# Properties

 Label 2352.2.a.w.1.1 Level $2352$ Weight $2$ Character 2352.1 Self dual yes Analytic conductor $18.781$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{19} -1.00000 q^{25} +1.00000 q^{27} +4.00000 q^{29} +9.00000 q^{31} +2.00000 q^{33} +3.00000 q^{37} -1.00000 q^{39} +10.0000 q^{41} -5.00000 q^{43} +2.00000 q^{45} -6.00000 q^{47} +12.0000 q^{53} +4.00000 q^{55} +1.00000 q^{57} -12.0000 q^{59} -10.0000 q^{61} -2.00000 q^{65} +5.00000 q^{67} +6.00000 q^{71} +3.00000 q^{73} -1.00000 q^{75} +1.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +4.00000 q^{87} -16.0000 q^{89} +9.00000 q^{93} +2.00000 q^{95} +6.00000 q^{97} +2.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 9.00000 1.61645 0.808224 0.588875i $$-0.200429\pi$$
0.808224 + 0.588875i $$0.200429\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 9.00000 0.933257
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 10.0000 0.901670
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 15.0000 1.33103 0.665517 0.746382i $$-0.268211\pi$$
0.665517 + 0.746382i $$0.268211\pi$$
$$128$$ 0 0
$$129$$ −5.00000 −0.440225
$$130$$ 0 0
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 2.00000 0.172133
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 18.0000 1.44579
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 4.00000 0.311400
$$166$$ 0 0
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ 16.0000 1.13995 0.569976 0.821661i $$-0.306952\pi$$
0.569976 + 0.821661i $$0.306952\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.00000 0.202721
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ 1.00000 0.0649570
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 0 0
$$263$$ −4.00000 −0.246651 −0.123325 0.992366i $$-0.539356\pi$$
−0.123325 + 0.992366i $$0.539356\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ −16.0000 −0.979184
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 13.0000 0.781094 0.390547 0.920583i $$-0.372286\pi$$
0.390547 + 0.920583i $$0.372286\pi$$
$$278$$ 0 0
$$279$$ 9.00000 0.538816
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 0 0
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 0 0
$$293$$ −8.00000 −0.467365 −0.233682 0.972313i $$-0.575078\pi$$
−0.233682 + 0.972313i $$0.575078\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −2.00000 −0.114897
$$304$$ 0 0
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ −17.0000 −0.970241 −0.485121 0.874447i $$-0.661224\pi$$
−0.485121 + 0.874447i $$0.661224\pi$$
$$308$$ 0 0
$$309$$ −7.00000 −0.398216
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ 9.00000 0.497701
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ 3.00000 0.164399
$$334$$ 0 0
$$335$$ 10.0000 0.546358
$$336$$ 0 0
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ 0 0
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 0 0
$$355$$ 12.0000 0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ −9.00000 −0.469796 −0.234898 0.972020i $$-0.575476\pi$$
−0.234898 + 0.972020i $$0.575476\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −3.00000 −0.154100 −0.0770498 0.997027i $$-0.524550\pi$$
−0.0770498 + 0.997027i $$0.524550\pi$$
$$380$$ 0 0
$$381$$ 15.0000 0.768473
$$382$$ 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
$$387$$ −5.00000 −0.254164
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −14.0000 −0.706207
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ 9.00000 0.451697 0.225849 0.974162i $$-0.427485\pi$$
0.225849 + 0.974162i $$0.427485\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ 0 0
$$403$$ −9.00000 −0.448322
$$404$$ 0 0
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ −3.00000 −0.146911
$$418$$ 0 0
