# Properties

 Label 112.2.a.a.1.1 Level $112$ Weight $2$ Character 112.1 Self dual yes Analytic conductor $0.894$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.894324502638$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

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

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 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
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 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
$$35$$ 4.00000 0.676123
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ −4.00000 −0.596285
$$46$$ 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
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 16.0000 1.92617
$$70$$ 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 $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 0 0
$$75$$ −22.0000 −2.54034
$$76$$ 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
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 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
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 0 0
$$105$$ −8.00000 −0.780720
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 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
$$115$$ 32.0000 2.98402
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 4.00000 0.360668
$$124$$ 0 0
$$125$$ −24.0000 −2.14663
$$126$$ 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
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 0 0
$$135$$ −16.0000 −1.37706
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ 0 0
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ 20.0000 1.58610
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 0 0
$$183$$ −8.00000 −0.591377
$$184$$ 0 0
$$185$$ 24.0000 1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 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
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −24.0000 −1.69283
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 8.00000 0.558744
$$206$$ 0 0
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 32.0000 2.18238
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ 28.0000 1.89206
$$220$$ 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
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ −16.0000 −1.04372
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 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
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ −4.00000 −0.255551
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 14.0000 0.883672 0.441836 0.897096i $$-0.354327\pi$$
0.441836 + 0.897096i $$0.354327\pi$$
$$252$$ 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
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 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
$$267$$ −20.0000 −1.22398
$$268$$ 0 0
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 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
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 10.0000 0.594438 0.297219 0.954809i $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ 0 0
$$285$$ 16.0000 0.947758
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ −24.0000 −1.37876
$$304$$ 0 0
$$305$$ −16.0000 −0.916157
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ −24.0000 −1.36531
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 4.00000 0.225374
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −20.0000 −1.10600
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ 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
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −64.0000 −3.44564
$$346$$ 0 0
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 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
$$357$$ −4.00000 −0.211702
$$358$$ 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
$$363$$ 22.0000 1.15470
$$364$$ 0 0
$$365$$ 56.0000 2.93117
$$366$$ 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
$$371$$ 10.0000 0.519174
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ 48.0000 2.47871
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 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
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 28.0000 1.41241
$$394$$ 0 0
$$395$$ −32.0000 −1.61009
$$396$$ 0 0
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 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
$$405$$ 44.0000 2.18638
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ −4.00000 −0.197305
$$412$$ 0 0
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ 36.0000 1.76293
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 4.00000 0.194487
$$424$$ 0 0
$$425$$ −22.0000 −1.06716
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 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
$$435$$ 16.0000 0.767141
$$436$$ 0 0
$$437$$ −16.0000 −0.765384
$$438$$ 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
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ −40.0000 −1.89618
$$446$$ 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
