# Properties

 Label 1344.2.a.u.1.1 Level $1344$ Weight $2$ Character 1344.1 Self dual yes Analytic conductor $10.732$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1344.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.46410 q^{11} -2.00000 q^{13} +3.46410 q^{15} +0.535898 q^{17} -6.92820 q^{19} +1.00000 q^{21} -1.46410 q^{23} +7.00000 q^{25} -1.00000 q^{27} +4.92820 q^{29} -10.9282 q^{31} -1.46410 q^{33} +3.46410 q^{35} +2.00000 q^{37} +2.00000 q^{39} +11.4641 q^{41} +8.00000 q^{43} -3.46410 q^{45} +10.9282 q^{47} +1.00000 q^{49} -0.535898 q^{51} +2.00000 q^{53} -5.07180 q^{55} +6.92820 q^{57} +1.07180 q^{59} +8.92820 q^{61} -1.00000 q^{63} +6.92820 q^{65} +2.92820 q^{67} +1.46410 q^{69} +9.46410 q^{71} +12.9282 q^{73} -7.00000 q^{75} -1.46410 q^{77} -10.9282 q^{79} +1.00000 q^{81} +4.00000 q^{83} -1.85641 q^{85} -4.92820 q^{87} +3.46410 q^{89} +2.00000 q^{91} +10.9282 q^{93} +24.0000 q^{95} -8.92820 q^{97} +1.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{7} + 2q^{9} - 4q^{11} - 4q^{13} + 8q^{17} + 2q^{21} + 4q^{23} + 14q^{25} - 2q^{27} - 4q^{29} - 8q^{31} + 4q^{33} + 4q^{37} + 4q^{39} + 16q^{41} + 16q^{43} + 8q^{47} + 2q^{49} - 8q^{51} + 4q^{53} - 24q^{55} + 16q^{59} + 4q^{61} - 2q^{63} - 8q^{67} - 4q^{69} + 12q^{71} + 12q^{73} - 14q^{75} + 4q^{77} - 8q^{79} + 2q^{81} + 8q^{83} + 24q^{85} + 4q^{87} + 4q^{91} + 8q^{93} + 48q^{95} - 4q^{97} - 4q^{99} + O(q^{100})$$

## 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
$$5$$ −3.46410 −1.54919 −0.774597 0.632456i $$-0.782047\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.46410 0.441443 0.220722 0.975337i $$-0.429159\pi$$
0.220722 + 0.975337i $$0.429159\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 3.46410 0.894427
$$16$$ 0 0
$$17$$ 0.535898 0.129974 0.0649872 0.997886i $$-0.479299\pi$$
0.0649872 + 0.997886i $$0.479299\pi$$
$$18$$ 0 0
$$19$$ −6.92820 −1.58944 −0.794719 0.606977i $$-0.792382\pi$$
−0.794719 + 0.606977i $$0.792382\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −1.46410 −0.305286 −0.152643 0.988281i $$-0.548779\pi$$
−0.152643 + 0.988281i $$0.548779\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 4.92820 0.915144 0.457572 0.889172i $$-0.348719\pi$$
0.457572 + 0.889172i $$0.348719\pi$$
$$30$$ 0 0
$$31$$ −10.9282 −1.96276 −0.981382 0.192068i $$-0.938481\pi$$
−0.981382 + 0.192068i $$0.938481\pi$$
$$32$$ 0 0
$$33$$ −1.46410 −0.254867
$$34$$ 0 0
$$35$$ 3.46410 0.585540
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 11.4641 1.79039 0.895196 0.445673i $$-0.147036\pi$$
0.895196 + 0.445673i $$0.147036\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ −3.46410 −0.516398
$$46$$ 0 0
$$47$$ 10.9282 1.59404 0.797021 0.603951i $$-0.206408\pi$$
0.797021 + 0.603951i $$0.206408\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −0.535898 −0.0750408
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ −5.07180 −0.683881
$$56$$ 0 0
$$57$$ 6.92820 0.917663
$$58$$ 0 0
$$59$$ 1.07180 0.139536 0.0697680 0.997563i $$-0.477774\pi$$
0.0697680 + 0.997563i $$0.477774\pi$$
$$60$$ 0 0
$$61$$ 8.92820 1.14314 0.571570 0.820554i $$-0.306335\pi$$
0.571570 + 0.820554i $$0.306335\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 6.92820 0.859338
$$66$$ 0 0
$$67$$ 2.92820 0.357737 0.178868 0.983873i $$-0.442756\pi$$
0.178868 + 0.983873i $$0.442756\pi$$
$$68$$ 0 0
$$69$$ 1.46410 0.176257
$$70$$ 0 0
$$71$$ 9.46410 1.12318 0.561591 0.827415i $$-0.310189\pi$$
0.561591 + 0.827415i $$0.310189\pi$$
$$72$$ 0 0
$$73$$ 12.9282 1.51313 0.756566 0.653917i $$-0.226876\pi$$
0.756566 + 0.653917i $$0.226876\pi$$
$$74$$ 0 0
$$75$$ −7.00000 −0.808290
$$76$$ 0 0
$$77$$ −1.46410 −0.166850
$$78$$ 0 0
$$79$$ −10.9282 −1.22952 −0.614759 0.788715i $$-0.710747\pi$$
−0.614759 + 0.788715i $$0.710747\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −1.85641 −0.201356
$$86$$ 0 0
$$87$$ −4.92820 −0.528359
$$88$$ 0 0
$$89$$ 3.46410 0.367194 0.183597 0.983002i $$-0.441226\pi$$
