# Properties

 Label 6240.2.a.bo.1.1 Level $6240$ Weight $2$ Character 6240.1 Self dual yes Analytic conductor $49.827$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6240.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$49.8266508613$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 6240.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -4.37228 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -4.37228 q^{7} +1.00000 q^{9} +4.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.372281 q^{17} -2.00000 q^{19} -4.37228 q^{21} -2.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} +4.37228 q^{33} +4.37228 q^{35} +4.37228 q^{37} -1.00000 q^{39} -0.372281 q^{41} +4.74456 q^{43} -1.00000 q^{45} +2.74456 q^{47} +12.1168 q^{49} +0.372281 q^{51} -7.62772 q^{53} -4.37228 q^{55} -2.00000 q^{57} -2.74456 q^{59} -9.11684 q^{61} -4.37228 q^{63} +1.00000 q^{65} +1.25544 q^{67} -2.37228 q^{69} +4.37228 q^{71} -6.00000 q^{73} +1.00000 q^{75} -19.1168 q^{77} -9.62772 q^{79} +1.00000 q^{81} -14.7446 q^{83} -0.372281 q^{85} -2.74456 q^{87} -13.1168 q^{89} +4.37228 q^{91} +6.74456 q^{93} +2.00000 q^{95} +9.11684 q^{97} +4.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} + 3 q^{33} + 3 q^{35} + 3 q^{37} - 2 q^{39} + 5 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{47} + 7 q^{49} - 5 q^{51} - 21 q^{53} - 3 q^{55} - 4 q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 2 q^{65} + 14 q^{67} + q^{69} + 3 q^{71} - 12 q^{73} + 2 q^{75} - 21 q^{77} - 25 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{85} + 6 q^{87} - 9 q^{89} + 3 q^{91} + 2 q^{93} + 4 q^{95} + q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 4 * q^19 - 3 * q^21 + q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 + 3 * q^33 + 3 * q^35 + 3 * q^37 - 2 * q^39 + 5 * q^41 - 2 * q^43 - 2 * q^45 - 6 * q^47 + 7 * q^49 - 5 * q^51 - 21 * q^53 - 3 * q^55 - 4 * q^57 + 6 * q^59 - q^61 - 3 * q^63 + 2 * q^65 + 14 * q^67 + q^69 + 3 * q^71 - 12 * q^73 + 2 * q^75 - 21 * q^77 - 25 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^85 + 6 * q^87 - 9 * q^89 + 3 * q^91 + 2 * q^93 + 4 * q^95 + q^97 + 3 * q^99

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.37228 −1.65257 −0.826284 0.563254i $$-0.809549\pi$$
−0.826284 + 0.563254i $$0.809549\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.37228 1.31829 0.659146 0.752015i $$-0.270918\pi$$
0.659146 + 0.752015i $$0.270918\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 0.372281 0.0902915 0.0451457 0.998980i $$-0.485625\pi$$
0.0451457 + 0.998980i $$0.485625\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −4.37228 −0.954110
$$22$$ 0 0
$$23$$ −2.37228 −0.494655 −0.247327 0.968932i $$-0.579552\pi$$
−0.247327 + 0.968932i $$0.579552\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.74456 −0.509652 −0.254826 0.966987i $$-0.582018\pi$$
−0.254826 + 0.966987i $$0.582018\pi$$
$$30$$ 0 0
$$31$$ 6.74456 1.21136 0.605680 0.795709i $$-0.292901\pi$$
0.605680 + 0.795709i $$0.292901\pi$$
