# Properties

 Label 1792.2.b.h.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 56) 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.h.897.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +8.00000 q^{15} -2.00000 q^{17} -2.00000i q^{19} +2.00000i q^{21} +8.00000 q^{23} -11.0000 q^{25} +4.00000i q^{27} -2.00000i q^{29} -4.00000 q^{31} -4.00000i q^{35} -6.00000i q^{37} +2.00000 q^{41} -8.00000i q^{43} +4.00000i q^{45} +4.00000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -10.0000i q^{53} +4.00000 q^{57} -6.00000i q^{59} -4.00000i q^{61} -1.00000 q^{63} -12.0000i q^{67} +16.0000i q^{69} +14.0000 q^{73} -22.0000i q^{75} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +8.00000i q^{85} +4.00000 q^{87} -10.0000 q^{89} -8.00000i q^{93} -8.00000 q^{95} -2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} + 16q^{15} - 4q^{17} + 16q^{23} - 22q^{25} - 8q^{31} + 4q^{41} + 8q^{47} + 2q^{49} + 8q^{57} - 2q^{63} + 28q^{73} + 16q^{79} - 22q^{81} + 8q^{87} - 20q^{89} - 16q^{95} - 4q^{97} + O(q^{100})$$

## 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
$$5$$ − 4.00000i − 1.78885i −0.447214 0.894427i $$-0.647584\pi$$
0.447214 0.894427i $$-0.352416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 8.00000 2.06559
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ − 2.00000i − 0.458831i −0.973329 0.229416i $$-0.926318\pi$$
0.973329 0.229416i $$-0.0736815\pi$$
$$20$$ 0 0
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ −11.0000 −2.20000
$$26$$ 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
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 4.00000i − 0.676123i
$$36$$ 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
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 4.00000i 0.596285i
$$46$$ 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
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ − 6.00000i − 0.781133i −0.920575 0.390567i $$-0.872279\pi$$
0.920575 0.390567i $$-0.127721\pi$$
$$60$$ 0 0
$$61$$ − 4.00000i − 0.512148i −0.966657 0.256074i $$-0.917571\pi$$
0.966657 0.256074i $$-0.0824290\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 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
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ − 22.0000i − 2.54034i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 8.00000i 0.867722i
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i 0.802225 + 0.597022i $$0.203650\pi$$
−0.802225 + 0.597022i $$0.796350\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 8.00000 0.780720
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 0 0
$$111$$ 12.0000 1.13899
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ − 32.0000i − 2.98402i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 4.00000i 0.360668i
$$124$$ 0 0
$$125$$ 24.0000i 2.14663i
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 16.0000 1.40872
$$130$$ 0 0
$$131$$ 14.0000i 1.22319i 0.791173 + 0.611593i $$0.209471\pi$$
−0.791173 + 0.611593i $$0.790529\pi$$
$$132$$ 0 0
$$133$$ − 2.00000i − 0.173422i
$$134$$ 0 0
$$135$$ 16.0000 1.37706
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ − 18.0000i − 1.52674i −0.645961 0.763370i $$-0.723543\pi$$
0.645961 0.763370i $$-0.276457\pi$$
$$140$$ 0 0
$$141$$ 8.00000i 0.673722i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 2.00000i 0.164957i
$$148$$ 0 0
$$149$$ − 2.00000i − 0.163846i −0.996639 0.0819232i $$-0.973894\pi$$
0.996639 0.0819232i $$-0.0261062\pi$$
$$150$$ 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
$$155$$ 16.0000i 1.28515i
