# Properties

 Label 4560.2.a.i.1.1 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +6.00000 q^{13} +1.00000 q^{15} -2.00000 q^{17} -1.00000 q^{19} -4.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.00000 q^{29} +8.00000 q^{31} +2.00000 q^{33} -4.00000 q^{35} +10.0000 q^{37} -6.00000 q^{39} -4.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{47} +9.00000 q^{49} +2.00000 q^{51} +12.0000 q^{53} +2.00000 q^{55} +1.00000 q^{57} +8.00000 q^{59} +2.00000 q^{61} +4.00000 q^{63} -6.00000 q^{65} -4.00000 q^{67} +6.00000 q^{69} -12.0000 q^{71} -10.0000 q^{73} -1.00000 q^{75} -8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{83} +2.00000 q^{85} -8.00000 q^{87} -12.0000 q^{89} +24.0000 q^{91} -8.00000 q^{93} +1.00000 q^{95} -14.0000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ −8.00000 −0.857690
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 24.0000 2.51588
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ 0 0
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 4.00000 0.360668
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 22.0000 1.87959 0.939793 0.341743i $$-0.111017\pi$$
0.939793 + 0.341743i $$0.111017\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ −9.00000 −0.742307
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 16.0000 1.21646 0.608229 0.793762i $$-0.291880\pi$$
0.608229 + 0.793762i $$0.291880\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 2.00000 0.144715 0.0723575 0.997379i $$-0.476948\pi$$
0.0723575 + 0.997379i $$0.476948\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 32.0000 2.24596
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 32.0000 2.17230
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −9.00000 −0.574989
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 0 0
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ 26.0000 1.64111 0.820553 0.571571i $$-0.193666\pi$$
0.820553 + 0.571571i $$0.193666\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 40.0000 2.48548
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 0 0
$$263$$ 22.0000 1.35658 0.678289 0.734795i $$-0.262722\pi$$
0.678289 + 0.734795i $$0.262722\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 12.0000 0.734388
$$268$$ 0 0
$$269$$ 4.00000 0.243884 0.121942 0.992537i $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ −24.0000 −1.45255
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ −16.0000 −0.944450
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ −36.0000 −2.08193
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 22.0000 1.24751 0.623753 0.781622i $$-0.285607\pi$$
0.623753 + 0.781622i $$0.285607\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ −4.00000 −0.225374
$$316$$ 0 0
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 0 0
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 0 0
$$327$$ 6.00000 0.331801
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ −6.00000 −0.323029
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 0 0
$$357$$ 8.00000 0.423405
$$358$$ 0 0
$$359$$ −22.0000 −1.16112 −0.580558 0.814219i $$-0.697165\pi$$
−0.580558 + 0.814219i $$0.697165\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 48.0000 2.49204
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 48.0000 2.47213
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 8.00000 0.407718
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −20.0000 −0.998752 −0.499376 0.866385i $$-0.666437\pi$$
−0.499376 + 0.866385i $$0.666437\pi$$
$$402$$ 0 0
$$403$$ 48.0000 2.39105
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ 0 0
$$413$$ 32.0000 1.57462
$$414$$ 0 0
$$415$$ 2.00000 0.0981761
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 0 0
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 0 0
$$433$$ 38.0000 1.82616 0.913082 0.407777i $$-0.133696\pi$$
0.913082 + 0.407777i $$0.133696\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ 16.0000 0.755087 0.377543 0.925992i $$-0.376769\pi$$
0.377543 + 0.925992i $$0.376769\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −24.0000 −1.12514
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 0 0