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ −31.0000 −1.48976 −0.744882 0.667196i $$-0.767494\pi$$
−0.744882 + 0.667196i $$0.767494\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ −32.0000 −1.51695
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 20.0000 0.941763
$$452$$ 0 0
$$453$$ 16.0000 0.751746
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −20.0000 −0.931493 −0.465746 0.884918i $$-0.654214\pi$$
−0.465746 + 0.884918i $$0.654214\pi$$
$$462$$ 0 0
$$463$$ 17.0000 0.790057 0.395029 0.918669i $$-0.370735\pi$$
0.395029 + 0.918669i $$0.370735\pi$$
$$464$$ 0 0
$$465$$ 18.0000 0.834730
$$466$$ 0 0
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.0000 0.544892
$$486$$ 0 0
$$487$$ −31.0000 −1.40474 −0.702372 0.711810i $$-0.747876\pi$$
−0.702372 + 0.711810i $$0.747876\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −37.0000 −1.65635 −0.828174 0.560471i $$-0.810620\pi$$
−0.828174 + 0.560471i $$0.810620\pi$$
$$500$$ 0 0
$$501$$ −14.0000 −0.625474
$$502$$ 0 0
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ −12.0000 −0.532939
$$508$$ 0 0
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −14.0000 −0.616914
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ −8.00000 −0.351161
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ 31.0000 1.35554 0.677768 0.735276i $$-0.262948\pi$$
0.677768 + 0.735276i $$0.262948\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19.0000 −0.816874 −0.408437 0.912787i $$-0.633926\pi$$
−0.408437 + 0.912787i $$0.633926\pi$$
$$542$$ 0 0
$$543$$ −13.0000 −0.557883
$$544$$ 0 0
$$545$$ 18.0000 0.771035
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ 5.00000 0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ 0 0
$$565$$ 20.0000 0.841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 0 0
$$573$$ −10.0000 −0.417756
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 24.0000 0.993978
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ 0 0
$$591$$ 16.0000 0.658152
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 9.00000 0.367118 0.183559 0.983009i $$-0.441238\pi$$
0.183559 + 0.983009i $$0.441238\pi$$
$$602$$ 0 0
$$603$$ 5.00000 0.203616
$$604$$ 0 0
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 0 0
$$615$$ 20.0000 0.806478
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −29.0000 −1.16561 −0.582804 0.812613i $$-0.698045\pi$$
−0.582804 + 0.812613i $$0.698045\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 2.00000 0.0798723
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 30.0000 1.19051
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ 0 0
$$645$$ −10.0000 −0.393750
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −28.0000 −1.09405
$$656$$ 0 0
$$657$$ 3.00000 0.117041
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 41.0000 1.59472 0.797358 0.603507i $$-0.206231\pi$$
0.797358 + 0.603507i $$0.206231\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −20.0000 −0.772091
$$672$$ 0 0
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −24.0000 −0.916993
$$686$$ 0 0
$$687$$ 19.0000 0.724895
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −37.0000 −1.40755 −0.703773 0.710425i $$-0.748503\pi$$
−0.703773 + 0.710425i $$0.748503\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −6.00000 −0.227593
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 3.00000 0.113147
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 1.00000 0.0375029
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ −6.00000 −0.224074
$$718$$ 0 0
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −14.0000 −0.520666
$$724$$ 0 0
$$725$$ −4.00000 −0.148556
$$726$$ 0 0
$$727$$ −13.0000 −0.482143 −0.241072 0.970507i $$-0.577499\pi$$
