# Properties

 Label 3200.2.f.m.449.2 Level $3200$ Weight $2$ Character 3200.449 Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.449 Dual form 3200.2.f.m.449.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.23607 q^{3} +2.00000 q^{9} +O(q^{10})$$ $$q-2.23607 q^{3} +2.00000 q^{9} +2.23607i q^{11} -4.00000 q^{13} -3.00000i q^{17} +2.23607i q^{19} +8.94427i q^{23} +2.23607 q^{27} -4.00000i q^{29} +8.94427 q^{31} -5.00000i q^{33} -8.00000 q^{37} +8.94427 q^{39} -5.00000 q^{41} -8.94427 q^{43} -8.94427i q^{47} +7.00000 q^{49} +6.70820i q^{51} +4.00000 q^{53} -5.00000i q^{57} +8.94427i q^{59} +8.00000i q^{61} +6.70820 q^{67} -20.0000i q^{69} -8.94427 q^{71} -9.00000i q^{73} -11.0000 q^{81} -6.70820 q^{83} +8.94427i q^{87} +15.0000 q^{89} -20.0000 q^{93} -2.00000i q^{97} +4.47214i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{9} + O(q^{10})$$ $$4 q + 8 q^{9} - 16 q^{13} - 32 q^{37} - 20 q^{41} + 28 q^{49} + 16 q^{53} - 44 q^{81} + 60 q^{89} - 80 q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ −2.23607 −1.29099 −0.645497 0.763763i $$-0.723350\pi$$
−0.645497 + 0.763763i $$0.723350\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 2.23607i 0.674200i 0.941469 + 0.337100i $$0.109446\pi$$
−0.941469 + 0.337100i $$0.890554\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ 2.23607i 0.512989i 0.966546 + 0.256495i $$0.0825676\pi$$
−0.966546 + 0.256495i $$0.917432\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.94427i 1.86501i 0.361158 + 0.932505i $$0.382382\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2.23607 0.430331
$$28$$ 0 0
$$29$$ − 4.00000i − 0.742781i −0.928477 0.371391i $$-0.878881\pi$$
0.928477 0.371391i $$-0.121119\pi$$
$$30$$ 0 0
$$31$$ 8.94427 1.60644 0.803219 0.595683i $$-0.203119\pi$$
0.803219 + 0.595683i $$0.203119\pi$$
$$32$$ 0 0
$$33$$ − 5.00000i − 0.870388i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 8.94427 1.43223
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −8.94427 −1.36399 −0.681994 0.731357i $$-0.738887\pi$$
−0.681994 + 0.731357i $$0.738887\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.94427i − 1.30466i −0.757937 0.652328i $$-0.773792\pi$$
0.757937 0.652328i $$-0.226208\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.70820i 0.939336i
$$52$$ 0 0
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 5.00000i − 0.662266i
$$58$$ 0 0
$$59$$ 8.94427i 1.16445i 0.813029 + 0.582223i $$0.197817\pi$$
−0.813029 + 0.582223i $$0.802183\pi$$
$$60$$ 0 0
$$61$$ 8.00000i 1.02430i 0.858898 + 0.512148i $$0.171150\pi$$
−0.858898 + 0.512148i $$0.828850\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.70820 0.819538 0.409769 0.912189i $$-0.365609\pi$$
0.409769 + 0.912189i $$0.365609\pi$$
$$68$$ 0 0
$$69$$ − 20.0000i − 2.40772i
$$70$$ 0 0
$$71$$ −8.94427 −1.06149 −0.530745 0.847532i $$-0.678088\pi$$
−0.530745 + 0.847532i $$0.678088\pi$$
$$72$$ 0 0
$$73$$ − 9.00000i − 1.05337i −0.850060 0.526685i $$-0.823435\pi$$
0.850060 0.526685i $$-0.176565\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −6.70820 −0.736321 −0.368161 0.929762i $$-0.620012\pi$$
−0.368161 + 0.929762i $$0.620012\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.94427i 0.958927i
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −20.0000 −2.07390
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 0 0
$$99$$ 4.47214i 0.449467i
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 8.94427i 0.881305i 0.897678 + 0.440653i $$0.145253\pi$$
