# Properties

 Label 9408.2.a.ea.1.2 Level 9408 Weight 2 Character 9408.1 Self dual yes Analytic conductor 75.123 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4704) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} -2.82843 q^{11} +1.41421 q^{13} +1.41421 q^{15} +1.41421 q^{17} -2.82843 q^{23} -3.00000 q^{25} +1.00000 q^{27} -4.00000 q^{31} -2.82843 q^{33} -4.00000 q^{37} +1.41421 q^{39} -1.41421 q^{41} -5.65685 q^{43} +1.41421 q^{45} -12.0000 q^{47} +1.41421 q^{51} -10.0000 q^{53} -4.00000 q^{55} +1.41421 q^{61} +2.00000 q^{65} -11.3137 q^{67} -2.82843 q^{69} +2.82843 q^{71} +12.7279 q^{73} -3.00000 q^{75} -11.3137 q^{79} +1.00000 q^{81} +4.00000 q^{83} +2.00000 q^{85} +7.07107 q^{89} -4.00000 q^{93} -9.89949 q^{97} -2.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} - 6q^{25} + 2q^{27} - 8q^{31} - 8q^{37} - 24q^{47} - 20q^{53} - 8q^{55} + 4q^{65} - 6q^{75} + 2q^{81} + 8q^{83} + 4q^{85} - 8q^{93} + O(q^{100})$$

## 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
$$5$$ 1.41421 0.632456 0.316228 0.948683i $$-0.397584\pi$$
0.316228 + 0.948683i $$0.397584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.82843 −0.852803 −0.426401 0.904534i $$-0.640219\pi$$
−0.426401 + 0.904534i $$0.640219\pi$$
$$12$$ 0 0
$$13$$ 1.41421 0.392232 0.196116 0.980581i $$-0.437167\pi$$
0.196116 + 0.980581i $$0.437167\pi$$
$$14$$ 0 0
$$15$$ 1.41421 0.365148
$$16$$ 0 0
$$17$$ 1.41421 0.342997 0.171499 0.985184i $$-0.445139\pi$$
0.171499 + 0.985184i $$0.445139\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −2.82843 −0.492366
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ 1.41421 0.226455
$$40$$ 0 0
$$41$$ −1.41421 −0.220863 −0.110432 0.993884i $$-0.535223\pi$$
−0.110432 + 0.993884i $$0.535223\pi$$
$$42$$ 0 0
$$43$$ −5.65685 −0.862662 −0.431331 0.902194i $$-0.641956\pi$$
−0.431331 + 0.902194i $$0.641956\pi$$
$$44$$ 0 0
$$45$$ 1.41421 0.210819
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 1.41421 0.198030
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 1.41421 0.181071 0.0905357 0.995893i $$-0.471142\pi$$
0.0905357 + 0.995893i $$0.471142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ −11.3137 −1.38219 −0.691095 0.722764i $$-0.742871\pi$$
−0.691095 + 0.722764i $$0.742871\pi$$
$$68$$ 0 0
$$69$$ −2.82843 −0.340503
$$70$$ 0 0
$$71$$ 2.82843 0.335673 0.167836 0.985815i $$-0.446322\pi$$
0.167836 + 0.985815i $$0.446322\pi$$
$$72$$ 0 0
$$73$$ 12.7279 1.48969 0.744845 0.667237i $$-0.232523\pi$$
0.744845 + 0.667237i $$0.232523\pi$$
$$74$$ 0 0
$$75$$ −3.00000 −0.346410
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.3137 −1.27289 −0.636446 0.771321i $$-0.719596\pi$$
−0.636446 + 0.771321i $$0.719596\pi$$
$$80$$ 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
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ −2.82843 −0.284268
$$100$$ 0 0
$$101$$ −7.07107 −0.703598 −0.351799 0.936076i $$-0.614430\pi$$
−0.351799 + 0.936076i $$0.614430\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 19.7990 1.91404 0.957020 0.290021i $$-0.0936623\pi$$
0.957020 + 0.290021i $$0.0936623\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 1.41421 0.130744
