# Properties

 Label 7800.2.a.bh.1.3 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3732.1 Defining polynomial: $$x^{3} - x^{2} - 13 x + 19$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.56943$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +4.96747 q^{7} +1.00000 q^{9} +0.430572 q^{11} +1.00000 q^{13} -3.00000 q^{17} +3.39804 q^{19} -4.96747 q^{21} +5.39804 q^{23} -1.00000 q^{27} -4.13886 q^{29} +0.430572 q^{31} -0.430572 q^{33} +8.53690 q^{37} -1.00000 q^{39} -6.53690 q^{41} +6.53690 q^{43} -12.1063 q^{47} +17.6758 q^{49} +3.00000 q^{51} +4.39804 q^{53} -3.39804 q^{57} +12.1063 q^{59} +6.13886 q^{61} +4.96747 q^{63} -9.56943 q^{67} -5.39804 q^{69} +9.67575 q^{71} -9.67575 q^{73} +2.13886 q^{77} +10.5369 q^{79} +1.00000 q^{81} +7.82861 q^{83} +4.13886 q^{87} +2.86114 q^{89} +4.96747 q^{91} -0.430572 q^{93} +5.93494 q^{97} +0.430572 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99} + O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.96747 1.87753 0.938763 0.344562i $$-0.111973\pi$$
0.938763 + 0.344562i $$0.111973\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.430572 0.129822 0.0649112 0.997891i $$-0.479324\pi$$
0.0649112 + 0.997891i $$0.479324\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 3.39804 0.779564 0.389782 0.920907i $$-0.372550\pi$$
0.389782 + 0.920907i $$0.372550\pi$$
$$20$$ 0 0
$$21$$ −4.96747 −1.08399
$$22$$ 0 0
$$23$$ 5.39804 1.12557 0.562785 0.826603i $$-0.309730\pi$$
0.562785 + 0.826603i $$0.309730\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.13886 −0.768566 −0.384283 0.923215i $$-0.625551\pi$$
−0.384283 + 0.923215i $$0.625551\pi$$
$$30$$ 0 0
$$31$$ 0.430572 0.0773331 0.0386665 0.999252i $$-0.487689\pi$$
0.0386665 + 0.999252i $$0.487689\pi$$
$$32$$ 0 0
$$33$$ −0.430572 −0.0749530
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.53690 1.40346 0.701729 0.712444i $$-0.252412\pi$$
0.701729 + 0.712444i $$0.252412\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.53690 −1.02089 −0.510446 0.859910i $$-0.670520\pi$$
−0.510446 + 0.859910i $$0.670520\pi$$
$$42$$ 0 0
$$43$$ 6.53690 0.996867 0.498434 0.866928i $$-0.333909\pi$$
0.498434 + 0.866928i $$0.333909\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.1063 −1.76589 −0.882944 0.469477i $$-0.844442\pi$$
−0.882944 + 0.469477i $$0.844442\pi$$
$$48$$ 0 0
$$49$$ 17.6758 2.52511
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ 4.39804 0.604118 0.302059 0.953289i $$-0.402326\pi$$
0.302059 + 0.953289i $$0.402326\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.39804 −0.450082
$$58$$ 0 0
$$59$$ 12.1063 1.57611 0.788055 0.615605i $$-0.211088\pi$$
0.788055 + 0.615605i $$0.211088\pi$$
$$60$$ 0 0
$$61$$ 6.13886 0.786000 0.393000 0.919538i $$-0.371437\pi$$
0.393000 + 0.919538i $$0.371437\pi$$
$$62$$ 0 0
$$63$$ 4.96747 0.625842
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.56943 −1.16909 −0.584546 0.811361i $$-0.698727\pi$$
−0.584546 + 0.811361i $$0.698727\pi$$
$$68$$ 0 0
$$69$$ −5.39804 −0.649848
$$70$$ 0 0
$$71$$ 9.67575 1.14830 0.574150 0.818750i $$-0.305333\pi$$
0.574150 + 0.818750i $$0.305333\pi$$
$$72$$ 0 0
$$73$$ −9.67575 −1.13246 −0.566231 0.824247i $$-0.691599\pi$$
−0.566231 + 0.824247i $$0.691599\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.13886 0.243745
$$78$$ 0 0
$$79$$ 10.5369 1.18549 0.592747 0.805389i $$-0.298043\pi$$
0.592747 + 0.805389i $$0.298043\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 7.82861 0.859302 0.429651 0.902995i $$-0.358637\pi$$
0.429651 + 0.902995i $$0.358637\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.13886 0.443732
$$88$$ 0 0
$$89$$ 2.86114 0.303281 0.151640 0.988436i $$-0.451544\pi$$
0.151640 + 0.988436i $$0.451544\pi$$
$$90$$ 0 0
$$91$$ 4.96747 0.520732
$$92$$ 0 0
$$93$$ −0.430572 −0.0446483
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.93494 0.602602 0.301301 0.953529i $$-0.402579\pi$$
