# Properties

 Label 8112.2.a.bv.1.1 Level $8112$ Weight $2$ Character 8112.1 Self dual yes Analytic conductor $64.775$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8112 = 2^{4} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.7746461197$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 8112.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{11} +6.00000 q^{17} -3.46410 q^{19} -3.46410 q^{21} -5.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +3.46410 q^{31} -3.46410 q^{33} -6.92820 q^{37} -6.92820 q^{41} -4.00000 q^{43} +3.46410 q^{47} +5.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} -3.46410 q^{57} +10.3923 q^{59} -2.00000 q^{61} -3.46410 q^{63} +10.3923 q^{67} -3.46410 q^{71} -5.00000 q^{75} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} +6.00000 q^{87} +6.92820 q^{89} +3.46410 q^{93} -13.8564 q^{97} -3.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 10 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{43} + 10 q^{49} + 12 q^{51} + 12 q^{53} - 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 10 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^43 + 10 * q^49 + 12 * q^51 + 12 * q^53 - 4 * q^61 - 10 * q^75 + 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ −3.46410 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ −3.46410 −0.794719 −0.397360 0.917663i $$-0.630073\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ −3.46410 −0.755929
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 3.46410 0.622171 0.311086 0.950382i $$-0.399307\pi$$
0.311086 + 0.950382i $$0.399307\pi$$
$$32$$ 0 0
$$33$$ −3.46410 −0.603023
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.92820 −1.13899 −0.569495 0.821995i $$-0.692861\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.92820 −1.08200 −0.541002 0.841021i $$-0.681955\pi$$
−0.541002 + 0.841021i $$0.681955\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.46410 −0.458831
$$58$$ 0 0
$$59$$ 10.3923 1.35296 0.676481 0.736460i $$-0.263504\pi$$
0.676481 + 0.736460i $$0.263504\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ −3.46410 −0.436436
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.3923 1.26962 0.634811 0.772667i $$-0.281078\pi$$
0.634811 + 0.772667i $$0.281078\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.46410 −0.411113 −0.205557 0.978645i $$-0.565900\pi$$
−0.205557 + 0.978645i $$0.565900\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ −5.00000 −0.577350
$$76$$ 0 0
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 3.46410 0.380235 0.190117 0.981761i $$-0.439113\pi$$
0.190117 + 0.981761i $$0.439113\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ 6.92820 0.734388 0.367194 0.930144i $$-0.380318\pi$$
0.367194 + 0.930144i $$0.380318\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.46410 0.359211
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −13.8564 −1.40690 −0.703452 0.710742i $$-0.748359\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 0 0
$$99$$ −3.46410 −0.348155
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 6.92820 0.663602 0.331801 0.943349i $$-0.392344\pi$$
0.331801 + 0.943349i $$0.392344\pi$$
$$110$$ 0 0
$$111$$ −6.92820 −0.657596
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −20.7846 −1.90532
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −6.92820 −0.624695
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 20.7846 1.77575 0.887875 0.460086i $$-0.152181\pi$$
0.887875 + 0.460086i $$0.152181\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 3.46410 0.291730
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 5.00000 0.412393
$$148$$ 0 0
$$149$$ 13.8564 1.13516 0.567581 0.823318i $$-0.307880\pi$$
0.567581 + 0.823318i $$0.307880\pi$$
$$150$$ 0 0
$$151$$ 10.3923 0.845714 0.422857 0.906196i $$-0.361027\pi$$
0.422857 + 0.906196i $$0.361027\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3.46410 0.271329 0.135665 0.990755i $$-0.456683\pi$$
0.135665 + 0.990755i $$0.456683\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −3.46410 −0.264906
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 17.3205 1.30931
$$176$$ 0 0
$$177$$ 10.3923 0.781133