0.183597 + 0.983002i $$0.441226\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 10.9282 1.13320
$$94$$ 0 0
$$95$$ 24.0000 2.46235
$$96$$ 0 0
$$97$$ −8.92820 −0.906522 −0.453261 0.891378i $$-0.649739\pi$$
−0.453261 + 0.891378i $$0.649739\pi$$
$$98$$ 0 0
$$99$$ 1.46410 0.147148
$$100$$ 0 0
$$101$$ 15.4641 1.53874 0.769368 0.638806i $$-0.220571\pi$$
0.769368 + 0.638806i $$0.220571\pi$$
$$102$$ 0 0
$$103$$ −10.9282 −1.07679 −0.538394 0.842693i $$-0.680969\pi$$
−0.538394 + 0.842693i $$0.680969\pi$$
$$104$$ 0 0
$$105$$ −3.46410 −0.338062
$$106$$ 0 0
$$107$$ −12.3923 −1.19801 −0.599005 0.800746i $$-0.704437\pi$$
−0.599005 + 0.800746i $$0.704437\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ −19.8564 −1.86793 −0.933967 0.357360i $$-0.883677\pi$$
−0.933967 + 0.357360i $$0.883677\pi$$
$$114$$ 0 0
$$115$$ 5.07180 0.472947
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ −0.535898 −0.0491257
$$120$$ 0 0
$$121$$ −8.85641 −0.805128
$$122$$ 0 0
$$123$$ −11.4641 −1.03368
$$124$$ 0 0
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ 2.92820 0.259836 0.129918 0.991525i $$-0.458529\pi$$
0.129918 + 0.991525i $$0.458529\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 6.92820 0.600751
$$134$$ 0 0
$$135$$ 3.46410 0.298142
$$136$$ 0 0
$$137$$ −8.92820 −0.762788 −0.381394 0.924413i $$-0.624556\pi$$
−0.381394 + 0.924413i $$0.624556\pi$$
$$138$$ 0 0
$$139$$ 17.8564 1.51456 0.757280 0.653090i $$-0.226528\pi$$
0.757280 + 0.653090i $$0.226528\pi$$
$$140$$ 0 0
$$141$$ −10.9282 −0.920321
$$142$$ 0 0
$$143$$ −2.92820 −0.244869
$$144$$ 0 0
$$145$$ −17.0718 −1.41774
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 21.8564 1.77865 0.889325 0.457277i $$-0.151175\pi$$
0.889325 + 0.457277i $$0.151175\pi$$
$$152$$ 0 0
$$153$$ 0.535898 0.0433248
$$154$$ 0 0
$$155$$ 37.8564 3.04070
$$156$$ 0 0
$$157$$ 0.928203 0.0740787 0.0370393 0.999314i $$-0.488207\pi$$
0.0370393 + 0.999314i $$0.488207\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 1.46410 0.115387
$$162$$ 0 0
$$163$$ −10.9282 −0.855963 −0.427981 0.903788i $$-0.640775\pi$$
−0.427981 + 0.903788i $$0.640775\pi$$
$$164$$ 0 0
$$165$$ 5.07180 0.394839
$$166$$ 0 0
$$167$$ −10.9282 −0.845650 −0.422825 0.906211i $$-0.638961\pi$$
−0.422825 + 0.906211i $$0.638961\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −6.92820 −0.529813
$$172$$ 0 0
$$173$$ −22.3923 −1.70246 −0.851228 0.524797i $$-0.824141\pi$$
−0.851228 + 0.524797i $$0.824141\pi$$
$$174$$ 0 0
$$175$$ −7.00000 −0.529150
$$176$$ 0 0
$$177$$ −1.07180 −0.0805612
$$178$$ 0 0
$$179$$ −9.46410 −0.707380 −0.353690 0.935363i $$-0.615073\pi$$
−0.353690 + 0.935363i $$0.615073\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ −8.92820 −0.659992
$$184$$ 0 0
$$185$$ −6.92820 −0.509372
$$186$$ 0 0
$$187$$ 0.784610 0.0573763
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 9.46410 0.684798 0.342399 0.939555i $$-0.388760\pi$$
0.342399 + 0.939555i $$0.388760\pi$$
$$192$$ 0 0
$$193$$ 15.8564 1.14137 0.570685 0.821169i $$-0.306678\pi$$
0.570685 + 0.821169i $$0.306678\pi$$
$$194$$ 0 0
$$195$$ −6.92820 −0.496139
$$196$$ 0 0
$$197$$ 7.85641 0.559746 0.279873 0.960037i $$-0.409708\pi$$
0.279873 + 0.960037i $$0.409708\pi$$
$$198$$ 0 0
$$199$$ 5.85641 0.415150 0.207575 0.978219i $$-0.433443\pi$$
0.207575 + 0.978219i $$0.433443\pi$$
$$200$$ 0 0
$$201$$ −2.92820 −0.206540
$$202$$ 0 0
$$203$$ −4.92820 −0.345892
$$204$$ 0 0
$$205$$ −39.7128 −2.77366
$$206$$ 0 0
$$207$$ −1.46410 −0.101762
$$208$$ 0 0
$$209$$ −10.1436 −0.701647
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ −9.46410 −0.648470
$$214$$ 0 0
$$215$$ −27.7128 −1.89000
$$216$$ 0 0
$$217$$ 10.9282 0.741855
$$218$$ 0 0
$$219$$ −12.9282 −0.873607
$$220$$ 0 0
$$221$$ −1.07180 −0.0720969
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 0 0
$$225$$ 7.00000 0.466667
$$226$$ 0 0
$$227$$ 22.9282 1.52180 0.760899 0.648870i $$-0.224758\pi$$