0.301301 + 0.953529i $$0.402579\pi$$
$$98$$ 0 0
$$99$$ 0.430572 0.0432742
$$100$$ 0 0
$$101$$ −16.3330 −1.62519 −0.812596 0.582827i $$-0.801946\pi$$
−0.812596 + 0.582827i $$0.801946\pi$$
$$102$$ 0 0
$$103$$ 8.53690 0.841165 0.420583 0.907254i $$-0.361826\pi$$
0.420583 + 0.907254i $$0.361826\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.6572 −1.12695 −0.563473 0.826134i $$-0.690535\pi$$
−0.563473 + 0.826134i $$0.690535\pi$$
$$108$$ 0 0
$$109$$ −3.67575 −0.352073 −0.176037 0.984384i $$-0.556328\pi$$
−0.176037 + 0.984384i $$0.556328\pi$$
$$110$$ 0 0
$$111$$ −8.53690 −0.810286
$$112$$ 0 0
$$113$$ 3.13886 0.295279 0.147639 0.989041i $$-0.452833\pi$$
0.147639 + 0.989041i $$0.452833\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −14.9024 −1.36610
$$120$$ 0 0
$$121$$ −10.8146 −0.983146
$$122$$ 0 0
$$123$$ 6.53690 0.589412
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −19.6758 −1.74594 −0.872970 0.487773i $$-0.837809\pi$$
−0.872970 + 0.487773i $$0.837809\pi$$
$$128$$ 0 0
$$129$$ −6.53690 −0.575542
$$130$$ 0 0
$$131$$ 3.39804 0.296888 0.148444 0.988921i $$-0.452573\pi$$
0.148444 + 0.988921i $$0.452573\pi$$
$$132$$ 0 0
$$133$$ 16.8797 1.46365
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 5.65723 0.483330 0.241665 0.970360i $$-0.422307\pi$$
0.241665 + 0.970360i $$0.422307\pi$$
$$138$$ 0 0
$$139$$ 17.8699 1.51570 0.757852 0.652427i $$-0.226249\pi$$
0.757852 + 0.652427i $$0.226249\pi$$
$$140$$ 0 0
$$141$$ 12.1063 1.01954
$$142$$ 0 0
$$143$$ 0.430572 0.0360063
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −17.6758 −1.45787
$$148$$ 0 0
$$149$$ 16.2126 1.32819 0.664096 0.747647i $$-0.268817\pi$$
0.664096 + 0.747647i $$0.268817\pi$$
$$150$$ 0 0
$$151$$ −5.89368 −0.479621 −0.239810 0.970820i $$-0.577085\pi$$
−0.239810 + 0.970820i $$0.577085\pi$$
$$152$$ 0 0
$$153$$ −3.00000 −0.242536
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.39804 0.351002 0.175501 0.984479i $$-0.443845\pi$$
0.175501 + 0.984479i $$0.443845\pi$$
$$158$$ 0 0
$$159$$ −4.39804 −0.348787
$$160$$ 0 0
$$161$$ 26.8146 2.11329
$$162$$ 0 0
$$163$$ −16.4718 −1.29017 −0.645087 0.764109i $$-0.723179\pi$$
−0.645087 + 0.764109i $$0.723179\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.5369 −0.970134 −0.485067 0.874477i $$-0.661205\pi$$
−0.485067 + 0.874477i $$0.661205\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 3.39804 0.259855
$$172$$ 0 0
$$173$$ 7.53690 0.573020 0.286510 0.958077i $$-0.407505\pi$$
0.286510 + 0.958077i $$0.407505\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.1063 −0.909967
$$178$$ 0 0
$$179$$ −25.6107 −1.91423 −0.957116 0.289703i $$-0.906443\pi$$
−0.957116 + 0.289703i $$0.906443\pi$$
$$180$$ 0 0
$$181$$ 9.25919 0.688230 0.344115 0.938928i $$-0.388179\pi$$
0.344115 + 0.938928i $$0.388179\pi$$
$$182$$ 0 0
$$183$$ −6.13886 −0.453797
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.29172 −0.0944597
$$188$$ 0 0
$$189$$ −4.96747 −0.361330
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ −6.47184 −0.465853 −0.232926 0.972494i $$-0.574830\pi$$
−0.232926 + 0.972494i $$0.574830\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.259187 0.0184663 0.00923314 0.999957i $$-0.497061\pi$$
0.00923314 + 0.999957i $$0.497061\pi$$
$$198$$ 0 0
$$199$$ −8.79608 −0.623538 −0.311769 0.950158i $$-0.600921\pi$$
−0.311769 + 0.950158i $$0.600921\pi$$
$$200$$ 0 0
$$201$$ 9.56943 0.674975
$$202$$ 0 0
$$203$$ −20.5596 −1.44300
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 5.39804 0.375190
$$208$$ 0 0
$$209$$ 1.46310 0.101205
$$210$$ 0 0
$$211$$ 19.6758 1.35453 0.677267 0.735737i $$-0.263164\pi$$
0.677267 + 0.735737i $$0.263164\pi$$
$$212$$ 0 0
$$213$$ −9.67575 −0.662972
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.13886 0.145195
$$218$$ 0 0
$$219$$ 9.67575 0.653827
$$220$$ 0 0
$$221$$ −3.00000 −0.201802
$$222$$ 0 0
