# Properties

 Label 3120.2.g.q.961.4 Level $3120$ Weight $2$ Character 3120.961 Analytic conductor $24.913$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 961.4 Root $$2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 3120.961 Dual form 3120.2.g.q.961.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000i q^{5} +4.60555i q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000i q^{5} +4.60555i q^{7} +1.00000 q^{9} +3.60555 q^{13} +1.00000i q^{15} +4.60555 q^{17} +4.60555i q^{19} +4.60555i q^{21} +1.39445 q^{23} -1.00000 q^{25} +1.00000 q^{27} +4.60555 q^{29} -6.00000i q^{31} -4.60555 q^{35} -9.21110i q^{37} +3.60555 q^{39} +3.21110i q^{41} -8.00000 q^{43} +1.00000i q^{45} +9.21110i q^{47} -14.2111 q^{49} +4.60555 q^{51} +6.00000 q^{53} +4.60555i q^{57} +9.21110i q^{59} -11.2111 q^{61} +4.60555i q^{63} +3.60555i q^{65} +3.21110i q^{67} +1.39445 q^{69} -9.21110i q^{71} -1.39445i q^{73} -1.00000 q^{75} +14.4222 q^{79} +1.00000 q^{81} -2.78890i q^{83} +4.60555i q^{85} +4.60555 q^{87} +15.2111i q^{89} +16.6056i q^{91} -6.00000i q^{93} -4.60555 q^{95} +1.39445i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{9} + 4 q^{17} + 20 q^{23} - 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{35} - 32 q^{43} - 28 q^{49} + 4 q^{51} + 24 q^{53} - 16 q^{61} + 20 q^{69} - 4 q^{75} + 4 q^{81} + 4 q^{87} - 4 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^9 + 4 * q^17 + 20 * q^23 - 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^35 - 32 * q^43 - 28 * q^49 + 4 * q^51 + 24 * q^53 - 16 * q^61 + 20 * q^69 - 4 * q^75 + 4 * q^81 + 4 * q^87 - 4 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$$.

 $$n$$ $$1951$$ $$2081$$ $$2341$$ $$2497$$ $$2641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ 4.60555i 1.74073i 0.492403 + 0.870367i $$0.336119\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 3.60555 1.00000
$$14$$ 0 0
$$15$$ 1.00000i 0.258199i
$$16$$ 0 0
$$17$$ 4.60555 1.11701 0.558505 0.829501i $$-0.311375\pi$$
0.558505 + 0.829501i $$0.311375\pi$$
$$18$$ 0 0
$$19$$ 4.60555i 1.05659i 0.849062 + 0.528293i $$0.177168\pi$$
−0.849062 + 0.528293i $$0.822832\pi$$
$$20$$ 0 0
$$21$$ 4.60555i 1.00501i
$$22$$ 0 0
$$23$$ 1.39445 0.290763 0.145381 0.989376i $$-0.453559\pi$$
0.145381 + 0.989376i $$0.453559\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ 0 0
$$31$$ − 6.00000i − 1.07763i −0.842424 0.538816i $$-0.818872\pi$$
0.842424 0.538816i $$-0.181128\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.60555 −0.778480
$$36$$ 0 0
$$37$$ − 9.21110i − 1.51430i −0.653243 0.757148i $$-0.726592\pi$$
0.653243 0.757148i $$-0.273408\pi$$
$$38$$ 0 0
$$39$$ 3.60555 0.577350
$$40$$ 0 0
$$41$$ 3.21110i 0.501490i 0.968053 + 0.250745i $$0.0806756\pi$$
−0.968053 + 0.250745i $$0.919324\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ 9.21110i 1.34358i 0.740743 + 0.671789i $$0.234474\pi$$
−0.740743 + 0.671789i $$0.765526\pi$$
$$48$$ 0 0
$$49$$ −14.2111 −2.03016
$$50$$ 0 0
$$51$$ 4.60555 0.644906
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.60555i 0.610020i
$$58$$ 0 0
$$59$$ 9.21110i 1.19918i 0.800306 + 0.599592i $$0.204670\pi$$
−0.800306 + 0.599592i $$0.795330\pi$$
$$60$$ 0 0
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ 0 0
$$63$$ 4.60555i 0.580245i
$$64$$ 0 0
$$65$$ 3.60555i 0.447214i
$$66$$ 0 0
$$67$$ 3.21110i 0.392299i 0.980574 + 0.196149i $$0.0628437\pi$$
−0.980574 + 0.196149i $$0.937156\pi$$
$$68$$ 0 0
$$69$$ 1.39445 0.167872
$$70$$ 0 0
$$71$$ − 9.21110i − 1.09316i −0.837408 0.546578i $$-0.815930\pi$$
0.837408 0.546578i $$-0.184070\pi$$
$$72$$ 0 0
$$73$$ − 1.39445i − 0.163208i −0.996665 0.0816039i $$-0.973996\pi$$
0.996665 0.0816039i $$-0.0260043\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 2.78890i − 0.306121i −0.988217 0.153061i $$-0.951087\pi$$
0.988217 0.153061i $$-0.0489130\pi$$
$$84$$ 0 0
$$85$$ 4.60555i 0.499542i
$$86$$ 0 0
$$87$$ 4.60555 0.493767
$$88$$ 0 0
$$89$$ 15.2111i 1.61237i 0.591661 + 0.806187i $$0.298472\pi$$