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 20.0000 1.58610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 2.00000i 0.152944i
$$172$$ 0 0
$$173$$ − 8.00000i − 0.608229i −0.952636 0.304114i $$-0.901639\pi$$
0.952636 0.304114i $$-0.0983605\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ − 4.00000i − 0.298974i −0.988764 0.149487i $$-0.952238\pi$$
0.988764 0.149487i $$-0.0477622\pi$$
$$180$$ 0 0
$$181$$ 8.00000i 0.594635i 0.954779 + 0.297318i $$0.0960920\pi$$
−0.954779 + 0.297318i $$0.903908\pi$$
$$182$$ 0 0
$$183$$ 8.00000 0.591377
$$184$$ 0 0
$$185$$ −24.0000 −1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 24.0000 1.69283
$$202$$ 0 0
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ − 8.00000i − 0.558744i
$$206$$ 0 0
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000i 1.37686i 0.725304 + 0.688428i $$0.241699\pi$$
−0.725304 + 0.688428i $$0.758301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −32.0000 −2.18238
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ 28.0000i 1.89206i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ 0 0
$$227$$ 14.0000i 0.929213i 0.885517 + 0.464606i $$0.153804\pi$$
−0.885517 + 0.464606i $$0.846196\pi$$
$$228$$ 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
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ − 16.0000i − 1.04372i
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ − 4.00000i − 0.255551i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 14.0000i 0.883672i 0.897096 + 0.441836i $$0.145673\pi$$
−0.897096 + 0.441836i $$0.854327\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −16.0000 −1.00196
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ − 6.00000i − 0.372822i
$$260$$ 0 0
$$261$$ 2.00000i 0.123797i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −40.0000 −2.45718
$$266$$ 0 0
$$267$$ − 20.0000i − 1.22398i
$$268$$ 0 0
$$269$$ 24.0000i 1.46331i 0.681677 + 0.731653i $$0.261251\pi$$
−0.681677 + 0.731653i $$0.738749\pi$$
$$270$$ 0 0
$$271$$ −32.0000 −1.94386 −0.971931 0.235267i $$-0.924404\pi$$
−0.971931 + 0.235267i $$0.924404\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 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
$$285$$ − 16.0000i − 0.947758i
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ − 4.00000i − 0.234484i
$$292$$ 0 0
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 8.00000i − 0.461112i
$$302$$ 0 0
$$303$$ −24.0000 −1.37876
$$304$$ 0 0
$$305$$ −16.0000 −0.916157
$$306$$ 0 0
$$307$$ − 2.00000i − 0.114146i −0.998370 0.0570730i $$-0.981823\pi$$
0.998370 0.0570730i $$-0.0181768\pi$$
$$308$$ 0 0
$$309$$ − 24.0000i − 1.36531i
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ 4.00000i 0.225374i
$$316$$ 0 0
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ 4.00000i 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.0000 1.10600
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ − 8.00000i − 0.439720i −0.975531 0.219860i $$-0.929440\pi$$
0.975531 0.219860i $$-0.0705600\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ −48.0000 −2.62252
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 12.0000i 0.651751i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 64.0000 3.44564
$$346$$ 0 0
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 0 0
$$349$$ − 8.00000i − 0.428230i −0.976808 0.214115i $$-0.931313\pi$$
0.976808 0.214115i $$-0.0686868\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 4.00000i − 0.211702i
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ 22.0000i 1.15470i
$$364$$ 0 0
$$365$$ − 56.0000i − 2.93117i
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ − 10.0000i − 0.519174i
