2 4 140.83 odd 4
1225.2.b.j.99.3 4 140.27 odd 4
2205.2.a.t.1.1 2 84.83 odd 2
2205.2.a.v.1.1 2 12.11 even 2
3920.2.a.br.1.1 2 7.6 odd 2
3920.2.a.bw.1.2 2 1.1 even 1 trivial