$$32$$ 0 0
$$33$$ 4.37228 0.761116
$$34$$ 0 0
$$35$$ 4.37228 0.739050
$$36$$ 0 0
$$37$$ 4.37228 0.718799 0.359399 0.933184i $$-0.382982\pi$$
0.359399 + 0.933184i $$0.382982\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −0.372281 −0.0581406 −0.0290703 0.999577i $$-0.509255\pi$$
−0.0290703 + 0.999577i $$0.509255\pi$$
$$42$$ 0 0
$$43$$ 4.74456 0.723539 0.361770 0.932268i $$-0.382173\pi$$
0.361770 + 0.932268i $$0.382173\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 2.74456 0.400336 0.200168 0.979762i $$-0.435851\pi$$
0.200168 + 0.979762i $$0.435851\pi$$
$$48$$ 0 0
$$49$$ 12.1168 1.73098
$$50$$ 0 0
$$51$$ 0.372281 0.0521298
$$52$$ 0 0
$$53$$ −7.62772 −1.04775 −0.523874 0.851796i $$-0.675514\pi$$
−0.523874 + 0.851796i $$0.675514\pi$$
$$54$$ 0 0
$$55$$ −4.37228 −0.589558
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −2.74456 −0.357312 −0.178656 0.983912i $$-0.557175\pi$$
−0.178656 + 0.983912i $$0.557175\pi$$
$$60$$ 0 0
$$61$$ −9.11684 −1.16729 −0.583646 0.812008i $$-0.698374\pi$$
−0.583646 + 0.812008i $$0.698374\pi$$
$$62$$ 0 0
$$63$$ −4.37228 −0.550856
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 1.25544 0.153376 0.0766880 0.997055i $$-0.475565\pi$$
0.0766880 + 0.997055i $$0.475565\pi$$
$$68$$ 0 0
$$69$$ −2.37228 −0.285589
$$70$$ 0 0
$$71$$ 4.37228 0.518894 0.259447 0.965757i $$-0.416460\pi$$
0.259447 + 0.965757i $$0.416460\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −19.1168 −2.17857
$$78$$ 0 0
$$79$$ −9.62772 −1.08320 −0.541601 0.840635i $$-0.682182\pi$$
−0.541601 + 0.840635i $$0.682182\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −14.7446 −1.61843 −0.809213 0.587515i $$-0.800106\pi$$
−0.809213 + 0.587515i $$0.800106\pi$$
$$84$$ 0 0
$$85$$ −0.372281 −0.0403796
$$86$$ 0 0
$$87$$ −2.74456 −0.294248
$$88$$ 0 0
$$89$$ −13.1168 −1.39038 −0.695191 0.718825i $$-0.744680\pi$$
−0.695191 + 0.718825i $$0.744680\pi$$
$$90$$ 0 0
$$91$$ 4.37228 0.458340
$$92$$ 0 0
$$93$$ 6.74456 0.699379
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 9.11684 0.925675 0.462838 0.886443i $$-0.346831\pi$$
0.462838 + 0.886443i $$0.346831\pi$$
$$98$$ 0 0
$$99$$ 4.37228 0.439431
$$100$$ 0 0
$$101$$ −5.25544 −0.522936 −0.261468 0.965212i $$-0.584207\pi$$
−0.261468 + 0.965212i $$0.584207\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 4.37228 0.426691
$$106$$ 0 0
$$107$$ 1.62772 0.157358 0.0786788 0.996900i $$-0.474930\pi$$
0.0786788 + 0.996900i $$0.474930\pi$$
$$108$$ 0 0
$$109$$ −9.25544 −0.886510 −0.443255 0.896396i $$-0.646176\pi$$
−0.443255 + 0.896396i $$0.646176\pi$$
$$110$$ 0 0
$$111$$ 4.37228 0.414999
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 2.37228 0.221216
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −1.62772 −0.149213
$$120$$ 0 0
$$121$$ 8.11684 0.737895
$$122$$ 0 0
$$123$$ −0.372281 −0.0335675
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 4.74456 0.417735
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 8.74456 0.758250
