# Properties

 Label 1792.2.b.f.897.2 Level $1792$ Weight $2$ Character 1792.897 Analytic conductor $14.309$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 897.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1792.897 Dual form 1792.2.b.f.897.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} +4.00000i q^{13} -2.00000 q^{17} +6.00000i q^{19} +2.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} +4.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} +8.00000 q^{33} +10.0000i q^{37} -8.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -2.00000i q^{53} -12.0000 q^{57} +10.0000i q^{59} +8.00000i q^{61} -1.00000 q^{63} +8.00000i q^{67} -16.0000i q^{69} +6.00000 q^{73} +10.0000i q^{75} -4.00000i q^{77} -16.0000 q^{79} -11.0000 q^{81} -2.00000i q^{83} +4.00000 q^{87} -18.0000 q^{89} +4.00000i q^{91} -8.00000i q^{93} -2.00000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} - 4q^{17} - 16q^{23} + 10q^{25} - 8q^{31} + 16q^{33} - 16q^{39} + 20q^{41} + 8q^{47} + 2q^{49} - 24q^{57} - 2q^{63} + 12q^{73} - 32q^{79} - 22q^{81} + 8q^{87} - 36q^{89} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 6.00000i 1.37649i 0.725476 + 0.688247i $$0.241620\pi$$
−0.725476 + 0.688247i $$0.758380\pi$$
$$20$$ 0 0
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ − 2.00000i − 0.371391i −0.982607 0.185695i $$-0.940546\pi$$
0.982607 0.185695i $$-0.0594537\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 8.00000 1.39262
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ − 4.00000i − 0.560112i
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −12.0000 −1.58944
$$58$$ 0 0
$$59$$ 10.0000i 1.30189i 0.759125 + 0.650945i $$0.225627\pi$$
−0.759125 + 0.650945i $$0.774373\pi$$
$$60$$ 0 0
$$61$$ 8.00000i 1.02430i 0.858898 + 0.512148i $$0.171150\pi$$
−0.858898 + 0.512148i $$0.828850\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ − 16.0000i − 1.92617i
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 10.0000i 1.15470i
$$76$$ 0 0
$$77$$ − 4.00000i − 0.455842i
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 2.00000i − 0.219529i −0.993958 0.109764i $$-0.964990\pi$$
0.993958 0.109764i $$-0.0350096\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 4.00000i 0.419314i
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 0 0
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 0 0
$$111$$ −20.0000 −1.89832
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ 20.0000i 1.80334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 6.00000i 0.524222i 0.965038 + 0.262111i $$0.0844187\pi$$
−0.965038 + 0.262111i $$0.915581\pi$$
$$132$$ 0 0
$$133$$ 6.00000i 0.520266i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ 8.00000i 0.673722i
$$142$$ 0 0
$$143$$ 16.0000 1.33799
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000i 0.164957i
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 12.0000i − 0.957704i −0.877896 0.478852i $$-0.841053\pi$$
0.877896 0.478852i $$-0.158947\pi$$
$$158$$ 0 0
$$159$$ 4.00000 0.317221
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −20.0000 −1.54765 −0.773823 0.633402i $$-0.781658\pi$$
−0.773823 + 0.633402i $$0.781658\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ − 6.00000i − 0.458831i
$$172$$ 0 0
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 0 0
$$177$$ −20.0000 −1.50329
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ − 12.0000i − 0.891953i −0.895045 0.445976i $$-0.852856\pi$$
0.895045 0.445976i $$-0.147144\pi$$
$$182$$ 0 0
$$183$$ −16.0000 −1.18275
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 0 0
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ − 8.00000i − 0.550743i −0.961338 0.275371i $$-0.911199\pi$$
0.961338 0.275371i $$-0.0888008\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ 12.0000i 0.810885i
$$220$$ 0 0
$$221$$ − 8.00000i − 0.538138i
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 22.0000i 1.46019i 0.683345 + 0.730096i $$0.260525\pi$$