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −20.7846 −1.51992
$$188$$ 0 0
$$189$$ −3.46410 −0.251976
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 10.3923 0.733017
$$202$$ 0 0
$$203$$ −20.7846 −1.45879
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ −3.46410 −0.237356
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 3.46410 0.231973 0.115987 0.993251i $$-0.462997\pi$$
0.115987 + 0.993251i $$0.462997\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 17.3205 1.14960 0.574801 0.818293i $$-0.305079\pi$$
0.574801 + 0.818293i $$0.305079\pi$$
$$228$$ 0 0
$$229$$ 6.92820 0.457829 0.228914 0.973447i $$-0.426482\pi$$
0.228914 + 0.973447i $$0.426482\pi$$
$$230$$ 0 0
$$231$$ 12.0000 0.789542
$$232$$ 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
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 10.3923 0.672222 0.336111 0.941822i $$-0.390888\pi$$
0.336111 + 0.941822i $$0.390888\pi$$
$$240$$ 0 0
$$241$$ −13.8564 −0.892570 −0.446285 0.894891i $$-0.647253\pi$$
−0.446285 + 0.894891i $$0.647253\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 3.46410 0.219529
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.92820 0.423999
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 10.3923 0.631288 0.315644 0.948878i $$-0.397780\pi$$
0.315644 + 0.948878i $$0.397780\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 17.3205 1.04447
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 3.46410 0.207390
$$280$$ 0 0
$$281$$ −6.92820 −0.413302 −0.206651 0.978415i $$-0.566256\pi$$
−0.206651 + 0.978415i $$0.566256\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −13.8564 −0.812277
$$292$$ 0 0
$$293$$ −27.7128 −1.61900 −0.809500 0.587120i $$-0.800262\pi$$
−0.809500 + 0.587120i $$0.800262\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.46410 −0.201008
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 13.8564 0.798670
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −10.3923 −0.593120 −0.296560 0.955014i $$-0.595840\pi$$
−0.296560 + 0.955014i $$0.595840\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 13.8564 0.778253 0.389127 0.921184i $$-0.372777\pi$$
0.389127 + 0.921184i $$0.372777\pi$$
$$318$$ 0 0
$$319$$ −20.7846 −1.16371
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −20.7846 −1.15649
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 6.92820 0.383131
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −3.46410 −0.190404 −0.0952021 0.995458i $$-0.530350\pi$$
−0.0952021 + 0.995458i $$0.530350\pi$$
$$332$$ 0 0
$$333$$ −6.92820 −0.379663
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 6.92820 0.374088
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −36.0000 −1.93258 −0.966291 0.257454i $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ 0 0
$$349$$ 6.92820 0.370858 0.185429 0.982658i $$-0.440632\pi$$
0.185429 + 0.982658i $$0.440632\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 34.6410 1.84376 0.921878 0.387481i $$-0.126655\pi$$
0.921878 + 0.387481i $$0.126655\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −20.7846 −1.10004
$$358$$ 0 0
$$359$$ 17.3205 0.914141 0.457071 0.889430i $$-0.348899\pi$$
0.457071 + 0.889430i $$0.348899\pi$$
$$360$$ 0 0
$$361$$ −7.00000 −0.368421
$$362$$ 0 0
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ −6.92820 −0.360668
$$370$$ 0 0
$$371$$ −20.7846 −1.07908
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −17.3205 −0.889695 −0.444847 0.895606i $$-0.646742\pi$$
−0.444847 + 0.895606i $$0.646742\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ −3.46410 −0.177007 −0.0885037 0.996076i $$-0.528208\pi$$
−0.0885037 + 0.996076i $$0.528208\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 34.6410 1.73858 0.869291 0.494300i $$-0.164576\pi$$
0.869291 + 0.494300i $$0.164576\pi$$
$$398$$ 0 0
$$399$$ 12.0000 0.600751
$$400$$ 0 0
$$401$$ −6.92820 −0.345978 −0.172989 0.984924i $$-0.555343\pi$$
−0.172989 + 0.984924i $$0.555343\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 27.7128 1.37031 0.685155 0.728397i $$-0.259734\pi$$
0.685155 + 0.728397i $$0.259734\pi$$
$$410$$ 0 0
$$411$$ 20.7846 1.02523
$$412$$ 0 0
