# Properties

 Label 1800.2.f.k.649.1 Level $1800$ Weight $2$ Character 1800.649 Analytic conductor $14.373$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1800.649 Dual form 1800.2.f.k.649.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.00000i q^{7} +O(q^{10})$$ $$q-5.00000i q^{7} +6.00000 q^{11} +3.00000i q^{13} +2.00000i q^{17} -1.00000 q^{19} -2.00000i q^{23} +6.00000 q^{29} +3.00000 q^{31} -6.00000i q^{37} -4.00000 q^{41} -11.0000i q^{43} +10.0000i q^{47} -18.0000 q^{49} -8.00000i q^{53} -6.00000 q^{59} +3.00000 q^{61} -1.00000i q^{67} +12.0000 q^{71} -10.0000i q^{73} -30.0000i q^{77} +8.00000 q^{79} -6.00000i q^{83} -16.0000 q^{89} +15.0000 q^{91} -7.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 12q^{11} - 2q^{19} + 12q^{29} + 6q^{31} - 8q^{41} - 36q^{49} - 12q^{59} + 6q^{61} + 24q^{71} + 16q^{79} - 32q^{89} + 30q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 5.00000i − 1.88982i −0.327327 0.944911i $$-0.606148\pi$$
0.327327 0.944911i $$-0.393852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ 3.00000i 0.832050i 0.909353 + 0.416025i $$0.136577\pi$$
−0.909353 + 0.416025i $$0.863423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 2.00000i − 0.417029i −0.978019 0.208514i $$-0.933137\pi$$
0.978019 0.208514i $$-0.0668628\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ − 11.0000i − 1.67748i −0.544529 0.838742i $$-0.683292\pi$$
0.544529 0.838742i $$-0.316708\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 0 0
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 8.00000i − 1.09888i −0.835532 0.549442i $$-0.814840\pi$$
0.835532 0.549442i $$-0.185160\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 1.00000i − 0.122169i −0.998133 0.0610847i $$-0.980544\pi$$
0.998133 0.0610847i $$-0.0194560\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 30.0000i − 3.41882i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 15.0000 1.57243
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.00000i − 0.710742i −0.934725 0.355371i $$-0.884354\pi$$
0.934725 0.355371i $$-0.115646\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 10.0000 0.916698
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 5.00000i 0.433555i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 18.0000i 1.50524i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 9.00000 0.732410 0.366205 0.930534i $$-0.380657\pi$$
0.366205 + 0.930534i $$0.380657\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.00000i 0.558661i 0.960195 + 0.279330i $$0.0901125\pi$$
−0.960195 + 0.279330i $$0.909888\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −10.0000 −0.788110
$$162$$ 0 0
$$163$$ 7.00000i 0.548282i 0.961689 + 0.274141i $$0.0883936\pi$$
−0.961689 + 0.274141i $$0.911606\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 0 0
$$169$$ 4.00000 0.307692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −19.0000 −1.41226 −0.706129 0.708083i $$-0.749560\pi$$
−0.706129 + 0.708083i $$0.749560\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ − 3.00000i − 0.215945i −0.994154 0.107972i $$-0.965564\pi$$
0.994154 0.107972i $$-0.0344358\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 0 0
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 30.0000i − 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 15.0000i − 1.01827i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 3.00000i − 0.200895i −0.994942 0.100447i $$-0.967973\pi$$
0.994942 0.100447i $$-0.0320274\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ 3.00000 0.198246 0.0991228 0.995075i $$-0.468396\pi$$
0.0991228 + 0.995075i $$0.468396\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 20.0000i 1.31024i 0.755523 + 0.655122i $$0.227383\pi$$
−0.755523 + 0.655122i $$0.772617\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3.00000i − 0.190885i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ − 12.0000i − 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.00000i 0.0600842i 0.999549 + 0.0300421i $$0.00956413\pi$$
−0.999549 + 0.0300421i $$0.990436\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ − 13.0000i − 0.772770i −0.922338 0.386385i $$-0.873724\pi$$
0.922338 0.386385i $$-0.126276\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.0000i 1.18056i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 2.00000i − 0.116841i −0.998292 0.0584206i $$-0.981394\pi$$