0.760899 + 0.648870i $$0.224758\pi$$
$$228$$ 0 0
$$229$$ 27.8564 1.84080 0.920402 0.390974i $$-0.127862\pi$$
0.920402 + 0.390974i $$0.127862\pi$$
$$230$$ 0 0
$$231$$ 1.46410 0.0963308
$$232$$ 0 0
$$233$$ 12.9282 0.846955 0.423477 0.905907i $$-0.360809\pi$$
0.423477 + 0.905907i $$0.360809\pi$$
$$234$$ 0 0
$$235$$ −37.8564 −2.46948
$$236$$ 0 0
$$237$$ 10.9282 0.709863
$$238$$ 0 0
$$239$$ 14.5359 0.940249 0.470125 0.882600i $$-0.344209\pi$$
0.470125 + 0.882600i $$0.344209\pi$$
$$240$$ 0 0
$$241$$ −3.07180 −0.197872 −0.0989359 0.995094i $$-0.531544\pi$$
−0.0989359 + 0.995094i $$0.531544\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −3.46410 −0.221313
$$246$$ 0 0
$$247$$ 13.8564 0.881662
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 9.07180 0.572607 0.286303 0.958139i $$-0.407573\pi$$
0.286303 + 0.958139i $$0.407573\pi$$
$$252$$ 0 0
$$253$$ −2.14359 −0.134767
$$254$$ 0 0
$$255$$ 1.85641 0.116253
$$256$$ 0 0
$$257$$ 11.4641 0.715111 0.357556 0.933892i $$-0.383610\pi$$
0.357556 + 0.933892i $$0.383610\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 4.92820 0.305048
$$262$$ 0 0
$$263$$ 12.3923 0.764142 0.382071 0.924133i $$-0.375211\pi$$
0.382071 + 0.924133i $$0.375211\pi$$
$$264$$ 0 0
$$265$$ −6.92820 −0.425596
$$266$$ 0 0
$$267$$ −3.46410 −0.212000
$$268$$ 0 0
$$269$$ −3.46410 −0.211210 −0.105605 0.994408i $$-0.533678\pi$$
−0.105605 + 0.994408i $$0.533678\pi$$
$$270$$ 0 0
$$271$$ 2.92820 0.177876 0.0889378 0.996037i $$-0.471653\pi$$
0.0889378 + 0.996037i $$0.471653\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ 10.2487 0.618021
$$276$$ 0 0
$$277$$ −0.143594 −0.00862770 −0.00431385 0.999991i $$-0.501373\pi$$
−0.00431385 + 0.999991i $$0.501373\pi$$
$$278$$ 0 0
$$279$$ −10.9282 −0.654254
$$280$$ 0 0
$$281$$ 12.9282 0.771232 0.385616 0.922659i $$-0.373989\pi$$
0.385616 + 0.922659i $$0.373989\pi$$
$$282$$ 0 0
$$283$$ 14.9282 0.887390 0.443695 0.896178i $$-0.353667\pi$$
0.443695 + 0.896178i $$0.353667\pi$$
$$284$$ 0 0
$$285$$ −24.0000 −1.42164
$$286$$ 0 0
$$287$$ −11.4641 −0.676705
$$288$$ 0 0
$$289$$ −16.7128 −0.983107
$$290$$ 0 0
$$291$$ 8.92820 0.523381
$$292$$ 0 0
$$293$$ −16.5359 −0.966037 −0.483019 0.875610i $$-0.660460\pi$$
−0.483019 + 0.875610i $$0.660460\pi$$
$$294$$ 0 0
$$295$$ −3.71281 −0.216168
$$296$$ 0 0
$$297$$ −1.46410 −0.0849558
$$298$$ 0 0
$$299$$ 2.92820 0.169342
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ −15.4641 −0.888389
$$304$$ 0 0
$$305$$ −30.9282 −1.77094
$$306$$ 0 0
$$307$$ −22.9282 −1.30858 −0.654291 0.756243i $$-0.727033\pi$$
−0.654291 + 0.756243i $$0.727033\pi$$
$$308$$ 0 0
$$309$$ 10.9282 0.621684
$$310$$ 0 0
$$311$$ 18.9282 1.07332 0.536660 0.843799i $$-0.319686\pi$$
0.536660 + 0.843799i $$0.319686\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ 3.46410 0.195180
$$316$$ 0 0
$$317$$ −0.143594 −0.00806502 −0.00403251 0.999992i $$-0.501284\pi$$
−0.00403251 + 0.999992i $$0.501284\pi$$
$$318$$ 0 0
$$319$$ 7.21539 0.403984
$$320$$ 0 0
$$321$$ 12.3923 0.691671
$$322$$ 0 0
$$323$$ −3.71281 −0.206586
$$324$$ 0 0
$$325$$ −14.0000 −0.776580
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ −10.9282 −0.602491
$$330$$ 0 0
$$331$$ −24.0000 −1.31916 −0.659580 0.751635i $$-0.729266\pi$$
−0.659580 + 0.751635i $$0.729266\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ −10.1436 −0.554204
$$336$$ 0 0
$$337$$ 7.85641 0.427966 0.213983 0.976837i $$-0.431356\pi$$
0.213983 + 0.976837i $$0.431356\pi$$
$$338$$ 0 0
$$339$$ 19.8564 1.07845
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −5.07180 −0.273056
$$346$$ 0 0
$$347$$ −23.3205 −1.25191 −0.625955 0.779859i $$-0.715291\pi$$
−0.625955 + 0.779859i $$0.715291\pi$$
$$348$$ 0 0
$$349$$ −10.7846 −0.577287 −0.288643 0.957437i $$-0.593204\pi$$
−0.288643 + 0.957437i $$0.593204\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 8.53590 0.454320 0.227160 0.973857i $$-0.427056\pi$$
0.227160 + 0.973857i $$0.427056\pi$$