$$413$$ −36.0000 −1.77144
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −34.6410 −1.68830 −0.844150 0.536107i $$-0.819894\pi$$
−0.844150 + 0.536107i $$0.819894\pi$$
$$422$$ 0 0
$$423$$ 3.46410 0.168430
$$424$$ 0 0
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ 6.92820 0.335279
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.2487 1.16802 0.584010 0.811747i $$-0.301483\pi$$
0.584010 + 0.811747i $$0.301483\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 5.00000 0.238095
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13.8564 0.655386
$$448$$ 0 0
$$449$$ −6.92820 −0.326962 −0.163481 0.986546i $$-0.552272\pi$$
−0.163481 + 0.986546i $$0.552272\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 10.3923 0.488273
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −27.7128 −1.29635 −0.648175 0.761491i $$-0.724468\pi$$
−0.648175 + 0.761491i $$0.724468\pi$$
$$458$$ 0 0
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −13.8564 −0.645357 −0.322679 0.946509i $$-0.604583\pi$$
−0.322679 + 0.946509i $$0.604583\pi$$
$$462$$ 0 0
$$463$$ −17.3205 −0.804952 −0.402476 0.915430i $$-0.631850\pi$$
−0.402476 + 0.915430i $$0.631850\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 13.8564 0.637118
$$474$$ 0 0
$$475$$ 17.3205 0.794719
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ −10.3923 −0.474837 −0.237418 0.971408i $$-0.576301\pi$$
−0.237418 + 0.971408i $$0.576301\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −38.1051 −1.72671 −0.863354 0.504599i $$-0.831640\pi$$
−0.863354 + 0.504599i $$0.831640\pi$$
$$488$$ 0 0
$$489$$ 3.46410 0.156652
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ 10.3923 0.465223 0.232612 0.972570i $$-0.425273\pi$$
0.232612 + 0.972570i $$0.425273\pi$$
$$500$$ 0 0
$$501$$ 17.3205 0.773823
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −41.5692 −1.84252 −0.921262 0.388943i $$-0.872840\pi$$
−0.921262 + 0.388943i $$0.872840\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −3.46410 −0.152944
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 17.3205 0.755929
$$526$$ 0 0
$$527$$ 20.7846 0.905392
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 10.3923 0.450988
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ −17.3205 −0.746047
$$540$$ 0 0
$$541$$ −6.92820 −0.297867 −0.148933 0.988847i $$-0.547584\pi$$
−0.148933 + 0.988847i $$0.547584\pi$$
$$542$$ 0 0
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −20.7846 −0.885454
$$552$$ 0 0
$$553$$ −27.7128 −1.17847
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −13.8564 −0.587115 −0.293557 0.955941i $$-0.594839\pi$$
−0.293557 + 0.955941i $$0.594839\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −20.7846 −0.877527
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.46410 −0.145479
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −20.7846 −0.860811
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10.3923 −0.428936 −0.214468 0.976731i $$-0.568802\pi$$
−0.214468 + 0.976731i $$0.568802\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.92820 −0.284507 −0.142254 0.989830i $$-0.545435\pi$$
−0.142254 + 0.989830i $$0.545435\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 10.3923 0.423207
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ −20.7846 −0.842235
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −20.7846 −0.839482 −0.419741 0.907644i $$-0.637879\pi$$
−0.419741 + 0.907644i $$0.637879\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.92820 −0.278919 −0.139459 0.990228i $$-0.544536\pi$$
−0.139459 + 0.990228i $$0.544536\pi$$
$$618$$ 0 0
$$619$$ 31.1769 1.25311 0.626553 0.779379i $$-0.284465\pi$$
0.626553 + 0.779379i $$0.284465\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ 0 0
$$629$$ −41.5692 −1.65747
$$630$$ 0 0
$$631$$ 38.1051 1.51694 0.758470 0.651707i $$-0.225947\pi$$
0.758470 + 0.651707i $$0.225947\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −3.46410 −0.137038
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ 10.3923 0.409832 0.204916 0.978780i $$-0.434308\pi$$
0.204916 + 0.978780i $$0.434308\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 20.7846 0.808428 0.404214 0.914665i $$-0.367545\pi$$