−0.241072 + 0.970507i $$0.577499\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.0000 0.368355
$$738$$ 0 0
$$739$$ 15.0000 0.551784 0.275892 0.961189i $$-0.411027\pi$$
0.275892 + 0.961189i $$0.411027\pi$$
$$740$$ 0 0
$$741$$ −1.00000 −0.0367359
$$742$$ 0 0
$$743$$ −42.0000 −1.54083 −0.770415 0.637542i $$-0.779951\pi$$
−0.770415 + 0.637542i $$0.779951\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ 0 0
$$753$$ −8.00000 −0.291536
$$754$$ 0 0
$$755$$ 32.0000 1.16460
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ −26.0000 −0.936367
$$772$$ 0 0
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ 0 0
$$775$$ −9.00000 −0.323290
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ 4.00000 0.142948
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ 0 0
$$787$$ 40.0000 1.42585 0.712923 0.701242i $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ 0 0
$$789$$ −4.00000 −0.142404
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 0 0
$$795$$ 24.0000 0.851192
$$796$$ 0 0
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −16.0000 −0.565332
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 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 $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 32.0000 1.12367 0.561836 0.827249i $$-0.310095\pi$$
0.561836 + 0.827249i $$0.310095\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ 0 0
$$829$$ −41.0000 −1.42399 −0.711994 0.702185i $$-0.752208\pi$$
−0.711994 + 0.702185i $$0.752208\pi$$
$$830$$ 0 0
$$831$$ 13.0000 0.450965
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −28.0000 −0.968980
$$836$$ 0 0
$$837$$ 9.00000 0.311086
$$838$$ 0 0
$$839$$ −44.0000 −1.51905 −0.759524 0.650479i $$-0.774568\pi$$
−0.759524 + 0.650479i $$0.774568\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ −4.00000 −0.137767
$$844$$ 0 0
$$845$$ −24.0000 −0.825625
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −11.0000 −0.377519
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −35.0000 −1.19838 −0.599189 0.800608i $$-0.704510\pi$$
−0.599189 + 0.800608i $$0.704510\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ 32.0000 1.09310 0.546550 0.837427i $$-0.315941\pi$$
0.546550 + 0.837427i $$0.315941\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 0 0
$$867$$ −17.0000 −0.577350
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ −5.00000 −0.169419
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ 0 0
$$879$$ −8.00000 −0.269833
$$880$$ 0 0
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 0 0
$$883$$ 13.0000 0.437485 0.218742 0.975783i $$-0.429805\pi$$
0.218742 + 0.975783i $$0.429805\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ −34.0000 −1.14161 −0.570804 0.821086i $$-0.693368\pi$$
−0.570804 + 0.821086i $$0.693368\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −26.0000 −0.864269
$$906$$ 0 0
$$907$$ 37.0000 1.22856 0.614282 0.789086i $$-0.289446\pi$$
0.614282 + 0.789086i $$0.289446\pi$$
$$908$$ 0 0
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ −20.0000 −0.661180
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −23.0000 −0.758700 −0.379350 0.925253i $$-0.623852\pi$$
−0.379350 + 0.925253i $$0.623852\pi$$
$$920$$ 0 0
$$921$$ −17.0000 −0.560169
$$922$$ 0 0
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 0 0
$$927$$ −7.00000 −0.229910
$$928$$ 0 0
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −6.00000 −0.196431
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −15.0000 −0.490029 −0.245014 0.969519i $$-0.578793\pi$$
−0.245014 + 0.969519i $$0.578793\pi$$
$$938$$ 0 0
$$939$$ 1.00000 0.0326338
$$940$$ 0 0
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 10.0000 0.324956 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$948$$ 0 0
$$949$$ −3.00000 −0.0973841