$$223$$ −14.1941 −0.950509 −0.475254 0.879848i $$-0.657644\pi$$
−0.475254 + 0.879848i $$0.657644\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −0.495634 −0.0328964 −0.0164482 0.999865i $$-0.505236\pi$$
−0.0164482 + 0.999865i $$0.505236\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −2.13886 −0.140726
$$232$$ 0 0
$$233$$ −11.1389 −0.729731 −0.364865 0.931060i $$-0.618885\pi$$
−0.364865 + 0.931060i $$0.618885\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −10.5369 −0.684445
$$238$$ 0 0
$$239$$ −17.9762 −1.16278 −0.581392 0.813624i $$-0.697492\pi$$
−0.581392 + 0.813624i $$0.697492\pi$$
$$240$$ 0 0
$$241$$ 30.4718 1.96286 0.981432 0.191812i $$-0.0614363\pi$$
0.981432 + 0.191812i $$0.0614363\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.39804 0.216212
$$248$$ 0 0
$$249$$ −7.82861 −0.496118
$$250$$ 0 0
$$251$$ 23.3330 1.47276 0.736382 0.676566i $$-0.236532\pi$$
0.736382 + 0.676566i $$0.236532\pi$$
$$252$$ 0 0
$$253$$ 2.32425 0.146124
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.6758 1.41447 0.707237 0.706976i $$-0.249941\pi$$
0.707237 + 0.706976i $$0.249941\pi$$
$$258$$ 0 0
$$259$$ 42.4068 2.63503
$$260$$ 0 0
$$261$$ −4.13886 −0.256189
$$262$$ 0 0
$$263$$ −4.53690 −0.279757 −0.139879 0.990169i $$-0.544671\pi$$
−0.139879 + 0.990169i $$0.544671\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.86114 −0.175099
$$268$$ 0 0
$$269$$ 12.8699 0.784690 0.392345 0.919818i $$-0.371664\pi$$
0.392345 + 0.919818i $$0.371664\pi$$
$$270$$ 0 0
$$271$$ −8.68976 −0.527865 −0.263933 0.964541i $$-0.585020\pi$$
−0.263933 + 0.964541i $$0.585020\pi$$
$$272$$ 0 0
$$273$$ −4.96747 −0.300645
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5.93494 −0.356596 −0.178298 0.983977i $$-0.557059\pi$$
−0.178298 + 0.983977i $$0.557059\pi$$
$$278$$ 0 0
$$279$$ 0.430572 0.0257777
$$280$$ 0 0
$$281$$ 11.2039 0.668370 0.334185 0.942508i $$-0.391539\pi$$
0.334185 + 0.942508i $$0.391539\pi$$
$$282$$ 0 0
$$283$$ 7.20392 0.428228 0.214114 0.976809i $$-0.431314\pi$$
0.214114 + 0.976809i $$0.431314\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −32.4718 −1.91675
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −5.93494 −0.347912
$$292$$ 0 0
$$293$$ −4.27771 −0.249907 −0.124953 0.992163i $$-0.539878\pi$$
−0.124953 + 0.992163i $$0.539878\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −0.430572 −0.0249843
$$298$$ 0 0
$$299$$ 5.39804 0.312177
$$300$$ 0 0
$$301$$ 32.4718 1.87165
$$302$$ 0 0
$$303$$ 16.3330 0.938305
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 14.8146 0.845514 0.422757 0.906243i $$-0.361062\pi$$
0.422757 + 0.906243i $$0.361062\pi$$
$$308$$ 0 0
$$309$$ −8.53690 −0.485647
$$310$$ 0 0
$$311$$ −17.3515 −0.983914 −0.491957 0.870620i $$-0.663718\pi$$
−0.491957 + 0.870620i $$0.663718\pi$$
$$312$$ 0 0
$$313$$ 22.3980 1.26601 0.633006 0.774147i $$-0.281821\pi$$
0.633006 + 0.774147i $$0.281821\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −27.2679 −1.53152 −0.765759 0.643127i $$-0.777637\pi$$
−0.765759 + 0.643127i $$0.777637\pi$$
$$318$$ 0 0
$$319$$ −1.78208 −0.0997771
$$320$$ 0 0
$$321$$ 11.6572 0.650643
$$322$$ 0 0
$$323$$ −10.1941 −0.567216
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 3.67575 0.203270
$$328$$ 0 0
$$329$$ −60.1378 −3.31550
$$330$$ 0 0
$$331$$ −10.8797 −0.598001 −0.299000 0.954253i $$-0.596653\pi$$
−0.299000 + 0.954253i $$0.596653\pi$$
$$332$$ 0 0
$$333$$ 8.53690 0.467819
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −34.2029 −1.86315 −0.931574 0.363551i $$-0.881564\pi$$
−0.931574 + 0.363551i $$0.881564\pi$$
$$338$$ 0 0
$$339$$ −3.13886 −0.170479
$$340$$ 0 0
$$341$$ 0.185393 0.0100396
$$342$$ 0 0
$$343$$ 53.0315 2.86343
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −28.2777 −1.51803 −0.759014 0.651075i $$-0.774318\pi$$
−0.759014 + 0.651075i $$0.774318\pi$$
$$348$$ 0 0