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −3.00000 −0.272727
$$122$$ 0 0
$$123$$ −1.41421 −0.127515
$$124$$ 0 0
$$125$$ −11.3137 −1.01193
$$126$$ 0 0
$$127$$ 11.3137 1.00393 0.501965 0.864888i $$-0.332611\pi$$
0.501965 + 0.864888i $$0.332611\pi$$
$$128$$ 0 0
$$129$$ −5.65685 −0.498058
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 1.41421 0.121716
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −11.3137 −0.920697 −0.460348 0.887738i $$-0.652275\pi$$
−0.460348 + 0.887738i $$0.652275\pi$$
$$152$$ 0 0
$$153$$ 1.41421 0.114332
$$154$$ 0 0
$$155$$ −5.65685 −0.454369
$$156$$ 0 0
$$157$$ 4.24264 0.338600 0.169300 0.985565i $$-0.445849\pi$$
0.169300 + 0.985565i $$0.445849\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 16.9706 1.32924 0.664619 0.747183i $$-0.268594\pi$$
0.664619 + 0.747183i $$0.268594\pi$$
$$164$$ 0 0
$$165$$ −4.00000 −0.311400
$$166$$ 0 0
$$167$$ −20.0000 −1.54765 −0.773823 0.633402i $$-0.781658\pi$$
−0.773823 + 0.633402i $$0.781658\pi$$
$$168$$ 0 0
$$169$$ −11.0000 −0.846154
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 7.07107 0.537603 0.268802 0.963196i $$-0.413372\pi$$
0.268802 + 0.963196i $$0.413372\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −14.1421 −1.05703 −0.528516 0.848923i $$-0.677252\pi$$
−0.528516 + 0.848923i $$0.677252\pi$$
$$180$$ 0 0
$$181$$ 15.5563 1.15629 0.578147 0.815933i $$-0.303776\pi$$
0.578147 + 0.815933i $$0.303776\pi$$
$$182$$ 0 0
$$183$$ 1.41421 0.104542
$$184$$ 0 0
$$185$$ −5.65685 −0.415900
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.48528 −0.613973 −0.306987 0.951714i $$-0.599321\pi$$
−0.306987 + 0.951714i $$0.599321\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −11.3137 −0.798007
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ −2.82843 −0.196589
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 5.65685 0.389434 0.194717 0.980859i $$-0.437621\pi$$
0.194717 + 0.980859i $$0.437621\pi$$
$$212$$ 0 0
$$213$$ 2.82843 0.193801
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 12.7279 0.860073
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −7.07107 −0.467269 −0.233635 0.972324i $$-0.575062\pi$$
−0.233635 + 0.972324i $$0.575062\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ −16.9706 −1.10704
$$236$$ 0 0
$$237$$ −11.3137 −0.734904
$$238$$ 0 0
$$239$$ 19.7990 1.28069 0.640345 0.768087i $$-0.278791\pi$$
0.640345 + 0.768087i $$0.278791\pi$$
$$240$$ 0 0
$$241$$ −1.41421 −0.0910975 −0.0455488 0.998962i $$-0.514504\pi$$
−0.0455488 + 0.998962i $$0.514504\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ 0 0
$$255$$ 2.00000 0.125245
$$256$$ 0 0
$$257$$ 9.89949 0.617514 0.308757 0.951141i $$-0.400087\pi$$
0.308757 + 0.951141i $$0.400087\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −14.1421 −0.872041 −0.436021 0.899937i $$-0.643613\pi$$
−0.436021 + 0.899937i $$0.643613\pi$$
$$264$$ 0 0
$$265$$ −14.1421 −0.868744
$$266$$ 0 0
$$267$$ 7.07107 0.432742
$$268$$ 0 0
$$269$$ 9.89949 0.603583 0.301791 0.953374i $$-0.402415\pi$$
0.301791 + 0.953374i $$0.402415\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 8.48528 0.511682
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 0 0
$$291$$ −9.89949 −0.580319