−0.591661 + 0.806187i $$0.701528\pi$$
$$90$$ 0 0
$$91$$ 16.6056i 1.74073i
$$92$$ 0 0
$$93$$ − 6.00000i − 0.622171i
$$94$$ 0 0
$$95$$ −4.60555 −0.472520
$$96$$ 0 0
$$97$$ 1.39445i 0.141585i 0.997491 + 0.0707924i $$0.0225528\pi$$
−0.997491 + 0.0707924i $$0.977447\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.39445 −0.735775 −0.367888 0.929870i $$-0.619919\pi$$
−0.367888 + 0.929870i $$0.619919\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −4.60555 −0.449456
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 1.39445i 0.133564i 0.997768 + 0.0667820i $$0.0212732\pi$$
−0.997768 + 0.0667820i $$0.978727\pi$$
$$110$$ 0 0
$$111$$ − 9.21110i − 0.874279i
$$112$$ 0 0
$$113$$ −13.8167 −1.29976 −0.649881 0.760036i $$-0.725181\pi$$
−0.649881 + 0.760036i $$0.725181\pi$$
$$114$$ 0 0
$$115$$ 1.39445i 0.130033i
$$116$$ 0 0
$$117$$ 3.60555 0.333333
$$118$$ 0 0
$$119$$ 21.2111i 1.94442i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 3.21110i 0.289535i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −1.21110 −0.107468 −0.0537340 0.998555i $$-0.517112\pi$$
−0.0537340 + 0.998555i $$0.517112\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −22.6056 −1.97506 −0.987528 0.157443i $$-0.949675\pi$$
−0.987528 + 0.157443i $$0.949675\pi$$
$$132$$ 0 0
$$133$$ −21.2111 −1.83924
$$134$$ 0 0
$$135$$ 1.00000i 0.0860663i
$$136$$ 0 0
$$137$$ 3.21110i 0.274343i 0.990547 + 0.137172i $$0.0438011\pi$$
−0.990547 + 0.137172i $$0.956199\pi$$
$$138$$ 0 0
$$139$$ −17.2111 −1.45983 −0.729913 0.683540i $$-0.760440\pi$$
−0.729913 + 0.683540i $$0.760440\pi$$
$$140$$ 0 0
$$141$$ 9.21110i 0.775715i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.60555i 0.382470i
$$146$$ 0 0
$$147$$ −14.2111 −1.17211
$$148$$ 0 0
$$149$$ 15.2111i 1.24614i 0.782165 + 0.623071i $$0.214115\pi$$
−0.782165 + 0.623071i $$0.785885\pi$$
$$150$$ 0 0
$$151$$ − 6.00000i − 0.488273i −0.969741 0.244137i $$-0.921495\pi$$
0.969741 0.244137i $$-0.0785045\pi$$
$$152$$ 0 0
$$153$$ 4.60555 0.372337
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 20.4222 1.62987 0.814935 0.579553i $$-0.196773\pi$$
0.814935 + 0.579553i $$0.196773\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 6.42221i 0.506141i
$$162$$ 0 0
$$163$$ − 24.4222i − 1.91289i −0.291905 0.956447i $$-0.594289\pi$$
0.291905 0.956447i $$-0.405711\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 9.21110i − 0.712777i −0.934338 0.356388i $$-0.884008\pi$$
0.934338 0.356388i $$-0.115992\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 4.60555i 0.352195i
$$172$$ 0 0
$$173$$ −12.4222 −0.944443 −0.472221 0.881480i $$-0.656548\pi$$
−0.472221 + 0.881480i $$0.656548\pi$$
$$174$$ 0 0
$$175$$ − 4.60555i − 0.348147i
$$176$$ 0 0
$$177$$ 9.21110i 0.692349i
$$178$$ 0 0
$$179$$ 19.8167 1.48117 0.740583 0.671965i $$-0.234549\pi$$
0.740583 + 0.671965i $$0.234549\pi$$
$$180$$ 0 0
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ 0 0
$$183$$ −11.2111 −0.828749
$$184$$ 0 0
$$185$$ 9.21110 0.677214
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.60555i 0.335005i
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 7.81665i 0.562655i 0.959612 + 0.281328i $$0.0907747\pi$$
−0.959612 + 0.281328i $$0.909225\pi$$
$$194$$ 0 0
$$195$$ 3.60555i 0.258199i
$$196$$ 0 0
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ −22.4222 −1.58947 −0.794734 0.606958i $$-0.792390\pi$$
−0.794734 + 0.606958i $$0.792390\pi$$
$$200$$ 0 0
$$201$$ 3.21110i 0.226494i
$$202$$ 0 0
$$203$$ 21.2111i 1.48873i
$$204$$ 0 0
$$205$$ −3.21110 −0.224273
$$206$$ 0 0
$$207$$ 1.39445 0.0969209
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 17.2111 1.18486 0.592431 0.805622i $$-0.298168\pi$$
0.592431 + 0.805622i $$0.298168\pi$$
$$212$$ 0 0
$$213$$ − 9.21110i − 0.631134i
$$214$$ 0 0
$$215$$ − 8.00000i − 0.545595i
$$216$$ 0 0
$$217$$ 27.6333 1.87587
$$218$$ 0 0
$$219$$ − 1.39445i − 0.0942281i
$$220$$ 0 0
$$221$$ 16.6056 1.11701
$$222$$ 0 0