$$354$$ 0 0
$$355$$ −32.7846 −1.74003
$$356$$ 0 0
$$357$$ 0.535898 0.0283628
$$358$$ 0 0
$$359$$ 7.32051 0.386362 0.193181 0.981163i $$-0.438120\pi$$
0.193181 + 0.981163i $$0.438120\pi$$
$$360$$ 0 0
$$361$$ 29.0000 1.52632
$$362$$ 0 0
$$363$$ 8.85641 0.464841
$$364$$ 0 0
$$365$$ −44.7846 −2.34413
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 11.4641 0.596797
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ −30.0000 −1.55334 −0.776671 0.629907i $$-0.783093\pi$$
−0.776671 + 0.629907i $$0.783093\pi$$
$$374$$ 0 0
$$375$$ 6.92820 0.357771
$$376$$ 0 0
$$377$$ −9.85641 −0.507631
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −2.92820 −0.150016
$$382$$ 0 0
$$383$$ 35.7128 1.82484 0.912420 0.409256i $$-0.134212\pi$$
0.912420 + 0.409256i $$0.134212\pi$$
$$384$$ 0 0
$$385$$ 5.07180 0.258483
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 0 0
$$389$$ −14.7846 −0.749609 −0.374805 0.927104i $$-0.622290\pi$$
−0.374805 + 0.927104i $$0.622290\pi$$
$$390$$ 0 0
$$391$$ −0.784610 −0.0396794
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 37.8564 1.90476
$$396$$ 0 0
$$397$$ −10.7846 −0.541264 −0.270632 0.962683i $$-0.587233\pi$$
−0.270632 + 0.962683i $$0.587233\pi$$
$$398$$ 0 0
$$399$$ −6.92820 −0.346844
$$400$$ 0 0
$$401$$ 4.92820 0.246103 0.123051 0.992400i $$-0.460732\pi$$
0.123051 + 0.992400i $$0.460732\pi$$
$$402$$ 0 0
$$403$$ 21.8564 1.08875
$$404$$ 0 0
$$405$$ −3.46410 −0.172133
$$406$$ 0 0
$$407$$ 2.92820 0.145146
$$408$$ 0 0
$$409$$ 4.92820 0.243684 0.121842 0.992550i $$-0.461120\pi$$
0.121842 + 0.992550i $$0.461120\pi$$
$$410$$ 0 0
$$411$$ 8.92820 0.440396
$$412$$ 0 0
$$413$$ −1.07180 −0.0527397
$$414$$ 0 0
$$415$$ −13.8564 −0.680184
$$416$$ 0 0
$$417$$ −17.8564 −0.874432
$$418$$ 0 0
$$419$$ −17.0718 −0.834012 −0.417006 0.908904i $$-0.636921\pi$$
−0.417006 + 0.908904i $$0.636921\pi$$
$$420$$ 0 0
$$421$$ 4.14359 0.201946 0.100973 0.994889i $$-0.467804\pi$$
0.100973 + 0.994889i $$0.467804\pi$$
$$422$$ 0 0
$$423$$ 10.9282 0.531347
$$424$$ 0 0
$$425$$ 3.75129 0.181964
$$426$$ 0 0
$$427$$ −8.92820 −0.432066
$$428$$ 0 0
$$429$$ 2.92820 0.141375
$$430$$ 0 0
$$431$$ −10.2487 −0.493663 −0.246832 0.969058i $$-0.579389\pi$$
−0.246832 + 0.969058i $$0.579389\pi$$
$$432$$ 0 0
$$433$$ −27.8564 −1.33869 −0.669347 0.742950i $$-0.733426\pi$$
−0.669347 + 0.742950i $$0.733426\pi$$
$$434$$ 0 0
$$435$$ 17.0718 0.818530
$$436$$ 0 0
$$437$$ 10.1436 0.485234
$$438$$ 0 0
$$439$$ 13.8564 0.661330 0.330665 0.943748i $$-0.392727\pi$$
0.330665 + 0.943748i $$0.392727\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 28.3923 1.34896 0.674480 0.738294i $$-0.264368\pi$$
0.674480 + 0.738294i $$0.264368\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ 7.85641 0.370767 0.185383 0.982666i $$-0.440647\pi$$
0.185383 + 0.982666i $$0.440647\pi$$
$$450$$ 0 0
$$451$$ 16.7846 0.790356
$$452$$ 0 0
$$453$$ −21.8564 −1.02690
$$454$$ 0 0
$$455$$ −6.92820 −0.324799
$$456$$ 0 0
$$457$$ 37.7128 1.76413 0.882065 0.471127i $$-0.156153\pi$$
0.882065 + 0.471127i $$0.156153\pi$$
$$458$$ 0 0
$$459$$ −0.535898 −0.0250136
$$460$$ 0 0
$$461$$ 27.1769 1.26576 0.632878 0.774252i $$-0.281874\pi$$
0.632878 + 0.774252i $$0.281874\pi$$
$$462$$ 0 0
$$463$$ 32.7846 1.52363 0.761815 0.647795i $$-0.224309\pi$$
0.761815 + 0.647795i $$0.224309\pi$$
$$464$$ 0 0
$$465$$ −37.8564 −1.75555
$$466$$ 0 0
$$467$$ −20.7846 −0.961797 −0.480899 0.876776i $$-0.659689\pi$$
−0.480899 + 0.876776i $$0.659689\pi$$
$$468$$ 0 0
$$469$$ −2.92820 −0.135212
$$470$$ 0 0
$$471$$ −0.928203 −0.0427693
$$472$$ 0 0
$$473$$ 11.7128 0.538556
$$474$$ 0 0
$$475$$ −48.4974 −2.22521
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −5.07180 −0.231736 −0.115868 0.993265i $$-0.536965\pi$$
−0.115868 + 0.993265i $$0.536965\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ −1.46410 −0.0666189
$$484$$ 0 0
$$485$$ 30.9282 1.40438
$$486$$ 0 0
$$487$$ −27.7128 −1.25579 −0.627894 0.778299i $$-0.716083\pi$$