$$292$$ 0 0
$$293$$ −29.6985 −1.73500 −0.867502 0.497434i $$-0.834276\pi$$
−0.867502 + 0.497434i $$0.834276\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.82843 −0.164122
$$298$$ 0 0
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −7.07107 −0.406222
$$304$$ 0 0
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 12.7279 0.719425 0.359712 0.933063i $$-0.382875\pi$$
0.359712 + 0.933063i $$0.382875\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 19.7990 1.10507
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −4.24264 −0.235339
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 33.9411 1.86557 0.932786 0.360429i $$-0.117370\pi$$
0.932786 + 0.360429i $$0.117370\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 0 0
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 11.3137 0.612672
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ −25.4558 −1.36654 −0.683271 0.730165i $$-0.739443\pi$$
−0.683271 + 0.730165i $$0.739443\pi$$
$$348$$ 0 0
$$349$$ 26.8701 1.43832 0.719161 0.694844i $$-0.244527\pi$$
0.719161 + 0.694844i $$0.244527\pi$$
$$350$$ 0 0
$$351$$ 1.41421 0.0754851
$$352$$ 0 0
$$353$$ −18.3848 −0.978523 −0.489261 0.872137i $$-0.662734\pi$$
−0.489261 + 0.872137i $$0.662734\pi$$
$$354$$ 0 0
$$355$$ 4.00000 0.212298
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 14.1421 0.746393 0.373197 0.927752i $$-0.378262\pi$$
0.373197 + 0.927752i $$0.378262\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −3.00000 −0.157459
$$364$$ 0 0
$$365$$ 18.0000 0.942163
$$366$$ 0 0
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ 0 0
$$369$$ −1.41421 −0.0736210
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ −11.3137 −0.584237
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 11.3137 0.579619
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −5.65685 −0.287554
$$388$$ 0 0
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ −38.1838 −1.91639 −0.958194 0.286119i $$-0.907635\pi$$
−0.958194 + 0.286119i $$0.907635\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ −5.65685 −0.281788
$$404$$ 0 0
$$405$$ 1.41421 0.0702728
$$406$$ 0 0
$$407$$ 11.3137 0.560800
$$408$$ 0 0
$$409$$ 29.6985 1.46850 0.734248 0.678882i $$-0.237535\pi$$
0.734248 + 0.678882i $$0.237535\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 5.65685 0.277684
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ −4.24264 −0.205798
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −2.82843 −0.136241 −0.0681203 0.997677i $$-0.521700\pi$$
−0.0681203 + 0.997677i $$0.521700\pi$$
$$432$$ 0 0
$$433$$ 4.24264 0.203888 0.101944 0.994790i $$-0.467494\pi$$
0.101944 + 0.994790i $$0.467494\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 19.7990 0.940678 0.470339 0.882486i $$-0.344132\pi$$
0.470339 + 0.882486i $$0.344132\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 0 0
$$453$$ −11.3137 −0.531564
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ 1.41421 0.0660098
$$460$$ 0 0
$$461$$ −4.24264 −0.197599 −0.0987997 0.995107i $$-0.531500\pi$$
−0.0987997 + 0.995107i $$0.531500\pi$$
$$462$$ 0 0
$$463$$ 39.5980 1.84027 0.920137 0.391596i $$-0.128077\pi$$
0.920137 + 0.391596i $$0.128077\pi$$