−0.683345 + 0.730096i $$0.739475\pi$$
$$228$$ 0 0
$$229$$ − 20.0000i − 1.32164i −0.750546 0.660819i $$-0.770209\pi$$
0.750546 0.660819i $$-0.229791\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 32.0000i − 2.07862i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ − 10.0000i − 0.631194i −0.948893 0.315597i $$-0.897795\pi$$
0.948893 0.315597i $$-0.102205\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 10.0000i 0.621370i
$$260$$ 0 0
$$261$$ 2.00000i 0.123797i
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 36.0000i − 2.20316i
$$268$$ 0 0
$$269$$ 12.0000i 0.731653i 0.930683 + 0.365826i $$0.119214\pi$$
−0.930683 + 0.365826i $$0.880786\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ − 20.0000i − 1.20605i
$$276$$ 0 0
$$277$$ 14.0000i 0.841178i 0.907251 + 0.420589i $$0.138177\pi$$
−0.907251 + 0.420589i $$0.861823\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 10.0000i 0.594438i 0.954809 + 0.297219i $$0.0960592\pi$$
−0.954809 + 0.297219i $$0.903941\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.0000 0.590281
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 4.00000i − 0.234484i
$$292$$ 0 0
$$293$$ 8.00000i 0.467365i 0.972313 + 0.233682i $$0.0750776\pi$$
−0.972313 + 0.233682i $$0.924922\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 16.0000 0.928414
$$298$$ 0 0
$$299$$ − 32.0000i − 1.85061i
$$300$$ 0 0
$$301$$ 4.00000i 0.230556i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 10.0000i − 0.570730i −0.958419 0.285365i $$-0.907885\pi$$
0.958419 0.285365i $$-0.0921148\pi$$
$$308$$ 0 0
$$309$$ 8.00000i 0.455104i
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 34.0000i 1.90963i 0.297200 + 0.954815i $$0.403947\pi$$
−0.297200 + 0.954815i $$0.596053\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 32.0000 1.78607
$$322$$ 0 0
$$323$$ − 12.0000i − 0.667698i
$$324$$ 0 0
$$325$$ 20.0000i 1.10940i
$$326$$ 0 0
$$327$$ 20.0000 1.10600
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ − 12.0000i − 0.659580i −0.944054 0.329790i $$-0.893022\pi$$
0.944054 0.329790i $$-0.106978\pi$$
$$332$$ 0 0
$$333$$ − 10.0000i − 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 12.0000i 0.651751i
$$340$$ 0 0
$$341$$ 16.0000i 0.866449i
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ − 20.0000i − 1.07058i −0.844670 0.535288i $$-0.820203\pi$$
0.844670 0.535288i $$-0.179797\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 4.00000i − 0.211702i
$$358$$ 0 0
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −17.0000 −0.894737
$$362$$ 0 0
$$363$$ − 10.0000i − 0.524864i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ − 2.00000i − 0.103835i
$$372$$ 0 0
$$373$$ 6.00000i 0.310668i 0.987862 + 0.155334i $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ 28.0000i 1.43826i 0.694874 + 0.719132i $$0.255460\pi$$
−0.694874 + 0.719132i $$0.744540\pi$$
$$380$$ 0 0
$$381$$ − 32.0000i − 1.63941i
$$382$$ 0 0
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4.00000i − 0.203331i
$$388$$ 0 0
$$389$$ − 6.00000i − 0.304212i −0.988364 0.152106i $$-0.951394\pi$$
0.988364 0.152106i $$-0.0486055\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 28.0000i 1.40528i 0.711546 + 0.702640i $$0.247995\pi$$
−0.711546 + 0.702640i $$0.752005\pi$$
$$398$$ 0 0
$$399$$ −12.0000 −0.600751
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 28.0000i 1.38114i
$$412$$ 0 0
$$413$$ 10.0000i 0.492068i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ − 6.00000i − 0.293119i −0.989202 0.146560i $$-0.953180\pi$$
0.989202 0.146560i $$-0.0468200\pi$$
$$420$$ 0 0
$$421$$ − 18.0000i − 0.877266i −0.898666 0.438633i $$-0.855463\pi$$
0.898666 0.438633i $$-0.144537\pi$$
$$422$$ 0 0
$$423$$ −4.00000 −0.194487
$$424$$ 0 0
$$425$$ −10.0000 −0.485071
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 0 0
$$429$$ 32.0000i 1.54497i
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 48.0000i − 2.29615i
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ − 40.0000i − 1.88353i