−0.897678 + 0.440653i $$0.854747\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.23607 −0.216169 −0.108084 0.994142i $$-0.534472\pi$$
−0.108084 + 0.994142i $$0.534472\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 17.8885 1.69791
$$112$$ 0 0
$$113$$ 1.00000i 0.0940721i 0.998893 + 0.0470360i $$0.0149776\pi$$
−0.998893 + 0.0470360i $$0.985022\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −8.00000 −0.739600
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 6.00000 0.545455
$$122$$ 0 0
$$123$$ 11.1803 1.00810
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.94427i − 0.793676i −0.917889 0.396838i $$-0.870108\pi$$
0.917889 0.396838i $$-0.129892\pi$$
$$128$$ 0 0
$$129$$ 20.0000 1.76090
$$130$$ 0 0
$$131$$ − 8.94427i − 0.781465i −0.920504 0.390732i $$-0.872222\pi$$
0.920504 0.390732i $$-0.127778\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 13.0000i − 1.11066i −0.831628 0.555332i $$-0.812591\pi$$
0.831628 0.555332i $$-0.187409\pi$$
$$138$$ 0 0
$$139$$ − 20.1246i − 1.70695i −0.521136 0.853474i $$-0.674492\pi$$
0.521136 0.853474i $$-0.325508\pi$$
$$140$$ 0 0
$$141$$ 20.0000i 1.68430i
$$142$$ 0 0
$$143$$ − 8.94427i − 0.747958i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −15.6525 −1.29099
$$148$$ 0 0
$$149$$ − 20.0000i − 1.63846i −0.573462 0.819232i $$-0.694400\pi$$
0.573462 0.819232i $$-0.305600\pi$$
$$150$$ 0 0
$$151$$ −17.8885 −1.45575 −0.727875 0.685710i $$-0.759492\pi$$
−0.727875 + 0.685710i $$0.759492\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −12.0000 −0.957704 −0.478852 0.877896i $$-0.658947\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 0 0
$$159$$ −8.94427 −0.709327
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.1803 0.875712 0.437856 0.899045i $$-0.355738\pi$$
0.437856 + 0.899045i $$0.355738\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.94427i 0.692129i 0.938211 + 0.346064i $$0.112482\pi$$
−0.938211 + 0.346064i $$0.887518\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 4.47214i 0.341993i
$$172$$ 0 0
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 20.0000i − 1.50329i
$$178$$ 0 0
$$179$$ − 15.6525i − 1.16992i −0.811062 0.584960i $$-0.801110\pi$$
0.811062 0.584960i $$-0.198890\pi$$
$$180$$ 0 0
$$181$$ − 20.0000i − 1.48659i −0.668965 0.743294i $$-0.733262\pi$$
0.668965 0.743294i $$-0.266738\pi$$
$$182$$ 0 0
$$183$$ − 17.8885i − 1.32236i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.70820 0.490552
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.94427 −0.647185 −0.323592 0.946197i $$-0.604891\pi$$
−0.323592 + 0.946197i $$0.604891\pi$$
$$192$$ 0 0
$$193$$ − 21.0000i − 1.51161i −0.654795 0.755807i $$-0.727245\pi$$
0.654795 0.755807i $$-0.272755\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ −17.8885 −1.26809 −0.634043 0.773298i $$-0.718606\pi$$
−0.634043 + 0.773298i $$0.718606\pi$$
$$200$$ 0 0
$$201$$ −15.0000 −1.05802
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 17.8885i 1.24334i
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ − 6.70820i − 0.461812i −0.972976 0.230906i $$-0.925831\pi$$
0.972976 0.230906i $$-0.0741690\pi$$
$$212$$ 0 0
$$213$$ 20.0000 1.37038
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 20.1246i 1.35990i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ − 17.8885i − 1.19791i −0.800784 0.598953i $$-0.795584\pi$$
0.800784 0.598953i $$-0.204416\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 8.94427 0.593652 0.296826 0.954932i $$-0.404072\pi$$