$$464$$ 0 0
$$465$$ −5.65685 −0.262330
$$466$$ 0 0
$$467$$ −16.0000 −0.740392 −0.370196 0.928954i $$-0.620709\pi$$
−0.370196 + 0.928954i $$0.620709\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.24264 0.195491
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 0 0
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ −5.65685 −0.257930
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −14.0000 −0.635707
$$486$$ 0 0
$$487$$ −11.3137 −0.512673 −0.256337 0.966588i $$-0.582516\pi$$
−0.256337 + 0.966588i $$0.582516\pi$$
$$488$$ 0 0
$$489$$ 16.9706 0.767435
$$490$$ 0 0
$$491$$ 14.1421 0.638226 0.319113 0.947717i $$-0.396615\pi$$
0.319113 + 0.947717i $$0.396615\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −11.3137 −0.506471 −0.253236 0.967405i $$-0.581495\pi$$
−0.253236 + 0.967405i $$0.581495\pi$$
$$500$$ 0 0
$$501$$ −20.0000 −0.893534
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ 0 0
$$507$$ −11.0000 −0.488527
$$508$$ 0 0
$$509$$ 32.5269 1.44173 0.720865 0.693075i $$-0.243745\pi$$
0.720865 + 0.693075i $$0.243745\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −16.9706 −0.747812
$$516$$ 0 0
$$517$$ 33.9411 1.49273
$$518$$ 0 0
$$519$$ 7.07107 0.310385
$$520$$ 0 0
$$521$$ −29.6985 −1.30111 −0.650557 0.759457i $$-0.725465\pi$$
−0.650557 + 0.759457i $$0.725465\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.65685 −0.246416
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −2.00000 −0.0866296
$$534$$ 0 0
$$535$$ 28.0000 1.21055
$$536$$ 0 0
$$537$$ −14.1421 −0.610278
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 0 0
$$543$$ 15.5563 0.667587
$$544$$ 0 0
$$545$$ 5.65685 0.242313
$$546$$ 0 0
$$547$$ 22.6274 0.967478 0.483739 0.875212i $$-0.339278\pi$$
0.483739 + 0.875212i $$0.339278\pi$$
$$548$$ 0 0
$$549$$ 1.41421 0.0603572
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −5.65685 −0.240120
$$556$$ 0 0
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ −16.0000 −0.674320 −0.337160 0.941447i $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ 0 0
$$565$$ −2.82843 −0.118993
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.0000 1.17382 0.586911 0.809652i $$-0.300344\pi$$
0.586911 + 0.809652i $$0.300344\pi$$
$$570$$ 0 0
$$571$$ −22.6274 −0.946928 −0.473464 0.880813i $$-0.656997\pi$$
−0.473464 + 0.880813i $$0.656997\pi$$
$$572$$ 0 0
$$573$$ −8.48528 −0.354478
$$574$$ 0 0
$$575$$ 8.48528 0.353861
$$576$$ 0 0
$$577$$ 21.2132 0.883117 0.441559 0.897232i $$-0.354426\pi$$
0.441559 + 0.897232i $$0.354426\pi$$
$$578$$ 0 0
$$579$$ −26.0000 −1.08052
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 28.2843 1.17141
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ −8.00000 −0.330195 −0.165098 0.986277i $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 29.6985 1.21957 0.609785 0.792567i $$-0.291256\pi$$
0.609785 + 0.792567i $$0.291256\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −31.1127 −1.27123 −0.635615 0.772006i $$-0.719253\pi$$
−0.635615 + 0.772006i $$0.719253\pi$$
$$600$$ 0 0
$$601$$ −15.5563 −0.634557 −0.317278 0.948332i $$-0.602769\pi$$
−0.317278 + 0.948332i $$0.602769\pi$$
$$602$$ 0 0
$$603$$ −11.3137 −0.460730
