# Properties

 Label 6012.2.a.d.1.4 Level $6012$ Weight $2$ Character 6012.1 Self dual yes Analytic conductor $48.006$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.826865.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 668) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.69135$$ of defining polynomial Character $$\chi$$ $$=$$ 6012.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{5} +4.48567 q^{7} +O(q^{10})$$ $$q-2.00000 q^{5} +4.48567 q^{7} +1.62500 q^{11} +3.38270 q^{13} +5.30425 q^{17} +6.97394 q^{19} -0.132692 q^{23} -1.00000 q^{25} +7.86837 q^{29} +10.5868 q^{31} -8.97134 q^{35} +1.72133 q^{37} -8.40828 q^{41} -6.76540 q^{43} -0.208543 q^{47} +13.1212 q^{49} -8.68695 q^{53} -3.25001 q^{55} -13.8152 q^{59} -3.06741 q^{61} -6.76540 q^{65} +7.51019 q^{67} -2.06752 q^{71} +5.43694 q^{73} +7.28924 q^{77} -17.5256 q^{79} +12.1481 q^{83} -10.6085 q^{85} +10.8020 q^{89} +15.1737 q^{91} -13.9479 q^{95} -4.21809 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 10q^{5} + 9q^{7} + O(q^{10})$$ $$5q - 10q^{5} + 9q^{7} - 5q^{11} - 4q^{13} + 2q^{17} + 5q^{19} - 6q^{23} - 5q^{25} + 5q^{29} + 9q^{31} - 18q^{35} + 8q^{37} + 4q^{41} + 8q^{43} - 13q^{47} + 14q^{49} + 2q^{53} + 10q^{55} - 4q^{59} + 11q^{61} + 8q^{65} + 28q^{67} - 2q^{71} + 8q^{73} + 12q^{77} - 10q^{79} - 2q^{83} - 4q^{85} + 17q^{89} - 12q^{91} - 10q^{95} - 27q^{97} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 4.48567 1.69542 0.847712 0.530457i $$-0.177979\pi$$
0.847712 + 0.530457i $$0.177979\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.62500 0.489957 0.244979 0.969528i $$-0.421219\pi$$
0.244979 + 0.969528i $$0.421219\pi$$
$$12$$ 0 0
$$13$$ 3.38270 0.938192 0.469096 0.883147i $$-0.344580\pi$$
0.469096 + 0.883147i $$0.344580\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.30425 1.28647 0.643235 0.765669i $$-0.277592\pi$$
0.643235 + 0.765669i $$0.277592\pi$$
$$18$$ 0 0
$$19$$ 6.97394 1.59993 0.799966 0.600045i $$-0.204851\pi$$
0.799966 + 0.600045i $$0.204851\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.132692 −0.0276682 −0.0138341 0.999904i $$-0.504404\pi$$
−0.0138341 + 0.999904i $$0.504404\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.86837 1.46112 0.730560 0.682849i $$-0.239259\pi$$
0.730560 + 0.682849i $$0.239259\pi$$
$$30$$ 0 0
$$31$$ 10.5868 1.90145 0.950726 0.310031i $$-0.100339\pi$$
0.950726 + 0.310031i $$0.100339\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −8.97134 −1.51643
$$36$$ 0 0
$$37$$ 1.72133 0.282985 0.141493 0.989939i $$-0.454810\pi$$
0.141493 + 0.989939i $$0.454810\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.40828 −1.31315 −0.656576 0.754259i $$-0.727996\pi$$
−0.656576 + 0.754259i $$0.727996\pi$$
$$42$$ 0 0
$$43$$ −6.76540 −1.03171 −0.515857 0.856675i $$-0.672526\pi$$
−0.515857 + 0.856675i $$0.672526\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.208543 −0.0304191 −0.0152095 0.999884i $$-0.504842\pi$$
−0.0152095 + 0.999884i $$0.504842\pi$$
$$48$$ 0 0
$$49$$ 13.1212 1.87446
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.68695 −1.19324 −0.596622 0.802522i $$-0.703491\pi$$
−0.596622 + 0.802522i $$0.703491\pi$$
$$54$$ 0 0
$$55$$ −3.25001 −0.438231
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −13.8152 −1.79859 −0.899293 0.437347i $$-0.855918\pi$$
−0.899293 + 0.437347i $$0.855918\pi$$
$$60$$ 0 0
$$61$$ −3.06741 −0.392742 −0.196371 0.980530i $$-0.562916\pi$$
−0.196371 + 0.980530i $$0.562916\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.76540 −0.839145
$$66$$ 0 0
$$67$$ 7.51019 0.917515 0.458758 0.888561i $$-0.348295\pi$$
0.458758 + 0.888561i $$0.348295\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.06752 −0.245370 −0.122685 0.992446i $$-0.539150\pi$$
−0.122685 + 0.992446i $$0.539150\pi$$
$$72$$ 0 0
$$73$$ 5.43694 0.636346 0.318173 0.948033i $$-0.396931\pi$$
0.318173 + 0.948033i $$0.396931\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 7.28924 0.830686