0.296826 + 0.954932i $$0.404072\pi$$
$$228$$ 0 0
$$229$$ − 16.0000i − 1.05731i −0.848837 0.528655i $$-0.822697\pi$$
0.848837 0.528655i $$-0.177303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −8.94427 −0.578557 −0.289278 0.957245i $$-0.593415\pi$$
−0.289278 + 0.957245i $$0.593415\pi$$
$$240$$ 0 0
$$241$$ 5.00000 0.322078 0.161039 0.986948i $$-0.448515\pi$$
0.161039 + 0.986948i $$0.448515\pi$$
$$242$$ 0 0
$$243$$ 17.8885 1.14755
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 8.94427i − 0.569110i
$$248$$ 0 0
$$249$$ 15.0000 0.950586
$$250$$ 0 0
$$251$$ − 29.0689i − 1.83481i −0.397953 0.917406i $$-0.630279\pi$$
0.397953 0.917406i $$-0.369721\pi$$
$$252$$ 0 0
$$253$$ −20.0000 −1.25739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ − 8.00000i − 0.495188i
$$262$$ 0 0
$$263$$ − 8.94427i − 0.551527i −0.961225 0.275764i $$-0.911069\pi$$
0.961225 0.275764i $$-0.0889307\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −33.5410 −2.05268
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ −26.8328 −1.62998 −0.814989 0.579477i $$-0.803257\pi$$
−0.814989 + 0.579477i $$0.803257\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −12.0000 −0.721010 −0.360505 0.932757i $$-0.617396\pi$$
−0.360505 + 0.932757i $$0.617396\pi$$
$$278$$ 0 0
$$279$$ 17.8885 1.07096
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 24.5967 1.46212 0.731062 0.682311i $$-0.239025\pi$$
0.731062 + 0.682311i $$0.239025\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 4.47214i 0.262161i
$$292$$ 0 0
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000i 0.290129i
$$298$$ 0 0
$$299$$ − 35.7771i − 2.06904i
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.23607 0.127619 0.0638096 0.997962i $$-0.479675\pi$$
0.0638096 + 0.997962i $$0.479675\pi$$
$$308$$ 0 0
$$309$$ − 20.0000i − 1.13776i
$$310$$ 0 0
$$311$$ 17.8885 1.01437 0.507183 0.861838i $$-0.330687\pi$$
0.507183 + 0.861838i $$0.330687\pi$$
$$312$$ 0 0
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 32.0000 1.79730 0.898650 0.438667i $$-0.144549\pi$$
0.898650 + 0.438667i $$0.144549\pi$$
$$318$$ 0 0
$$319$$ 8.94427 0.500783
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ 0 0
$$323$$ 6.70820 0.373254
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.23607i 0.122905i 0.998110 + 0.0614527i $$0.0195733\pi$$
−0.998110 + 0.0614527i $$0.980427\pi$$
$$332$$ 0 0
$$333$$ −16.0000 −0.876795
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 33.0000i − 1.79762i −0.438334 0.898812i $$-0.644431\pi$$
0.438334 0.898812i $$-0.355569\pi$$
$$338$$ 0 0
$$339$$ − 2.23607i − 0.121447i
$$340$$ 0 0
$$341$$ 20.0000i 1.08306i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 33.5410 1.80058 0.900288 0.435294i $$-0.143356\pi$$
0.900288 + 0.435294i $$0.143356\pi$$
$$348$$ 0 0
$$349$$ 24.0000i 1.28469i 0.766415 + 0.642345i $$0.222038\pi$$
−0.766415 + 0.642345i $$0.777962\pi$$
$$350$$ 0 0
$$351$$ −8.94427 −0.477410
$$352$$ 0 0
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −17.8885 −0.944121 −0.472061 0.881566i $$-0.656490\pi$$
−0.472061 + 0.881566i $$0.656490\pi$$
$$360$$ 0 0
$$361$$ 14.0000 0.736842
$$362$$ 0 0
$$363$$ −13.4164 −0.704179
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 16.0000i 0.824042i
$$378$$ 0 0
$$379$$ 29.0689i 1.49317i 0.665291 + 0.746584i $$0.268307\pi$$
−0.665291 + 0.746584i $$0.731693\pi$$
$$380$$ 0 0
$$381$$ 20.0000i 1.02463i
$$382$$ 0 0
$$383$$ 17.8885i 0.914062i 0.889451 + 0.457031i $$0.151087\pi$$