$$223$$ − 1.81665i − 0.121652i −0.998148 0.0608261i $$-0.980627\pi$$
0.998148 0.0608261i $$-0.0193735\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 0 0
$$229$$ − 19.8167i − 1.30952i −0.755836 0.654761i $$-0.772769\pi$$
0.755836 0.654761i $$-0.227231\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.81665 −0.119013 −0.0595065 0.998228i $$-0.518953\pi$$
−0.0595065 + 0.998228i $$0.518953\pi$$
$$234$$ 0 0
$$235$$ −9.21110 −0.600866
$$236$$ 0 0
$$237$$ 14.4222 0.936823
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ − 6.42221i − 0.413691i −0.978374 0.206845i $$-0.933680\pi$$
0.978374 0.206845i $$-0.0663197\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ − 14.2111i − 0.907914i
$$246$$ 0 0
$$247$$ 16.6056i 1.05659i
$$248$$ 0 0
$$249$$ − 2.78890i − 0.176739i
$$250$$ 0 0
$$251$$ 13.3944 0.845450 0.422725 0.906258i $$-0.361074\pi$$
0.422725 + 0.906258i $$0.361074\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 4.60555i 0.288411i
$$256$$ 0 0
$$257$$ −28.6056 −1.78437 −0.892183 0.451675i $$-0.850827\pi$$
−0.892183 + 0.451675i $$0.850827\pi$$
$$258$$ 0 0
$$259$$ 42.4222 2.63599
$$260$$ 0 0
$$261$$ 4.60555 0.285076
$$262$$ 0 0
$$263$$ 7.81665 0.481996 0.240998 0.970526i $$-0.422525\pi$$
0.240998 + 0.970526i $$0.422525\pi$$
$$264$$ 0 0
$$265$$ 6.00000i 0.368577i
$$266$$ 0 0
$$267$$ 15.2111i 0.930904i
$$268$$ 0 0
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ 0 0
$$271$$ 0.422205i 0.0256471i 0.999918 + 0.0128236i $$0.00408198\pi$$
−0.999918 + 0.0128236i $$0.995918\pi$$
$$272$$ 0 0
$$273$$ 16.6056i 1.00501i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −16.4222 −0.986715 −0.493357 0.869827i $$-0.664230\pi$$
−0.493357 + 0.869827i $$0.664230\pi$$
$$278$$ 0 0
$$279$$ − 6.00000i − 0.359211i
$$280$$ 0 0
$$281$$ 27.2111i 1.62328i 0.584159 + 0.811639i $$0.301424\pi$$
−0.584159 + 0.811639i $$0.698576\pi$$
$$282$$ 0 0
$$283$$ −10.4222 −0.619536 −0.309768 0.950812i $$-0.600251\pi$$
−0.309768 + 0.950812i $$0.600251\pi$$
$$284$$ 0 0
$$285$$ −4.60555 −0.272809
$$286$$ 0 0
$$287$$ −14.7889 −0.872961
$$288$$ 0 0
$$289$$ 4.21110 0.247712
$$290$$ 0 0
$$291$$ 1.39445i 0.0817440i
$$292$$ 0 0
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ −9.21110 −0.536291
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.02776 0.290763
$$300$$ 0 0
$$301$$ − 36.8444i − 2.12368i
$$302$$ 0 0
$$303$$ −7.39445 −0.424800
$$304$$ 0 0
$$305$$ − 11.2111i − 0.641946i
$$306$$ 0 0
$$307$$ 8.78890i 0.501609i 0.968038 + 0.250804i $$0.0806951\pi$$
−0.968038 + 0.250804i $$0.919305\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 3.57779 0.202229 0.101114 0.994875i $$-0.467759\pi$$
0.101114 + 0.994875i $$0.467759\pi$$
$$314$$ 0 0
$$315$$ −4.60555 −0.259493
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 21.2111i 1.18022i
$$324$$ 0 0
$$325$$ −3.60555 −0.200000
$$326$$ 0 0
$$327$$ 1.39445i 0.0771132i
$$328$$ 0 0
$$329$$ −42.4222 −2.33881
$$330$$ 0 0
$$331$$ 16.6056i 0.912724i 0.889794 + 0.456362i $$0.150848\pi$$
−0.889794 + 0.456362i $$0.849152\pi$$
$$332$$ 0 0
$$333$$ − 9.21110i − 0.504765i
$$334$$ 0 0
$$335$$ −3.21110 −0.175441
$$336$$ 0 0
$$337$$ 13.6333 0.742654 0.371327 0.928502i $$-0.378903\pi$$
0.371327 + 0.928502i $$0.378903\pi$$
$$338$$ 0 0
$$339$$ −13.8167 −0.750418
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 33.2111i − 1.79323i
$$344$$ 0 0
$$345$$ 1.39445i 0.0750746i
$$346$$ 0 0
$$347$$ 27.6333 1.48343 0.741717 0.670713i $$-0.234012\pi$$
0.741717 + 0.670713i $$0.234012\pi$$
$$348$$ 0 0
$$349$$ − 7.81665i − 0.418416i −0.977871 0.209208i $$-0.932911\pi$$
0.977871 0.209208i $$-0.0670886\pi$$
$$350$$ 0 0
$$351$$ 3.60555 0.192450
$$352$$ 0 0
$$353$$ − 8.78890i − 0.467786i −0.972262 0.233893i $$-0.924853\pi$$
0.972262 0.233893i $$-0.0751465\pi$$
$$354$$ 0 0
$$355$$ 9.21110 0.488875
$$356$$ 0 0
$$357$$ 21.2111i 1.12261i
$$358$$ 0 0
$$359$$ − 15.6333i − 0.825094i −0.910936 0.412547i $$-0.864639\pi$$
0.910936 0.412547i $$-0.135361\pi$$
$$360$$ 0 0
$$361$$ −2.21110 −0.116374