−0.627894 + 0.778299i $$0.716083\pi$$
$$488$$ 0 0
$$489$$ 10.9282 0.494190
$$490$$ 0 0
$$491$$ 6.53590 0.294961 0.147480 0.989065i $$-0.452884\pi$$
0.147480 + 0.989065i $$0.452884\pi$$
$$492$$ 0 0
$$493$$ 2.64102 0.118945
$$494$$ 0 0
$$495$$ −5.07180 −0.227960
$$496$$ 0 0
$$497$$ −9.46410 −0.424523
$$498$$ 0 0
$$499$$ 21.8564 0.978427 0.489214 0.872164i $$-0.337284\pi$$
0.489214 + 0.872164i $$0.337284\pi$$
$$500$$ 0 0
$$501$$ 10.9282 0.488236
$$502$$ 0 0
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −53.5692 −2.38380
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ 40.2487 1.78399 0.891996 0.452043i $$-0.149304\pi$$
0.891996 + 0.452043i $$0.149304\pi$$
$$510$$ 0 0
$$511$$ −12.9282 −0.571910
$$512$$ 0 0
$$513$$ 6.92820 0.305888
$$514$$ 0 0
$$515$$ 37.8564 1.66815
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 22.3923 0.982913
$$520$$ 0 0
$$521$$ 30.3923 1.33151 0.665756 0.746170i $$-0.268109\pi$$
0.665756 + 0.746170i $$0.268109\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 0 0
$$525$$ 7.00000 0.305505
$$526$$ 0 0
$$527$$ −5.85641 −0.255109
$$528$$ 0 0
$$529$$ −20.8564 −0.906800
$$530$$ 0 0
$$531$$ 1.07180 0.0465120
$$532$$ 0 0
$$533$$ −22.9282 −0.993131
$$534$$ 0 0
$$535$$ 42.9282 1.85595
$$536$$ 0 0
$$537$$ 9.46410 0.408406
$$538$$ 0 0
$$539$$ 1.46410 0.0630633
$$540$$ 0 0
$$541$$ 7.85641 0.337773 0.168887 0.985635i $$-0.445983\pi$$
0.168887 + 0.985635i $$0.445983\pi$$
$$542$$ 0 0
$$543$$ −6.00000 −0.257485
$$544$$ 0 0
$$545$$ −6.92820 −0.296772
$$546$$ 0 0
$$547$$ −10.9282 −0.467256 −0.233628 0.972326i $$-0.575060\pi$$
−0.233628 + 0.972326i $$0.575060\pi$$
$$548$$ 0 0
$$549$$ 8.92820 0.381046
$$550$$ 0 0
$$551$$ −34.1436 −1.45457
$$552$$ 0 0
$$553$$ 10.9282 0.464714
$$554$$ 0 0
$$555$$ 6.92820 0.294086
$$556$$ 0 0
$$557$$ 4.14359 0.175570 0.0877848 0.996139i $$-0.472021\pi$$
0.0877848 + 0.996139i $$0.472021\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ −0.784610 −0.0331262
$$562$$ 0 0
$$563$$ −33.0718 −1.39381 −0.696905 0.717163i $$-0.745440\pi$$
−0.696905 + 0.717163i $$0.745440\pi$$
$$564$$ 0 0
$$565$$ 68.7846 2.89379
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −24.9282 −1.04504 −0.522522 0.852626i $$-0.675009\pi$$
−0.522522 + 0.852626i $$0.675009\pi$$
$$570$$ 0 0
$$571$$ −16.7846 −0.702414 −0.351207 0.936298i $$-0.614229\pi$$
−0.351207 + 0.936298i $$0.614229\pi$$
$$572$$ 0 0
$$573$$ −9.46410 −0.395369
$$574$$ 0 0
$$575$$ −10.2487 −0.427401
$$576$$ 0 0
$$577$$ 23.8564 0.993155 0.496578 0.867992i $$-0.334590\pi$$
0.496578 + 0.867992i $$0.334590\pi$$
$$578$$ 0 0
$$579$$ −15.8564 −0.658970
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 2.92820 0.121274
$$584$$ 0 0
$$585$$ 6.92820 0.286446
$$586$$ 0 0
$$587$$ 20.7846 0.857873 0.428936 0.903335i $$-0.358888\pi$$
0.428936 + 0.903335i $$0.358888\pi$$
$$588$$ 0 0
$$589$$ 75.7128 3.11969
$$590$$ 0 0
$$591$$ −7.85641 −0.323169
$$592$$ 0 0
$$593$$ 8.53590 0.350527 0.175264 0.984522i $$-0.443922\pi$$
0.175264 + 0.984522i $$0.443922\pi$$
$$594$$ 0 0
$$595$$ 1.85641 0.0761052
$$596$$ 0 0
$$597$$ −5.85641 −0.239687
$$598$$ 0 0
$$599$$ −28.3923 −1.16008 −0.580039 0.814589i $$-0.696963\pi$$
−0.580039 + 0.814589i $$0.696963\pi$$
$$600$$ 0 0
$$601$$ 43.5692 1.77723 0.888613 0.458658i $$-0.151670\pi$$
0.888613 + 0.458658i $$0.151670\pi$$
$$602$$ 0 0
$$603$$ 2.92820 0.119246
$$604$$ 0 0
$$605$$ 30.6795 1.24730
$$606$$ 0 0
$$607$$ 13.8564 0.562414 0.281207 0.959647i $$-0.409265\pi$$
0.281207 + 0.959647i $$0.409265\pi$$
$$608$$ 0 0
$$609$$ 4.92820 0.199701
$$610$$ 0 0
$$611$$ −21.8564 −0.884216
$$612$$ 0 0
$$613$$ −27.8564 −1.12511 −0.562555 0.826760i $$-0.690181\pi$$
−0.562555 + 0.826760i $$0.690181\pi$$
$$614$$ 0 0
$$615$$ 39.7128 1.60138
$$616$$ 0 0
$$617$$ 18.7846 0.756240 0.378120 0.925757i $$-0.376571\pi$$
0.378120 + 0.925757i $$0.376571\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 1.46410 0.0587524