−0.889451 + 0.457031i $$0.848913\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −17.8885 −0.909326
$$388$$ 0 0
$$389$$ 20.0000i 1.01404i 0.861934 + 0.507020i $$0.169253\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 0 0
$$391$$ 26.8328 1.35699
$$392$$ 0 0
$$393$$ 20.0000i 1.00887i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 0 0
$$403$$ −35.7771 −1.78218
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 17.8885i − 0.886702i
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 29.0689i 1.43386i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 45.0000i 2.20366i
$$418$$ 0 0
$$419$$ 20.1246i 0.983152i 0.870835 + 0.491576i $$0.163579\pi$$
−0.870835 + 0.491576i $$0.836421\pi$$
$$420$$ 0 0
$$421$$ − 32.0000i − 1.55958i −0.626038 0.779792i $$-0.715325\pi$$
0.626038 0.779792i $$-0.284675\pi$$
$$422$$ 0 0
$$423$$ − 17.8885i − 0.869771i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 20.0000i 0.965609i
$$430$$ 0 0
$$431$$ −17.8885 −0.861661 −0.430830 0.902433i $$-0.641779\pi$$
−0.430830 + 0.902433i $$0.641779\pi$$
$$432$$ 0 0
$$433$$ 11.0000i 0.528626i 0.964437 + 0.264313i $$0.0851452\pi$$
−0.964437 + 0.264313i $$0.914855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −20.0000 −0.956730
$$438$$ 0 0
$$439$$ −26.8328 −1.28066 −0.640330 0.768100i $$-0.721202\pi$$
−0.640330 + 0.768100i $$0.721202\pi$$
$$440$$ 0 0
$$441$$ 14.0000 0.666667
$$442$$ 0 0
$$443$$ −11.1803 −0.531194 −0.265597 0.964084i $$-0.585569\pi$$
−0.265597 + 0.964084i $$0.585569\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 44.7214i 2.11525i
$$448$$ 0 0
$$449$$ 11.0000 0.519122 0.259561 0.965727i $$-0.416422\pi$$
0.259561 + 0.965727i $$0.416422\pi$$
$$450$$ 0 0
$$451$$ − 11.1803i − 0.526462i
$$452$$ 0 0
$$453$$ 40.0000 1.87936
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 37.0000i − 1.73079i −0.501093 0.865393i $$-0.667069\pi$$
0.501093 0.865393i $$-0.332931\pi$$
$$458$$ 0 0
$$459$$ − 6.70820i − 0.313112i
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 35.7771i 1.66270i 0.555748 + 0.831351i $$0.312432\pi$$
−0.555748 + 0.831351i $$0.687568\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.94427 0.413892 0.206946 0.978352i $$-0.433648\pi$$
0.206946 + 0.978352i $$0.433648\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 26.8328 1.23639
$$472$$ 0 0
$$473$$ − 20.0000i − 0.919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 0 0
$$479$$ 26.8328 1.22602 0.613011 0.790074i $$-0.289958\pi$$
0.613011 + 0.790074i $$0.289958\pi$$
$$480$$ 0 0
$$481$$ 32.0000 1.45907
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 26.8328i − 1.21591i −0.793971 0.607955i $$-0.791990\pi$$
0.793971 0.607955i $$-0.208010\pi$$
$$488$$ 0 0
$$489$$ −25.0000 −1.13054
$$490$$ 0 0
$$491$$ 26.8328i 1.21095i 0.795865 + 0.605474i $$0.207016\pi$$
−0.795865 + 0.605474i $$0.792984\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 26.8328i − 1.20120i −0.799549 0.600601i $$-0.794928\pi$$
0.799549 0.600601i $$-0.205072\pi$$
$$500$$ 0 0
$$501$$ − 20.0000i − 0.893534i
$$502$$ 0 0
$$503$$ 17.8885i 0.797611i 0.917036 + 0.398805i $$0.130575\pi$$
−0.917036 + 0.398805i $$0.869425\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −6.70820 −0.297922
$$508$$ 0 0
$$509$$ 4.00000i 0.177297i 0.996063 + 0.0886484i $$0.0282548\pi$$
−0.996063 + 0.0886484i $$0.971745\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 5.00000i 0.220755i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 20.0000 0.879599