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −7.48913 −0.639839 −0.319920 0.947445i $$-0.603656\pi$$
−0.319920 + 0.947445i $$0.603656\pi$$
$$138$$ 0 0
$$139$$ 3.11684 0.264367 0.132184 0.991225i $$-0.457801\pi$$
0.132184 + 0.991225i $$0.457801\pi$$
$$140$$ 0 0
$$141$$ 2.74456 0.231134
$$142$$ 0 0
$$143$$ −4.37228 −0.365629
$$144$$ 0 0
$$145$$ 2.74456 0.227924
$$146$$ 0 0
$$147$$ 12.1168 0.999380
$$148$$ 0 0
$$149$$ −13.1168 −1.07457 −0.537287 0.843400i $$-0.680551\pi$$
−0.537287 + 0.843400i $$0.680551\pi$$
$$150$$ 0 0
$$151$$ 20.2337 1.64659 0.823297 0.567611i $$-0.192132\pi$$
0.823297 + 0.567611i $$0.192132\pi$$
$$152$$ 0 0
$$153$$ 0.372281 0.0300972
$$154$$ 0 0
$$155$$ −6.74456 −0.541736
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −7.62772 −0.604917
$$160$$ 0 0
$$161$$ 10.3723 0.817450
$$162$$ 0 0
$$163$$ −17.8614 −1.39901 −0.699507 0.714626i $$-0.746597\pi$$
−0.699507 + 0.714626i $$0.746597\pi$$
$$164$$ 0 0
$$165$$ −4.37228 −0.340382
$$166$$ 0 0
$$167$$ 2.74456 0.212381 0.106190 0.994346i $$-0.466135\pi$$
0.106190 + 0.994346i $$0.466135\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −11.4891 −0.873502 −0.436751 0.899582i $$-0.643871\pi$$
−0.436751 + 0.899582i $$0.643871\pi$$
$$174$$ 0 0
$$175$$ −4.37228 −0.330513
$$176$$ 0 0
$$177$$ −2.74456 −0.206294
$$178$$ 0 0
$$179$$ 3.25544 0.243323 0.121661 0.992572i $$-0.461178\pi$$
0.121661 + 0.992572i $$0.461178\pi$$
$$180$$ 0 0
$$181$$ −5.11684 −0.380332 −0.190166 0.981752i $$-0.560903\pi$$
−0.190166 + 0.981752i $$0.560903\pi$$
$$182$$ 0 0
$$183$$ −9.11684 −0.673936
$$184$$ 0 0
$$185$$ −4.37228 −0.321457
$$186$$ 0 0
$$187$$ 1.62772 0.119031
$$188$$ 0 0
$$189$$ −4.37228 −0.318037
$$190$$ 0 0
$$191$$ 8.74456 0.632734 0.316367 0.948637i $$-0.397537\pi$$
0.316367 + 0.948637i $$0.397537\pi$$
$$192$$ 0 0
$$193$$ −21.1168 −1.52002 −0.760012 0.649909i $$-0.774807\pi$$
−0.760012 + 0.649909i $$0.774807\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 3.48913 0.248590 0.124295 0.992245i $$-0.460333\pi$$
0.124295 + 0.992245i $$0.460333\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 1.25544 0.0885517
$$202$$ 0 0
$$203$$ 12.0000 0.842235
$$204$$ 0 0
$$205$$ 0.372281 0.0260013
$$206$$ 0 0
$$207$$ −2.37228 −0.164885
$$208$$ 0 0
$$209$$ −8.74456 −0.604874
$$210$$ 0 0
$$211$$ −21.4891 −1.47937 −0.739686 0.672952i $$-0.765026\pi$$
−0.739686 + 0.672952i $$0.765026\pi$$
$$212$$ 0 0
$$213$$ 4.37228 0.299584
$$214$$ 0 0
$$215$$ −4.74456 −0.323576
$$216$$ 0 0
$$217$$ −29.4891 −2.00185
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ −0.372281 −0.0250424
$$222$$ 0 0
$$223$$ −20.2337 −1.35495 −0.677474 0.735547i $$-0.736925\pi$$
−0.677474 + 0.735547i $$0.736925\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 0 0
$$229$$ 24.2337 1.60141 0.800704 0.599061i $$-0.204459\pi$$
0.800704 + 0.599061i $$0.204459\pi$$
$$230$$ 0 0
$$231$$ −19.1168 −1.25780
$$232$$ 0 0
$$233$$ −16.3723 −1.07258 −0.536292 0.844033i $$-0.680175\pi$$
−0.536292 + 0.844033i $$0.680175\pi$$
$$234$$ 0 0
$$235$$ −2.74456 −0.179036