$$465$$ 8.00000 0.370991
$$466$$ 0 0
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ 42.0000 1.91903 0.959514 0.281659i $$-0.0908848\pi$$
0.959514 + 0.281659i $$0.0908848\pi$$
$$480$$ 0 0
$$481$$ 60.0000 2.73576
$$482$$ 0 0
$$483$$ 24.0000 1.09204
$$484$$ 0 0
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ 14.0000 0.631811 0.315906 0.948791i $$-0.397692\pi$$
0.315906 + 0.948791i $$0.397692\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ −48.0000 −2.15309
$$498$$ 0 0
$$499$$ −44.0000 −1.96971 −0.984855 0.173379i $$-0.944532\pi$$
−0.984855 + 0.173379i $$0.944532\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ −26.0000 −1.15928 −0.579641 0.814872i $$-0.696807\pi$$
−0.579641 + 0.814872i $$0.696807\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −23.0000 −1.02147
$$508$$ 0 0
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ −40.0000 −1.76950
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ −16.0000 −0.702322
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 0 0
$$525$$ −4.00000 −0.174574
$$526$$ 0 0
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ 0 0
$$537$$ −16.0000 −0.690451
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ −18.0000 −0.772454
$$544$$ 0 0
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 0 0
$$555$$ 10.0000 0.424476
$$556$$ 0 0
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 4.00000 0.167984
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ −2.00000 −0.0835512
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ 0 0
$$579$$ 26.0000 1.08052
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ 0 0
$$587$$ 6.00000 0.247647 0.123823 0.992304i $$-0.460484\pi$$
0.123823 + 0.992304i $$0.460484\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 34.0000 1.38689 0.693444 0.720510i $$-0.256092\pi$$
0.693444 + 0.720510i $$0.256092\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ −32.0000 −1.29671
$$610$$ 0 0
$$611$$ −36.0000 −1.45640
$$612$$ 0 0
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 0 0
$$615$$ −4.00000 −0.161296
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 0 0
$$623$$ −48.0000 −1.92308
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −2.00000 −0.0798723
$$628$$ 0 0
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 54.0000 2.13956
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ 50.0000 1.96570 0.982851 0.184399i $$-0.0590339\pi$$
0.982851 + 0.184399i $$0.0590339\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ 0 0
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 4.00000 0.155113
$$666$$ 0 0
$$667$$ −48.0000 −1.85857
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 0 0
$$679$$ −56.0000 −2.14908
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ −22.0000 −0.840577
$$686$$ 0 0
$$687$$ −22.0000 −0.839352
$$688$$ 0 0
$$689$$ 72.0000 2.74298
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ −8.00000 −0.303895
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 8.00000 0.303022
$$698$$ 0 0
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ 0 0
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ −48.0000 −1.79761
$$714$$ 0 0
$$715$$ 12.0000 0.448775
$$716$$ 0 0
$$717$$ 6.00000 0.224074
$$718$$ 0 0
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ 10.0000 0.371904
$$724$$ 0 0
$$725$$ 8.00000 0.297113
$$726$$ 0 0
$$727$$ 48.0000 1.78022 0.890111 0.455744i $$-0.150627\pi$$
0.890111 + 0.455744i $$0.150627\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ 0 0
$$735$$ 9.00000 0.331970
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ −2.00000 −0.0731762
$$748$$ 0 0
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ −26.0000 −0.947493
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 0 0
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ −50.0000 −1.81250 −0.906249 0.422744i $$-0.861067\pi$$
−0.906249 + 0.422744i $$0.861067\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 48.0000 1.73318
$$768$$ 0 0
$$769$$ −42.0000 −1.51456 −0.757279 0.653091i $$-0.773472\pi$$
−0.757279 + 0.653091i $$0.773472\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 0 0