$$518$$ 0 0
$$519$$ 53.6656 2.35566
$$520$$ 0 0
$$521$$ 3.00000 0.131432 0.0657162 0.997838i $$-0.479067\pi$$
0.0657162 + 0.997838i $$0.479067\pi$$
$$522$$ 0 0
$$523$$ −33.5410 −1.46665 −0.733323 0.679880i $$-0.762032\pi$$
−0.733323 + 0.679880i $$0.762032\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 26.8328i − 1.16886i
$$528$$ 0 0
$$529$$ −57.0000 −2.47826
$$530$$ 0 0
$$531$$ 17.8885i 0.776297i
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 35.0000i 1.51036i
$$538$$ 0 0
$$539$$ 15.6525i 0.674200i
$$540$$ 0 0
$$541$$ − 40.0000i − 1.71973i −0.510518 0.859867i $$-0.670546\pi$$
0.510518 0.859867i $$-0.329454\pi$$
$$542$$ 0 0
$$543$$ 44.7214i 1.91918i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.1246 0.860466 0.430233 0.902718i $$-0.358431\pi$$
0.430233 + 0.902718i $$0.358431\pi$$
$$548$$ 0 0
$$549$$ 16.0000i 0.682863i
$$550$$ 0 0
$$551$$ 8.94427 0.381039
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 0 0
$$559$$ 35.7771 1.51321
$$560$$ 0 0
$$561$$ −15.0000 −0.633300
$$562$$ 0 0
$$563$$ −8.94427 −0.376956 −0.188478 0.982077i $$-0.560355\pi$$
−0.188478 + 0.982077i $$0.560355\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −11.0000 −0.461144 −0.230572 0.973055i $$-0.574060\pi$$
−0.230572 + 0.973055i $$0.574060\pi$$
$$570$$ 0 0
$$571$$ − 44.7214i − 1.87153i −0.352623 0.935765i $$-0.614710\pi$$
0.352623 0.935765i $$-0.385290\pi$$
$$572$$ 0 0
$$573$$ 20.0000 0.835512
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.0000i 0.957503i 0.877951 + 0.478751i $$0.158910\pi$$
−0.877951 + 0.478751i $$0.841090\pi$$
$$578$$ 0 0
$$579$$ 46.9574i 1.95148i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8.94427i 0.370434i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −6.70820 −0.276877 −0.138439 0.990371i $$-0.544208\pi$$
−0.138439 + 0.990371i $$0.544208\pi$$
$$588$$ 0 0
$$589$$ 20.0000i 0.824086i
$$590$$ 0 0
$$591$$ 26.8328 1.10375
$$592$$ 0 0
$$593$$ 9.00000i 0.369586i 0.982777 + 0.184793i $$0.0591614\pi$$
−0.982777 + 0.184793i $$0.940839\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 40.0000 1.63709
$$598$$ 0 0
$$599$$ −35.7771 −1.46181 −0.730906 0.682478i $$-0.760902\pi$$
−0.730906 + 0.682478i $$0.760902\pi$$
$$600$$ 0 0
$$601$$ 25.0000 1.01977 0.509886 0.860242i $$-0.329688\pi$$
0.509886 + 0.860242i $$0.329688\pi$$
$$602$$ 0 0
$$603$$ 13.4164 0.546358
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 17.8885i − 0.726074i −0.931775 0.363037i $$-0.881740\pi$$
0.931775 0.363037i $$-0.118260\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 35.7771i 1.44739i
$$612$$ 0 0
$$613$$ −4.00000 −0.161558 −0.0807792 0.996732i $$-0.525741\pi$$
−0.0807792 + 0.996732i $$0.525741\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 0 0
$$619$$ 8.94427i 0.359501i 0.983712 + 0.179750i $$0.0575290\pi$$
−0.983712 + 0.179750i $$0.942471\pi$$
$$620$$ 0 0
$$621$$ 20.0000i 0.802572i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 11.1803 0.446500
$$628$$ 0 0
$$629$$ 24.0000i 0.956943i
$$630$$ 0 0
$$631$$ 8.94427 0.356066 0.178033 0.984025i $$-0.443027\pi$$
0.178033 + 0.984025i $$0.443027\pi$$
$$632$$ 0 0
$$633$$ 15.0000i 0.596196i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −28.0000 −1.10940
$$638$$ 0 0
$$639$$ −17.8885 −0.707660
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −8.94427 −0.352728 −0.176364 0.984325i $$-0.556434\pi$$
−0.176364 + 0.984325i $$0.556434\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 17.8885i − 0.703271i −0.936137 0.351636i $$-0.885626\pi$$