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 1.39445 0.0729888
$$366$$ 0 0
$$367$$ 19.6333 1.02485 0.512425 0.858732i $$-0.328747\pi$$
0.512425 + 0.858732i $$0.328747\pi$$
$$368$$ 0 0
$$369$$ 3.21110i 0.167163i
$$370$$ 0 0
$$371$$ 27.6333i 1.43465i
$$372$$ 0 0
$$373$$ −20.4222 −1.05742 −0.528711 0.848802i $$-0.677324\pi$$
−0.528711 + 0.848802i $$0.677324\pi$$
$$374$$ 0 0
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 0 0
$$377$$ 16.6056 0.855229
$$378$$ 0 0
$$379$$ − 35.0278i − 1.79925i −0.436658 0.899627i $$-0.643838\pi$$
0.436658 0.899627i $$-0.356162\pi$$
$$380$$ 0 0
$$381$$ −1.21110 −0.0620467
$$382$$ 0 0
$$383$$ 27.6333i 1.41200i 0.708214 + 0.705998i $$0.249501\pi$$
−0.708214 + 0.705998i $$0.750499\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ 4.60555 0.233511 0.116755 0.993161i $$-0.462751\pi$$
0.116755 + 0.993161i $$0.462751\pi$$
$$390$$ 0 0
$$391$$ 6.42221 0.324785
$$392$$ 0 0
$$393$$ −22.6056 −1.14030
$$394$$ 0 0
$$395$$ 14.4222i 0.725660i
$$396$$ 0 0
$$397$$ − 3.63331i − 0.182350i −0.995835 0.0911752i $$-0.970938\pi$$
0.995835 0.0911752i $$-0.0290623\pi$$
$$398$$ 0 0
$$399$$ −21.2111 −1.06188
$$400$$ 0 0
$$401$$ 8.78890i 0.438897i 0.975624 + 0.219448i $$0.0704257\pi$$
−0.975624 + 0.219448i $$0.929574\pi$$
$$402$$ 0 0
$$403$$ − 21.6333i − 1.07763i
$$404$$ 0 0
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ − 14.7889i − 0.731264i −0.930760 0.365632i $$-0.880853\pi$$
0.930760 0.365632i $$-0.119147\pi$$
$$410$$ 0 0
$$411$$ 3.21110i 0.158392i
$$412$$ 0 0
$$413$$ −42.4222 −2.08746
$$414$$ 0 0
$$415$$ 2.78890 0.136902
$$416$$ 0 0
$$417$$ −17.2111 −0.842831
$$418$$ 0 0
$$419$$ −4.18335 −0.204370 −0.102185 0.994765i $$-0.532583\pi$$
−0.102185 + 0.994765i $$0.532583\pi$$
$$420$$ 0 0
$$421$$ − 19.8167i − 0.965805i −0.875674 0.482902i $$-0.839583\pi$$
0.875674 0.482902i $$-0.160417\pi$$
$$422$$ 0 0
$$423$$ 9.21110i 0.447859i
$$424$$ 0 0
$$425$$ −4.60555 −0.223402
$$426$$ 0 0
$$427$$ − 51.6333i − 2.49871i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 12.0000i − 0.578020i −0.957326 0.289010i $$-0.906674\pi$$
0.957326 0.289010i $$-0.0933260\pi$$
$$432$$ 0 0
$$433$$ 19.2111 0.923227 0.461613 0.887081i $$-0.347271\pi$$
0.461613 + 0.887081i $$0.347271\pi$$
$$434$$ 0 0
$$435$$ 4.60555i 0.220819i
$$436$$ 0 0
$$437$$ 6.42221i 0.307216i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −14.2111 −0.676719
$$442$$ 0 0
$$443$$ −15.6333 −0.742761 −0.371380 0.928481i $$-0.621115\pi$$
−0.371380 + 0.928481i $$0.621115\pi$$
$$444$$ 0 0
$$445$$ −15.2111 −0.721075
$$446$$ 0 0
$$447$$ 15.2111i 0.719460i
$$448$$ 0 0
$$449$$ 33.6333i 1.58725i 0.608405 + 0.793627i $$0.291810\pi$$
−0.608405 + 0.793627i $$0.708190\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 6.00000i − 0.281905i
$$454$$ 0 0
$$455$$ −16.6056 −0.778480
$$456$$ 0 0
$$457$$ 38.2389i 1.78874i 0.447330 + 0.894369i $$0.352375\pi$$
−0.447330 + 0.894369i $$0.647625\pi$$
$$458$$ 0 0
$$459$$ 4.60555 0.214969
$$460$$ 0 0
$$461$$ − 33.6333i − 1.56646i −0.621733 0.783230i $$-0.713571\pi$$
0.621733 0.783230i $$-0.286429\pi$$
$$462$$ 0 0
$$463$$ 31.3944i 1.45902i 0.683968 + 0.729512i $$0.260253\pi$$
−0.683968 + 0.729512i $$0.739747\pi$$
$$464$$ 0 0
$$465$$ 6.00000 0.278243
$$466$$ 0 0
$$467$$ 30.4222 1.40777 0.703886 0.710313i $$-0.251447\pi$$
0.703886 + 0.710313i $$0.251447\pi$$
$$468$$ 0 0
$$469$$ −14.7889 −0.682888
$$470$$ 0 0
$$471$$ 20.4222 0.941006
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 4.60555i − 0.211317i
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ − 5.57779i − 0.254856i −0.991848 0.127428i $$-0.959328\pi$$
0.991848 0.127428i $$-0.0406722\pi$$
$$480$$ 0 0
$$481$$ − 33.2111i − 1.51430i
$$482$$ 0 0
$$483$$ 6.42221i 0.292220i
$$484$$ 0 0
$$485$$ −1.39445 −0.0633187
$$486$$ 0 0
$$487$$ 0.972244i 0.0440566i 0.999757 + 0.0220283i $$0.00701239\pi$$
−0.999757 + 0.0220283i $$0.992988\pi$$
$$488$$ 0 0
$$489$$ − 24.4222i − 1.10441i
$$490$$ 0 0