$$604$$ 0 0
$$605$$ −4.24264 −0.172488
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.9706 −0.686555
$$612$$ 0 0
$$613$$ 44.0000 1.77714 0.888572 0.458738i $$-0.151698\pi$$
0.888572 + 0.458738i $$0.151698\pi$$
$$614$$ 0 0
$$615$$ −2.00000 −0.0806478
$$616$$ 0 0
$$617$$ 36.0000 1.44931 0.724653 0.689114i $$-0.242000\pi$$
0.724653 + 0.689114i $$0.242000\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −2.82843 −0.113501
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −5.65685 −0.225554
$$630$$ 0 0
$$631$$ −16.9706 −0.675587 −0.337794 0.941220i $$-0.609681\pi$$
−0.337794 + 0.941220i $$0.609681\pi$$
$$632$$ 0 0
$$633$$ 5.65685 0.224840
$$634$$ 0 0
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 2.82843 0.111891
$$640$$ 0 0
$$641$$ 4.00000 0.157991 0.0789953 0.996875i $$-0.474829\pi$$
0.0789953 + 0.996875i $$0.474829\pi$$
$$642$$ 0 0
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8.00000 −0.313064 −0.156532 0.987673i $$-0.550031\pi$$
−0.156532 + 0.987673i $$0.550031\pi$$
$$654$$ 0 0
$$655$$ −5.65685 −0.221032
$$656$$ 0 0
$$657$$ 12.7279 0.496564
$$658$$ 0 0
$$659$$ 19.7990 0.771259 0.385630 0.922654i $$-0.373984\pi$$
0.385630 + 0.922654i $$0.373984\pi$$
$$660$$ 0 0
$$661$$ −29.6985 −1.15514 −0.577569 0.816342i $$-0.695998\pi$$
−0.577569 + 0.816342i $$0.695998\pi$$
$$662$$ 0 0
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ −32.0000 −1.23351 −0.616755 0.787155i $$-0.711553\pi$$
−0.616755 + 0.787155i $$0.711553\pi$$
$$674$$ 0 0
$$675$$ −3.00000 −0.115470
$$676$$ 0 0
$$677$$ 46.6690 1.79364 0.896819 0.442398i $$-0.145872\pi$$
0.896819 + 0.442398i $$0.145872\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 19.7990 0.757587 0.378794 0.925481i $$-0.376339\pi$$
0.378794 + 0.925481i $$0.376339\pi$$
$$684$$ 0 0
$$685$$ 16.9706 0.648412
$$686$$ 0 0
$$687$$ −7.07107 −0.269778
$$688$$ 0 0
$$689$$ −14.1421 −0.538772
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 28.2843 1.07288
$$696$$ 0 0
$$697$$ −2.00000 −0.0757554
$$698$$ 0 0
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ 16.0000 0.604312 0.302156 0.953259i $$-0.402294\pi$$
0.302156 + 0.953259i $$0.402294\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −16.9706 −0.639148
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −36.0000 −1.35201 −0.676004 0.736898i $$-0.736290\pi$$
−0.676004 + 0.736898i $$0.736290\pi$$
$$710$$ 0 0
$$711$$ −11.3137 −0.424297
$$712$$ 0 0
$$713$$ 11.3137 0.423702
$$714$$ 0 0
$$715$$ −5.65685 −0.211554
$$716$$ 0 0
$$717$$ 19.7990 0.739407
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −1.41421 −0.0525952
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −29.6985 −1.09694 −0.548469 0.836171i $$-0.684789\pi$$
−0.548469 + 0.836171i $$0.684789\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ 0 0
$$739$$ 16.9706 0.624272 0.312136 0.950037i $$-0.398955\pi$$
0.312136 + 0.950037i $$0.398955\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −36.7696 −1.34894 −0.674472 0.738300i $$-0.735629\pi$$
−0.674472 + 0.738300i $$0.735629\pi$$
$$744$$ 0 0
$$745$$ 8.48528 0.310877
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.2843 1.03211 0.516054 0.856556i $$-0.327400\pi$$