0.936137 0.351636i $$-0.114374\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 18.0000i − 0.702247i
$$658$$ 0 0
$$659$$ 6.70820i 0.261315i 0.991428 + 0.130657i $$0.0417087\pi$$
−0.991428 + 0.130657i $$0.958291\pi$$
$$660$$ 0 0
$$661$$ − 28.0000i − 1.08907i −0.838737 0.544537i $$-0.816705\pi$$
0.838737 0.544537i $$-0.183295\pi$$
$$662$$ 0 0
$$663$$ − 26.8328i − 1.04210i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 35.7771 1.38529
$$668$$ 0 0
$$669$$ 40.0000i 1.54649i
$$670$$ 0 0
$$671$$ −17.8885 −0.690580
$$672$$ 0 0
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8.00000 0.307465 0.153732 0.988113i $$-0.450871\pi$$
0.153732 + 0.988113i $$0.450871\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ −15.6525 −0.598925 −0.299463 0.954108i $$-0.596807\pi$$
−0.299463 + 0.954108i $$0.596807\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 35.7771i 1.36498i
$$688$$ 0 0
$$689$$ −16.0000 −0.609551
$$690$$ 0 0
$$691$$ 15.6525i 0.595448i 0.954652 + 0.297724i $$0.0962276\pi$$
−0.954652 + 0.297724i $$0.903772\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15.0000i 0.568166i
$$698$$ 0 0
$$699$$ 13.4164i 0.507455i
$$700$$ 0 0
$$701$$ 12.0000i 0.453234i 0.973984 + 0.226617i $$0.0727665\pi$$
−0.973984 + 0.226617i $$0.927233\pi$$
$$702$$ 0 0
$$703$$ − 17.8885i − 0.674679i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 16.0000i 0.600893i 0.953799 + 0.300446i $$0.0971356\pi$$
−0.953799 + 0.300446i $$0.902864\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 80.0000i 2.99602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.0000 0.746914
$$718$$ 0 0
$$719$$ 17.8885 0.667130 0.333565 0.942727i $$-0.391748\pi$$
0.333565 + 0.942727i $$0.391748\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −11.1803 −0.415801
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.8328i 0.995174i 0.867414 + 0.497587i $$0.165780\pi$$
−0.867414 + 0.497587i $$0.834220\pi$$
$$728$$ 0 0
$$729$$ −7.00000 −0.259259
$$730$$ 0 0
$$731$$ 26.8328i 0.992448i
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15.0000i 0.552532i
$$738$$ 0 0
$$739$$ − 26.8328i − 0.987061i −0.869728 0.493531i $$-0.835706\pi$$
0.869728 0.493531i $$-0.164294\pi$$
$$740$$ 0 0
$$741$$ 20.0000i 0.734718i
$$742$$ 0 0
$$743$$ 8.94427i 0.328134i 0.986449 + 0.164067i $$0.0524612\pi$$
−0.986449 + 0.164067i $$0.947539\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −13.4164 −0.490881
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26.8328 −0.979143 −0.489572 0.871963i $$-0.662847\pi$$
−0.489572 + 0.871963i $$0.662847\pi$$
$$752$$ 0 0
$$753$$ 65.0000i 2.36873i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ 0 0
$$759$$ 44.7214 1.62328
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 35.7771i − 1.29184i
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ − 40.2492i − 1.44954i
$$772$$ 0 0
$$773$$ −4.00000 −0.143870 −0.0719350 0.997409i $$-0.522917\pi$$
−0.0719350 + 0.997409i $$0.522917\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 11.1803i − 0.400577i
$$780$$ 0 0
$$781$$ − 20.0000i − 0.715656i
$$782$$ 0 0
$$783$$ − 8.94427i − 0.319642i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −26.8328 −0.956487 −0.478243 0.878227i $$-0.658726\pi$$
−0.478243 + 0.878227i $$0.658726\pi$$
$$788$$ 0 0
$$789$$ 20.0000i 0.712019i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 32.0000i − 1.13635i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 52.0000 1.84193 0.920967 0.389640i $$-0.127401\pi$$