$$372$$ 0 0
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 0 0
$$375$$ −48.0000 −2.47871
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ − 16.0000i − 0.819705i
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000i 0.406663i
$$388$$ 0 0
$$389$$ 10.0000i 0.507020i 0.967333 + 0.253510i $$0.0815851\pi$$
−0.967333 + 0.253510i $$0.918415\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ −28.0000 −1.41241
$$394$$ 0 0
$$395$$ − 32.0000i − 1.61009i
$$396$$ 0 0
$$397$$ 8.00000i 0.401508i 0.979642 + 0.200754i $$0.0643393\pi$$
−0.979642 + 0.200754i $$0.935661\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 44.0000i 2.18638i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ − 4.00000i − 0.197305i
$$412$$ 0 0
$$413$$ − 6.00000i − 0.295241i
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ 36.0000 1.76293
$$418$$ 0 0
$$419$$ 26.0000i 1.27018i 0.772437 + 0.635092i $$0.219038\pi$$
−0.772437 + 0.635092i $$0.780962\pi$$
$$420$$ 0 0
$$421$$ 22.0000i 1.07221i 0.844150 + 0.536107i $$0.180106\pi$$
−0.844150 + 0.536107i $$0.819894\pi$$
$$422$$ 0 0
$$423$$ −4.00000 −0.194487
$$424$$ 0 0
$$425$$ 22.0000 1.06716
$$426$$ 0 0
$$427$$ − 4.00000i − 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 0 0
$$435$$ − 16.0000i − 0.767141i
$$436$$ 0 0
$$437$$ − 16.0000i − 0.765384i
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 0 0
$$445$$ 40.0000i 1.89618i
$$446$$ 0 0
$$447$$ 4.00000 0.189194
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 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
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ −32.0000 −1.48396
$$466$$ 0 0
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 0 0
$$469$$ − 12.0000i − 0.554109i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 22.0000i 1.00943i
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 0 0
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 16.0000i 0.728025i
$$484$$ 0 0
$$485$$ 8.00000i 0.363261i
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ −32.0000 −1.44709
$$490$$ 0 0
$$491$$ 36.0000i 1.62466i 0.583200 + 0.812329i $$0.301800\pi$$
−0.583200 + 0.812329i $$0.698200\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 0 0
$$501$$ 24.0000i 1.07224i
$$502$$ 0 0
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ 48.0000 2.13597
$$506$$ 0 0
$$507$$ 26.0000i 1.15470i
$$508$$ 0 0
$$509$$ 24.0000i 1.06378i 0.846813 + 0.531891i $$0.178518\pi$$
−0.846813 + 0.531891i $$0.821482\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 0 0
$$513$$ 8.00000 0.353209
$$514$$ 0 0
$$515$$ 48.0000i 2.11513i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.0000 0.702322
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 34.0000i 1.48672i 0.668894 + 0.743358i $$0.266768\pi$$
−0.668894 + 0.743358i $$0.733232\pi$$
$$524$$ 0 0
$$525$$ − 22.0000i − 0.960159i
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 6.00000i 0.260378i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 48.0000 2.07522
$$536$$ 0 0
$$537$$ 8.00000 0.345225
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 22.0000i − 0.945854i −0.881102 0.472927i $$-0.843197\pi$$
0.881102 0.472927i $$-0.156803\pi$$
$$542$$ 0 0
$$543$$ −16.0000 −0.686626
$$544$$ 0 0
$$545$$ −40.0000 −1.71341
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ 4.00000i 0.170716i
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ − 48.0000i − 2.03749i
$$556$$ 0 0
$$557$$ 26.0000i 1.10166i 0.834619 + 0.550828i $$0.185688\pi$$
−0.834619 + 0.550828i $$0.814312\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 34.0000i 1.43293i 0.697623 + 0.716465i $$0.254241\pi$$