0.404214 + 0.914665i $$0.367545\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 3.46410 0.133930
$$670$$ 0 0
$$671$$ 6.92820 0.267460
$$672$$ 0 0
$$673$$ 46.0000 1.77317 0.886585 0.462566i $$-0.153071\pi$$
0.886585 + 0.462566i $$0.153071\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 48.0000 1.84207
$$680$$ 0 0
$$681$$ 17.3205 0.663723
$$682$$ 0 0
$$683$$ −31.1769 −1.19295 −0.596476 0.802631i $$-0.703433\pi$$
−0.596476 + 0.802631i $$0.703433\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.92820 0.264327
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −45.0333 −1.71315 −0.856574 0.516024i $$-0.827412\pi$$
−0.856574 + 0.516024i $$0.827412\pi$$
$$692$$ 0 0
$$693$$ 12.0000 0.455842
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −41.5692 −1.57455
$$698$$ 0 0
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 20.7846 0.781686
$$708$$ 0 0
$$709$$ 6.92820 0.260194 0.130097 0.991501i $$-0.458471\pi$$
0.130097 + 0.991501i $$0.458471\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 10.3923 0.388108
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −27.7128 −1.03208
$$722$$ 0 0
$$723$$ −13.8564 −0.515325
$$724$$ 0 0
$$725$$ −30.0000 −1.11417
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 34.6410 1.27950 0.639748 0.768585i $$-0.279039\pi$$
0.639748 + 0.768585i $$0.279039\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36.0000 −1.32608
$$738$$ 0 0
$$739$$ −38.1051 −1.40172 −0.700860 0.713299i $$-0.747200\pi$$
−0.700860 + 0.713299i $$0.747200\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −3.46410 −0.127086 −0.0635428 0.997979i $$-0.520240\pi$$
−0.0635428 + 0.997979i $$0.520240\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 3.46410 0.126745
$$748$$ 0 0
$$749$$ 41.5692 1.51891
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.4974 1.75803 0.879015 0.476794i $$-0.158201\pi$$
0.879015 + 0.476794i $$0.158201\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 27.7128 0.999350 0.499675 0.866213i $$-0.333453\pi$$
0.499675 + 0.866213i $$0.333453\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 0 0
$$773$$ −13.8564 −0.498380 −0.249190 0.968455i $$-0.580164\pi$$
−0.249190 + 0.968455i $$0.580164\pi$$
$$774$$ 0 0
$$775$$ −17.3205 −0.622171
$$776$$ 0 0
$$777$$ 24.0000 0.860995
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −10.3923 −0.370446 −0.185223 0.982697i $$-0.559301\pi$$
−0.185223 + 0.982697i $$0.559301\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 20.7846 0.739016
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 20.7846 0.735307
$$800$$ 0 0
$$801$$ 6.92820 0.244796
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 38.1051 1.33805 0.669026 0.743239i $$-0.266712\pi$$
0.669026 + 0.743239i $$0.266712\pi$$
$$812$$ 0 0
$$813$$ 10.3923 0.364474
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 13.8564 0.484774
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 13.8564 0.483592 0.241796 0.970327i $$-0.422264\pi$$
0.241796 + 0.970327i $$0.422264\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ 17.3205 0.603023
$$826$$ 0 0
$$827$$ 24.2487 0.843210 0.421605 0.906780i $$-0.361467\pi$$
0.421605 + 0.906780i $$0.361467\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 30.0000 1.03944
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.46410 0.119737
$$838$$ 0 0
$$839$$ 3.46410 0.119594 0.0597970 0.998211i $$-0.480955\pi$$
0.0597970 + 0.998211i $$0.480955\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −6.92820 −0.238620
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.46410 −0.119028
$$848$$ 0 0
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 20.7846 0.711651 0.355826 0.934552i $$-0.384200\pi$$
0.355826 + 0.934552i $$0.384200\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ −31.1769 −1.06127 −0.530637 0.847599i $$-0.678047\pi$$
−0.530637 + 0.847599i $$0.678047\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ −27.7128 −0.940093
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −13.8564 −0.468968
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 48.4974 1.63764 0.818821 0.574049i $$-0.194628\pi$$