$$453$$ 32.0000 1.50349
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 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
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 22.0000 1.00943
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 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
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 8.00000 0.363261
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 0 0
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ −48.0000 −2.13597
$$506$$ 0 0
$$507$$ 26.0000 1.15470
$$508$$ 0 0
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 0 0
$$513$$ 8.00000 0.353209
$$514$$ 0 0
$$515$$ −48.0000 −2.11513
$$516$$ 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 $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 0 0
$$525$$ 22.0000 0.960159
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 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
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 0 0
$$543$$ −16.0000 −0.686626
$$544$$ 0 0
$$545$$ −40.0000 −1.71341
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ −48.0000 −2.03749
$$556$$ 0 0
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −34.0000 −1.43293 −0.716465 0.697623i $$-0.754241\pi$$
−0.716465 + 0.697623i $$0.754241\pi$$
$$564$$ 0 0
$$565$$ −24.0000 −1.00969
$$566$$ 0 0
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ −16.0000 −0.668410
$$574$$ 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
$$579$$ 36.0000 1.49611
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10.0000 −0.412744 −0.206372 0.978474i $$-0.566166\pi$$
−0.206372 + 0.978474i $$0.566166\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 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
$$595$$ −8.00000 −0.327968
$$596$$ 0 0
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 0 0
$$605$$ 44.0000 1.78885
$$606$$ 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
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 0 0
$$615$$ −16.0000 −0.645182
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −6.00000 −0.241160 −0.120580 0.992704i $$-0.538475\pi$$
−0.120580 + 0.992704i $$0.538475\pi$$
$$620$$ 0 0
$$621$$ −32.0000 −1.28412
$$622$$ 0 0
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 40.0000 1.58986
$$634$$ 0 0
$$635$$ 32.0000 1.26988
$$636$$ 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
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ −64.0000 −2.52000
$$646$$ 0 0
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 0 0
$$653$$ −22.0000 −0.860927 −0.430463 0.902608i $$-0.641650\pi$$
−0.430463 + 0.902608i $$0.641650\pi$$
$$654$$ 0 0
$$655$$ 56.0000 2.18810
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 20.0000 0.777910 0.388955 0.921257i $$-0.372836\pi$$
0.388955 + 0.921257i $$0.372836\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ 0 0
$$669$$ −48.0000 −1.85579
$$670$$ 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
$$675$$ 44.0000 1.69356
$$676$$ 0 0
$$677$$ −24.0000 −0.922395 −0.461197 0.887298i $$-0.652580\pi$$
−0.461197 + 0.887298i $$0.652580\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −8.00000 −0.305664
$$686$$ 0 0
$$687$$ 32.0000 1.22088
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 72.0000 2.73112
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ −52.0000 −1.96682
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 0 0
$$705$$ 32.0000 1.20519
$$706$$ 0 0
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −32.0000 −1.19506
$$718$$ 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
$$723$$ 4.00000 0.148762
$$724$$ 0 0
$$725$$ 22.0000 0.817059
$$726$$ 0 0
$$727$$ −20.0000 −0.741759 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 0 0
$$733$$ 52.0000 1.92066 0.960332 0.278859i $$-0.0899564\pi$$
0.960332 + 0.278859i $$0.0899564\pi$$
$$734$$ 0 0
$$735$$ 8.00000 0.295084
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 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
$$755$$ 64.0000 2.32920
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ −10.0000 −0.362024
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ 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
$$771$$ −36.0000 −1.29651
$$772$$ 0 0
$$773$$ 4.00000 0.143870 0.0719350 0.997409i $$-0.477083\pi$$
0.0719350 + 0.997409i $$0.477083\pi$$
$$774$$ 0 0
$$775$$ −44.0000 −1.58053
$$776$$ 0 0
$$777$$ −12.0000 −0.430498
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 8.00000 0.285897
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −80.0000 −2.83731
$$796$$ 0 0
$$797$$ −24.0000 −0.850124 −0.425062 0.905164i $$-0.639748\pi$$
−0.425062 + 0.905164i $$0.639748\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −32.0000 −1.12785
$$806$$ 0 0
$$807$$ 48.0000 1.68968
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −30.0000 −1.05344 −0.526721 0.850038i $$-0.676579\pi$$
−0.526721 + 0.850038i $$0.676579\pi$$
$$812$$ 0 0
$$813$$ 64.0000 2.24458
$$814$$ 0 0
$$815$$ 64.0000 2.24182
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ −44.0000 −1.52634