0.516054 + 0.856556i $$0.327400\pi$$
$$752$$ 0 0
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −28.0000 −1.01768 −0.508839 0.860862i $$-0.669925\pi$$
−0.508839 + 0.860862i $$0.669925\pi$$
$$758$$ 0 0
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ −43.8406 −1.58922 −0.794611 0.607119i $$-0.792325\pi$$
−0.794611 + 0.607119i $$0.792325\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −18.3848 −0.662972 −0.331486 0.943460i $$-0.607550\pi$$
−0.331486 + 0.943460i $$0.607550\pi$$
$$770$$ 0 0
$$771$$ 9.89949 0.356522
$$772$$ 0 0
$$773$$ −15.5563 −0.559523 −0.279761 0.960070i $$-0.590255\pi$$
−0.279761 + 0.960070i $$0.590255\pi$$
$$774$$ 0 0
$$775$$ 12.0000 0.431053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6.00000 0.214149
$$786$$ 0 0
$$787$$ −20.0000 −0.712923 −0.356462 0.934310i $$-0.616017\pi$$
−0.356462 + 0.934310i $$0.616017\pi$$
$$788$$ 0 0
$$789$$ −14.1421 −0.503473
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2.00000 0.0710221
$$794$$ 0 0
$$795$$ −14.1421 −0.501570
$$796$$ 0 0
$$797$$ −18.3848 −0.651222 −0.325611 0.945504i $$-0.605570\pi$$
−0.325611 + 0.945504i $$0.605570\pi$$
$$798$$ 0 0
$$799$$ −16.9706 −0.600375
$$800$$ 0 0
$$801$$ 7.07107 0.249844
$$802$$ 0 0
$$803$$ −36.0000 −1.27041
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 9.89949 0.348479
$$808$$ 0 0
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ −20.0000 −0.701431
$$814$$ 0 0
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26.0000 0.907406 0.453703 0.891153i $$-0.350103\pi$$
0.453703 + 0.891153i $$0.350103\pi$$
$$822$$ 0 0
$$823$$ −33.9411 −1.18311 −0.591557 0.806263i $$-0.701486\pi$$
−0.591557 + 0.806263i $$0.701486\pi$$
$$824$$ 0 0
$$825$$ 8.48528 0.295420
$$826$$ 0 0
$$827$$ 19.7990 0.688478 0.344239 0.938882i $$-0.388137\pi$$
0.344239 + 0.938882i $$0.388137\pi$$
$$828$$ 0 0
$$829$$ −38.1838 −1.32618 −0.663089 0.748541i $$-0.730755\pi$$
−0.663089 + 0.748541i $$0.730755\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −28.2843 −0.978818
$$836$$ 0 0
$$837$$ −4.00000 −0.138260
$$838$$ 0 0
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 12.0000 0.413302
$$844$$ 0 0
$$845$$ −15.5563 −0.535155
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 11.3137 0.387829
$$852$$ 0 0
$$853$$ 55.1543 1.88845 0.944224 0.329304i $$-0.106814\pi$$
0.944224 + 0.329304i $$0.106814\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −9.89949 −0.338160 −0.169080 0.985602i $$-0.554080\pi$$
−0.169080 + 0.985602i $$0.554080\pi$$
$$858$$ 0 0
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −8.48528 −0.288842 −0.144421 0.989516i $$-0.546132\pi$$
−0.144421 + 0.989516i $$0.546132\pi$$
$$864$$ 0 0
$$865$$ 10.0000 0.340010
$$866$$ 0 0
$$867$$ −15.0000 −0.509427
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ −9.89949 −0.335047
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 4.00000 0.135070 0.0675352 0.997717i $$-0.478487\pi$$
0.0675352 + 0.997717i $$0.478487\pi$$
$$878$$ 0 0
$$879$$ −29.6985 −1.00171
$$880$$ 0 0
$$881$$ −15.5563 −0.524107 −0.262053 0.965053i $$-0.584400\pi$$
−0.262053 + 0.965053i $$0.584400\pi$$
$$882$$ 0 0
$$883$$ 11.3137 0.380737 0.190368 0.981713i $$-0.439032\pi$$