$$349$$ 27.3330 1.46310 0.731550 0.681787i $$-0.238797\pi$$
0.731550 + 0.681787i $$0.238797\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −3.18539 −0.169541 −0.0847707 0.996400i $$-0.527016\pi$$
−0.0847707 + 0.996400i $$0.527016\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 14.9024 0.788719
$$358$$ 0 0
$$359$$ 14.1063 0.744503 0.372252 0.928132i $$-0.378586\pi$$
0.372252 + 0.928132i $$0.378586\pi$$
$$360$$ 0 0
$$361$$ −7.45331 −0.392280
$$362$$ 0 0
$$363$$ 10.8146 0.567620
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.5922 −1.33590 −0.667950 0.744206i $$-0.732828\pi$$
−0.667950 + 0.744206i $$0.732828\pi$$
$$368$$ 0 0
$$369$$ −6.53690 −0.340297
$$370$$ 0 0
$$371$$ 21.8471 1.13425
$$372$$ 0 0
$$373$$ −15.4068 −0.797733 −0.398866 0.917009i $$-0.630596\pi$$
−0.398866 + 0.917009i $$0.630596\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.13886 −0.213162
$$378$$ 0 0
$$379$$ 21.7821 1.11887 0.559435 0.828874i $$-0.311018\pi$$
0.559435 + 0.828874i $$0.311018\pi$$
$$380$$ 0 0
$$381$$ 19.6758 1.00802
$$382$$ 0 0
$$383$$ −4.47184 −0.228500 −0.114250 0.993452i $$-0.536447\pi$$
−0.114250 + 0.993452i $$0.536447\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 6.53690 0.332289
$$388$$ 0 0
$$389$$ 13.4166 0.680247 0.340123 0.940381i $$-0.389531\pi$$
0.340123 + 0.940381i $$0.389531\pi$$
$$390$$ 0 0
$$391$$ −16.1941 −0.818972
$$392$$ 0 0
$$393$$ −3.39804 −0.171409
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 36.7310 1.84348 0.921739 0.387812i $$-0.126769\pi$$
0.921739 + 0.387812i $$0.126769\pi$$
$$398$$ 0 0
$$399$$ −16.8797 −0.845040
$$400$$ 0 0
$$401$$ 27.8514 1.39083 0.695415 0.718608i $$-0.255221\pi$$
0.695415 + 0.718608i $$0.255221\pi$$
$$402$$ 0 0
$$403$$ 0.430572 0.0214483
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.67575 0.182200
$$408$$ 0 0
$$409$$ 25.8514 1.27827 0.639134 0.769096i $$-0.279293\pi$$
0.639134 + 0.769096i $$0.279293\pi$$
$$410$$ 0 0
$$411$$ −5.65723 −0.279050
$$412$$ 0 0
$$413$$ 60.1378 2.95919
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −17.8699 −0.875092
$$418$$ 0 0
$$419$$ 28.5369 1.39412 0.697059 0.717013i $$-0.254491\pi$$
0.697059 + 0.717013i $$0.254491\pi$$
$$420$$ 0 0
$$421$$ 4.51837 0.220212 0.110106 0.993920i $$-0.464881\pi$$
0.110106 + 0.993920i $$0.464881\pi$$
$$422$$ 0 0
$$423$$ −12.1063 −0.588630
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 30.4946 1.47574
$$428$$ 0 0
$$429$$ −0.430572 −0.0207882
$$430$$ 0 0
$$431$$ 34.1941 1.64707 0.823537 0.567263i $$-0.191998\pi$$
0.823537 + 0.567263i $$0.191998\pi$$
$$432$$ 0 0
$$433$$ 11.6572 0.560211 0.280105 0.959969i $$-0.409631\pi$$
0.280105 + 0.959969i $$0.409631\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 18.3428 0.877454
$$438$$ 0 0
$$439$$ −36.6194 −1.74775 −0.873875 0.486151i $$-0.838401\pi$$
−0.873875 + 0.486151i $$0.838401\pi$$
$$440$$ 0 0
$$441$$ 17.6758 0.841702
$$442$$ 0 0
$$443$$ 4.25919 0.202360 0.101180 0.994868i $$-0.467738\pi$$
0.101180 + 0.994868i $$0.467738\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −16.2126 −0.766832
$$448$$ 0 0
$$449$$ 24.9437 1.17716 0.588582 0.808437i $$-0.299686\pi$$
0.588582 + 0.808437i $$0.299686\pi$$
$$450$$ 0 0
$$451$$ −2.81461 −0.132535
$$452$$ 0 0
$$453$$ 5.89368 0.276909
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −7.05527 −0.330032 −0.165016 0.986291i $$-0.552768\pi$$
−0.165016 + 0.986291i $$0.552768\pi$$
$$458$$ 0 0
$$459$$ 3.00000 0.140028
$$460$$ 0 0
$$461$$ −5.35150 −0.249244 −0.124622 0.992204i $$-0.539772\pi$$
−0.124622 + 0.992204i $$0.539772\pi$$
$$462$$ 0 0
$$463$$ 20.0878 0.933559 0.466780 0.884374i $$-0.345414\pi$$
0.466780 + 0.884374i $$0.345414\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9.00873 0.416874 0.208437 0.978036i $$-0.433162\pi$$
0.208437 + 0.978036i $$0.433162\pi$$
$$468$$ 0 0
$$469$$ −47.5358 −2.19500
$$470$$ 0 0
$$471$$ −4.39804 −0.202651
$$472$$ 0 0