$$491$$ −7.81665 −0.352761 −0.176380 0.984322i $$-0.556439\pi$$
−0.176380 + 0.984322i $$0.556439\pi$$
$$492$$ 0 0
$$493$$ 21.2111 0.955300
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 42.4222 1.90290
$$498$$ 0 0
$$499$$ − 23.0278i − 1.03086i −0.856930 0.515432i $$-0.827631\pi$$
0.856930 0.515432i $$-0.172369\pi$$
$$500$$ 0 0
$$501$$ − 9.21110i − 0.411522i
$$502$$ 0 0
$$503$$ −23.4500 −1.04558 −0.522791 0.852461i $$-0.675109\pi$$
−0.522791 + 0.852461i $$0.675109\pi$$
$$504$$ 0 0
$$505$$ − 7.39445i − 0.329049i
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ − 33.6333i − 1.49077i −0.666634 0.745385i $$-0.732266\pi$$
0.666634 0.745385i $$-0.267734\pi$$
$$510$$ 0 0
$$511$$ 6.42221 0.284102
$$512$$ 0 0
$$513$$ 4.60555i 0.203340i
$$514$$ 0 0
$$515$$ 4.00000i 0.176261i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12.4222 −0.545274
$$520$$ 0 0
$$521$$ 21.6333 0.947772 0.473886 0.880586i $$-0.342851\pi$$
0.473886 + 0.880586i $$0.342851\pi$$
$$522$$ 0 0
$$523$$ −32.8444 −1.43619 −0.718093 0.695947i $$-0.754985\pi$$
−0.718093 + 0.695947i $$0.754985\pi$$
$$524$$ 0 0
$$525$$ − 4.60555i − 0.201003i
$$526$$ 0 0
$$527$$ − 27.6333i − 1.20373i
$$528$$ 0 0
$$529$$ −21.0555 −0.915457
$$530$$ 0 0
$$531$$ 9.21110i 0.399728i
$$532$$ 0 0
$$533$$ 11.5778i 0.501490i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 19.8167 0.855152
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6.97224i 0.299760i 0.988704 + 0.149880i $$0.0478888\pi$$
−0.988704 + 0.149880i $$0.952111\pi$$
$$542$$ 0 0
$$543$$ 8.42221 0.361431
$$544$$ 0 0
$$545$$ −1.39445 −0.0597316
$$546$$ 0 0
$$547$$ −14.4222 −0.616649 −0.308324 0.951281i $$-0.599768\pi$$
−0.308324 + 0.951281i $$0.599768\pi$$
$$548$$ 0 0
$$549$$ −11.2111 −0.478478
$$550$$ 0 0
$$551$$ 21.2111i 0.903623i
$$552$$ 0 0
$$553$$ 66.4222i 2.82456i
$$554$$ 0 0
$$555$$ 9.21110 0.390990
$$556$$ 0 0
$$557$$ 11.5778i 0.490567i 0.969451 + 0.245283i $$0.0788810\pi$$
−0.969451 + 0.245283i $$0.921119\pi$$
$$558$$ 0 0
$$559$$ −28.8444 −1.21999
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −34.0555 −1.43527 −0.717634 0.696420i $$-0.754775\pi$$
−0.717634 + 0.696420i $$0.754775\pi$$
$$564$$ 0 0
$$565$$ − 13.8167i − 0.581271i
$$566$$ 0 0
$$567$$ 4.60555i 0.193415i
$$568$$ 0 0
$$569$$ 33.6333 1.40998 0.704991 0.709216i $$-0.250951\pi$$
0.704991 + 0.709216i $$0.250951\pi$$
$$570$$ 0 0
$$571$$ 30.0555 1.25778 0.628892 0.777493i $$-0.283509\pi$$
0.628892 + 0.777493i $$0.283509\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ −1.39445 −0.0581525
$$576$$ 0 0
$$577$$ − 37.3944i − 1.55675i −0.627799 0.778376i $$-0.716044\pi$$
0.627799 0.778376i $$-0.283956\pi$$
$$578$$ 0 0
$$579$$ 7.81665i 0.324849i
$$580$$ 0 0
$$581$$ 12.8444 0.532876
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 3.60555i 0.149071i
$$586$$ 0 0
$$587$$ 6.42221i 0.265073i 0.991178 + 0.132536i $$0.0423121\pi$$
−0.991178 + 0.132536i $$0.957688\pi$$
$$588$$ 0 0
$$589$$ 27.6333 1.13861
$$590$$ 0 0
$$591$$ − 6.00000i − 0.246807i
$$592$$ 0 0
$$593$$ − 24.4222i − 1.00290i −0.865187 0.501450i $$-0.832800\pi$$
0.865187 0.501450i $$-0.167200\pi$$
$$594$$ 0 0
$$595$$ −21.2111 −0.869570
$$596$$ 0 0
$$597$$ −22.4222 −0.917680
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ 0 0
$$603$$ 3.21110i 0.130766i
$$604$$ 0 0
$$605$$ 11.0000i 0.447214i
$$606$$ 0 0
$$607$$ −17.2111 −0.698577 −0.349289 0.937015i $$-0.613577\pi$$
−0.349289 + 0.937015i $$0.613577\pi$$
$$608$$ 0 0
$$609$$ 21.2111i 0.859517i
$$610$$ 0 0
$$611$$ 33.2111i 1.34358i
$$612$$ 0 0
$$613$$ − 33.2111i − 1.34138i −0.741736 0.670692i $$-0.765997\pi$$
0.741736 0.670692i $$-0.234003\pi$$
$$614$$ 0 0
$$615$$ −3.21110 −0.129484
$$616$$ 0 0
$$617$$ 12.4222i 0.500099i 0.968233 + 0.250050i $$0.0804469\pi$$
−0.968233 + 0.250050i $$0.919553\pi$$
$$618$$ 0 0
$$619$$ − 25.8167i − 1.03766i −0.854878 0.518829i $$-0.826368\pi$$
0.854878 0.518829i $$-0.173632\pi$$
$$620$$ 0 0
$$621$$ 1.39445 0.0559573
$$622$$ 0 0