$$452$$ 0 0
$$453$$ 32.0000i 1.50349i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ − 8.00000i − 0.373408i
$$460$$ 0 0
$$461$$ − 28.0000i − 1.30409i −0.758180 0.652045i $$-0.773911\pi$$
0.758180 0.652045i $$-0.226089\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ 0 0
$$469$$ 8.00000i 0.369406i
$$470$$ 0 0
$$471$$ 24.0000 1.10586
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 30.0000i 1.37649i
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ 0 0
$$483$$ − 16.0000i − 0.728025i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ 0 0
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 8.00000i 0.361035i 0.983572 + 0.180517i $$0.0577772\pi$$
−0.983572 + 0.180517i $$0.942223\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 8.00000i 0.358129i 0.983837 + 0.179065i $$0.0573071\pi$$
−0.983837 + 0.179065i $$0.942693\pi$$
$$500$$ 0 0
$$501$$ − 40.0000i − 1.78707i
$$502$$ 0 0
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 6.00000i − 0.266469i
$$508$$ 0 0
$$509$$ − 20.0000i − 0.886484i −0.896402 0.443242i $$-0.853828\pi$$
0.896402 0.443242i $$-0.146172\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 0 0
$$513$$ −24.0000 −1.05963
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16.0000i − 0.703679i
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 0 0
$$525$$ 10.0000i 0.436436i
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ − 10.0000i − 0.433963i
$$532$$ 0 0
$$533$$ 40.0000i 1.73259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 4.00000i − 0.172292i
$$540$$ 0 0
$$541$$ − 30.0000i − 1.28980i −0.764267 0.644900i $$-0.776899\pi$$
0.764267 0.644900i $$-0.223101\pi$$
$$542$$ 0 0
$$543$$ 24.0000 1.02994
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 0 0
$$549$$ − 8.00000i − 0.341432i
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ − 46.0000i − 1.93867i −0.245745 0.969334i $$-0.579033\pi$$
0.245745 0.969334i $$-0.420967\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ − 4.00000i − 0.167395i −0.996491 0.0836974i $$-0.973327\pi$$
0.996491 0.0836974i $$-0.0266729\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −40.0000 −1.66812
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 0 0
$$579$$ 28.0000i 1.16364i
$$580$$ 0 0
$$581$$ − 2.00000i − 0.0829740i
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 0 0
$$589$$ − 24.0000i − 0.988903i
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 46.0000 1.87638 0.938190 0.346122i $$-0.112502\pi$$
0.938190 + 0.346122i $$0.112502\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 48.0000 1.94826 0.974130 0.225989i $$-0.0725612\pi$$
0.974130 + 0.225989i $$0.0725612\pi$$
$$608$$ 0 0
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 16.0000i 0.647291i
$$612$$ 0 0
$$613$$ − 22.0000i − 0.888572i −0.895885 0.444286i $$-0.853457\pi$$
0.895885 0.444286i $$-0.146543\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ − 22.0000i − 0.884255i −0.896952 0.442127i $$-0.854224\pi$$
0.896952 0.442127i $$-0.145776\pi$$
$$620$$ 0 0
$$621$$ − 32.0000i − 1.28412i
$$622$$ 0 0
$$623$$ −18.0000 −0.721155
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 48.0000i 1.91694i
$$628$$ 0 0
$$629$$ − 20.0000i − 0.797452i
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 0 0
$$633$$ 16.0000 0.635943
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.00000i 0.158486i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 40.0000 1.57014
$$650$$ 0 0
$$651$$ − 8.00000i − 0.313545i
$$652$$ 0 0
$$653$$ 38.0000i 1.48705i 0.668705 + 0.743527i $$0.266849\pi$$
−0.668705 + 0.743527i $$0.733151\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ − 12.0000i − 0.467454i −0.972302 0.233727i $$-0.924908\pi$$
0.972302 0.233727i $$-0.0750921\pi$$
$$660$$ 0 0
$$661$$ 24.0000i 0.933492i 0.884391 + 0.466746i $$0.154574\pi$$
−0.884391 + 0.466746i $$0.845426\pi$$
$$662$$ 0 0
$$663$$ 16.0000 0.621389
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000i 0.619522i