$$78$$ 0 0
$$79$$ −17.5256 −1.97178 −0.985892 0.167383i $$-0.946468\pi$$
−0.985892 + 0.167383i $$0.946468\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.1481 1.33343 0.666714 0.745314i $$-0.267700\pi$$
0.666714 + 0.745314i $$0.267700\pi$$
$$84$$ 0 0
$$85$$ −10.6085 −1.15065
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.8020 1.14501 0.572506 0.819900i $$-0.305971\pi$$
0.572506 + 0.819900i $$0.305971\pi$$
$$90$$ 0 0
$$91$$ 15.1737 1.59063
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −13.9479 −1.43102
$$96$$ 0 0
$$97$$ −4.21809 −0.428282 −0.214141 0.976803i $$-0.568695\pi$$
−0.214141 + 0.976803i $$0.568695\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.33291 0.232133 0.116066 0.993241i $$-0.462971\pi$$
0.116066 + 0.993241i $$0.462971\pi$$
$$102$$ 0 0
$$103$$ −11.3121 −1.11461 −0.557307 0.830306i $$-0.688165\pi$$
−0.557307 + 0.830306i $$0.688165\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.0942 −1.07251 −0.536257 0.844055i $$-0.680162\pi$$
−0.536257 + 0.844055i $$0.680162\pi$$
$$108$$ 0 0
$$109$$ −2.77633 −0.265924 −0.132962 0.991121i $$-0.542449\pi$$
−0.132962 + 0.991121i $$0.542449\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 17.2579 1.62348 0.811742 0.584017i $$-0.198520\pi$$
0.811742 + 0.584017i $$0.198520\pi$$
$$114$$ 0 0
$$115$$ 0.265384 0.0247472
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 23.7931 2.18111
$$120$$ 0 0
$$121$$ −8.35936 −0.759942
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −8.31768 −0.738075 −0.369037 0.929415i $$-0.620313\pi$$
−0.369037 + 0.929415i $$0.620313\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.84309 0.510513 0.255257 0.966873i $$-0.417840\pi$$
0.255257 + 0.966873i $$0.417840\pi$$
$$132$$ 0 0
$$133$$ 31.2828 2.71256
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.6762 −1.16843 −0.584217 0.811598i $$-0.698598\pi$$
−0.584217 + 0.811598i $$0.698598\pi$$
$$138$$ 0 0
$$139$$ 10.2998 0.873618 0.436809 0.899554i $$-0.356109\pi$$
0.436809 + 0.899554i $$0.356109\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 5.49690 0.459674
$$144$$ 0 0
$$145$$ −15.7367 −1.30687
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 7.84309 0.642531 0.321266 0.946989i $$-0.395892\pi$$
0.321266 + 0.946989i $$0.395892\pi$$
$$150$$ 0 0
$$151$$ −15.5232 −1.26326 −0.631632 0.775268i $$-0.717615\pi$$
−0.631632 + 0.775268i $$0.717615\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −21.1737 −1.70071
$$156$$ 0 0
$$157$$ 17.4174 1.39006 0.695030 0.718980i $$-0.255391\pi$$
0.695030 + 0.718980i $$0.255391\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.595212 −0.0469093
$$162$$ 0 0
$$163$$ 23.9912 1.87914 0.939568 0.342363i $$-0.111227\pi$$
0.939568 + 0.342363i $$0.111227\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000 0.0773823
$$168$$ 0 0
$$169$$ −1.55733 −0.119795
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1.68232 0.127904 0.0639522 0.997953i $$-0.479629\pi$$
0.0639522 + 0.997953i $$0.479629\pi$$
$$174$$ 0 0
$$175$$ −4.48567 −0.339085
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0.760672 0.0568553 0.0284276 0.999596i $$-0.490950\pi$$
0.0284276 + 0.999596i $$0.490950\pi$$
$$180$$ 0 0
$$181$$ −6.35665 −0.472486 −0.236243 0.971694i $$-0.575916\pi$$
−0.236243 + 0.971694i $$0.575916\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3.44267 −0.253110
$$186$$ 0 0
$$187$$ 8.61943 0.630315
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.1989 −0.810327 −0.405163 0.914244i $$-0.632785\pi$$
−0.405163 + 0.914244i $$0.632785\pi$$
$$192$$ 0 0
$$193$$ −14.9471 −1.07592 −0.537959 0.842971i $$-0.680804\pi$$
−0.537959 + 0.842971i $$0.680804\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0775 −1.28797 −0.643984 0.765039i $$-0.722720\pi$$
−0.643984 + 0.765039i $$0.722720\pi$$
$$198$$ 0 0
$$199$$ −3.67617 −0.260596 −0.130298 0.991475i $$-0.541593\pi$$
−0.130298 + 0.991475i $$0.541593\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 35.2949 2.47722