$$623$$ −70.0555 −2.80671
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 42.4222i − 1.69148i
$$630$$ 0 0
$$631$$ − 3.21110i − 0.127832i −0.997955 0.0639160i $$-0.979641\pi$$
0.997955 0.0639160i $$-0.0203590\pi$$
$$632$$ 0 0
$$633$$ 17.2111 0.684080
$$634$$ 0 0
$$635$$ − 1.21110i − 0.0480611i
$$636$$ 0 0
$$637$$ −51.2389 −2.03016
$$638$$ 0 0
$$639$$ − 9.21110i − 0.364386i
$$640$$ 0 0
$$641$$ 0.422205 0.0166761 0.00833805 0.999965i $$-0.497346\pi$$
0.00833805 + 0.999965i $$0.497346\pi$$
$$642$$ 0 0
$$643$$ − 9.63331i − 0.379901i −0.981794 0.189950i $$-0.939167\pi$$
0.981794 0.189950i $$-0.0608327\pi$$
$$644$$ 0 0
$$645$$ − 8.00000i − 0.315000i
$$646$$ 0 0
$$647$$ 34.6056 1.36048 0.680242 0.732987i $$-0.261875\pi$$
0.680242 + 0.732987i $$0.261875\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 27.6333 1.08303
$$652$$ 0 0
$$653$$ 39.2111 1.53445 0.767225 0.641379i $$-0.221637\pi$$
0.767225 + 0.641379i $$0.221637\pi$$
$$654$$ 0 0
$$655$$ − 22.6056i − 0.883272i
$$656$$ 0 0
$$657$$ − 1.39445i − 0.0544026i
$$658$$ 0 0
$$659$$ 26.2389 1.02212 0.511060 0.859545i $$-0.329253\pi$$
0.511060 + 0.859545i $$0.329253\pi$$
$$660$$ 0 0
$$661$$ 50.2389i 1.95407i 0.213090 + 0.977033i $$0.431647\pi$$
−0.213090 + 0.977033i $$0.568353\pi$$
$$662$$ 0 0
$$663$$ 16.6056 0.644906
$$664$$ 0 0
$$665$$ − 21.2111i − 0.822531i
$$666$$ 0 0
$$667$$ 6.42221 0.248669
$$668$$ 0 0
$$669$$ − 1.81665i − 0.0702359i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −37.6333 −1.45066 −0.725329 0.688403i $$-0.758312\pi$$
−0.725329 + 0.688403i $$0.758312\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −28.0555 −1.07826 −0.539130 0.842222i $$-0.681247\pi$$
−0.539130 + 0.842222i $$0.681247\pi$$
$$678$$ 0 0
$$679$$ −6.42221 −0.246462
$$680$$ 0 0
$$681$$ − 24.0000i − 0.919682i
$$682$$ 0 0
$$683$$ − 9.21110i − 0.352453i −0.984350 0.176227i $$-0.943611\pi$$
0.984350 0.176227i $$-0.0563891\pi$$
$$684$$ 0 0
$$685$$ −3.21110 −0.122690
$$686$$ 0 0
$$687$$ − 19.8167i − 0.756053i
$$688$$ 0 0
$$689$$ 21.6333 0.824163
$$690$$ 0 0
$$691$$ − 20.2389i − 0.769922i −0.922933 0.384961i $$-0.874215\pi$$
0.922933 0.384961i $$-0.125785\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 17.2111i − 0.652854i
$$696$$ 0 0
$$697$$ 14.7889i 0.560169i
$$698$$ 0 0
$$699$$ −1.81665 −0.0687122
$$700$$ 0 0
$$701$$ 47.0278 1.77621 0.888107 0.459637i $$-0.152020\pi$$
0.888107 + 0.459637i $$0.152020\pi$$
$$702$$ 0 0
$$703$$ 42.4222 1.59998
$$704$$ 0 0
$$705$$ −9.21110 −0.346910
$$706$$ 0 0
$$707$$ − 34.0555i − 1.28079i
$$708$$ 0 0
$$709$$ − 1.39445i − 0.0523696i −0.999657 0.0261848i $$-0.991664\pi$$
0.999657 0.0261848i $$-0.00833584\pi$$
$$710$$ 0 0
$$711$$ 14.4222 0.540875
$$712$$ 0 0
$$713$$ − 8.36669i − 0.313335i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 0 0
$$721$$ 18.4222i 0.686079i
$$722$$ 0 0
$$723$$ − 6.42221i − 0.238844i
$$724$$ 0 0
$$725$$ −4.60555 −0.171046
$$726$$ 0 0
$$727$$ 14.4222 0.534890 0.267445 0.963573i $$-0.413821\pi$$
0.267445 + 0.963573i $$0.413821\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −36.8444 −1.36274
$$732$$ 0 0
$$733$$ − 34.0555i − 1.25787i −0.777458 0.628935i $$-0.783491\pi$$
0.777458 0.628935i $$-0.216509\pi$$
$$734$$ 0 0
$$735$$ − 14.2111i − 0.524184i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 20.2389i − 0.744498i −0.928133 0.372249i $$-0.878587\pi$$
0.928133 0.372249i $$-0.121413\pi$$
$$740$$ 0 0
$$741$$ 16.6056i 0.610020i
$$742$$ 0 0
$$743$$ − 36.8444i − 1.35169i −0.737044 0.675845i $$-0.763779\pi$$
0.737044 0.675845i $$-0.236221\pi$$
$$744$$ 0 0
$$745$$ −15.2111 −0.557292
$$746$$ 0 0
$$747$$ − 2.78890i − 0.102040i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 10.4222 0.380312 0.190156 0.981754i $$-0.439101\pi$$
0.190156 + 0.981754i $$0.439101\pi$$
$$752$$ 0 0
$$753$$ 13.3944 0.488121
$$754$$ 0 0
$$755$$ 6.00000 0.218362
$$756$$ 0 0
$$757$$ 12.7889 0.464820 0.232410 0.972618i $$-0.425339\pi$$