$$668$$ 0 0
$$669$$ 48.0000i 1.85579i
$$670$$ 0 0
$$671$$ 32.0000 1.23535
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 0 0
$$675$$ 20.0000i 0.769800i
$$676$$ 0 0
$$677$$ 36.0000i 1.38359i 0.722093 + 0.691796i $$0.243180\pi$$
−0.722093 + 0.691796i $$0.756820\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −44.0000 −1.68608
$$682$$ 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
$$687$$ 40.0000 1.52610
$$688$$ 0 0
$$689$$ 8.00000 0.304776
$$690$$ 0 0
$$691$$ 22.0000i 0.836919i 0.908235 + 0.418460i $$0.137430\pi$$
−0.908235 + 0.418460i $$0.862570\pi$$
$$692$$ 0 0
$$693$$ 4.00000i 0.151947i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ 0 0
$$699$$ 12.0000i 0.453882i
$$700$$ 0 0
$$701$$ 6.00000i 0.226617i 0.993560 + 0.113308i $$0.0361448\pi$$
−0.993560 + 0.113308i $$0.963855\pi$$
$$702$$ 0 0
$$703$$ −60.0000 −2.26294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 26.0000i 0.976450i 0.872718 + 0.488225i $$0.162356\pi$$
−0.872718 + 0.488225i $$0.837644\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 48.0000i 1.79259i
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 0 0
$$723$$ 28.0000i 1.04133i
$$724$$ 0 0
$$725$$ − 10.0000i − 0.371391i
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ − 8.00000i − 0.295891i
$$732$$ 0 0
$$733$$ − 8.00000i − 0.295487i −0.989026 0.147743i $$-0.952799\pi$$
0.989026 0.147743i $$-0.0472010\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ 0 0
$$739$$ − 4.00000i − 0.147142i −0.997290 0.0735712i $$-0.976560\pi$$
0.997290 0.0735712i $$-0.0234396\pi$$
$$740$$ 0 0
$$741$$ − 48.0000i − 1.76332i
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 2.00000i 0.0731762i
$$748$$ 0 0
$$749$$ − 16.0000i − 0.584627i
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 20.0000 0.728841
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ −64.0000 −2.32305
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ − 10.0000i − 0.362024i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −40.0000 −1.44432
$$768$$ 0 0
$$769$$ 6.00000 0.216366 0.108183 0.994131i $$-0.465497\pi$$
0.108183 + 0.994131i $$0.465497\pi$$
$$770$$ 0 0
$$771$$ 4.00000i 0.144056i
$$772$$ 0 0
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ −20.0000 −0.718421
$$776$$ 0 0
$$777$$ −20.0000 −0.717496
$$778$$ 0 0
$$779$$ 60.0000i 2.14972i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 8.00000 0.285897
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 46.0000i 1.63972i 0.572562 + 0.819861i $$0.305950\pi$$
−0.572562 + 0.819861i $$0.694050\pi$$
$$788$$ 0 0
$$789$$ 32.0000i 1.13923i
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −32.0000 −1.13635
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 20.0000i − 0.708436i −0.935163 0.354218i $$-0.884747\pi$$
0.935163 0.354218i $$-0.115253\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ − 24.0000i − 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −24.0000 −0.844840
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ 34.0000i 1.19390i 0.802278 + 0.596951i $$0.203621\pi$$
−0.802278 + 0.596951i $$0.796379\pi$$
$$812$$ 0 0
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 0 0
$$819$$ − 4.00000i − 0.139771i
$$820$$ 0 0
$$821$$ − 2.00000i − 0.0698005i −0.999391 0.0349002i $$-0.988889\pi$$
0.999391 0.0349002i $$-0.0111113\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 0 0
$$825$$ 40.0000 1.39262
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ − 24.0000i − 0.833554i −0.909009 0.416777i $$-0.863160\pi$$
0.909009 0.416777i $$-0.136840\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 0 0
$$839$$ 52.0000 1.79524 0.897620 0.440771i $$-0.145295\pi$$
0.897620 + 0.440771i $$0.145295\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ 12.0000i 0.413302i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ − 80.0000i − 2.74236i
$$852$$ 0 0
$$853$$ − 36.0000i − 1.23262i −0.787505 0.616308i $$-0.788628\pi$$