$$622$$ 0 0
$$623$$ −3.46410 −0.138786
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 10.1436 0.405096
$$628$$ 0 0
$$629$$ 1.07180 0.0427353
$$630$$ 0 0
$$631$$ 26.9282 1.07199 0.535997 0.844220i $$-0.319936\pi$$
0.535997 + 0.844220i $$0.319936\pi$$
$$632$$ 0 0
$$633$$ −16.0000 −0.635943
$$634$$ 0 0
$$635$$ −10.1436 −0.402536
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 9.46410 0.374394
$$640$$ 0 0
$$641$$ −6.78461 −0.267976 −0.133988 0.990983i $$-0.542778\pi$$
−0.133988 + 0.990983i $$0.542778\pi$$
$$642$$ 0 0
$$643$$ 12.7846 0.504176 0.252088 0.967704i $$-0.418883\pi$$
0.252088 + 0.967704i $$0.418883\pi$$
$$644$$ 0 0
$$645$$ 27.7128 1.09119
$$646$$ 0 0
$$647$$ −5.07180 −0.199393 −0.0996965 0.995018i $$-0.531787\pi$$
−0.0996965 + 0.995018i $$0.531787\pi$$
$$648$$ 0 0
$$649$$ 1.56922 0.0615972
$$650$$ 0 0
$$651$$ −10.9282 −0.428310
$$652$$ 0 0
$$653$$ 20.9282 0.818984 0.409492 0.912314i $$-0.365706\pi$$
0.409492 + 0.912314i $$0.365706\pi$$
$$654$$ 0 0
$$655$$ 41.5692 1.62424
$$656$$ 0 0
$$657$$ 12.9282 0.504377
$$658$$ 0 0
$$659$$ −14.5359 −0.566238 −0.283119 0.959085i $$-0.591369\pi$$
−0.283119 + 0.959085i $$0.591369\pi$$
$$660$$ 0 0
$$661$$ −26.7846 −1.04180 −0.520900 0.853618i $$-0.674404\pi$$
−0.520900 + 0.853618i $$0.674404\pi$$
$$662$$ 0 0
$$663$$ 1.07180 0.0416251
$$664$$ 0 0
$$665$$ −24.0000 −0.930680
$$666$$ 0 0
$$667$$ −7.21539 −0.279381
$$668$$ 0 0
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ 13.0718 0.504631
$$672$$ 0 0
$$673$$ 31.8564 1.22797 0.613987 0.789316i $$-0.289565\pi$$
0.613987 + 0.789316i $$0.289565\pi$$
$$674$$ 0 0
$$675$$ −7.00000 −0.269430
$$676$$ 0 0
$$677$$ 1.60770 0.0617887 0.0308944 0.999523i $$-0.490164\pi$$
0.0308944 + 0.999523i $$0.490164\pi$$
$$678$$ 0 0
$$679$$ 8.92820 0.342633
$$680$$ 0 0
$$681$$ −22.9282 −0.878611
$$682$$ 0 0
$$683$$ 18.2487 0.698268 0.349134 0.937073i $$-0.386476\pi$$
0.349134 + 0.937073i $$0.386476\pi$$
$$684$$ 0 0
$$685$$ 30.9282 1.18171
$$686$$ 0 0
$$687$$ −27.8564 −1.06279
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 41.8564 1.59229 0.796146 0.605104i $$-0.206869\pi$$
0.796146 + 0.605104i $$0.206869\pi$$
$$692$$ 0 0
$$693$$ −1.46410 −0.0556166
$$694$$ 0 0
$$695$$ −61.8564 −2.34635
$$696$$ 0 0
$$697$$ 6.14359 0.232705
$$698$$ 0 0
$$699$$ −12.9282 −0.488990
$$700$$ 0 0
$$701$$ −28.6410 −1.08176 −0.540878 0.841101i $$-0.681908\pi$$
−0.540878 + 0.841101i $$0.681908\pi$$
$$702$$ 0 0
$$703$$ −13.8564 −0.522604
$$704$$ 0 0
$$705$$ 37.8564 1.42575
$$706$$ 0 0
$$707$$ −15.4641 −0.581587
$$708$$ 0 0
$$709$$ −17.7128 −0.665219 −0.332609 0.943065i $$-0.607929\pi$$
−0.332609 + 0.943065i $$0.607929\pi$$
$$710$$ 0 0
$$711$$ −10.9282 −0.409840
$$712$$ 0 0
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 10.1436 0.379349
$$716$$ 0 0
$$717$$ −14.5359 −0.542853
$$718$$ 0 0
$$719$$ −13.8564 −0.516757 −0.258378 0.966044i $$-0.583188\pi$$
−0.258378 + 0.966044i $$0.583188\pi$$
$$720$$ 0 0
$$721$$ 10.9282 0.406988
$$722$$ 0 0
$$723$$ 3.07180 0.114241
$$724$$ 0 0
$$725$$ 34.4974 1.28120
$$726$$ 0 0
$$727$$ 21.0718 0.781510 0.390755 0.920495i $$-0.372214\pi$$
0.390755 + 0.920495i $$0.372214\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 4.28719 0.158567
$$732$$ 0 0
$$733$$ −5.71281 −0.211008 −0.105504 0.994419i $$-0.533646\pi$$
−0.105504 + 0.994419i $$0.533646\pi$$
$$734$$ 0 0
$$735$$ 3.46410 0.127775
$$736$$ 0 0
$$737$$ 4.28719 0.157921
$$738$$ 0 0
$$739$$ 37.0718 1.36371 0.681854 0.731488i $$-0.261174\pi$$
0.681854 + 0.731488i $$0.261174\pi$$
$$740$$ 0 0
$$741$$ −13.8564 −0.509028
$$742$$ 0 0
$$743$$ 11.6077 0.425845 0.212923 0.977069i $$-0.431702\pi$$
0.212923 + 0.977069i $$0.431702\pi$$
$$744$$ 0 0
$$745$$ −62.3538 −2.28447
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 12.3923 0.452805
$$750$$ 0 0
$$751$$ 27.7128 1.01125 0.505627 0.862752i $$-0.331261\pi$$
0.505627 + 0.862752i $$0.331261\pi$$
$$752$$ 0 0