0.818821 + 0.574049i $$0.194628\pi$$
$$878$$ 0 0
$$879$$ −27.7128 −0.934730
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 27.7128 0.929458
$$890$$ 0 0
$$891$$ −3.46410 −0.116052
$$892$$ 0 0
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 20.7846 0.693206
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 13.8564 0.461112
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −41.5692 −1.37274
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ −10.3923 −0.342438
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 34.6410 1.13899
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ −20.7846 −0.681921 −0.340960 0.940078i $$-0.610752\pi$$
−0.340960 + 0.940078i $$0.610752\pi$$
$$930$$ 0 0
$$931$$ −17.3205 −0.567657
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 51.9615 1.68852 0.844261 0.535932i $$-0.180040\pi$$
0.844261 + 0.535932i $$0.180040\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 13.8564 0.449325
$$952$$ 0 0
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −20.7846 −0.671871
$$958$$ 0 0
$$959$$ −72.0000 −2.32500
$$960$$ 0 0
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −10.3923 −0.334194 −0.167097 0.985940i $$-0.553439\pi$$
−0.167097 + 0.985940i $$0.553439\pi$$
$$968$$ 0 0
$$969$$ −20.7846 −0.667698
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −13.8564 −0.444216
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 48.4974 1.55157 0.775785 0.630997i $$-0.217354\pi$$
0.775785 + 0.630997i $$0.217354\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 6.92820 0.221201
$$982$$ 0 0
$$983$$ −51.9615 −1.65732 −0.828658 0.559756i $$-0.810895\pi$$
−0.828658 + 0.559756i $$0.810895\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −12.0000 −0.381964
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −3.46410 −0.109930
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 0 0
$$999$$ −6.92820 −0.219199
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 8112.2.a.bv.1.1 2
4.3 odd 2 507.2.a.f.1.2 2
12.11 even 2 1521.2.a.l.1.1 2
13.5 odd 4 624.2.c.e.337.1 2
13.8 odd 4 624.2.c.e.337.2 2
13.12 even 2 inner 8112.2.a.bv.1.2 2
39.5 even 4 1872.2.c.e.1585.1 2
39.8 even 4 1872.2.c.e.1585.2 2
52.3 odd 6 507.2.e.e.22.1 4
52.7 even 12 507.2.j.c.361.1 2
52.11 even 12 507.2.j.a.316.1 2
52.15 even 12 507.2.j.c.316.1 2
52.19 even 12 507.2.j.a.361.1 2
52.23 odd 6 507.2.e.e.22.2 4
52.31 even 4 39.2.b.a.25.1 2
52.35 odd 6 507.2.e.e.484.1 4
52.43 odd 6 507.2.e.e.484.2 4
52.47 even 4 39.2.b.a.25.2 yes 2
52.51 odd 2 507.2.a.f.1.1 2
104.5 odd 4 2496.2.c.d.961.1 2
104.21 odd 4 2496.2.c.d.961.2 2
104.83 even 4 2496.2.c.k.961.2 2
104.99 even 4 2496.2.c.k.961.1 2
156.47 odd 4 117.2.b.a.64.1 2
156.83 odd 4 117.2.b.a.64.2 2
156.155 even 2 1521.2.a.l.1.2 2
260.47 odd 4 975.2.h.f.649.2 4
260.83 odd 4 975.2.h.f.649.1 4
260.99 even 4 975.2.b.d.376.1 2
260.187 odd 4 975.2.h.f.649.4 4
260.203 odd 4 975.2.h.f.649.3 4
260.239 even 4 975.2.b.d.376.2 2
364.83 odd 4 1911.2.c.d.883.1 2
364.307 odd 4 1911.2.c.d.883.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.b.a.25.1 2 52.31 even 4
39.2.b.a.25.2 yes 2 52.47 even 4
117.2.b.a.64.1 2 156.47 odd 4
117.2.b.a.64.2 2 156.83 odd 4
507.2.a.f.1.1 2 52.51 odd 2
507.2.a.f.1.2 2 4.3 odd 2
507.2.e.e.22.1 4 52.3 odd 6
507.2.e.e.22.2 4 52.23 odd 6
507.2.e.e.484.1 4 52.35 odd 6
507.2.e.e.484.2 4 52.43 odd 6
507.2.j.a.316.1 2 52.11 even 12
507.2.j.a.361.1 2 52.19 even 12
507.2.j.c.316.1 2 52.15 even 12
507.2.j.c.361.1 2 52.7 even 12
624.2.c.e.337.1 2 13.5 odd 4
624.2.c.e.337.2 2 13.8 odd 4
975.2.b.d.376.1 2 260.99 even 4
975.2.b.d.376.2 2 260.239 even 4
975.2.h.f.649.1 4 260.83 odd 4
975.2.h.f.649.2 4 260.47 odd 4
975.2.h.f.649.3 4 260.203 odd 4
975.2.h.f.649.4 4 260.187 odd 4
1521.2.a.l.1.1 2 12.11 even 2
1521.2.a.l.1.2 2 156.155 even 2
1872.2.c.e.1585.1 2 39.5 even 4
1872.2.c.e.1585.2 2 39.8 even 4
1911.2.c.d.883.1 2 364.83 odd 4
1911.2.c.d.883.2 2 364.307 odd 4
2496.2.c.d.961.1 2 104.5 odd 4
2496.2.c.d.961.2 2 104.21 odd 4
2496.2.c.k.961.1 2 104.99 even 4
2496.2.c.k.961.2 2 104.83 even 4
8112.2.a.bv.1.1 2 1.1 even 1 trivial
8112.2.a.bv.1.2 2 13.12 even 2 inner