$$473$$ 2.81461 0.129416
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4.39804 0.201373
$$478$$ 0 0
$$479$$ 19.5044 0.891177 0.445589 0.895238i $$-0.352994\pi$$
0.445589 + 0.895238i $$0.352994\pi$$
$$480$$ 0 0
$$481$$ 8.53690 0.389249
$$482$$ 0 0
$$483$$ −26.8146 −1.22011
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −14.9860 −0.679080 −0.339540 0.940592i $$-0.610271\pi$$
−0.339540 + 0.940592i $$0.610271\pi$$
$$488$$ 0 0
$$489$$ 16.4718 0.744882
$$490$$ 0 0
$$491$$ 22.0000 0.992846 0.496423 0.868081i $$-0.334646\pi$$
0.496423 + 0.868081i $$0.334646\pi$$
$$492$$ 0 0
$$493$$ 12.4166 0.559214
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 48.0640 2.15597
$$498$$ 0 0
$$499$$ −12.5596 −0.562247 −0.281123 0.959672i $$-0.590707\pi$$
−0.281123 + 0.959672i $$0.590707\pi$$
$$500$$ 0 0
$$501$$ 12.5369 0.560107
$$502$$ 0 0
$$503$$ −29.6758 −1.32318 −0.661588 0.749867i $$-0.730117\pi$$
−0.661588 + 0.749867i $$0.730117\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 30.3417 1.34487 0.672436 0.740155i $$-0.265248\pi$$
0.672436 + 0.740155i $$0.265248\pi$$
$$510$$ 0 0
$$511$$ −48.0640 −2.12623
$$512$$ 0 0
$$513$$ −3.39804 −0.150027
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −5.21265 −0.229252
$$518$$ 0 0
$$519$$ −7.53690 −0.330833
$$520$$ 0 0
$$521$$ −7.72229 −0.338320 −0.169160 0.985589i $$-0.554105\pi$$
−0.169160 + 0.985589i $$0.554105\pi$$
$$522$$ 0 0
$$523$$ 27.6572 1.20937 0.604683 0.796466i $$-0.293300\pi$$
0.604683 + 0.796466i $$0.293300\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.29172 −0.0562681
$$528$$ 0 0
$$529$$ 6.13886 0.266907
$$530$$ 0 0
$$531$$ 12.1063 0.525370
$$532$$ 0 0
$$533$$ −6.53690 −0.283144
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 25.6107 1.10518
$$538$$ 0 0
$$539$$ 7.61069 0.327816
$$540$$ 0 0
$$541$$ −13.6758 −0.587967 −0.293983 0.955811i $$-0.594981\pi$$
−0.293983 + 0.955811i $$0.594981\pi$$
$$542$$ 0 0
$$543$$ −9.25919 −0.397350
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 14.1291 0.604115 0.302058 0.953290i $$-0.402327\pi$$
0.302058 + 0.953290i $$0.402327\pi$$
$$548$$ 0 0
$$549$$ 6.13886 0.262000
$$550$$ 0 0
$$551$$ −14.0640 −0.599147
$$552$$ 0 0
$$553$$ 52.3417 2.22580
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10.4718 −0.443706 −0.221853 0.975080i $$-0.571210\pi$$
−0.221853 + 0.975080i $$0.571210\pi$$
$$558$$ 0 0
$$559$$ 6.53690 0.276481
$$560$$ 0 0
$$561$$ 1.29172 0.0545363
$$562$$ 0 0
$$563$$ 5.95346 0.250909 0.125454 0.992099i $$-0.459961\pi$$
0.125454 + 0.992099i $$0.459961\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.96747 0.208614
$$568$$ 0 0
$$569$$ −22.4631 −0.941702 −0.470851 0.882213i $$-0.656053\pi$$
−0.470851 + 0.882213i $$0.656053\pi$$
$$570$$ 0 0
$$571$$ −33.2679 −1.39222 −0.696110 0.717936i $$-0.745087\pi$$
−0.696110 + 0.717936i $$0.745087\pi$$
$$572$$ 0 0
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −14.4904 −0.603242 −0.301621 0.953428i $$-0.597528\pi$$
−0.301621 + 0.953428i $$0.597528\pi$$
$$578$$ 0 0
$$579$$ 6.47184 0.268960
$$580$$ 0 0
$$581$$ 38.8884 1.61336
$$582$$ 0 0
$$583$$ 1.89368 0.0784280
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 19.3742 0.799661 0.399830 0.916589i $$-0.369069\pi$$
0.399830 + 0.916589i $$0.369069\pi$$
$$588$$ 0 0
$$589$$ 1.46310 0.0602861
$$590$$ 0 0
$$591$$ −0.259187 −0.0106615
$$592$$ 0 0
$$593$$ 26.9437 1.10644 0.553222 0.833034i $$-0.313398\pi$$
0.553222 + 0.833034i $$0.313398\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.79608 0.360000
$$598$$ 0 0
$$599$$ 3.91641 0.160020 0.0800102 0.996794i $$-0.474505\pi$$
0.0800102 + 0.996794i $$0.474505\pi$$
$$600$$ 0 0
$$601$$ 9.87967 0.403000 0.201500 0.979489i $$-0.435418\pi$$
0.201500 + 0.979489i $$0.435418\pi$$
$$602$$ 0 0
$$603$$ −9.56943 −0.389697
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 27.0087 1.09625 0.548125 0.836396i $$-0.315342\pi$$