$$753$$ −9.07180 −0.330595
$$754$$ 0 0
$$755$$ −75.7128 −2.75547
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ 2.14359 0.0778075
$$760$$ 0 0
$$761$$ 27.4641 0.995573 0.497786 0.867300i $$-0.334146\pi$$
0.497786 + 0.867300i $$0.334146\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ −1.85641 −0.0671185
$$766$$ 0 0
$$767$$ −2.14359 −0.0774007
$$768$$ 0 0
$$769$$ 4.14359 0.149422 0.0747109 0.997205i $$-0.476197\pi$$
0.0747109 + 0.997205i $$0.476197\pi$$
$$770$$ 0 0
$$771$$ −11.4641 −0.412870
$$772$$ 0 0
$$773$$ 5.32051 0.191365 0.0956827 0.995412i $$-0.469497\pi$$
0.0956827 + 0.995412i $$0.469497\pi$$
$$774$$ 0 0
$$775$$ −76.4974 −2.74787
$$776$$ 0 0
$$777$$ 2.00000 0.0717496
$$778$$ 0 0
$$779$$ −79.4256 −2.84572
$$780$$ 0 0
$$781$$ 13.8564 0.495821
$$782$$ 0 0
$$783$$ −4.92820 −0.176120
$$784$$ 0 0
$$785$$ −3.21539 −0.114762
$$786$$ 0 0
$$787$$ −23.7128 −0.845270 −0.422635 0.906300i $$-0.638895\pi$$
−0.422635 + 0.906300i $$0.638895\pi$$
$$788$$ 0 0
$$789$$ −12.3923 −0.441178
$$790$$ 0 0
$$791$$ 19.8564 0.706013
$$792$$ 0 0
$$793$$ −17.8564 −0.634100
$$794$$ 0 0
$$795$$ 6.92820 0.245718
$$796$$ 0 0
$$797$$ −6.39230 −0.226427 −0.113214 0.993571i $$-0.536114\pi$$
−0.113214 + 0.993571i $$0.536114\pi$$
$$798$$ 0 0
$$799$$ 5.85641 0.207185
$$800$$ 0 0
$$801$$ 3.46410 0.122398
$$802$$ 0 0
$$803$$ 18.9282 0.667962
$$804$$ 0 0
$$805$$ −5.07180 −0.178757
$$806$$ 0 0
$$807$$ 3.46410 0.121942
$$808$$ 0 0
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 0 0
$$811$$ 37.5692 1.31923 0.659617 0.751602i $$-0.270719\pi$$
0.659617 + 0.751602i $$0.270719\pi$$
$$812$$ 0 0
$$813$$ −2.92820 −0.102697
$$814$$ 0 0
$$815$$ 37.8564 1.32605
$$816$$ 0 0
$$817$$ −55.4256 −1.93910
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 12.1436 0.423814 0.211907 0.977290i $$-0.432033\pi$$
0.211907 + 0.977290i $$0.432033\pi$$
$$822$$ 0 0
$$823$$ −21.0718 −0.734517 −0.367258 0.930119i $$-0.619704\pi$$
−0.367258 + 0.930119i $$0.619704\pi$$
$$824$$ 0 0
$$825$$ −10.2487 −0.356814
$$826$$ 0 0
$$827$$ 37.1769 1.29277 0.646384 0.763012i $$-0.276280\pi$$
0.646384 + 0.763012i $$0.276280\pi$$
$$828$$ 0 0
$$829$$ −21.7128 −0.754117 −0.377059 0.926189i $$-0.623064\pi$$
−0.377059 + 0.926189i $$0.623064\pi$$
$$830$$ 0 0
$$831$$ 0.143594 0.00498120
$$832$$ 0 0
$$833$$ 0.535898 0.0185678
$$834$$ 0 0
$$835$$ 37.8564 1.31007
$$836$$ 0 0
$$837$$ 10.9282 0.377734
$$838$$ 0 0
$$839$$ −10.9282 −0.377283 −0.188642 0.982046i $$-0.560408\pi$$
−0.188642 + 0.982046i $$0.560408\pi$$
$$840$$ 0 0
$$841$$ −4.71281 −0.162511
$$842$$ 0 0
$$843$$ −12.9282 −0.445271
$$844$$ 0 0
$$845$$ 31.1769 1.07252
$$846$$ 0 0
$$847$$ 8.85641 0.304310
$$848$$ 0 0
$$849$$ −14.9282 −0.512335
$$850$$ 0 0
$$851$$ −2.92820 −0.100378
$$852$$ 0 0
$$853$$ −23.0718 −0.789963 −0.394982 0.918689i $$-0.629249\pi$$
−0.394982 + 0.918689i $$0.629249\pi$$
$$854$$ 0 0
$$855$$ 24.0000 0.820783
$$856$$ 0 0
$$857$$ 55.1769 1.88481 0.942404 0.334477i $$-0.108560\pi$$
0.942404 + 0.334477i $$0.108560\pi$$
$$858$$ 0 0
$$859$$ 38.9282 1.32821 0.664107 0.747638i $$-0.268812\pi$$
0.664107 + 0.747638i $$0.268812\pi$$
$$860$$ 0 0
$$861$$ 11.4641 0.390696
$$862$$ 0 0
$$863$$ −9.46410 −0.322162 −0.161081 0.986941i $$-0.551498\pi$$
−0.161081 + 0.986941i $$0.551498\pi$$
$$864$$ 0 0
$$865$$ 77.5692 2.63743
$$866$$ 0 0
$$867$$ 16.7128 0.567597
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −5.85641 −0.198437
$$872$$ 0 0
$$873$$ −8.92820 −0.302174
$$874$$ 0 0
$$875$$ 6.92820 0.234216
$$876$$ 0 0
$$877$$ −8.14359 −0.274990 −0.137495 0.990502i $$-0.543905\pi$$
−0.137495 + 0.990502i $$0.543905\pi$$
$$878$$ 0 0
$$879$$ 16.5359 0.557742
$$880$$ 0 0
$$881$$ −13.3205 −0.448779 −0.224390 0.974500i $$-0.572039\pi$$
−0.224390 + 0.974500i $$0.572039\pi$$
$$882$$ 0 0
$$883$$ 33.5692 1.12969 0.564847 0.825196i $$-0.308935\pi$$
0.564847 + 0.825196i $$0.308935\pi$$
$$884$$ 0 0
$$885$$ 3.71281 0.124805
$$886$$ 0 0