−0.697623 + 0.716465i $$0.745759\pi$$
$$564$$ 0 0
$$565$$ − 24.0000i − 1.00969i
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ 16.0000i 0.669579i 0.942293 + 0.334790i $$0.108665\pi$$
−0.942293 + 0.334790i $$0.891335\pi$$
$$572$$ 0 0
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ −88.0000 −3.66985
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 0 0
$$579$$ − 36.0000i − 1.49611i
$$580$$ 0 0
$$581$$ 6.00000i 0.248922i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 10.0000i − 0.412744i −0.978474 0.206372i $$-0.933834\pi$$
0.978474 0.206372i $$-0.0661657\pi$$
$$588$$ 0 0
$$589$$ 8.00000i 0.329634i
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ 0 0
$$593$$ 42.0000 1.72473 0.862367 0.506284i $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ 0 0
$$595$$ 8.00000i 0.327968i
$$596$$ 0 0
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ − 44.0000i − 1.78885i
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ 0 0
$$615$$ 16.0000 0.645182
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ − 6.00000i − 0.241160i −0.992704 0.120580i $$-0.961525\pi$$
0.992704 0.120580i $$-0.0384755\pi$$
$$620$$ 0 0
$$621$$ 32.0000i 1.28412i
$$622$$ 0 0
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ −40.0000 −1.58986
$$634$$ 0 0
$$635$$ 32.0000i 1.26988i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 0 0
$$645$$ − 64.0000i − 2.52000i
$$646$$ 0 0
$$647$$ −36.0000 −1.41531 −0.707653 0.706560i $$-0.750246\pi$$
−0.707653 + 0.706560i $$0.750246\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ − 8.00000i − 0.313545i
$$652$$ 0 0
$$653$$ 22.0000i 0.860927i 0.902608 + 0.430463i $$0.141650\pi$$
−0.902608 + 0.430463i $$0.858350\pi$$
$$654$$ 0 0
$$655$$ 56.0000 2.18810
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ − 40.0000i − 1.55818i −0.626913 0.779089i $$-0.715682\pi$$
0.626913 0.779089i $$-0.284318\pi$$
$$660$$ 0 0
$$661$$ 20.0000i 0.777910i 0.921257 + 0.388955i $$0.127164\pi$$
−0.921257 + 0.388955i $$0.872836\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ − 16.0000i − 0.619522i
$$668$$ 0 0
$$669$$ 48.0000i 1.85579i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ 0 0
$$675$$ − 44.0000i − 1.69356i
$$676$$ 0 0
$$677$$ − 24.0000i − 0.922395i −0.887298 0.461197i $$-0.847420\pi$$
0.887298 0.461197i $$-0.152580\pi$$
$$678$$ 0 0
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ 8.00000i 0.305664i
$$686$$ 0 0
$$687$$ 32.0000 1.22088
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 14.0000i 0.532585i 0.963892 + 0.266293i $$0.0857987\pi$$
−0.963892 + 0.266293i $$0.914201\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −72.0000 −2.73112
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ − 52.0000i − 1.96682i
$$700$$ 0 0
$$701$$ − 10.0000i − 0.377695i −0.982006 0.188847i $$-0.939525\pi$$
0.982006 0.188847i $$-0.0604752\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ 32.0000 1.20519
$$706$$ 0 0
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ 10.0000i 0.375558i 0.982211 + 0.187779i $$0.0601289\pi$$
−0.982211 + 0.187779i $$0.939871\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 32.0000i 1.19506i
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ 0 0
$$723$$ − 4.00000i − 0.148762i
$$724$$ 0 0
$$725$$ 22.0000i 0.817059i
$$726$$ 0 0
$$727$$ 20.0000 0.741759 0.370879 0.928681i $$-0.379056\pi$$
0.370879 + 0.928681i $$0.379056\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 16.0000i 0.591781i
$$732$$ 0 0
$$733$$ − 52.0000i − 1.92066i −0.278859 0.960332i $$-0.589956\pi$$
0.278859 0.960332i $$-0.410044\pi$$
$$734$$ 0 0
$$735$$ 8.00000 0.295084