0.190368 + 0.981713i $$0.439032\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.82843 −0.0947559
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −20.0000 −0.668526
$$896$$ 0 0
$$897$$ −4.00000 −0.133556
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −14.1421 −0.471143
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 22.0000 0.731305
$$906$$ 0 0
$$907$$ 11.3137 0.375666 0.187833 0.982201i $$-0.439854\pi$$
0.187833 + 0.982201i $$0.439854\pi$$
$$908$$ 0 0
$$909$$ −7.07107 −0.234533
$$910$$ 0 0
$$911$$ −25.4558 −0.843390 −0.421695 0.906738i $$-0.638565\pi$$
−0.421695 + 0.906738i $$0.638565\pi$$
$$912$$ 0 0
$$913$$ −11.3137 −0.374429
$$914$$ 0 0
$$915$$ 2.00000 0.0661180
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.9706 0.559807 0.279904 0.960028i $$-0.409697\pi$$
0.279904 + 0.960028i $$0.409697\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 0 0
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ 12.0000 0.394558
$$926$$ 0 0
$$927$$ −12.0000 −0.394132
$$928$$ 0 0
$$929$$ 41.0122 1.34557 0.672783 0.739840i $$-0.265099\pi$$
0.672783 + 0.739840i $$0.265099\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −12.0000 −0.392862
$$934$$ 0 0
$$935$$ −5.65685 −0.184999
$$936$$ 0 0
$$937$$ −26.8701 −0.877807 −0.438903 0.898534i $$-0.644633\pi$$
−0.438903 + 0.898534i $$0.644633\pi$$
$$938$$ 0 0
$$939$$ 12.7279 0.415360
$$940$$ 0 0
$$941$$ −29.6985 −0.968143 −0.484071 0.875028i $$-0.660843\pi$$
−0.484071 + 0.875028i $$0.660843\pi$$
$$942$$ 0 0
$$943$$ 4.00000 0.130258
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −53.7401 −1.74632 −0.873160 0.487435i $$-0.837933\pi$$
−0.873160 + 0.487435i $$0.837933\pi$$
$$948$$ 0 0
$$949$$ 18.0000 0.584305
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 58.0000 1.87880 0.939402 0.342817i $$-0.111381\pi$$
0.939402 + 0.342817i $$0.111381\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 19.7990 0.638014
$$964$$ 0 0
$$965$$ −36.7696 −1.18365
$$966$$ 0 0
$$967$$ −45.2548 −1.45530 −0.727649 0.685950i $$-0.759387\pi$$
−0.727649 + 0.685950i $$0.759387\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −4.24264 −0.135873
$$976$$ 0 0
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 0 0
$$985$$ −8.48528 −0.270364
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −5.65685 −0.179696 −0.0898479 0.995955i $$-0.528638\pi$$
−0.0898479 + 0.995955i $$0.528638\pi$$
$$992$$ 0 0
$$993$$ 33.9411 1.07709
$$994$$ 0 0
$$995$$ −11.3137 −0.358669
$$996$$ 0 0
$$997$$ 4.24264 0.134366 0.0671829 0.997741i $$-0.478599\pi$$
0.0671829 + 0.997741i $$0.478599\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 9408.2.a.ea.1.2 2
4.3 odd 2 9408.2.a.dl.1.2 2
7.6 odd 2 9408.2.a.dl.1.1 2
8.3 odd 2 4704.2.a.bq.1.1 yes 2
8.5 even 2 4704.2.a.bj.1.1 2
28.27 even 2 inner 9408.2.a.ea.1.1 2
56.13 odd 2 4704.2.a.bq.1.2 yes 2
56.27 even 2 4704.2.a.bj.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bj.1.1 2 8.5 even 2
4704.2.a.bj.1.2 yes 2 56.27 even 2
4704.2.a.bq.1.1 yes 2 8.3 odd 2
4704.2.a.bq.1.2 yes 2 56.13 odd 2
9408.2.a.dl.1.1 2 7.6 odd 2
9408.2.a.dl.1.2 2 4.3 odd 2
9408.2.a.ea.1.1 2 28.27 even 2 inner
9408.2.a.ea.1.2 2 1.1 even 1 trivial