0.548125 + 0.836396i $$0.315342\pi$$
$$608$$ 0 0
$$609$$ 20.5596 0.833119
$$610$$ 0 0
$$611$$ −12.1063 −0.489769
$$612$$ 0 0
$$613$$ 16.2777 0.657451 0.328725 0.944426i $$-0.393381\pi$$
0.328725 + 0.944426i $$0.393381\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4.99127 −0.200941 −0.100470 0.994940i $$-0.532035\pi$$
−0.100470 + 0.994940i $$0.532035\pi$$
$$618$$ 0 0
$$619$$ −27.8884 −1.12093 −0.560465 0.828178i $$-0.689377\pi$$
−0.560465 + 0.828178i $$0.689377\pi$$
$$620$$ 0 0
$$621$$ −5.39804 −0.216616
$$622$$ 0 0
$$623$$ 14.2126 0.569418
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −1.46310 −0.0584307
$$628$$ 0 0
$$629$$ −25.6107 −1.02117
$$630$$ 0 0
$$631$$ −4.66702 −0.185791 −0.0928956 0.995676i $$-0.529612\pi$$
−0.0928956 + 0.995676i $$0.529612\pi$$
$$632$$ 0 0
$$633$$ −19.6758 −0.782041
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17.6758 0.700339
$$638$$ 0 0
$$639$$ 9.67575 0.382767
$$640$$ 0 0
$$641$$ −21.1476 −0.835280 −0.417640 0.908613i $$-0.637143\pi$$
−0.417640 + 0.908613i $$0.637143\pi$$
$$642$$ 0 0
$$643$$ 34.7495 1.37039 0.685194 0.728360i $$-0.259717\pi$$
0.685194 + 0.728360i $$0.259717\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −4.45331 −0.175078 −0.0875389 0.996161i $$-0.527900\pi$$
−0.0875389 + 0.996161i $$0.527900\pi$$
$$648$$ 0 0
$$649$$ 5.21265 0.204614
$$650$$ 0 0
$$651$$ −2.13886 −0.0838283
$$652$$ 0 0
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −9.67575 −0.377487
$$658$$ 0 0
$$659$$ 35.4806 1.38213 0.691063 0.722794i $$-0.257143\pi$$
0.691063 + 0.722794i $$0.257143\pi$$
$$660$$ 0 0
$$661$$ −1.33298 −0.0518469 −0.0259235 0.999664i $$-0.508253\pi$$
−0.0259235 + 0.999664i $$0.508253\pi$$
$$662$$ 0 0
$$663$$ 3.00000 0.116510
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −22.3417 −0.865075
$$668$$ 0 0
$$669$$ 14.1941 0.548777
$$670$$ 0 0
$$671$$ 2.64322 0.102040
$$672$$ 0 0
$$673$$ 31.6845 1.22135 0.610674 0.791882i $$-0.290899\pi$$
0.610674 + 0.791882i $$0.290899\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 23.2039 0.891799 0.445899 0.895083i $$-0.352884\pi$$
0.445899 + 0.895083i $$0.352884\pi$$
$$678$$ 0 0
$$679$$ 29.4816 1.13140
$$680$$ 0 0
$$681$$ 0.495634 0.0189927
$$682$$ 0 0
$$683$$ −13.0965 −0.501125 −0.250562 0.968100i $$-0.580616\pi$$
−0.250562 + 0.968100i $$0.580616\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ 0 0
$$689$$ 4.39804 0.167552
$$690$$ 0 0
$$691$$ −38.2539 −1.45525 −0.727624 0.685976i $$-0.759375\pi$$
−0.727624 + 0.685976i $$0.759375\pi$$
$$692$$ 0 0
$$693$$ 2.13886 0.0812484
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 19.6107 0.742808
$$698$$ 0 0
$$699$$ 11.1389 0.421310
$$700$$ 0 0
$$701$$ 34.1563 1.29007 0.645033 0.764155i $$-0.276844\pi$$
0.645033 + 0.764155i $$0.276844\pi$$
$$702$$ 0 0
$$703$$ 29.0087 1.09409
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −81.1336 −3.05134
$$708$$ 0 0
$$709$$ 21.3330 0.801177 0.400588 0.916258i $$-0.368806\pi$$
0.400588 + 0.916258i $$0.368806\pi$$
$$710$$ 0 0
$$711$$ 10.5369 0.395165
$$712$$ 0 0
$$713$$ 2.32425 0.0870438
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 17.9762 0.671334
$$718$$ 0 0
$$719$$ 0.0185238 0.000690822 0 0.000345411 1.00000i $$-0.499890\pi$$
0.000345411 1.00000i $$0.499890\pi$$
$$720$$ 0 0
$$721$$ 42.4068 1.57931
$$722$$ 0 0
$$723$$ −30.4718 −1.13326
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.6670 0.469794 0.234897 0.972020i $$-0.424525\pi$$
0.234897 + 0.972020i $$0.424525\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −19.6107 −0.725328
$$732$$ 0 0
$$733$$ 8.51837 0.314633 0.157317 0.987548i $$-0.449716\pi$$
0.157317 + 0.987548i $$0.449716\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.12033 −0.151774
$$738$$ 0 0
$$739$$ 11.9122 0.438197 0.219099 0.975703i $$-0.429688\pi$$
0.219099 + 0.975703i $$0.429688\pi$$
$$740$$ 0 0