0.232410 + 0.972618i $$0.425339\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.6333i 1.21921i 0.792707 + 0.609603i $$0.208671\pi$$
−0.792707 + 0.609603i $$0.791329\pi$$
$$762$$ 0 0
$$763$$ −6.42221 −0.232499
$$764$$ 0 0
$$765$$ 4.60555i 0.166514i
$$766$$ 0 0
$$767$$ 33.2111i 1.19918i
$$768$$ 0 0
$$769$$ − 12.8444i − 0.463181i −0.972813 0.231591i $$-0.925607\pi$$
0.972813 0.231591i $$-0.0743930\pi$$
$$770$$ 0 0
$$771$$ −28.6056 −1.03020
$$772$$ 0 0
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ 0 0
$$775$$ 6.00000i 0.215526i
$$776$$ 0 0
$$777$$ 42.4222 1.52189
$$778$$ 0 0
$$779$$ −14.7889 −0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 4.60555 0.164589
$$784$$ 0 0
$$785$$ 20.4222i 0.728900i
$$786$$ 0 0
$$787$$ − 49.2666i − 1.75617i −0.478509 0.878083i $$-0.658823\pi$$
0.478509 0.878083i $$-0.341177\pi$$
$$788$$ 0 0
$$789$$ 7.81665 0.278280
$$790$$ 0 0
$$791$$ − 63.6333i − 2.26254i
$$792$$ 0 0
$$793$$ −40.4222 −1.43543
$$794$$ 0 0
$$795$$ 6.00000i 0.212798i
$$796$$ 0 0
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 42.4222i 1.50079i
$$800$$ 0 0
$$801$$ 15.2111i 0.537458i
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −6.42221 −0.226353
$$806$$ 0 0
$$807$$ 25.8167 0.908789
$$808$$ 0 0
$$809$$ −6.84441 −0.240637 −0.120318 0.992735i $$-0.538392\pi$$
−0.120318 + 0.992735i $$0.538392\pi$$
$$810$$ 0 0
$$811$$ − 32.2389i − 1.13206i −0.824385 0.566030i $$-0.808479\pi$$
0.824385 0.566030i $$-0.191521\pi$$
$$812$$ 0 0
$$813$$ 0.422205i 0.0148074i
$$814$$ 0 0
$$815$$ 24.4222 0.855473
$$816$$ 0 0
$$817$$ − 36.8444i − 1.28902i
$$818$$ 0 0
$$819$$ 16.6056i 0.580245i
$$820$$ 0 0
$$821$$ − 3.21110i − 0.112068i −0.998429 0.0560341i $$-0.982154\pi$$
0.998429 0.0560341i $$-0.0178456\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27.6333i 0.960904i 0.877021 + 0.480452i $$0.159527\pi$$
−0.877021 + 0.480452i $$0.840473\pi$$
$$828$$ 0 0
$$829$$ 46.8444 1.62697 0.813487 0.581583i $$-0.197567\pi$$
0.813487 + 0.581583i $$0.197567\pi$$
$$830$$ 0 0
$$831$$ −16.4222 −0.569680
$$832$$ 0 0
$$833$$ −65.4500 −2.26771
$$834$$ 0 0
$$835$$ 9.21110 0.318763
$$836$$ 0 0
$$837$$ − 6.00000i − 0.207390i
$$838$$ 0 0
$$839$$ 18.4222i 0.636005i 0.948090 + 0.318003i $$0.103012\pi$$
−0.948090 + 0.318003i $$0.896988\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ 0 0
$$843$$ 27.2111i 0.937200i
$$844$$ 0 0
$$845$$ 13.0000i 0.447214i
$$846$$ 0 0
$$847$$ 50.6611i 1.74073i
$$848$$ 0 0
$$849$$ −10.4222 −0.357689
$$850$$ 0 0
$$851$$ − 12.8444i − 0.440301i
$$852$$ 0 0
$$853$$ − 14.7889i − 0.506362i −0.967419 0.253181i $$-0.918523\pi$$
0.967419 0.253181i $$-0.0814769\pi$$
$$854$$ 0 0
$$855$$ −4.60555 −0.157507
$$856$$ 0 0
$$857$$ −23.0278 −0.786613 −0.393307 0.919407i $$-0.628669\pi$$
−0.393307 + 0.919407i $$0.628669\pi$$
$$858$$ 0 0
$$859$$ −25.2111 −0.860192 −0.430096 0.902783i $$-0.641520\pi$$
−0.430096 + 0.902783i $$0.641520\pi$$
$$860$$ 0 0
$$861$$ −14.7889 −0.504004
$$862$$ 0 0
$$863$$ − 51.6333i − 1.75762i −0.477173 0.878809i $$-0.658339\pi$$
0.477173 0.878809i $$-0.341661\pi$$
$$864$$ 0 0
$$865$$ − 12.4222i − 0.422368i
$$866$$ 0 0
$$867$$ 4.21110 0.143017
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.5778i 0.392299i
$$872$$ 0 0
$$873$$ 1.39445i 0.0471949i
$$874$$ 0 0
$$875$$ 4.60555 0.155696
$$876$$ 0 0
$$877$$ − 24.8444i − 0.838936i −0.907770 0.419468i $$-0.862217\pi$$
0.907770 0.419468i $$-0.137783\pi$$
$$878$$ 0 0
$$879$$ 18.0000i 0.607125i
$$880$$ 0 0
$$881$$ 39.2111 1.32106 0.660528 0.750802i $$-0.270333\pi$$
0.660528 + 0.750802i $$0.270333\pi$$
$$882$$ 0 0
$$883$$ −9.57779 −0.322318 −0.161159 0.986928i $$-0.551523\pi$$
−0.161159 + 0.986928i $$0.551523\pi$$
$$884$$ 0 0
$$885$$ −9.21110 −0.309628
$$886$$ 0 0
$$887$$ −6.97224 −0.234105 −0.117053 0.993126i $$-0.537345\pi$$
−0.117053 + 0.993126i $$0.537345\pi$$
$$888$$ 0 0
$$889$$ − 5.57779i − 0.187073i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −42.4222 −1.41960
$$894$$ 0 0
$$895$$ 19.8167i 0.662398i
$$896$$ 0 0
$$897$$ 5.02776 0.167872