0.920967 + 0.389640i $$0.127401\pi$$
$$798$$ 0 0
$$799$$ −26.8328 −0.949277
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 20.1246 0.710182
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 26.8328i 0.942228i 0.882072 + 0.471114i $$0.156148\pi$$
−0.882072 + 0.471114i $$0.843852\pi$$
$$812$$ 0 0
$$813$$ 60.0000 2.10429
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 20.0000i − 0.699711i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 28.0000i 0.977207i 0.872506 + 0.488603i $$0.162493\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 0 0
$$823$$ 35.7771i 1.24711i 0.781779 + 0.623555i $$0.214312\pi$$
−0.781779 + 0.623555i $$0.785688\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −24.5967 −0.855313 −0.427656 0.903941i $$-0.640661\pi$$
−0.427656 + 0.903941i $$0.640661\pi$$
$$828$$ 0 0
$$829$$ 20.0000i 0.694629i 0.937749 + 0.347314i $$0.112906\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 0 0
$$831$$ 26.8328 0.930820
$$832$$ 0 0
$$833$$ − 21.0000i − 0.727607i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ −35.7771 −1.23516 −0.617581 0.786507i $$-0.711887\pi$$
−0.617581 + 0.786507i $$0.711887\pi$$
$$840$$ 0 0
$$841$$ 13.0000 0.448276
$$842$$ 0 0
$$843$$ 22.3607 0.770143
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −55.0000 −1.88760
$$850$$ 0 0
$$851$$ − 71.5542i − 2.45285i
$$852$$ 0 0
$$853$$ −4.00000 −0.136957 −0.0684787 0.997653i $$-0.521815\pi$$
−0.0684787 + 0.997653i $$0.521815\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.0000i 0.922302i 0.887322 + 0.461151i $$0.152563\pi$$
−0.887322 + 0.461151i $$0.847437\pi$$
$$858$$ 0 0
$$859$$ − 38.0132i − 1.29699i −0.761218 0.648496i $$-0.775398\pi$$
0.761218 0.648496i $$-0.224602\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 26.8328i − 0.913400i −0.889621 0.456700i $$-0.849031\pi$$
0.889621 0.456700i $$-0.150969\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −17.8885 −0.607527
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −26.8328 −0.909195
$$872$$ 0 0
$$873$$ − 4.00000i − 0.135379i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 0 0
$$879$$ −53.6656 −1.81010
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 33.5410 1.12875 0.564373 0.825520i $$-0.309118\pi$$
0.564373 + 0.825520i $$0.309118\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 17.8885i 0.600639i 0.953839 + 0.300319i $$0.0970932\pi$$
−0.953839 + 0.300319i $$0.902907\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 24.5967i − 0.824022i
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 80.0000i 2.67112i
$$898$$ 0 0
$$899$$ − 35.7771i − 1.19323i
$$900$$ 0 0
$$901$$ − 12.0000i − 0.399778i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 8.94427 0.296990 0.148495 0.988913i $$-0.452557\pi$$
0.148495 + 0.988913i $$0.452557\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 35.7771 1.18535 0.592674 0.805443i $$-0.298072\pi$$
0.592674 + 0.805443i $$0.298072\pi$$
$$912$$ 0 0
$$913$$ − 15.0000i − 0.496428i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.94427 0.295044 0.147522 0.989059i $$-0.452870\pi$$
0.147522 + 0.989059i $$0.452870\pi$$
$$920$$ 0 0
$$921$$ −5.00000 −0.164756
$$922$$ 0 0
$$923$$ 35.7771 1.17762
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 17.8885i 0.587537i
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ 15.6525i 0.512989i
$$932$$ 0 0
$$933$$ −40.0000 −1.30954