$$204$$ 0 0
$$205$$ 16.8166 1.17452
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 11.3327 0.783899
$$210$$ 0 0
$$211$$ −10.2297 −0.704243 −0.352122 0.935954i $$-0.614540\pi$$
−0.352122 + 0.935954i $$0.614540\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 13.5308 0.922793
$$216$$ 0 0
$$217$$ 47.4891 3.22377
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 17.9427 1.20696
$$222$$ 0 0
$$223$$ −22.1032 −1.48014 −0.740070 0.672530i $$-0.765208\pi$$
−0.740070 + 0.672530i $$0.765208\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −22.2050 −1.47380 −0.736898 0.676003i $$-0.763710\pi$$
−0.736898 + 0.676003i $$0.763710\pi$$
$$228$$ 0 0
$$229$$ 1.66030 0.109716 0.0548580 0.998494i $$-0.482529\pi$$
0.0548580 + 0.998494i $$0.482529\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 19.0524 1.24817 0.624083 0.781358i $$-0.285473\pi$$
0.624083 + 0.781358i $$0.285473\pi$$
$$234$$ 0 0
$$235$$ 0.417085 0.0272076
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5.68706 −0.367865 −0.183933 0.982939i $$-0.558883\pi$$
−0.183933 + 0.982939i $$0.558883\pi$$
$$240$$ 0 0
$$241$$ −15.9721 −1.02885 −0.514427 0.857534i $$-0.671995\pi$$
−0.514427 + 0.857534i $$0.671995\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −26.2425 −1.67657
$$246$$ 0 0
$$247$$ 23.5908 1.50104
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 30.9878 1.95593 0.977967 0.208760i $$-0.0669429\pi$$
0.977967 + 0.208760i $$0.0669429\pi$$
$$252$$ 0 0
$$253$$ −0.215625 −0.0135562
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3.32611 0.207477 0.103739 0.994605i $$-0.466919\pi$$
0.103739 + 0.994605i $$0.466919\pi$$
$$258$$ 0 0
$$259$$ 7.72133 0.479780
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −8.47433 −0.522549 −0.261275 0.965265i $$-0.584143\pi$$
−0.261275 + 0.965265i $$0.584143\pi$$
$$264$$ 0 0
$$265$$ 17.3739 1.06727
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2.94140 0.179340 0.0896702 0.995972i $$-0.471419\pi$$
0.0896702 + 0.995972i $$0.471419\pi$$
$$270$$ 0 0
$$271$$ 12.7654 0.775443 0.387721 0.921777i $$-0.373262\pi$$
0.387721 + 0.921777i $$0.373262\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.62500 −0.0979915
$$276$$ 0 0
$$277$$ −6.54097 −0.393009 −0.196505 0.980503i $$-0.562959\pi$$
−0.196505 + 0.980503i $$0.562959\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.83281 −0.347956 −0.173978 0.984750i $$-0.555662\pi$$
−0.173978 + 0.984750i $$0.555662\pi$$
$$282$$ 0 0
$$283$$ 6.70270 0.398434 0.199217 0.979955i $$-0.436160\pi$$
0.199217 + 0.979955i $$0.436160\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −37.7168 −2.22635
$$288$$ 0 0
$$289$$ 11.1350 0.655003
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0.113390 0.00662433 0.00331217 0.999995i $$-0.498946\pi$$
0.00331217 + 0.999995i $$0.498946\pi$$
$$294$$ 0 0
$$295$$ 27.6304 1.60870
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −0.448857 −0.0259581
$$300$$ 0 0
$$301$$ −30.3474 −1.74919
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 6.13482 0.351279
$$306$$ 0 0
$$307$$ −19.7204 −1.12550 −0.562751 0.826627i $$-0.690257\pi$$
−0.562751 + 0.826627i $$0.690257\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 28.2858 1.60394 0.801970 0.597364i $$-0.203785\pi$$
0.801970 + 0.597364i $$0.203785\pi$$
$$312$$ 0 0
$$313$$ −2.95285 −0.166905 −0.0834526 0.996512i $$-0.526595\pi$$
−0.0834526 + 0.996512i $$0.526595\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.5101 1.32046 0.660230 0.751064i $$-0.270459\pi$$
0.660230 + 0.751064i $$0.270459\pi$$
$$318$$ 0 0
$$319$$ 12.7861 0.715886
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 36.9915 2.05826
$$324$$ 0 0
$$325$$ −3.38270 −0.187638
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −0.935453 −0.0515732
$$330$$ 0 0
$$331$$ −3.53388 −0.194240 −0.0971198 0.995273i $$-0.530963\pi$$