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 48.0000 1.66111
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 12.0000 0.413302
$$844$$ 0 0
$$845$$ 52.0000 1.78885
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ −32.0000 −1.09566 −0.547830 0.836590i $$-0.684546\pi$$
−0.547830 + 0.836590i $$0.684546\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ −4.00000 −0.136320
$$862$$ 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
$$867$$ 26.0000 0.883006
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 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
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ −48.0000 −1.61350
$$886$$ 0 0
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 20.0000 0.666297
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 0 0
$$905$$ −32.0000 −1.06372
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ 12.0000 0.398015
$$910$$ 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
$$915$$ 32.0000 1.05789
$$916$$ 0 0
$$917$$ 14.0000 0.462321
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −66.0000 −2.17007
$$926$$ 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
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ −48.0000 −1.57145
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 0 0
$$939$$ −28.0000 −0.913745
$$940$$ 0 0
$$941$$ −12.0000 −0.391189 −0.195594 0.980685i $$-0.562664\pi$$
−0.195594 + 0.980685i $$0.562664\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 0 0
$$945$$ 16.0000 0.520480
$$946$$ 0 0
$$947$$ −40.0000 −1.29983 −0.649913 0.760009i $$-0.725195\pi$$
−0.649913 + 0.760009i $$0.725195\pi$$
$$948$$ 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 $$-0.448217\pi$$
0.161966 + 0.986796i $$0.448217\pi$$
$$954$$ 0 0
$$955$$ −32.0000 −1.03550
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2.00000 −0.0645834
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 0 0
$$965$$ 72.0000 2.31776
$$966$$ 0 0
$$967$$ 24.0000 0.771788 0.385894 0.922543i $$-0.373893\pi$$
0.385894 + 0.922543i $$0.373893\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 26.0000 0.834380 0.417190 0.908819i $$-0.363015\pi$$
0.417190 + 0.908819i $$0.363015\pi$$
$$972$$ 0 0
$$973$$ 18.0000 0.577054
$$974$$ 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
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 0 0
$$985$$ 72.0000 2.29411
$$986$$ 0 0
$$987$$ 8.00000 0.254643
$$988$$ 0 0
$$989$$ 64.0000 2.03508
$$990$$ 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
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 52.0000 1.64686 0.823428 0.567420i $$-0.192059\pi$$
0.823428 + 0.567420i $$0.192059\pi$$
$$998$$ 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 112.2.a.a.1.1 1
3.2 odd 2 1008.2.a.m.1.1 1
4.3 odd 2 56.2.a.b.1.1 1
5.2 odd 4 2800.2.g.g.449.2 2
5.3 odd 4 2800.2.g.g.449.1 2
5.4 even 2 2800.2.a.bd.1.1 1
7.2 even 3 784.2.i.j.753.1 2
7.3 odd 6 784.2.i.b.177.1 2
7.4 even 3 784.2.i.j.177.1 2
7.5 odd 6 784.2.i.b.753.1 2
7.6 odd 2 784.2.a.i.1.1 1
8.3 odd 2 448.2.a.c.1.1 1
8.5 even 2 448.2.a.h.1.1 1
12.11 even 2 504.2.a.h.1.1 1
16.3 odd 4 1792.2.b.a.897.2 2
16.5 even 4 1792.2.b.h.897.2 2
16.11 odd 4 1792.2.b.a.897.1 2
16.13 even 4 1792.2.b.h.897.1 2
20.3 even 4 1400.2.g.b.449.2 2
20.7 even 4 1400.2.g.b.449.1 2
20.19 odd 2 1400.2.a.a.1.1 1
21.20 even 2 7056.2.a.c.1.1 1
24.5 odd 2 4032.2.a.a.1.1 1
24.11 even 2 4032.2.a.d.1.1 1
28.3 even 6 392.2.i.e.177.1 2
28.11 odd 6 392.2.i.a.177.1 2
28.19 even 6 392.2.i.e.361.1 2
28.23 odd 6 392.2.i.a.361.1 2
28.27 even 2 392.2.a.b.1.1 1
44.43 even 2 6776.2.a.h.1.1 1
52.51 odd 2 9464.2.a.h.1.1 1
56.13 odd 2 3136.2.a.c.1.1 1
56.27 even 2 3136.2.a.w.1.1 1
84.11 even 6 3528.2.s.a.3313.1 2
84.23 even 6 3528.2.s.a.361.1 2
84.47 odd 6 3528.2.s.ba.361.1 2
84.59 odd 6 3528.2.s.ba.3313.1 2
84.83 odd 2 3528.2.a.b.1.1 1
140.139 even 2 9800.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 1 4.3 odd 2
112.2.a.a.1.1 1 1.1 even 1 trivial
392.2.a.b.1.1 1 28.27 even 2
392.2.i.a.177.1 2 28.11 odd 6
392.2.i.a.361.1 2 28.23 odd 6
392.2.i.e.177.1 2 28.3 even 6
392.2.i.e.361.1 2 28.19 even 6
448.2.a.c.1.1 1 8.3 odd 2
448.2.a.h.1.1 1 8.5 even 2
504.2.a.h.1.1 1 12.11 even 2
784.2.a.i.1.1 1 7.6 odd 2
784.2.i.b.177.1 2 7.3 odd 6
784.2.i.b.753.1 2 7.5 odd 6
784.2.i.j.177.1 2 7.4 even 3
784.2.i.j.753.1 2 7.2 even 3
1008.2.a.m.1.1 1 3.2 odd 2
1400.2.a.a.1.1 1 20.19 odd 2
1400.2.g.b.449.1 2 20.7 even 4
1400.2.g.b.449.2 2 20.3 even 4
1792.2.b.a.897.1 2 16.11 odd 4
1792.2.b.a.897.2 2 16.3 odd 4
1792.2.b.h.897.1 2 16.13 even 4
1792.2.b.h.897.2 2 16.5 even 4
2800.2.a.bd.1.1 1 5.4 even 2
2800.2.g.g.449.1 2 5.3 odd 4
2800.2.g.g.449.2 2 5.2 odd 4
3136.2.a.c.1.1 1 56.13 odd 2
3136.2.a.w.1.1 1 56.27 even 2
3528.2.a.b.1.1 1 84.83 odd 2
3528.2.s.a.361.1 2 84.23 even 6
3528.2.s.a.3313.1 2 84.11 even 6
3528.2.s.ba.361.1 2 84.47 odd 6
3528.2.s.ba.3313.1 2 84.59 odd 6
4032.2.a.a.1.1 1 24.5 odd 2
4032.2.a.d.1.1 1 24.11 even 2
6776.2.a.h.1.1 1 44.43 even 2
7056.2.a.c.1.1 1 21.20 even 2
9464.2.a.h.1.1 1 52.51 odd 2
9800.2.a.bj.1.1 1 140.139 even 2