$$236$$ 0 0
$$237$$ −9.62772 −0.625388
$$238$$ 0 0
$$239$$ −14.6060 −0.944782 −0.472391 0.881389i $$-0.656609\pi$$
−0.472391 + 0.881389i $$0.656609\pi$$
$$240$$ 0 0
$$241$$ 6.74456 0.434455 0.217228 0.976121i $$-0.430299\pi$$
0.217228 + 0.976121i $$0.430299\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −12.1168 −0.774117
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ −14.7446 −0.934399
$$250$$ 0 0
$$251$$ 18.9783 1.19790 0.598948 0.800788i $$-0.295585\pi$$
0.598948 + 0.800788i $$0.295585\pi$$
$$252$$ 0 0
$$253$$ −10.3723 −0.652100
$$254$$ 0 0
$$255$$ −0.372281 −0.0233132
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ −19.1168 −1.18786
$$260$$ 0 0
$$261$$ −2.74456 −0.169884
$$262$$ 0 0
$$263$$ −2.51087 −0.154827 −0.0774136 0.996999i $$-0.524666\pi$$
−0.0774136 + 0.996999i $$0.524666\pi$$
$$264$$ 0 0
$$265$$ 7.62772 0.468567
$$266$$ 0 0
$$267$$ −13.1168 −0.802738
$$268$$ 0 0
$$269$$ 25.7228 1.56835 0.784174 0.620541i $$-0.213087\pi$$
0.784174 + 0.620541i $$0.213087\pi$$
$$270$$ 0 0
$$271$$ −3.48913 −0.211949 −0.105975 0.994369i $$-0.533796\pi$$
−0.105975 + 0.994369i $$0.533796\pi$$
$$272$$ 0 0
$$273$$ 4.37228 0.264623
$$274$$ 0 0
$$275$$ 4.37228 0.263658
$$276$$ 0 0
$$277$$ −21.2554 −1.27712 −0.638558 0.769574i $$-0.720469\pi$$
−0.638558 + 0.769574i $$0.720469\pi$$
$$278$$ 0 0
$$279$$ 6.74456 0.403786
$$280$$ 0 0
$$281$$ 3.48913 0.208144 0.104072 0.994570i $$-0.466813\pi$$
0.104072 + 0.994570i $$0.466813\pi$$
$$282$$ 0 0
$$283$$ −25.4891 −1.51517 −0.757586 0.652736i $$-0.773621\pi$$
−0.757586 + 0.652736i $$0.773621\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 1.62772 0.0960812
$$288$$ 0 0
$$289$$ −16.8614 −0.991847
$$290$$ 0 0
$$291$$ 9.11684 0.534439
$$292$$ 0 0
$$293$$ −2.74456 −0.160339 −0.0801695 0.996781i $$-0.525546\pi$$
−0.0801695 + 0.996781i $$0.525546\pi$$
$$294$$ 0 0
$$295$$ 2.74456 0.159795
$$296$$ 0 0
$$297$$ 4.37228 0.253705
$$298$$ 0 0
$$299$$ 2.37228 0.137193
$$300$$ 0 0
$$301$$ −20.7446 −1.19570
$$302$$ 0 0
$$303$$ −5.25544 −0.301917
$$304$$ 0 0
$$305$$ 9.11684 0.522029
$$306$$ 0 0
$$307$$ 9.11684 0.520326 0.260163 0.965565i $$-0.416224\pi$$
0.260163 + 0.965565i $$0.416224\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −31.7228 −1.79884 −0.899418 0.437090i $$-0.856009\pi$$
−0.899418 + 0.437090i $$0.856009\pi$$
$$312$$ 0 0
$$313$$ −28.9783 −1.63795 −0.818974 0.573831i $$-0.805457\pi$$
−0.818974 + 0.573831i $$0.805457\pi$$
$$314$$ 0 0
$$315$$ 4.37228 0.246350
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 1.62772 0.0908504
$$322$$ 0 0
$$323$$ −0.744563 −0.0414286
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −9.25544 −0.511827
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 22.0000 1.20923 0.604615 0.796518i $$-0.293327\pi$$
0.604615 + 0.796518i $$0.293327\pi$$
$$332$$ 0 0
$$333$$ 4.37228 0.239600
$$334$$ 0 0
$$335$$ −1.25544 −0.0685919
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 29.4891 1.59693
$$342$$ 0 0
$$343$$ −22.3723 −1.20799
$$344$$ 0 0