$$950$$ 0 0
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ −20.0000 −0.647185
$$956$$ 0 0
$$957$$ 8.00000 0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 0 0
$$963$$ 8.00000 0.257796
$$964$$ 0 0
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ −19.0000 −0.610999 −0.305499 0.952192i $$-0.598823\pi$$
−0.305499 + 0.952192i $$0.598823\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 1.00000 0.0320256
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ 9.00000 0.287348
$$982$$ 0 0
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 32.0000 1.01960
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ 25.0000 0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.0000 −0.601736 −0.300868 0.953666i $$-0.597276\pi$$
−0.300868 + 0.953666i $$0.597276\pi$$
$$998$$ 0 0
$$999$$ 3.00000 0.0949158
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 2352.2.a.w.1.1 1
3.2 odd 2 7056.2.a.m.1.1 1
4.3 odd 2 147.2.a.b.1.1 1
7.2 even 3 2352.2.q.c.1537.1 2
7.3 odd 6 336.2.q.f.289.1 2
7.4 even 3 2352.2.q.c.961.1 2
7.5 odd 6 336.2.q.f.193.1 2
7.6 odd 2 2352.2.a.d.1.1 1
8.3 odd 2 9408.2.a.bz.1.1 1
8.5 even 2 9408.2.a.k.1.1 1
12.11 even 2 441.2.a.a.1.1 1
20.19 odd 2 3675.2.a.c.1.1 1
21.5 even 6 1008.2.s.d.865.1 2
21.17 even 6 1008.2.s.d.289.1 2
21.20 even 2 7056.2.a.bp.1.1 1
28.3 even 6 21.2.e.a.16.1 yes 2
28.11 odd 6 147.2.e.a.79.1 2
28.19 even 6 21.2.e.a.4.1 2
28.23 odd 6 147.2.e.a.67.1 2
28.27 even 2 147.2.a.c.1.1 1
56.3 even 6 1344.2.q.m.961.1 2
56.5 odd 6 1344.2.q.c.193.1 2
56.13 odd 2 9408.2.a.cv.1.1 1
56.19 even 6 1344.2.q.m.193.1 2
56.27 even 2 9408.2.a.bg.1.1 1
56.45 odd 6 1344.2.q.c.961.1 2
84.11 even 6 441.2.e.e.226.1 2
84.23 even 6 441.2.e.e.361.1 2
84.47 odd 6 63.2.e.b.46.1 2
84.59 odd 6 63.2.e.b.37.1 2
84.83 odd 2 441.2.a.b.1.1 1
140.3 odd 12 525.2.r.e.499.2 4
140.19 even 6 525.2.i.e.151.1 2
140.47 odd 12 525.2.r.e.424.2 4
140.59 even 6 525.2.i.e.226.1 2
140.87 odd 12 525.2.r.e.499.1 4
140.103 odd 12 525.2.r.e.424.1 4
140.139 even 2 3675.2.a.a.1.1 1
252.31 even 6 567.2.g.a.541.1 2
252.47 odd 6 567.2.g.f.109.1 2
252.59 odd 6 567.2.g.f.541.1 2
252.103 even 6 567.2.h.f.298.1 2
252.115 even 6 567.2.h.f.352.1 2
252.131 odd 6 567.2.h.a.298.1 2
252.187 even 6 567.2.g.a.109.1 2
252.227 odd 6 567.2.h.a.352.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 28.19 even 6
21.2.e.a.16.1 yes 2 28.3 even 6
63.2.e.b.37.1 2 84.59 odd 6
63.2.e.b.46.1 2 84.47 odd 6
147.2.a.b.1.1 1 4.3 odd 2
147.2.a.c.1.1 1 28.27 even 2
147.2.e.a.67.1 2 28.23 odd 6
147.2.e.a.79.1 2 28.11 odd 6
336.2.q.f.193.1 2 7.5 odd 6
336.2.q.f.289.1 2 7.3 odd 6
441.2.a.a.1.1 1 12.11 even 2
441.2.a.b.1.1 1 84.83 odd 2
441.2.e.e.226.1 2 84.11 even 6
441.2.e.e.361.1 2 84.23 even 6
525.2.i.e.151.1 2 140.19 even 6
525.2.i.e.226.1 2 140.59 even 6
525.2.r.e.424.1 4 140.103 odd 12
525.2.r.e.424.2 4 140.47 odd 12
525.2.r.e.499.1 4 140.87 odd 12
525.2.r.e.499.2 4 140.3 odd 12
567.2.g.a.109.1 2 252.187 even 6
567.2.g.a.541.1 2 252.31 even 6
567.2.g.f.109.1 2 252.47 odd 6
567.2.g.f.541.1 2 252.59 odd 6
567.2.h.a.298.1 2 252.131 odd 6
567.2.h.a.352.1 2 252.227 odd 6
567.2.h.f.298.1 2 252.103 even 6
567.2.h.f.352.1 2 252.115 even 6
1008.2.s.d.289.1 2 21.17 even 6
1008.2.s.d.865.1 2 21.5 even 6
1344.2.q.c.193.1 2 56.5 odd 6
1344.2.q.c.961.1 2 56.45 odd 6
1344.2.q.m.193.1 2 56.19 even 6
1344.2.q.m.961.1 2 56.3 even 6
2352.2.a.d.1.1 1 7.6 odd 2
2352.2.a.w.1.1 1 1.1 even 1 trivial
2352.2.q.c.961.1 2 7.4 even 3
2352.2.q.c.1537.1 2 7.2 even 3
3675.2.a.a.1.1 1 140.139 even 2
3675.2.a.c.1.1 1 20.19 odd 2
7056.2.a.m.1.1 1 3.2 odd 2
7056.2.a.bp.1.1 1 21.20 even 2
9408.2.a.k.1.1 1 8.5 even 2
9408.2.a.bg.1.1 1 56.27 even 2
9408.2.a.bz.1.1 1 8.3 odd 2
9408.2.a.cv.1.1 1 56.13 odd 2