−0.0971198 + 0.995273i $$0.530963\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −15.0204 −0.820651
$$336$$ 0 0
$$337$$ −30.7676 −1.67602 −0.838008 0.545657i $$-0.816280\pi$$
−0.838008 + 0.545657i $$0.816280\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 17.2037 0.931631
$$342$$ 0 0
$$343$$ 27.4579 1.48259
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −8.95468 −0.480713 −0.240356 0.970685i $$-0.577264\pi$$
−0.240356 + 0.970685i $$0.577264\pi$$
$$348$$ 0 0
$$349$$ 3.10711 0.166320 0.0831599 0.996536i $$-0.473499\pi$$
0.0831599 + 0.996536i $$0.473499\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.0170 0.639600 0.319800 0.947485i $$-0.396384\pi$$
0.319800 + 0.947485i $$0.396384\pi$$
$$354$$ 0 0
$$355$$ 4.13504 0.219465
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 28.7949 1.51974 0.759869 0.650077i $$-0.225263\pi$$
0.759869 + 0.650077i $$0.225263\pi$$
$$360$$ 0 0
$$361$$ 29.6359 1.55978
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.8739 −0.569165
$$366$$ 0 0
$$367$$ 7.28255 0.380146 0.190073 0.981770i $$-0.439128\pi$$
0.190073 + 0.981770i $$0.439128\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −38.9668 −2.02306
$$372$$ 0 0
$$373$$ 25.2500 1.30740 0.653698 0.756756i $$-0.273217\pi$$
0.653698 + 0.756756i $$0.273217\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 26.6163 1.37081
$$378$$ 0 0
$$379$$ −15.2733 −0.784535 −0.392268 0.919851i $$-0.628309\pi$$
−0.392268 + 0.919851i $$0.628309\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.83494 −0.144859 −0.0724293 0.997374i $$-0.523075\pi$$
−0.0724293 + 0.997374i $$0.523075\pi$$
$$384$$ 0 0
$$385$$ −14.5785 −0.742988
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 17.8121 0.903110 0.451555 0.892243i $$-0.350869\pi$$
0.451555 + 0.892243i $$0.350869\pi$$
$$390$$ 0 0
$$391$$ −0.703831 −0.0355942
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 35.0512 1.76362
$$396$$ 0 0
$$397$$ 18.5585 0.931425 0.465712 0.884936i $$-0.345798\pi$$
0.465712 + 0.884936i $$0.345798\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −19.8040 −0.988967 −0.494483 0.869187i $$-0.664643\pi$$
−0.494483 + 0.869187i $$0.664643\pi$$
$$402$$ 0 0
$$403$$ 35.8121 1.78393
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.79717 0.138651
$$408$$ 0 0
$$409$$ 24.2604 1.19960 0.599799 0.800151i $$-0.295247\pi$$
0.599799 + 0.800151i $$0.295247\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −61.9704 −3.04937
$$414$$ 0 0
$$415$$ −24.2962 −1.19265
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 16.5947 0.810704 0.405352 0.914161i $$-0.367149\pi$$
0.405352 + 0.914161i $$0.367149\pi$$
$$420$$ 0 0
$$421$$ −24.1760 −1.17827 −0.589134 0.808035i $$-0.700531\pi$$
−0.589134 + 0.808035i $$0.700531\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −5.30425 −0.257294
$$426$$ 0 0
$$427$$ −13.7594 −0.665864
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −0.233761 −0.0112598 −0.00562992 0.999984i $$-0.501792\pi$$
−0.00562992 + 0.999984i $$0.501792\pi$$
$$432$$ 0 0
$$433$$ 31.0595 1.49263 0.746313 0.665595i $$-0.231822\pi$$
0.746313 + 0.665595i $$0.231822\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −0.925386 −0.0442672
$$438$$ 0 0
$$439$$ −20.3994 −0.973610 −0.486805 0.873511i $$-0.661838\pi$$
−0.486805 + 0.873511i $$0.661838\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −0.445779 −0.0211796 −0.0105898 0.999944i $$-0.503371\pi$$
−0.0105898 + 0.999944i $$0.503371\pi$$
$$444$$ 0 0
$$445$$ −21.6041 −1.02413
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −23.8874 −1.12732 −0.563659 0.826007i $$-0.690607\pi$$
−0.563659 + 0.826007i $$0.690607\pi$$
$$450$$ 0 0
$$451$$ −13.6635 −0.643389
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −30.3474 −1.42271
$$456$$ 0 0
$$457$$ 11.8064 0.552280 0.276140 0.961117i $$-0.410945\pi$$
0.276140 + 0.961117i $$0.410945\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0913 −0.656295 −0.328148 0.944626i $$-0.606424\pi$$