$$887$$ −26.9282 −0.904161 −0.452080 0.891977i $$-0.649318\pi$$
−0.452080 + 0.891977i $$0.649318\pi$$
$$888$$ 0 0
$$889$$ −2.92820 −0.0982088
$$890$$ 0 0
$$891$$ 1.46410 0.0490492
$$892$$ 0 0
$$893$$ −75.7128 −2.53363
$$894$$ 0 0
$$895$$ 32.7846 1.09587
$$896$$ 0 0
$$897$$ −2.92820 −0.0977699
$$898$$ 0 0
$$899$$ −53.8564 −1.79621
$$900$$ 0 0
$$901$$ 1.07180 0.0357067
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −20.7846 −0.690904
$$906$$ 0 0
$$907$$ 43.7128 1.45146 0.725730 0.687980i $$-0.241502\pi$$
0.725730 + 0.687980i $$0.241502\pi$$
$$908$$ 0 0
$$909$$ 15.4641 0.512912
$$910$$ 0 0
$$911$$ 29.1769 0.966674 0.483337 0.875434i $$-0.339425\pi$$
0.483337 + 0.875434i $$0.339425\pi$$
$$912$$ 0 0
$$913$$ 5.85641 0.193819
$$914$$ 0 0
$$915$$ 30.9282 1.02245
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 22.9282 0.755510
$$922$$ 0 0
$$923$$ −18.9282 −0.623029
$$924$$ 0 0
$$925$$ 14.0000 0.460317
$$926$$ 0 0
$$927$$ −10.9282 −0.358929
$$928$$ 0 0
$$929$$ 13.6077 0.446454 0.223227 0.974766i $$-0.428341\pi$$
0.223227 + 0.974766i $$0.428341\pi$$
$$930$$ 0 0
$$931$$ −6.92820 −0.227063
$$932$$ 0 0
$$933$$ −18.9282 −0.619682
$$934$$ 0 0
$$935$$ −2.71797 −0.0888870
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 36.5359 1.19104 0.595518 0.803342i $$-0.296947\pi$$
0.595518 + 0.803342i $$0.296947\pi$$
$$942$$ 0 0
$$943$$ −16.7846 −0.546582
$$944$$ 0 0
$$945$$ −3.46410 −0.112687
$$946$$ 0 0
$$947$$ −26.2487 −0.852969 −0.426484 0.904495i $$-0.640248\pi$$
−0.426484 + 0.904495i $$0.640248\pi$$
$$948$$ 0 0
$$949$$ −25.8564 −0.839334
$$950$$ 0 0
$$951$$ 0.143594 0.00465634
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 0 0
$$955$$ −32.7846 −1.06089
$$956$$ 0 0
$$957$$ −7.21539 −0.233240
$$958$$ 0 0
$$959$$ 8.92820 0.288307
$$960$$ 0 0
$$961$$ 88.4256 2.85244
$$962$$ 0 0
$$963$$ −12.3923 −0.399336
$$964$$ 0 0
$$965$$ −54.9282 −1.76820
$$966$$ 0 0
$$967$$ 34.9282 1.12322 0.561608 0.827404i $$-0.310183\pi$$
0.561608 + 0.827404i $$0.310183\pi$$
$$968$$ 0 0
$$969$$ 3.71281 0.119273
$$970$$ 0 0
$$971$$ −1.85641 −0.0595749 −0.0297875 0.999556i $$-0.509483\pi$$
−0.0297875 + 0.999556i $$0.509483\pi$$
$$972$$ 0 0
$$973$$ −17.8564 −0.572450
$$974$$ 0 0
$$975$$ 14.0000 0.448359
$$976$$ 0 0
$$977$$ 38.4974 1.23164 0.615821 0.787886i $$-0.288825\pi$$
0.615821 + 0.787886i $$0.288825\pi$$
$$978$$ 0 0
$$979$$ 5.07180 0.162095
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ −3.71281 −0.118420 −0.0592102 0.998246i $$-0.518858\pi$$
−0.0592102 + 0.998246i $$0.518858\pi$$
$$984$$ 0 0
$$985$$ −27.2154 −0.867154
$$986$$ 0 0
$$987$$ 10.9282 0.347849
$$988$$ 0 0
$$989$$ −11.7128 −0.372446
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 24.0000 0.761617
$$994$$ 0 0
$$995$$ −20.2872 −0.643147
$$996$$ 0 0
$$997$$ −31.0718 −0.984054 −0.492027 0.870580i $$-0.663744\pi$$
−0.492027 + 0.870580i $$0.663744\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 1344.2.a.u.1.1 2
3.2 odd 2 4032.2.a.br.1.2 2
4.3 odd 2 1344.2.a.v.1.1 2
7.6 odd 2 9408.2.a.dx.1.2 2
8.3 odd 2 672.2.a.i.1.2 2
8.5 even 2 672.2.a.j.1.2 yes 2
12.11 even 2 4032.2.a.bs.1.2 2
16.3 odd 4 5376.2.c.bh.2689.4 4
16.5 even 4 5376.2.c.bn.2689.3 4
16.11 odd 4 5376.2.c.bh.2689.1 4
16.13 even 4 5376.2.c.bn.2689.2 4
24.5 odd 2 2016.2.a.s.1.1 2
24.11 even 2 2016.2.a.t.1.1 2
28.27 even 2 9408.2.a.do.1.2 2
56.13 odd 2 4704.2.a.bm.1.1 2
56.27 even 2 4704.2.a.bn.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.2 2 8.3 odd 2
672.2.a.j.1.2 yes 2 8.5 even 2
1344.2.a.u.1.1 2 1.1 even 1 trivial
1344.2.a.v.1.1 2 4.3 odd 2
2016.2.a.s.1.1 2 24.5 odd 2
2016.2.a.t.1.1 2 24.11 even 2
4032.2.a.br.1.2 2 3.2 odd 2
4032.2.a.bs.1.2 2 12.11 even 2
4704.2.a.bm.1.1 2 56.13 odd 2
4704.2.a.bn.1.1 2 56.27 even 2
5376.2.c.bh.2689.1 4 16.11 odd 4
5376.2.c.bh.2689.4 4 16.3 odd 4
5376.2.c.bn.2689.2 4 16.13 even 4
5376.2.c.bn.2689.3 4 16.5 even 4
9408.2.a.do.1.2 2 28.27 even 2
9408.2.a.dx.1.2 2 7.6 odd 2