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 23.0000i − 0.751377i −0.926746 0.375689i $$-0.877406\pi$$
0.926746 0.375689i $$-0.122594\pi$$
$$938$$ 0 0
$$939$$ − 13.4164i − 0.437828i
$$940$$ 0 0
$$941$$ − 60.0000i − 1.95594i −0.208736 0.977972i $$-0.566935\pi$$
0.208736 0.977972i $$-0.433065\pi$$
$$942$$ 0 0
$$943$$ − 44.7214i − 1.45633i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.94427 0.290650 0.145325 0.989384i $$-0.453577\pi$$
0.145325 + 0.989384i $$0.453577\pi$$
$$948$$ 0 0
$$949$$ 36.0000i 1.16861i
$$950$$ 0 0
$$951$$ −71.5542 −2.32030
$$952$$ 0 0
$$953$$ 21.0000i 0.680257i 0.940379 + 0.340128i $$0.110471\pi$$
−0.940379 + 0.340128i $$0.889529\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −20.0000 −0.646508
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 49.0000 1.58065
$$962$$ 0 0
$$963$$ −4.47214 −0.144113
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 53.6656i 1.72577i 0.505400 + 0.862885i $$0.331345\pi$$
−0.505400 + 0.862885i $$0.668655\pi$$
$$968$$ 0 0
$$969$$ −15.0000 −0.481869
$$970$$ 0 0
$$971$$ − 51.4296i − 1.65045i −0.564802 0.825227i $$-0.691047\pi$$
0.564802 0.825227i $$-0.308953\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 27.0000i − 0.863807i −0.901920 0.431903i $$-0.857842\pi$$
0.901920 0.431903i $$-0.142158\pi$$
$$978$$ 0 0
$$979$$ 33.5410i 1.07198i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 17.8885i − 0.570556i −0.958445 0.285278i $$-0.907914\pi$$
0.958445 0.285278i $$-0.0920859\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 80.0000i − 2.54385i
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ − 5.00000i − 0.158670i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −8.00000 −0.253363 −0.126681 0.991943i $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ 0 0
$$999$$ −17.8885 −0.565968
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 3200.2.f.m.449.2 4
4.3 odd 2 inner 3200.2.f.m.449.3 4
5.2 odd 4 3200.2.d.p.1601.4 yes 4
5.3 odd 4 3200.2.d.o.1601.1 4
5.4 even 2 3200.2.f.n.449.4 4
8.3 odd 2 3200.2.f.n.449.2 4
8.5 even 2 3200.2.f.n.449.3 4
20.3 even 4 3200.2.d.o.1601.3 yes 4
20.7 even 4 3200.2.d.p.1601.2 yes 4
20.19 odd 2 3200.2.f.n.449.1 4
40.3 even 4 3200.2.d.o.1601.2 yes 4
40.13 odd 4 3200.2.d.o.1601.4 yes 4
40.19 odd 2 inner 3200.2.f.m.449.4 4
40.27 even 4 3200.2.d.p.1601.3 yes 4
40.29 even 2 inner 3200.2.f.m.449.1 4
40.37 odd 4 3200.2.d.p.1601.1 yes 4
80.3 even 4 6400.2.a.bs.1.1 2
80.13 odd 4 6400.2.a.bs.1.2 2
80.27 even 4 6400.2.a.bt.1.1 2
80.37 odd 4 6400.2.a.bt.1.2 2
80.43 even 4 6400.2.a.bq.1.2 2
80.53 odd 4 6400.2.a.bq.1.1 2
80.67 even 4 6400.2.a.br.1.2 2
80.77 odd 4 6400.2.a.br.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.o.1601.1 4 5.3 odd 4
3200.2.d.o.1601.2 yes 4 40.3 even 4
3200.2.d.o.1601.3 yes 4 20.3 even 4
3200.2.d.o.1601.4 yes 4 40.13 odd 4
3200.2.d.p.1601.1 yes 4 40.37 odd 4
3200.2.d.p.1601.2 yes 4 20.7 even 4
3200.2.d.p.1601.3 yes 4 40.27 even 4
3200.2.d.p.1601.4 yes 4 5.2 odd 4
3200.2.f.m.449.1 4 40.29 even 2 inner
3200.2.f.m.449.2 4 1.1 even 1 trivial
3200.2.f.m.449.3 4 4.3 odd 2 inner
3200.2.f.m.449.4 4 40.19 odd 2 inner
3200.2.f.n.449.1 4 20.19 odd 2
3200.2.f.n.449.2 4 8.3 odd 2
3200.2.f.n.449.3 4 8.5 even 2
3200.2.f.n.449.4 4 5.4 even 2
6400.2.a.bq.1.1 2 80.53 odd 4
6400.2.a.bq.1.2 2 80.43 even 4
6400.2.a.br.1.1 2 80.77 odd 4
6400.2.a.br.1.2 2 80.67 even 4
6400.2.a.bs.1.1 2 80.3 even 4
6400.2.a.bs.1.2 2 80.13 odd 4
6400.2.a.bt.1.1 2 80.27 even 4
6400.2.a.bt.1.2 2 80.37 odd 4