−0.328148 + 0.944626i $$0.606424\pi$$
$$462$$ 0 0
$$463$$ 14.6663 0.681602 0.340801 0.940135i $$-0.389302\pi$$
0.340801 + 0.940135i $$0.389302\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −30.8445 −1.42731 −0.713656 0.700496i $$-0.752962\pi$$
−0.713656 + 0.700496i $$0.752962\pi$$
$$468$$ 0 0
$$469$$ 33.6882 1.55558
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −10.9938 −0.505496
$$474$$ 0 0
$$475$$ −6.97394 −0.319987
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −4.40615 −0.201322 −0.100661 0.994921i $$-0.532096\pi$$
−0.100661 + 0.994921i $$0.532096\pi$$
$$480$$ 0 0
$$481$$ 5.82275 0.265495
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 8.43618 0.383067
$$486$$ 0 0
$$487$$ 24.2128 1.09719 0.548594 0.836089i $$-0.315163\pi$$
0.548594 + 0.836089i $$0.315163\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.9516 1.48709 0.743543 0.668689i $$-0.233144\pi$$
0.743543 + 0.668689i $$0.233144\pi$$
$$492$$ 0 0
$$493$$ 41.7358 1.87969
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −9.27422 −0.416006
$$498$$ 0 0
$$499$$ 9.79619 0.438538 0.219269 0.975664i $$-0.429633\pi$$
0.219269 + 0.975664i $$0.429633\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −14.0414 −0.626075 −0.313038 0.949741i $$-0.601347\pi$$
−0.313038 + 0.949741i $$0.601347\pi$$
$$504$$ 0 0
$$505$$ −4.66581 −0.207626
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 30.9992 1.37402 0.687008 0.726650i $$-0.258924\pi$$
0.687008 + 0.726650i $$0.258924\pi$$
$$510$$ 0 0
$$511$$ 24.3883 1.07888
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 22.6242 0.996941
$$516$$ 0 0
$$517$$ −0.338883 −0.0149040
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −23.3877 −1.02463 −0.512317 0.858796i $$-0.671213\pi$$
−0.512317 + 0.858796i $$0.671213\pi$$
$$522$$ 0 0
$$523$$ −14.2130 −0.621490 −0.310745 0.950493i $$-0.600579\pi$$
−0.310745 + 0.950493i $$0.600579\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 56.1552 2.44616
$$528$$ 0 0
$$529$$ −22.9824 −0.999234
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −28.4427 −1.23199
$$534$$ 0 0
$$535$$ 22.1883 0.959285
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 21.3221 0.918407
$$540$$ 0 0
$$541$$ −3.61867 −0.155579 −0.0777893 0.996970i $$-0.524786\pi$$
−0.0777893 + 0.996970i $$0.524786\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5.55267 0.237850
$$546$$ 0 0
$$547$$ 16.1945 0.692426 0.346213 0.938156i $$-0.387467\pi$$
0.346213 + 0.938156i $$0.387467\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 54.8736 2.33769
$$552$$ 0 0
$$553$$ −78.6141 −3.34301
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.39689 0.228674 0.114337 0.993442i $$-0.463526\pi$$
0.114337 + 0.993442i $$0.463526\pi$$
$$558$$ 0 0
$$559$$ −22.8853 −0.967946
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −38.3990 −1.61833 −0.809163 0.587585i $$-0.800079\pi$$
−0.809163 + 0.587585i $$0.800079\pi$$
$$564$$ 0 0
$$565$$ −34.5157 −1.45209
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 7.14674 0.299607 0.149803 0.988716i $$-0.452136\pi$$
0.149803 + 0.988716i $$0.452136\pi$$
$$570$$ 0 0
$$571$$ 9.09003 0.380406 0.190203 0.981745i $$-0.439085\pi$$
0.190203 + 0.981745i $$0.439085\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0.132692 0.00553363
$$576$$ 0 0
$$577$$ −5.92103 −0.246496 −0.123248 0.992376i $$-0.539331\pi$$
−0.123248 + 0.992376i $$0.539331\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 54.4924 2.26073
$$582$$ 0 0
$$583$$ −14.1163 −0.584639
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −27.2003 −1.12267 −0.561337 0.827587i $$-0.689713\pi$$
−0.561337 + 0.827587i $$0.689713\pi$$
$$588$$ 0 0
$$589$$ 73.8320 3.04220
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −16.3314 −0.670648 −0.335324 0.942103i $$-0.608846\pi$$
−0.335324 + 0.942103i $$0.608846\pi$$
$$594$$ 0 0
$$595$$ −47.5862 −1.95084
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 14.6823 0.599902 0.299951 0.953955i $$-0.403030\pi$$