$$345$$ 2.37228 0.127719
$$346$$ 0 0
$$347$$ 35.1168 1.88517 0.942585 0.333965i $$-0.108387\pi$$
0.942585 + 0.333965i $$0.108387\pi$$
$$348$$ 0 0
$$349$$ 20.9783 1.12294 0.561470 0.827497i $$-0.310236\pi$$
0.561470 + 0.827497i $$0.310236\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −9.25544 −0.492617 −0.246309 0.969191i $$-0.579218\pi$$
−0.246309 + 0.969191i $$0.579218\pi$$
$$354$$ 0 0
$$355$$ −4.37228 −0.232057
$$356$$ 0 0
$$357$$ −1.62772 −0.0861480
$$358$$ 0 0
$$359$$ 9.25544 0.488483 0.244242 0.969714i $$-0.421461\pi$$
0.244242 + 0.969714i $$0.421461\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 8.11684 0.426024
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 24.7446 1.29166 0.645828 0.763483i $$-0.276512\pi$$
0.645828 + 0.763483i $$0.276512\pi$$
$$368$$ 0 0
$$369$$ −0.372281 −0.0193802
$$370$$ 0 0
$$371$$ 33.3505 1.73147
$$372$$ 0 0
$$373$$ −15.4891 −0.801997 −0.400998 0.916079i $$-0.631337\pi$$
−0.400998 + 0.916079i $$0.631337\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 2.74456 0.141352
$$378$$ 0 0
$$379$$ −8.23369 −0.422936 −0.211468 0.977385i $$-0.567824\pi$$
−0.211468 + 0.977385i $$0.567824\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 27.4891 1.40463 0.702314 0.711867i $$-0.252150\pi$$
0.702314 + 0.711867i $$0.252150\pi$$
$$384$$ 0 0
$$385$$ 19.1168 0.974285
$$386$$ 0 0
$$387$$ 4.74456 0.241180
$$388$$ 0 0
$$389$$ −17.2554 −0.874885 −0.437443 0.899246i $$-0.644116\pi$$
−0.437443 + 0.899246i $$0.644116\pi$$
$$390$$ 0 0
$$391$$ −0.883156 −0.0446631
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 9.62772 0.484423
$$396$$ 0 0
$$397$$ −14.6060 −0.733053 −0.366526 0.930408i $$-0.619453\pi$$
−0.366526 + 0.930408i $$0.619453\pi$$
$$398$$ 0 0
$$399$$ 8.74456 0.437776
$$400$$ 0 0
$$401$$ 11.4891 0.573740 0.286870 0.957970i $$-0.407385\pi$$
0.286870 + 0.957970i $$0.407385\pi$$
$$402$$ 0 0
$$403$$ −6.74456 −0.335971
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 19.1168 0.947587
$$408$$ 0 0
$$409$$ −28.2337 −1.39607 −0.698033 0.716066i $$-0.745941\pi$$
−0.698033 + 0.716066i $$0.745941\pi$$
$$410$$ 0 0
$$411$$ −7.48913 −0.369411
$$412$$ 0 0
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 14.7446 0.723782
$$416$$ 0 0
$$417$$ 3.11684 0.152633
$$418$$ 0 0
$$419$$ −0.744563 −0.0363743 −0.0181871 0.999835i $$-0.505789\pi$$
−0.0181871 + 0.999835i $$0.505789\pi$$
$$420$$ 0 0
$$421$$ 13.2554 0.646030 0.323015 0.946394i $$-0.395303\pi$$
0.323015 + 0.946394i $$0.395303\pi$$
$$422$$ 0 0
$$423$$ 2.74456 0.133445
$$424$$ 0 0
$$425$$ 0.372281 0.0180583
$$426$$ 0 0
$$427$$ 39.8614 1.92903
$$428$$ 0 0
$$429$$ −4.37228 −0.211096
$$430$$ 0 0
$$431$$ −20.2337 −0.974622 −0.487311 0.873228i $$-0.662022\pi$$
−0.487311 + 0.873228i $$0.662022\pi$$
$$432$$ 0 0
$$433$$ −23.4891 −1.12882 −0.564408 0.825496i $$-0.690895\pi$$
−0.564408 + 0.825496i $$0.690895\pi$$
$$434$$ 0 0
$$435$$ 2.74456 0.131592
$$436$$ 0 0
$$437$$ 4.74456 0.226963
$$438$$ 0 0
$$439$$ 7.11684 0.339668 0.169834 0.985473i $$-0.445677\pi$$