0.998292 0.0584206i $$-0.0186065\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −55.0000 −3.17015
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 13.0000i 0.741949i 0.928643 + 0.370975i $$0.120976\pi$$
−0.928643 + 0.370975i $$0.879024\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 29.0000i 1.63918i 0.572953 + 0.819588i $$0.305798\pi$$
−0.572953 + 0.819588i $$0.694202\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.0000i 0.898650i 0.893368 + 0.449325i $$0.148335\pi$$
−0.893368 + 0.449325i $$0.851665\pi$$
$$318$$ 0 0
$$319$$ 36.0000 2.01561
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 2.00000i − 0.111283i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 50.0000 2.75659
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 23.0000i − 1.25289i −0.779466 0.626445i $$-0.784509\pi$$
0.779466 0.626445i $$-0.215491\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ 55.0000i 2.96972i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000i 0.678594i 0.940679 + 0.339297i $$0.110189\pi$$
−0.940679 + 0.339297i $$0.889811\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −40.0000 −2.07670
$$372$$ 0 0
$$373$$ 25.0000i 1.29445i 0.762299 + 0.647225i $$0.224071\pi$$
−0.762299 + 0.647225i $$0.775929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 18.0000i 0.927047i
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.0000i 0.552074i 0.961147 + 0.276037i $$0.0890213\pi$$
−0.961147 + 0.276037i $$0.910979\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 9.00000i 0.448322i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 36.0000i − 1.78445i
$$408$$ 0 0
$$409$$ 11.0000 0.543915 0.271957 0.962309i $$-0.412329\pi$$
0.271957 + 0.962309i $$0.412329\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 30.0000i 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 32.0000 1.56330 0.781651 0.623716i $$-0.214378\pi$$
0.781651 + 0.623716i $$0.214378\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 15.0000i − 0.725901i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −22.0000 −1.05970 −0.529851 0.848091i $$-0.677752\pi$$
−0.529851 + 0.848091i $$0.677752\pi$$
$$432$$ 0 0
$$433$$ − 19.0000i − 0.913082i −0.889702 0.456541i $$-0.849088\pi$$
0.889702 0.456541i $$-0.150912\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.00000i 0.0956730i
$$438$$ 0 0
$$439$$ 5.00000 0.238637 0.119318 0.992856i $$-0.461929\pi$$
0.119318 + 0.992856i $$0.461929\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4.00000 −0.188772 −0.0943858 0.995536i $$-0.530089\pi$$
−0.0943858 + 0.995536i $$0.530089\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 16.0000 0.745194 0.372597 0.927993i $$-0.378467\pi$$
0.372597 + 0.927993i $$0.378467\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.0000i 0.647843i 0.946084 + 0.323921i $$0.105001\pi$$
−0.946084 + 0.323921i $$0.894999\pi$$
$$468$$ 0 0
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 66.0000i − 3.03468i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 18.0000 0.820729
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 3.00000i − 0.135943i −0.997687 0.0679715i $$-0.978347\pi$$
0.997687 0.0679715i $$-0.0216527\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 60.0000i − 2.69137i
$$498$$ 0 0
$$499$$ −17.0000 −0.761025 −0.380512 0.924776i $$-0.624252\pi$$
−0.380512 + 0.924776i $$0.624252\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 36.0000i 1.60516i 0.596544 + 0.802580i $$0.296540\pi$$
−0.596544 + 0.802580i $$0.703460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −50.0000 −2.21187
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 60.0000i 2.63880i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 0 0
$$523$$ − 21.0000i − 0.918266i −0.888368 0.459133i $$-0.848160\pi$$
0.888368 0.459133i $$-0.151840\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000i 0.261364i
$$528$$ 0 0
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −108.000 −4.65189
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 16.0000i 0.684111i 0.939680 + 0.342055i $$0.111123\pi$$
−0.939680 + 0.342055i $$0.888877\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ − 40.0000i − 1.70097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14.0000i 0.593199i 0.955002 + 0.296600i $$0.0958526\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$558$$ 0 0