0.299951 + 0.953955i $$0.403030\pi$$
$$600$$ 0 0
$$601$$ −20.3773 −0.831206 −0.415603 0.909546i $$-0.636430\pi$$
−0.415603 + 0.909546i $$0.636430\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 16.7187 0.679713
$$606$$ 0 0
$$607$$ −1.72706 −0.0700992 −0.0350496 0.999386i $$-0.511159\pi$$
−0.0350496 + 0.999386i $$0.511159\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −0.705437 −0.0285389
$$612$$ 0 0
$$613$$ −9.43357 −0.381018 −0.190509 0.981685i $$-0.561014\pi$$
−0.190509 + 0.981685i $$0.561014\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 19.2934 0.776724 0.388362 0.921507i $$-0.373041\pi$$
0.388362 + 0.921507i $$0.373041\pi$$
$$618$$ 0 0
$$619$$ −11.7698 −0.473067 −0.236533 0.971623i $$-0.576011\pi$$
−0.236533 + 0.971623i $$0.576011\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 48.4543 1.94128
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 9.13038 0.364052
$$630$$ 0 0
$$631$$ 35.7072 1.42148 0.710742 0.703453i $$-0.248360\pi$$
0.710742 + 0.703453i $$0.248360\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16.6354 0.660154
$$636$$ 0 0
$$637$$ 44.3852 1.75861
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 32.4396 1.28129 0.640644 0.767838i $$-0.278667\pi$$
0.640644 + 0.767838i $$0.278667\pi$$
$$642$$ 0 0
$$643$$ 24.8629 0.980496 0.490248 0.871583i $$-0.336906\pi$$
0.490248 + 0.871583i $$0.336906\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.5888 0.888056 0.444028 0.896013i $$-0.353549\pi$$
0.444028 + 0.896013i $$0.353549\pi$$
$$648$$ 0 0
$$649$$ −22.4498 −0.881230
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −22.9597 −0.898481 −0.449241 0.893411i $$-0.648305\pi$$
−0.449241 + 0.893411i $$0.648305\pi$$
$$654$$ 0 0
$$655$$ −11.6862 −0.456617
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.23673 0.0481760 0.0240880 0.999710i $$-0.492332\pi$$
0.0240880 + 0.999710i $$0.492332\pi$$
$$660$$ 0 0
$$661$$ 12.5706 0.488940 0.244470 0.969657i $$-0.421386\pi$$
0.244470 + 0.969657i $$0.421386\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −62.5656 −2.42619
$$666$$ 0 0
$$667$$ −1.04407 −0.0404265
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.98455 −0.192427
$$672$$ 0 0
$$673$$ −48.5661 −1.87208 −0.936042 0.351888i $$-0.885540\pi$$
−0.936042 + 0.351888i $$0.885540\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3.90041 −0.149905 −0.0749525 0.997187i $$-0.523881\pi$$
−0.0749525 + 0.997187i $$0.523881\pi$$
$$678$$ 0 0
$$679$$ −18.9210 −0.726120
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 17.2389 0.659626 0.329813 0.944046i $$-0.393014\pi$$
0.329813 + 0.944046i $$0.393014\pi$$
$$684$$ 0 0
$$685$$ 27.3523 1.04508
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −29.3853 −1.11949
$$690$$ 0 0
$$691$$ −26.1641 −0.995330 −0.497665 0.867369i $$-0.665809\pi$$
−0.497665 + 0.867369i $$0.665809\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −20.5996 −0.781388
$$696$$ 0 0
$$697$$ −44.5996 −1.68933
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31.0982 −1.17456 −0.587282 0.809382i $$-0.699802\pi$$
−0.587282 + 0.809382i $$0.699802\pi$$
$$702$$ 0 0
$$703$$ 12.0045 0.452758
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10.4646 0.393564
$$708$$ 0 0
$$709$$ −9.23408 −0.346793 −0.173397 0.984852i $$-0.555474\pi$$
−0.173397 + 0.984852i $$0.555474\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.40479 −0.0526097
$$714$$ 0 0
$$715$$ −10.9938 −0.411145
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −33.3348 −1.24318 −0.621590 0.783343i $$-0.713513\pi$$
−0.621590 + 0.783343i $$0.713513\pi$$
$$720$$ 0 0
$$721$$ −50.7424 −1.88974
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.86837 −0.292224
$$726$$ 0 0
$$727$$ 2.10327 0.0780061 0.0390030 0.999239i $$-0.487582\pi$$
0.0390030 + 0.999239i $$0.487582\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −35.8854 −1.32727
$$732$$ 0 0