0.169834 + 0.985473i $$0.445677\pi$$
$$440$$ 0 0
$$441$$ 12.1168 0.576993
$$442$$ 0 0
$$443$$ −15.8614 −0.753598 −0.376799 0.926295i $$-0.622975\pi$$
−0.376799 + 0.926295i $$0.622975\pi$$
$$444$$ 0 0
$$445$$ 13.1168 0.621798
$$446$$ 0 0
$$447$$ −13.1168 −0.620405
$$448$$ 0 0
$$449$$ 1.11684 0.0527071 0.0263536 0.999653i $$-0.491610\pi$$
0.0263536 + 0.999653i $$0.491610\pi$$
$$450$$ 0 0
$$451$$ −1.62772 −0.0766463
$$452$$ 0 0
$$453$$ 20.2337 0.950662
$$454$$ 0 0
$$455$$ −4.37228 −0.204976
$$456$$ 0 0
$$457$$ −8.37228 −0.391639 −0.195819 0.980640i $$-0.562737\pi$$
−0.195819 + 0.980640i $$0.562737\pi$$
$$458$$ 0 0
$$459$$ 0.372281 0.0173766
$$460$$ 0 0
$$461$$ −9.86141 −0.459291 −0.229646 0.973274i $$-0.573757\pi$$
−0.229646 + 0.973274i $$0.573757\pi$$
$$462$$ 0 0
$$463$$ −1.39403 −0.0647861 −0.0323931 0.999475i $$-0.510313\pi$$
−0.0323931 + 0.999475i $$0.510313\pi$$
$$464$$ 0 0
$$465$$ −6.74456 −0.312772
$$466$$ 0 0
$$467$$ −2.37228 −0.109776 −0.0548880 0.998493i $$-0.517480\pi$$
−0.0548880 + 0.998493i $$0.517480\pi$$
$$468$$ 0 0
$$469$$ −5.48913 −0.253464
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 20.7446 0.953836
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ −7.62772 −0.349249
$$478$$ 0 0
$$479$$ 40.0951 1.83199 0.915996 0.401188i $$-0.131403\pi$$
0.915996 + 0.401188i $$0.131403\pi$$
$$480$$ 0 0
$$481$$ −4.37228 −0.199359
$$482$$ 0 0
$$483$$ 10.3723 0.471955
$$484$$ 0 0
$$485$$ −9.11684 −0.413975
$$486$$ 0 0
$$487$$ 32.8397 1.48811 0.744053 0.668120i $$-0.232901\pi$$
0.744053 + 0.668120i $$0.232901\pi$$
$$488$$ 0 0
$$489$$ −17.8614 −0.807721
$$490$$ 0 0
$$491$$ 4.74456 0.214119 0.107060 0.994253i $$-0.465856\pi$$
0.107060 + 0.994253i $$0.465856\pi$$
$$492$$ 0 0
$$493$$ −1.02175 −0.0460173
$$494$$ 0 0
$$495$$ −4.37228 −0.196519
$$496$$ 0 0
$$497$$ −19.1168 −0.857508
$$498$$ 0 0
$$499$$ 8.97825 0.401922 0.200961 0.979599i $$-0.435594\pi$$
0.200961 + 0.979599i $$0.435594\pi$$
$$500$$ 0 0
$$501$$ 2.74456 0.122618
$$502$$ 0 0
$$503$$ 18.5109 0.825359 0.412680 0.910876i $$-0.364593\pi$$
0.412680 + 0.910876i $$0.364593\pi$$
$$504$$ 0 0
$$505$$ 5.25544 0.233864
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −37.1168 −1.64518 −0.822588 0.568638i $$-0.807470\pi$$
−0.822588 + 0.568638i $$0.807470\pi$$
$$510$$ 0 0
$$511$$ 26.2337 1.16051
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ −11.4891 −0.504317
$$520$$ 0 0
$$521$$ 31.4891 1.37956 0.689782 0.724017i $$-0.257706\pi$$
0.689782 + 0.724017i $$0.257706\pi$$
$$522$$ 0 0
$$523$$ 43.7228 1.91187 0.955933 0.293586i $$-0.0948488\pi$$
0.955933 + 0.293586i $$0.0948488\pi$$
$$524$$ 0 0
$$525$$ −4.37228 −0.190822
$$526$$ 0 0
$$527$$ 2.51087 0.109375
$$528$$ 0 0
$$529$$ −17.3723 −0.755317
$$530$$ 0 0
$$531$$ −2.74456 −0.119104
$$532$$ 0 0
$$533$$ 0.372281 0.0161253
$$534$$ 0 0
$$535$$ −1.62772 −0.0703724
$$536$$ 0 0
$$537$$ 3.25544 0.140482
$$538$$ 0 0
$$539$$ 52.9783 2.28193
$$540$$ 0 0
$$541$$ −15.4891 −0.665930 −0.332965 0.942939i $$-0.608049\pi$$