$$559$$ 33.0000 1.39575
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 1.00000 0.0418487 0.0209243 0.999781i $$-0.493339\pi$$
0.0209243 + 0.999781i $$0.493339\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 43.0000i 1.79011i 0.445952 + 0.895057i $$0.352865\pi$$
−0.445952 + 0.895057i $$0.647135\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −30.0000 −1.24461
$$582$$ 0 0
$$583$$ − 48.0000i − 1.98796i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 24.0000i − 0.990586i −0.868726 0.495293i $$-0.835061\pi$$
0.868726 0.495293i $$-0.164939\pi$$
$$588$$ 0 0
$$589$$ −3.00000 −0.123613
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4.00000i 0.164260i 0.996622 + 0.0821302i $$0.0261723\pi$$
−0.996622 + 0.0821302i $$0.973828\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 32.0000i 1.28827i 0.764911 + 0.644136i $$0.222783\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 0 0
$$619$$ 19.0000 0.763674 0.381837 0.924230i $$-0.375291\pi$$
0.381837 + 0.924230i $$0.375291\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 80.0000i 3.20513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 31.0000 1.23409 0.617045 0.786928i $$-0.288330\pi$$
0.617045 + 0.786928i $$0.288330\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 54.0000i − 2.13956i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −28.0000 −1.09073 −0.545363 0.838200i $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 12.0000i − 0.464642i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 18.0000 0.694882
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ −35.0000 −1.34318
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 8.00000i − 0.303022i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ 6.00000i 0.226294i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 40.0000i − 1.50435i
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 6.00000i − 0.224702i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 13.0000i − 0.482143i −0.970507 0.241072i $$-0.922501\pi$$
0.970507 0.241072i $$-0.0774989\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 22.0000 0.813699
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 6.00000i − 0.221013i
$$738$$ 0 0
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 12.0000i − 0.440237i −0.975473 0.220119i $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 40.0000 1.46157
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 3.00000i − 0.109037i −0.998513 0.0545184i $$-0.982638\pi$$
0.998513 0.0545184i $$-0.0173624\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 52.0000 1.88500 0.942499 0.334208i $$-0.108469\pi$$
0.942499 + 0.334208i $$0.108469\pi$$
$$762$$ 0 0
$$763$$ − 35.0000i − 1.26709i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 18.0000i − 0.649942i
$$768$$ 0 0
$$769$$ −27.0000 −0.973645 −0.486822 0.873501i $$-0.661844\pi$$
−0.486822 + 0.873501i $$0.661844\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 28.0000i 1.00709i 0.863969 + 0.503545i $$0.167971\pi$$
−0.863969 + 0.503545i $$0.832029\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ 72.0000 2.57636
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 3.00000i − 0.106938i −0.998569 0.0534692i $$-0.982972\pi$$
0.998569 0.0534692i $$-0.0170279\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 60.0000 2.13335
$$792$$ 0 0
$$793$$ 9.00000i 0.319599i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 28.0000i − 0.991811i −0.868377 0.495905i $$-0.834836\pi$$
0.868377 0.495905i $$-0.165164\pi$$
$$798$$ 0 0
$$799$$ −20.0000 −0.707549
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 60.0000i − 2.11735i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −24.0000 −0.843795 −0.421898 0.906644i $$-0.638636\pi$$
−0.421898 + 0.906644i $$0.638636\pi$$
$$810$$ 0 0
$$811$$ 9.00000 0.316033 0.158016 0.987436i $$-0.449490\pi$$
0.158016 + 0.987436i $$0.449490\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 11.0000i 0.384841i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 49.0000i 1.70803i 0.520246 + 0.854016i $$0.325840\pi$$
−0.520246 + 0.854016i $$0.674160\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.0000i 1.11275i 0.830932 + 0.556375i $$0.187808\pi$$