$$733$$ −34.0965 −1.25938 −0.629691 0.776846i $$-0.716818\pi$$
−0.629691 + 0.776846i $$0.716818\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 12.2041 0.449543
$$738$$ 0 0
$$739$$ −11.1479 −0.410081 −0.205040 0.978754i $$-0.565733\pi$$
−0.205040 + 0.978754i $$0.565733\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 23.4849 0.861579 0.430789 0.902452i $$-0.358235\pi$$
0.430789 + 0.902452i $$0.358235\pi$$
$$744$$ 0 0
$$745$$ −15.6862 −0.574697
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −49.7648 −1.81837
$$750$$ 0 0
$$751$$ 31.1954 1.13834 0.569169 0.822221i $$-0.307265\pi$$
0.569169 + 0.822221i $$0.307265\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 31.0465 1.12990
$$756$$ 0 0
$$757$$ −19.3753 −0.704208 −0.352104 0.935961i $$-0.614534\pi$$
−0.352104 + 0.935961i $$0.614534\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.7008 −1.25790 −0.628951 0.777445i $$-0.716515\pi$$
−0.628951 + 0.777445i $$0.716515\pi$$
$$762$$ 0 0
$$763$$ −12.4537 −0.450855
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −46.7327 −1.68742
$$768$$ 0 0
$$769$$ −25.6173 −0.923785 −0.461892 0.886936i $$-0.652829\pi$$
−0.461892 + 0.886936i $$0.652829\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 25.4065 0.913808 0.456904 0.889516i $$-0.348958\pi$$
0.456904 + 0.889516i $$0.348958\pi$$
$$774$$ 0 0
$$775$$ −10.5868 −0.380291
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −58.6389 −2.10096
$$780$$ 0 0
$$781$$ −3.35973 −0.120221
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −34.8348 −1.24331
$$786$$ 0 0
$$787$$ 34.5483 1.23151 0.615757 0.787936i $$-0.288850\pi$$
0.615757 + 0.787936i $$0.288850\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 77.4131 2.75249
$$792$$ 0 0
$$793$$ −10.3761 −0.368467
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −43.1106 −1.52706 −0.763529 0.645774i $$-0.776535\pi$$
−0.763529 + 0.645774i $$0.776535\pi$$
$$798$$ 0 0
$$799$$ −1.10616 −0.0391332
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 8.83505 0.311782
$$804$$ 0 0
$$805$$ 1.19042 0.0419569
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −24.4150 −0.858386 −0.429193 0.903213i $$-0.641202\pi$$
−0.429193 + 0.903213i $$0.641202\pi$$
$$810$$ 0 0
$$811$$ −39.1406 −1.37441 −0.687206 0.726463i $$-0.741163\pi$$
−0.687206 + 0.726463i $$0.741163\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −47.9824 −1.68075
$$816$$ 0 0
$$817$$ −47.1815 −1.65067
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26.3557 −0.919822 −0.459911 0.887965i $$-0.652119\pi$$
−0.459911 + 0.887965i $$0.652119\pi$$
$$822$$ 0 0
$$823$$ 11.0365 0.384709 0.192354 0.981326i $$-0.438388\pi$$
0.192354 + 0.981326i $$0.438388\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 11.5066 0.400123 0.200062 0.979783i $$-0.435886\pi$$
0.200062 + 0.979783i $$0.435886\pi$$
$$828$$ 0 0
$$829$$ −17.9151 −0.622217 −0.311108 0.950374i $$-0.600700\pi$$
−0.311108 + 0.950374i $$0.600700\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 69.5983 2.41144
$$834$$ 0 0
$$835$$ −2.00000 −0.0692129
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −46.8888 −1.61878 −0.809390 0.587271i $$-0.800202\pi$$
−0.809390 + 0.587271i $$0.800202\pi$$
$$840$$ 0 0
$$841$$ 32.9113 1.13487
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 3.11467 0.107148
$$846$$ 0 0
$$847$$ −37.4973 −1.28842
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −0.228407 −0.00782969
$$852$$ 0 0
$$853$$ −7.97457 −0.273044 −0.136522 0.990637i $$-0.543592\pi$$
−0.136522 + 0.990637i $$0.543592\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −38.3453 −1.30985 −0.654925 0.755694i $$-0.727300\pi$$
−0.654925 + 0.755694i $$0.727300\pi$$
$$858$$ 0 0
$$859$$ 55.7156 1.90099 0.950497 0.310735i $$-0.100575\pi$$
0.950497 + 0.310735i $$0.100575\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −35.5096 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$864$$ 0 0
$$865$$ −3.36464 −0.114401