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000i 0.588570i 0.955718 + 0.294285i $$0.0950814\pi$$
−0.955718 + 0.294285i $$0.904919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 0 0
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ −28.0000 −1.02038
$$754$$ 0 0
$$755$$ − 64.0000i − 2.32920i
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ − 10.0000i − 0.362024i
$$764$$ 0 0
$$765$$ − 8.00000i − 0.289241i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 36.0000i 1.29651i
$$772$$ 0 0
$$773$$ 4.00000i 0.143870i 0.997409 + 0.0719350i $$0.0229174\pi$$
−0.997409 + 0.0719350i $$0.977083\pi$$
$$774$$ 0 0
$$775$$ 44.0000 1.58053
$$776$$ 0 0
$$777$$ 12.0000 0.430498
$$778$$ 0 0
$$779$$ − 4.00000i − 0.143315i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 8.00000 0.285897
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 80.0000i − 2.83731i
$$796$$ 0 0
$$797$$ 24.0000i 0.850124i 0.905164 + 0.425062i $$0.139748\pi$$
−0.905164 + 0.425062i $$0.860252\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ − 32.0000i − 1.12785i
$$806$$ 0 0
$$807$$ −48.0000 −1.68968
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ − 30.0000i − 1.05344i −0.850038 0.526721i $$-0.823421\pi$$
0.850038 0.526721i $$-0.176579\pi$$
$$812$$ 0 0
$$813$$ − 64.0000i − 2.24458i
$$814$$ 0 0
$$815$$ 64.0000 2.24182
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 0 0
$$829$$ − 20.0000i − 0.694629i −0.937749 0.347314i $$-0.887094\pi$$
0.937749 0.347314i $$-0.112906\pi$$
$$830$$ 0 0
$$831$$ −44.0000 −1.52634
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ − 48.0000i − 1.66111i
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 0 0
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ 12.0000i 0.413302i
$$844$$ 0 0
$$845$$ − 52.0000i − 1.78885i
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ − 48.0000i − 1.64542i
$$852$$ 0 0
$$853$$ − 32.0000i − 1.09566i −0.836590 0.547830i $$-0.815454\pi$$
0.836590 0.547830i $$-0.184546\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 14.0000i 0.477674i 0.971060 + 0.238837i $$0.0767661\pi$$
−0.971060 + 0.238837i $$0.923234\pi$$
$$860$$ 0 0
$$861$$ 4.00000i 0.136320i
$$862$$ 0 0
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ −32.0000 −1.08803
$$866$$ 0 0
$$867$$ − 26.0000i − 0.883006i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 24.0000i 0.811348i
$$876$$ 0 0
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ 0 0
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 0 0
$$885$$ − 48.0000i − 1.61350i
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 8.00000i − 0.267710i
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 8.00000i 0.266815i
$$900$$ 0 0
$$901$$ 20.0000i 0.666297i
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 0 0
$$905$$ 32.0000 1.06372
$$906$$ 0 0
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 0 0
$$909$$ − 12.0000i − 0.398015i
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ − 32.0000i − 1.05789i
$$916$$ 0 0
$$917$$ 14.0000i 0.462321i
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 66.0000i 2.17007i
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ − 2.00000i − 0.0655474i
$$932$$ 0 0
$$933$$ − 48.0000i − 1.57145i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ 0 0
$$939$$ − 28.0000i − 0.913745i
$$940$$ 0 0
$$941$$ 12.0000i 0.391189i 0.980685 + 0.195594i $$0.0626636\pi$$
−0.980685 + 0.195594i $$0.937336\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 0 0
$$945$$ 16.0000 0.520480
$$946$$ 0 0
$$947$$ 40.0000i 1.29983i 0.760009 + 0.649913i $$0.225195\pi$$