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ −40.0000 −1.43499
$$778$$ 0 0
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 0 0
$$789$$ −22.0000 −0.783221
$$790$$ 0 0
$$791$$ 32.0000 1.13779
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 0 0
$$795$$ 12.0000 0.425596
$$796$$ 0 0
$$797$$ −16.0000 −0.566749 −0.283375 0.959009i $$-0.591454\pi$$
−0.283375 + 0.959009i $$0.591454\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ 0 0
$$803$$ 20.0000 0.705785
$$804$$ 0 0
$$805$$ 24.0000 0.845889
$$806$$ 0 0
$$807$$ −4.00000 −0.140807
$$808$$ 0 0
$$809$$ −46.0000 −1.61727 −0.808637 0.588308i $$-0.799794\pi$$
−0.808637 + 0.588308i $$0.799794\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 4.00000 0.140286
$$814$$ 0 0
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 24.0000 0.838628
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 0 0
$$825$$ 2.00000 0.0696311
$$826$$ 0 0
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ −8.00000 −0.276520
$$838$$ 0 0
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −23.0000 −0.791224
$$846$$ 0 0
$$847$$ −28.0000 −0.962091
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −60.0000 −2.05677
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −48.0000 −1.63965 −0.819824 0.572615i $$-0.805929\pi$$
−0.819824 + 0.572615i $$0.805929\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ 0 0
$$863$$ −4.00000 −0.136162 −0.0680808 0.997680i $$-0.521688\pi$$
−0.0680808 + 0.997680i $$0.521688\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ −4.00000 −0.135225
$$876$$ 0 0
$$877$$ −6.00000 −0.202606 −0.101303 0.994856i $$-0.532301\pi$$
−0.101303 + 0.994856i $$0.532301\pi$$
$$878$$ 0 0
$$879$$ −16.0000 −0.539667
$$880$$ 0 0
$$881$$ 22.0000 0.741199 0.370599 0.928793i $$-0.379152\pi$$
0.370599 + 0.928793i $$0.379152\pi$$
$$882$$ 0 0
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 0 0
$$897$$ 36.0000 1.20201
$$898$$ 0 0
$$899$$ 64.0000 2.13452
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 0 0
$$905$$ −18.0000 −0.598340
$$906$$ 0 0
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 4.00000 0.132381
$$914$$ 0 0
$$915$$ 2.00000 0.0661180
$$916$$ 0 0
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 0 0
$$923$$ −72.0000 −2.36991
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −9.00000 −0.294963
$$932$$ 0 0
$$933$$ −22.0000 −0.720248
$$934$$ 0 0
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ −30.0000 −0.980057 −0.490029 0.871706i $$-0.663014\pi$$
−0.490029 + 0.871706i $$0.663014\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ −4.00000 −0.130396 −0.0651981 0.997872i $$-0.520768\pi$$
−0.0651981 + 0.997872i $$0.520768\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ −38.0000 −1.23483 −0.617417 0.786636i $$-0.711821\pi$$
−0.617417 + 0.786636i $$0.711821\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ −12.0000 −0.388718 −0.194359 0.980930i $$-0.562263\pi$$
−0.194359 + 0.980930i $$0.562263\pi$$
$$954$$ 0 0
$$955$$ −2.00000 −0.0647185
$$956$$ 0 0
$$957$$ 16.0000 0.517207
$$958$$ 0 0
$$959$$ 88.0000 2.84167
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ 0 0
$$965$$ 26.0000 0.836970
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 0 0
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 0 0
$$973$$ −48.0000 −1.53881
$$974$$ 0 0
$$975$$ −6.00000 −0.192154
$$976$$ 0 0
$$977$$ 8.00000 0.255943 0.127971 0.991778i $$-0.459153\pi$$
0.127971 + 0.991778i $$0.459153\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 0 0
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 24.0000 0.763928
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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 4560.2.a.i.1.1 1
4.3 odd 2 1140.2.a.b.1.1 1
12.11 even 2 3420.2.a.a.1.1 1
20.3 even 4 5700.2.f.i.3649.2 2
20.7 even 4 5700.2.f.i.3649.1 2
20.19 odd 2 5700.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.b.1.1 1 4.3 odd 2
3420.2.a.a.1.1 1 12.11 even 2
4560.2.a.i.1.1 1 1.1 even 1 trivial
5700.2.a.h.1.1 1 20.19 odd 2
5700.2.f.i.3649.1 2 20.7 even 4
5700.2.f.i.3649.2 2 20.3 even 4