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −28.4792 −0.966090
$$870$$ 0 0
$$871$$ 25.4047 0.860806
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 53.8281 1.81972
$$876$$ 0 0
$$877$$ 1.96237 0.0662646 0.0331323 0.999451i $$-0.489452\pi$$
0.0331323 + 0.999451i $$0.489452\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25.9608 0.874642 0.437321 0.899305i $$-0.355927\pi$$
0.437321 + 0.899305i $$0.355927\pi$$
$$882$$ 0 0
$$883$$ 32.1139 1.08072 0.540360 0.841434i $$-0.318288\pi$$
0.540360 + 0.841434i $$0.318288\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −6.96553 −0.233880 −0.116940 0.993139i $$-0.537308\pi$$
−0.116940 + 0.993139i $$0.537308\pi$$
$$888$$ 0 0
$$889$$ −37.3104 −1.25135
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −1.45436 −0.0486684
$$894$$ 0 0
$$895$$ −1.52134 −0.0508529
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 83.3012 2.77825
$$900$$ 0 0
$$901$$ −46.0777 −1.53507
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.7133 0.422604
$$906$$ 0 0
$$907$$ 34.2026 1.13568 0.567839 0.823140i $$-0.307780\pi$$
0.567839 + 0.823140i $$0.307780\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 33.5867 1.11278 0.556388 0.830922i $$-0.312187\pi$$
0.556388 + 0.830922i $$0.312187\pi$$
$$912$$ 0 0
$$913$$ 19.7407 0.653323
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 26.2102 0.865537
$$918$$ 0 0
$$919$$ 44.8672 1.48003 0.740017 0.672588i $$-0.234817\pi$$
0.740017 + 0.672588i $$0.234817\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −6.99381 −0.230204
$$924$$ 0 0
$$925$$ −1.72133 −0.0565971
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 12.6760 0.415887 0.207943 0.978141i $$-0.433323\pi$$
0.207943 + 0.978141i $$0.433323\pi$$
$$930$$ 0 0
$$931$$ 91.5068 2.99902
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −17.2389 −0.563771
$$936$$ 0 0
$$937$$ 3.52060 0.115013 0.0575065 0.998345i $$-0.481685\pi$$
0.0575065 + 0.998345i $$0.481685\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −54.2981 −1.77007 −0.885034 0.465526i $$-0.845865\pi$$
−0.885034 + 0.465526i $$0.845865\pi$$
$$942$$ 0 0
$$943$$ 1.11571 0.0363325
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −35.3015 −1.14715 −0.573573 0.819155i $$-0.694443\pi$$
−0.573573 + 0.819155i $$0.694443\pi$$
$$948$$ 0 0
$$949$$ 18.3915 0.597015
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −41.2048 −1.33475 −0.667377 0.744720i $$-0.732583\pi$$
−0.667377 + 0.744720i $$0.732583\pi$$
$$954$$ 0 0
$$955$$ 22.3979 0.724778
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −61.3468 −1.98099
$$960$$ 0 0
$$961$$ 81.0812 2.61552
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 29.8943 0.962330
$$966$$ 0 0
$$967$$ −16.5718 −0.532913 −0.266457 0.963847i $$-0.585853\pi$$
−0.266457 + 0.963847i $$0.585853\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 41.8901 1.34432 0.672158 0.740408i $$-0.265368\pi$$
0.672158 + 0.740408i $$0.265368\pi$$
$$972$$ 0 0
$$973$$ 46.2015 1.48115
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −54.2586 −1.73589 −0.867944 0.496663i $$-0.834559\pi$$
−0.867944 + 0.496663i $$0.834559\pi$$
$$978$$ 0 0
$$979$$ 17.5533 0.561007
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 40.0351 1.27692 0.638460 0.769655i $$-0.279572\pi$$
0.638460 + 0.769655i $$0.279572\pi$$
$$984$$ 0 0
$$985$$ 36.1550 1.15199
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0.897714 0.0285456
$$990$$ 0 0
$$991$$ 6.86359 0.218029 0.109015 0.994040i $$-0.465230\pi$$
0.109015 + 0.994040i $$0.465230\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 7.35233 0.233085
$$996$$ 0 0
$$997$$ 26.6578 0.844260 0.422130 0.906535i $$-0.361283\pi$$
0.422130 + 0.906535i $$0.361283\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 6012.2.a.d.1.4 5
3.2 odd 2 668.2.a.b.1.2 5
12.11 even 2 2672.2.a.j.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.2 5 3.2 odd 2
2672.2.a.j.1.4 5 12.11 even 2
6012.2.a.d.1.4 5 1.1 even 1 trivial