−0.830932 + 0.556375i $$0.812192\pi$$
$$828$$ 0 0
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 36.0000i − 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 125.000i − 4.29505i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ − 55.0000i − 1.88316i −0.336784 0.941582i $$-0.609339\pi$$
0.336784 0.941582i $$-0.390661\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 52.0000i 1.77629i 0.459567 + 0.888143i $$0.348005\pi$$
−0.459567 + 0.888143i $$0.651995\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 8.00000i − 0.272323i −0.990687 0.136162i $$-0.956523\pi$$
0.990687 0.136162i $$-0.0434766\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 17.0000i − 0.574049i −0.957923 0.287025i $$-0.907334\pi$$
0.957923 0.287025i $$-0.0926662\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 40.0000 1.34763 0.673817 0.738898i $$-0.264654\pi$$
0.673817 + 0.738898i $$0.264654\pi$$
$$882$$ 0 0
$$883$$ − 19.0000i − 0.639401i −0.947519 0.319700i $$-0.896418\pi$$
0.947519 0.319700i $$-0.103582\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 42.0000i − 1.41022i −0.709097 0.705111i $$-0.750897\pi$$
0.709097 0.705111i $$-0.249103\pi$$
$$888$$ 0 0
$$889$$ 40.0000 1.34156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 10.0000i − 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 18.0000 0.600334
$$900$$ 0 0
$$901$$ 16.0000 0.533037
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 14.0000 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$912$$ 0 0
$$913$$ − 36.0000i − 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 80.0000i 2.64183i
$$918$$ 0 0
$$919$$ −1.00000 −0.0329870 −0.0164935 0.999864i $$-0.505250\pi$$
−0.0164935 + 0.999864i $$0.505250\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 36.0000i 1.18495i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.0000i 0.751377i 0.926746 + 0.375689i $$0.122594\pi$$
−0.926746 + 0.375689i $$0.877406\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ 0 0
$$943$$ 8.00000i 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000i 0.0649913i 0.999472 + 0.0324956i $$0.0103455\pi$$
−0.999472 + 0.0324956i $$0.989654\pi$$
$$948$$ 0 0
$$949$$ 30.0000 0.973841
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −46.0000 −1.47621 −0.738105 0.674686i $$-0.764279\pi$$
−0.738105 + 0.674686i $$0.764279\pi$$
$$972$$ 0 0
$$973$$ − 60.0000i − 1.92351i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 10.0000i − 0.319928i −0.987123 0.159964i $$-0.948862\pi$$
0.987123 0.159964i $$-0.0511379\pi$$
$$978$$ 0 0
$$979$$ −96.0000 −3.06817
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −22.0000 −0.699559
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 54.0000i − 1.71020i −0.518465 0.855099i $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1800.2.f.k.649.1 2
3.2 odd 2 600.2.f.a.49.1 2
4.3 odd 2 3600.2.f.b.2449.2 2
5.2 odd 4 1800.2.a.x.1.1 1
5.3 odd 4 1800.2.a.a.1.1 1
5.4 even 2 inner 1800.2.f.k.649.2 2
12.11 even 2 1200.2.f.i.49.2 2
15.2 even 4 600.2.a.e.1.1 1
15.8 even 4 600.2.a.f.1.1 yes 1
15.14 odd 2 600.2.f.a.49.2 2
20.3 even 4 3600.2.a.bq.1.1 1
20.7 even 4 3600.2.a.a.1.1 1
20.19 odd 2 3600.2.f.b.2449.1 2
24.5 odd 2 4800.2.f.bj.3649.2 2
24.11 even 2 4800.2.f.a.3649.1 2
60.23 odd 4 1200.2.a.i.1.1 1
60.47 odd 4 1200.2.a.j.1.1 1
60.59 even 2 1200.2.f.i.49.1 2
120.29 odd 2 4800.2.f.bj.3649.1 2
120.53 even 4 4800.2.a.b.1.1 1
120.59 even 2 4800.2.f.a.3649.2 2
120.77 even 4 4800.2.a.ct.1.1 1
120.83 odd 4 4800.2.a.cs.1.1 1
120.107 odd 4 4800.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.e.1.1 1 15.2 even 4
600.2.a.f.1.1 yes 1 15.8 even 4
600.2.f.a.49.1 2 3.2 odd 2
600.2.f.a.49.2 2 15.14 odd 2
1200.2.a.i.1.1 1 60.23 odd 4
1200.2.a.j.1.1 1 60.47 odd 4
1200.2.f.i.49.1 2 60.59 even 2
1200.2.f.i.49.2 2 12.11 even 2
1800.2.a.a.1.1 1 5.3 odd 4
1800.2.a.x.1.1 1 5.2 odd 4
1800.2.f.k.649.1 2 1.1 even 1 trivial
1800.2.f.k.649.2 2 5.4 even 2 inner
3600.2.a.a.1.1 1 20.7 even 4
3600.2.a.bq.1.1 1 20.3 even 4
3600.2.f.b.2449.1 2 20.19 odd 2
3600.2.f.b.2449.2 2 4.3 odd 2
4800.2.a.a.1.1 1 120.107 odd 4
4800.2.a.b.1.1 1 120.53 even 4
4800.2.a.cs.1.1 1 120.83 odd 4
4800.2.a.ct.1.1 1 120.77 even 4
4800.2.f.a.3649.1 2 24.11 even 2
4800.2.f.a.3649.2 2 120.59 even 2
4800.2.f.bj.3649.1 2 120.29 odd 2
4800.2.f.bj.3649.2 2 24.5 odd 2