$$741$$ −3.39804 −0.124830
$$742$$ 0 0
$$743$$ −21.3568 −0.783504 −0.391752 0.920071i $$-0.628131\pi$$
−0.391752 + 0.920071i $$0.628131\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 7.82861 0.286434
$$748$$ 0 0
$$749$$ −57.9069 −2.11587
$$750$$ 0 0
$$751$$ 18.9902 0.692963 0.346481 0.938057i $$-0.387376\pi$$
0.346481 + 0.938057i $$0.387376\pi$$
$$752$$ 0 0
$$753$$ −23.3330 −0.850301
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.72229 0.0989433 0.0494716 0.998776i $$-0.484246\pi$$
0.0494716 + 0.998776i $$0.484246\pi$$
$$758$$ 0 0
$$759$$ −2.32425 −0.0843649
$$760$$ 0 0
$$761$$ 3.52816 0.127896 0.0639479 0.997953i $$-0.479631\pi$$
0.0639479 + 0.997953i $$0.479631\pi$$
$$762$$ 0 0
$$763$$ −18.2592 −0.661027
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.1063 0.437134
$$768$$ 0 0
$$769$$ −20.0651 −0.723565 −0.361782 0.932263i $$-0.617832\pi$$
−0.361782 + 0.932263i $$0.617832\pi$$
$$770$$ 0 0
$$771$$ −22.6758 −0.816647
$$772$$ 0 0
$$773$$ −29.8048 −1.07200 −0.536002 0.844216i $$-0.680066\pi$$
−0.536002 + 0.844216i $$0.680066\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −42.4068 −1.52133
$$778$$ 0 0
$$779$$ −22.2126 −0.795851
$$780$$ 0 0
$$781$$ 4.16611 0.149075
$$782$$ 0 0
$$783$$ 4.13886 0.147911
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −13.8471 −0.493597 −0.246799 0.969067i $$-0.579379\pi$$
−0.246799 + 0.969067i $$0.579379\pi$$
$$788$$ 0 0
$$789$$ 4.53690 0.161518
$$790$$ 0 0
$$791$$ 15.5922 0.554394
$$792$$ 0 0
$$793$$ 6.13886 0.217997
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −14.5271 −0.514576 −0.257288 0.966335i $$-0.582829\pi$$
−0.257288 + 0.966335i $$0.582829\pi$$
$$798$$ 0 0
$$799$$ 36.3190 1.28487
$$800$$ 0 0
$$801$$ 2.86114 0.101094
$$802$$ 0 0
$$803$$ −4.16611 −0.147019
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.8699 −0.453041
$$808$$ 0 0
$$809$$ −0.342772 −0.0120512 −0.00602560 0.999982i $$-0.501918\pi$$
−0.00602560 + 0.999982i $$0.501918\pi$$
$$810$$ 0 0
$$811$$ 17.5880 0.617597 0.308798 0.951128i $$-0.400073\pi$$
0.308798 + 0.951128i $$0.400073\pi$$
$$812$$ 0 0
$$813$$ 8.68976 0.304763
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 22.2126 0.777122
$$818$$ 0 0
$$819$$ 4.96747 0.173577
$$820$$ 0 0
$$821$$ 5.22244 0.182264 0.0911322 0.995839i $$-0.470951\pi$$
0.0911322 + 0.995839i $$0.470951\pi$$
$$822$$ 0 0
$$823$$ 1.69428 0.0590587 0.0295294 0.999564i $$-0.490599\pi$$
0.0295294 + 0.999564i $$0.490599\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −21.2637 −0.739411 −0.369706 0.929149i $$-0.620541\pi$$
−0.369706 + 0.929149i $$0.620541\pi$$
$$828$$ 0 0
$$829$$ −0.804816 −0.0279524 −0.0139762 0.999902i $$-0.504449\pi$$
−0.0139762 + 0.999902i $$0.504449\pi$$
$$830$$ 0 0
$$831$$ 5.93494 0.205881
$$832$$ 0 0
$$833$$ −53.0273 −1.83729
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −0.430572 −0.0148828
$$838$$ 0 0
$$839$$ −43.9535 −1.51744 −0.758721 0.651416i $$-0.774175\pi$$
−0.758721 + 0.651416i $$0.774175\pi$$
$$840$$ 0 0
$$841$$ −11.8699 −0.409306
$$842$$ 0 0
$$843$$ −11.2039 −0.385883
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −53.7212 −1.84588
$$848$$ 0 0
$$849$$ −7.20392 −0.247238
$$850$$ 0 0
$$851$$ 46.0825 1.57969
$$852$$ 0 0
$$853$$ 49.4620 1.69355 0.846774 0.531953i $$-0.178542\pi$$
0.846774 + 0.531953i $$0.178542\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 49.7398 1.69908 0.849539 0.527526i $$-0.176880\pi$$
0.849539 + 0.527526i $$0.176880\pi$$
$$858$$ 0 0
$$859$$ −30.9622 −1.05642 −0.528208 0.849115i $$-0.677136\pi$$
−0.528208 + 0.849115i $$0.677136\pi$$
$$860$$ 0 0
$$861$$ 32.4718 1.10664
$$862$$ 0 0
$$863$$ 38.4480 1.30879 0.654393 0.756154i $$-0.272924\pi$$
0.654393 + 0.756154i $$0.272924\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 4.53690 0.153904
$$870$$ 0 0
$$871$$ −9.56943 −0.324248
$$872$$ 0 0
$$873$$ 5.93494 0.200867