$$898$$ 0 0
$$899$$ − 27.6333i − 0.921622i
$$900$$ 0 0
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ − 36.8444i − 1.22611i
$$904$$ 0 0
$$905$$ 8.42221i 0.279964i
$$906$$ 0 0
$$907$$ −21.5778 −0.716479 −0.358239 0.933630i $$-0.616623\pi$$
−0.358239 + 0.933630i $$0.616623\pi$$
$$908$$ 0 0
$$909$$ −7.39445 −0.245258
$$910$$ 0 0
$$911$$ 27.6333 0.915532 0.457766 0.889073i $$-0.348650\pi$$
0.457766 + 0.889073i $$0.348650\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ − 11.2111i − 0.370628i
$$916$$ 0 0
$$917$$ − 104.111i − 3.43805i
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 8.78890i 0.289604i
$$922$$ 0 0
$$923$$ − 33.2111i − 1.09316i
$$924$$ 0 0
$$925$$ 9.21110i 0.302859i
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 39.2111i 1.28647i 0.765667 + 0.643237i $$0.222409\pi$$
−0.765667 + 0.643237i $$0.777591\pi$$
$$930$$ 0 0
$$931$$ − 65.4500i − 2.14504i
$$932$$ 0 0
$$933$$ −12.0000 −0.392862
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.3667 −0.338665 −0.169333 0.985559i $$-0.554161\pi$$
−0.169333 + 0.985559i $$0.554161\pi$$
$$938$$ 0 0
$$939$$ 3.57779 0.116757
$$940$$ 0 0
$$941$$ − 54.0000i − 1.76035i −0.474650 0.880175i $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 0 0
$$943$$ 4.47772i 0.145815i
$$944$$ 0 0
$$945$$ −4.60555 −0.149819
$$946$$ 0 0
$$947$$ 15.6333i 0.508014i 0.967202 + 0.254007i $$0.0817487\pi$$
−0.967202 + 0.254007i $$0.918251\pi$$
$$948$$ 0 0
$$949$$ − 5.02776i − 0.163208i
$$950$$ 0 0
$$951$$ − 18.0000i − 0.583690i
$$952$$ 0 0
$$953$$ −20.2389 −0.655601 −0.327800 0.944747i $$-0.606307\pi$$
−0.327800 + 0.944747i $$0.606307\pi$$
$$954$$ 0 0
$$955$$ 12.0000i 0.388311i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −14.7889 −0.477558
$$960$$ 0 0
$$961$$ −5.00000 −0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −7.81665 −0.251627
$$966$$ 0 0
$$967$$ 8.23886i 0.264944i 0.991187 + 0.132472i $$0.0422914\pi$$
−0.991187 + 0.132472i $$0.957709\pi$$
$$968$$ 0 0
$$969$$ 21.2111i 0.681399i
$$970$$ 0 0
$$971$$ 53.0278 1.70174 0.850871 0.525375i $$-0.176075\pi$$
0.850871 + 0.525375i $$0.176075\pi$$
$$972$$ 0 0
$$973$$ − 79.2666i − 2.54117i
$$974$$ 0 0
$$975$$ −3.60555 −0.115470
$$976$$ 0 0
$$977$$ 18.8444i 0.602886i 0.953484 + 0.301443i $$0.0974683\pi$$
−0.953484 + 0.301443i $$0.902532\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1.39445i 0.0445213i
$$982$$ 0 0
$$983$$ 42.4222i 1.35306i 0.736416 + 0.676529i $$0.236517\pi$$
−0.736416 + 0.676529i $$0.763483\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ −42.4222 −1.35031
$$988$$ 0 0
$$989$$ −11.1556 −0.354727
$$990$$ 0 0
$$991$$ 22.4222 0.712265 0.356132 0.934436i $$-0.384095\pi$$
0.356132 + 0.934436i $$0.384095\pi$$
$$992$$ 0 0
$$993$$ 16.6056i 0.526961i
$$994$$ 0 0
$$995$$ − 22.4222i − 0.710832i
$$996$$ 0 0
$$997$$ −16.4222 −0.520096 −0.260048 0.965596i $$-0.583738\pi$$
−0.260048 + 0.965596i $$0.583738\pi$$
$$998$$ 0 0
$$999$$ − 9.21110i − 0.291426i
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 3120.2.g.q.961.4 4
4.3 odd 2 390.2.b.c.181.3 yes 4
12.11 even 2 1170.2.b.d.181.1 4
13.12 even 2 inner 3120.2.g.q.961.1 4
20.3 even 4 1950.2.f.n.649.1 4
20.7 even 4 1950.2.f.m.649.4 4
20.19 odd 2 1950.2.b.k.1351.2 4
52.31 even 4 5070.2.a.bf.1.2 2
52.47 even 4 5070.2.a.z.1.1 2
52.51 odd 2 390.2.b.c.181.2 4
156.155 even 2 1170.2.b.d.181.4 4
260.103 even 4 1950.2.f.m.649.2 4
260.207 even 4 1950.2.f.n.649.3 4
260.259 odd 2 1950.2.b.k.1351.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 52.51 odd 2
390.2.b.c.181.3 yes 4 4.3 odd 2
1170.2.b.d.181.1 4 12.11 even 2
1170.2.b.d.181.4 4 156.155 even 2
1950.2.b.k.1351.2 4 20.19 odd 2
1950.2.b.k.1351.3 4 260.259 odd 2
1950.2.f.m.649.2 4 260.103 even 4
1950.2.f.m.649.4 4 20.7 even 4
1950.2.f.n.649.1 4 20.3 even 4
1950.2.f.n.649.3 4 260.207 even 4
3120.2.g.q.961.1 4 13.12 even 2 inner
3120.2.g.q.961.4 4 1.1 even 1 trivial
5070.2.a.z.1.1 2 52.47 even 4
5070.2.a.bf.1.2 2 52.31 even 4