−0.760009 + 0.649913i $$0.774805\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ −10.0000 −0.323932 −0.161966 0.986796i $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ 0 0
$$955$$ − 32.0000i − 1.03550i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2.00000 −0.0645834
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 72.0000i 2.31776i
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ 26.0000i 0.834380i 0.908819 + 0.417190i $$0.136985\pi$$
−0.908819 + 0.417190i $$0.863015\pi$$
$$972$$ 0 0
$$973$$ − 18.0000i − 0.577054i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000i 0.319275i
$$982$$ 0 0
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ −72.0000 −2.29411
$$986$$ 0 0
$$987$$ 8.00000i 0.254643i
$$988$$ 0 0
$$989$$ − 64.0000i − 2.03508i
$$990$$ 0 0
$$991$$ 48.0000 1.52477 0.762385 0.647124i $$-0.224028\pi$$
0.762385 + 0.647124i $$0.224028\pi$$
$$992$$ 0 0
$$993$$ 16.0000 0.507745
$$994$$ 0 0
$$995$$ 16.0000i 0.507234i
$$996$$ 0 0
$$997$$ 52.0000i 1.64686i 0.567420 + 0.823428i $$0.307941\pi$$
−0.567420 + 0.823428i $$0.692059\pi$$
$$998$$ 0 0
$$999$$ 24.0000 0.759326
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.h.897.2 2
4.3 odd 2 1792.2.b.a.897.1 2
8.3 odd 2 1792.2.b.a.897.2 2
8.5 even 2 inner 1792.2.b.h.897.1 2
16.3 odd 4 56.2.a.b.1.1 1
16.5 even 4 448.2.a.h.1.1 1
16.11 odd 4 448.2.a.c.1.1 1
16.13 even 4 112.2.a.a.1.1 1
48.5 odd 4 4032.2.a.a.1.1 1
48.11 even 4 4032.2.a.d.1.1 1
48.29 odd 4 1008.2.a.m.1.1 1
48.35 even 4 504.2.a.h.1.1 1
80.3 even 4 1400.2.g.b.449.2 2
80.13 odd 4 2800.2.g.g.449.1 2
80.19 odd 4 1400.2.a.a.1.1 1
80.29 even 4 2800.2.a.bd.1.1 1
80.67 even 4 1400.2.g.b.449.1 2
80.77 odd 4 2800.2.g.g.449.2 2
112.3 even 12 392.2.i.e.177.1 2
112.13 odd 4 784.2.a.i.1.1 1
112.19 even 12 392.2.i.e.361.1 2
112.27 even 4 3136.2.a.w.1.1 1
112.45 odd 12 784.2.i.b.177.1 2
112.51 odd 12 392.2.i.a.361.1 2
112.61 odd 12 784.2.i.b.753.1 2
112.67 odd 12 392.2.i.a.177.1 2
112.69 odd 4 3136.2.a.c.1.1 1
112.83 even 4 392.2.a.b.1.1 1
112.93 even 12 784.2.i.j.753.1 2
112.109 even 12 784.2.i.j.177.1 2
176.131 even 4 6776.2.a.h.1.1 1
208.51 odd 4 9464.2.a.h.1.1 1
336.83 odd 4 3528.2.a.b.1.1 1
336.125 even 4 7056.2.a.c.1.1 1
336.131 odd 12 3528.2.s.ba.361.1 2
336.179 even 12 3528.2.s.a.3313.1 2
336.227 odd 12 3528.2.s.ba.3313.1 2
336.275 even 12 3528.2.s.a.361.1 2
560.419 even 4 9800.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 1 16.3 odd 4
112.2.a.a.1.1 1 16.13 even 4
392.2.a.b.1.1 1 112.83 even 4
392.2.i.a.177.1 2 112.67 odd 12
392.2.i.a.361.1 2 112.51 odd 12
392.2.i.e.177.1 2 112.3 even 12
392.2.i.e.361.1 2 112.19 even 12
448.2.a.c.1.1 1 16.11 odd 4
448.2.a.h.1.1 1 16.5 even 4
504.2.a.h.1.1 1 48.35 even 4
784.2.a.i.1.1 1 112.13 odd 4
784.2.i.b.177.1 2 112.45 odd 12
784.2.i.b.753.1 2 112.61 odd 12
784.2.i.j.177.1 2 112.109 even 12
784.2.i.j.753.1 2 112.93 even 12
1008.2.a.m.1.1 1 48.29 odd 4
1400.2.a.a.1.1 1 80.19 odd 4
1400.2.g.b.449.1 2 80.67 even 4
1400.2.g.b.449.2 2 80.3 even 4
1792.2.b.a.897.1 2 4.3 odd 2
1792.2.b.a.897.2 2 8.3 odd 2
1792.2.b.h.897.1 2 8.5 even 2 inner
1792.2.b.h.897.2 2 1.1 even 1 trivial
2800.2.a.bd.1.1 1 80.29 even 4
2800.2.g.g.449.1 2 80.13 odd 4
2800.2.g.g.449.2 2 80.77 odd 4
3136.2.a.c.1.1 1 112.69 odd 4
3136.2.a.w.1.1 1 112.27 even 4
3528.2.a.b.1.1 1 336.83 odd 4
3528.2.s.a.361.1 2 336.275 even 12
3528.2.s.a.3313.1 2 336.179 even 12
3528.2.s.ba.361.1 2 336.131 odd 12
3528.2.s.ba.3313.1 2 336.227 odd 12
4032.2.a.a.1.1 1 48.5 odd 4
4032.2.a.d.1.1 1 48.11 even 4
6776.2.a.h.1.1 1 176.131 even 4
7056.2.a.c.1.1 1 336.125 even 4
9464.2.a.h.1.1 1 208.51 odd 4
9800.2.a.bj.1.1 1 560.419 even 4