0.787505 0.616308i $$-0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ − 26.0000i − 0.887109i −0.896248 0.443554i $$-0.853717\pi$$
0.896248 0.443554i $$-0.146283\pi$$
$$860$$ 0 0
$$861$$ 20.0000i 0.681598i
$$862$$ 0 0
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 26.0000i − 0.883006i
$$868$$ 0 0
$$869$$ 64.0000i 2.17105i
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 0 0
$$879$$ −16.0000 −0.539667
$$880$$ 0 0
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ 0 0
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −28.0000 −0.940148 −0.470074 0.882627i $$-0.655773\pi$$
−0.470074 + 0.882627i $$0.655773\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 44.0000i 1.47406i
$$892$$ 0 0
$$893$$ 24.0000i 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 64.0000 2.13690
$$898$$ 0 0
$$899$$ 8.00000i 0.266815i
$$900$$ 0 0
$$901$$ 4.00000i 0.133259i
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 6.00000i 0.198137i
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 50.0000i 1.64399i
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 6.00000i 0.196642i
$$932$$ 0 0
$$933$$ 48.0000i 1.57145i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ 52.0000i 1.69696i
$$940$$ 0 0
$$941$$ 40.0000i 1.30396i 0.758235 + 0.651981i $$0.226062\pi$$
−0.758235 + 0.651981i $$0.773938\pi$$
$$942$$ 0 0
$$943$$ −80.0000 −2.60516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 0 0
$$949$$ 24.0000i 0.779073i
$$950$$ 0 0
$$951$$ −68.0000 −2.20505
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 16.0000i − 0.517207i
$$958$$ 0 0
$$959$$ 14.0000 0.452084
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 16.0000i 0.515593i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 42.0000i 1.34784i 0.738802 + 0.673922i $$0.235392\pi$$
−0.738802 + 0.673922i $$0.764608\pi$$
$$972$$ 0 0
$$973$$ − 10.0000i − 0.320585i
$$974$$ 0 0
$$975$$ −40.0000 −1.28103
$$976$$ 0 0
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ 72.0000i 2.30113i
$$980$$ 0 0
$$981$$ 10.0000i 0.319275i
$$982$$ 0 0
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 8.00000i 0.254643i
$$988$$ 0 0
$$989$$ − 32.0000i − 1.01754i
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 24.0000 0.761617
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 1792.2.b.f.897.2 2
4.3 odd 2 1792.2.b.b.897.1 2
8.3 odd 2 1792.2.b.b.897.2 2
8.5 even 2 inner 1792.2.b.f.897.1 2
16.3 odd 4 224.2.a.b.1.1 yes 1
16.5 even 4 448.2.a.f.1.1 1
16.11 odd 4 448.2.a.b.1.1 1
16.13 even 4 224.2.a.a.1.1 1
48.5 odd 4 4032.2.a.p.1.1 1
48.11 even 4 4032.2.a.z.1.1 1
48.29 odd 4 2016.2.a.e.1.1 1
48.35 even 4 2016.2.a.g.1.1 1
80.19 odd 4 5600.2.a.c.1.1 1
80.29 even 4 5600.2.a.t.1.1 1
112.3 even 12 1568.2.i.j.961.1 2
112.13 odd 4 1568.2.a.h.1.1 1
112.19 even 12 1568.2.i.j.1537.1 2
112.27 even 4 3136.2.a.y.1.1 1
112.45 odd 12 1568.2.i.c.961.1 2
112.51 odd 12 1568.2.i.b.1537.1 2
112.61 odd 12 1568.2.i.c.1537.1 2
112.67 odd 12 1568.2.i.b.961.1 2
112.69 odd 4 3136.2.a.f.1.1 1
112.83 even 4 1568.2.a.b.1.1 1
112.93 even 12 1568.2.i.k.1537.1 2
112.109 even 12 1568.2.i.k.961.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.a.1.1 1 16.13 even 4
224.2.a.b.1.1 yes 1 16.3 odd 4
448.2.a.b.1.1 1 16.11 odd 4
448.2.a.f.1.1 1 16.5 even 4
1568.2.a.b.1.1 1 112.83 even 4
1568.2.a.h.1.1 1 112.13 odd 4
1568.2.i.b.961.1 2 112.67 odd 12
1568.2.i.b.1537.1 2 112.51 odd 12
1568.2.i.c.961.1 2 112.45 odd 12
1568.2.i.c.1537.1 2 112.61 odd 12
1568.2.i.j.961.1 2 112.3 even 12
1568.2.i.j.1537.1 2 112.19 even 12
1568.2.i.k.961.1 2 112.109 even 12
1568.2.i.k.1537.1 2 112.93 even 12
1792.2.b.b.897.1 2 4.3 odd 2
1792.2.b.b.897.2 2 8.3 odd 2
1792.2.b.f.897.1 2 8.5 even 2 inner
1792.2.b.f.897.2 2 1.1 even 1 trivial
2016.2.a.e.1.1 1 48.29 odd 4
2016.2.a.g.1.1 1 48.35 even 4
3136.2.a.f.1.1 1 112.69 odd 4
3136.2.a.y.1.1 1 112.27 even 4
4032.2.a.p.1.1 1 48.5 odd 4
4032.2.a.z.1.1 1 48.11 even 4
5600.2.a.c.1.1 1 80.19 odd 4
5600.2.a.t.1.1 1 80.29 even 4