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.6572 −0.461172 −0.230586 0.973052i $$-0.574064\pi$$
−0.230586 + 0.973052i $$0.574064\pi$$
$$878$$ 0 0
$$879$$ 4.27771 0.144284
$$880$$ 0 0
$$881$$ −12.8059 −0.431441 −0.215720 0.976455i $$-0.569210\pi$$
−0.215720 + 0.976455i $$0.569210\pi$$
$$882$$ 0 0
$$883$$ 3.50964 0.118109 0.0590544 0.998255i $$-0.481191\pi$$
0.0590544 + 0.998255i $$0.481191\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −41.3700 −1.38907 −0.694535 0.719459i $$-0.744390\pi$$
−0.694535 + 0.719459i $$0.744390\pi$$
$$888$$ 0 0
$$889$$ −97.7387 −3.27805
$$890$$ 0 0
$$891$$ 0.430572 0.0144247
$$892$$ 0 0
$$893$$ −41.1378 −1.37662
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.39804 −0.180235
$$898$$ 0 0
$$899$$ −1.78208 −0.0594356
$$900$$ 0 0
$$901$$ −13.1941 −0.439560
$$902$$ 0 0
$$903$$ −32.4718 −1.08060
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −25.4620 −0.845453 −0.422727 0.906257i $$-0.638927\pi$$
−0.422727 + 0.906257i $$0.638927\pi$$
$$908$$ 0 0
$$909$$ −16.3330 −0.541731
$$910$$ 0 0
$$911$$ −53.1378 −1.76053 −0.880267 0.474479i $$-0.842637\pi$$
−0.880267 + 0.474479i $$0.842637\pi$$
$$912$$ 0 0
$$913$$ 3.37079 0.111557
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.8797 0.557416
$$918$$ 0 0
$$919$$ −8.86114 −0.292302 −0.146151 0.989262i $$-0.546689\pi$$
−0.146151 + 0.989262i $$0.546689\pi$$
$$920$$ 0 0
$$921$$ −14.8146 −0.488158
$$922$$ 0 0
$$923$$ 9.67575 0.318481
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.53690 0.280388
$$928$$ 0 0
$$929$$ −6.51837 −0.213861 −0.106930 0.994267i $$-0.534102\pi$$
−0.106930 + 0.994267i $$0.534102\pi$$
$$930$$ 0 0
$$931$$ 60.0629 1.96848
$$932$$ 0 0
$$933$$ 17.3515 0.568063
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.58343 0.117066 0.0585328 0.998285i $$-0.481358\pi$$
0.0585328 + 0.998285i $$0.481358\pi$$
$$938$$ 0 0
$$939$$ −22.3980 −0.730932
$$940$$ 0 0
$$941$$ −53.5456 −1.74554 −0.872769 0.488134i $$-0.837678\pi$$
−0.872769 + 0.488134i $$0.837678\pi$$
$$942$$ 0 0
$$943$$ −35.2864 −1.14908
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 33.9947 1.10468 0.552340 0.833619i $$-0.313735\pi$$
0.552340 + 0.833619i $$0.313735\pi$$
$$948$$ 0 0
$$949$$ −9.67575 −0.314088
$$950$$ 0 0
$$951$$ 27.2679 0.884223
$$952$$ 0 0
$$953$$ −18.4631 −0.598079 −0.299039 0.954241i $$-0.596666\pi$$
−0.299039 + 0.954241i $$0.596666\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1.78208 0.0576064
$$958$$ 0 0
$$959$$ 28.1021 0.907464
$$960$$ 0 0
$$961$$ −30.8146 −0.994020
$$962$$ 0 0
$$963$$ −11.6572 −0.375649
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20.9209 0.672772 0.336386 0.941724i $$-0.390795\pi$$
0.336386 + 0.941724i $$0.390795\pi$$
$$968$$ 0 0
$$969$$ 10.1941 0.327482
$$970$$ 0 0
$$971$$ 23.0087 0.738385 0.369193 0.929353i $$-0.379634\pi$$
0.369193 + 0.929353i $$0.379634\pi$$
$$972$$ 0 0
$$973$$ 88.7681 2.84577
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.5184 0.400498 0.200249 0.979745i $$-0.435825\pi$$
0.200249 + 0.979745i $$0.435825\pi$$
$$978$$ 0 0
$$979$$ 1.23193 0.0393727
$$980$$ 0 0
$$981$$ −3.67575 −0.117358
$$982$$ 0 0
$$983$$ −60.1703 −1.91914 −0.959568 0.281478i $$-0.909175\pi$$
−0.959568 + 0.281478i $$0.909175\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 60.1378 1.91421
$$988$$ 0 0
$$989$$ 35.2864 1.12204
$$990$$ 0 0
$$991$$ −59.5827 −1.89271 −0.946353 0.323134i $$-0.895263\pi$$
−0.946353 + 0.323134i $$0.895263\pi$$
$$992$$ 0 0
$$993$$ 10.8797 0.345256
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 60.1193 1.90400 0.951998 0.306103i $$-0.0990254\pi$$
0.951998 + 0.306103i $$0.0990254\pi$$
$$998$$ 0 0
$$999$$ −8.53690 −0.270095
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 7800.2.a.bh.1.3 3
5.4 even 2 7800.2.a.bs.1.1 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bh.1.3 3 1.1 even 1 trivial
7800.2.a.bs.1.1 yes 3 5.4 even 2