−0.332965 + 0.942939i $$0.608049\pi$$
$$542$$ 0 0
$$543$$ −5.11684 −0.219585
$$544$$ 0 0
$$545$$ 9.25544 0.396459
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 0 0
$$549$$ −9.11684 −0.389097
$$550$$ 0 0
$$551$$ 5.48913 0.233845
$$552$$ 0 0
$$553$$ 42.0951 1.79007
$$554$$ 0 0
$$555$$ −4.37228 −0.185593
$$556$$ 0 0
$$557$$ 14.7446 0.624747 0.312374 0.949959i $$-0.398876\pi$$
0.312374 + 0.949959i $$0.398876\pi$$
$$558$$ 0 0
$$559$$ −4.74456 −0.200674
$$560$$ 0 0
$$561$$ 1.62772 0.0687223
$$562$$ 0 0
$$563$$ 31.1168 1.31142 0.655709 0.755013i $$-0.272370\pi$$
0.655709 + 0.755013i $$0.272370\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ −4.37228 −0.183619
$$568$$ 0 0
$$569$$ −8.97825 −0.376388 −0.188194 0.982132i $$-0.560263\pi$$
−0.188194 + 0.982132i $$0.560263\pi$$
$$570$$ 0 0
$$571$$ −16.6060 −0.694938 −0.347469 0.937691i $$-0.612959\pi$$
−0.347469 + 0.937691i $$0.612959\pi$$
$$572$$ 0 0
$$573$$ 8.74456 0.365309
$$574$$ 0 0
$$575$$ −2.37228 −0.0989310
$$576$$ 0 0
$$577$$ −9.86141 −0.410536 −0.205268 0.978706i $$-0.565807\pi$$
−0.205268 + 0.978706i $$0.565807\pi$$
$$578$$ 0 0
$$579$$ −21.1168 −0.877586
$$580$$ 0 0
$$581$$ 64.4674 2.67456
$$582$$ 0 0
$$583$$ −33.3505 −1.38124
$$584$$ 0 0
$$585$$ 1.00000 0.0413449
$$586$$ 0 0
$$587$$ 20.5109 0.846574 0.423287 0.905996i $$-0.360876\pi$$
0.423287 + 0.905996i $$0.360876\pi$$
$$588$$ 0 0
$$589$$ −13.4891 −0.555810
$$590$$ 0 0
$$591$$ 3.48913 0.143523
$$592$$ 0 0
$$593$$ −29.7228 −1.22057 −0.610285 0.792182i $$-0.708945\pi$$
−0.610285 + 0.792182i $$0.708945\pi$$
$$594$$ 0 0
$$595$$ 1.62772 0.0667300
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −13.8614 −0.565419 −0.282709 0.959206i $$-0.591233\pi$$
−0.282709 + 0.959206i $$0.591233\pi$$
$$602$$ 0 0
$$603$$ 1.25544 0.0511254
$$604$$ 0 0
$$605$$ −8.11684 −0.329997
$$606$$ 0 0
$$607$$ −43.7228 −1.77465 −0.887327 0.461141i $$-0.847440\pi$$
−0.887327 + 0.461141i $$0.847440\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ −2.74456 −0.111033
$$612$$ 0 0
$$613$$ −3.35053 −0.135327 −0.0676634 0.997708i $$-0.521554\pi$$
−0.0676634 + 0.997708i $$0.521554\pi$$
$$614$$ 0 0
$$615$$ 0.372281 0.0150118
$$616$$ 0 0
$$617$$ 22.4674 0.904502 0.452251 0.891891i $$-0.350621\pi$$
0.452251 + 0.891891i $$0.350621\pi$$
$$618$$ 0 0
$$619$$ −1.25544 −0.0504603 −0.0252301 0.999682i $$-0.508032\pi$$
−0.0252301 + 0.999682i $$0.508032\pi$$
$$620$$ 0 0
$$621$$ −2.37228 −0.0951964
$$622$$ 0 0
$$623$$ 57.3505 2.29770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −8.74456 −0.349224
$$628$$ 0 0
$$629$$ 1.62772 0.0649014
$$630$$ 0 0
$$631$$ 36.2337 1.44244 0.721220 0.692706i $$-0.243582\pi$$
0.721220 + 0.692706i $$0.243582\pi$$
$$632$$ 0 0
$$633$$ −21.4891 −0.854116
$$634$$ 0 0
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ −12.1168 −0.480087
$$638$$ 0 0
$$639$$ 4.37228 0.172965
$$640$$ 0 0
$$641$$ 40.9783 1.61854 0.809272 0.587434i $$-0.199862\pi$$
0.809272 + 0.587434i $$0.199862\pi$$
$$642$$ 0 0
$$643$$ −2.88316 −0.113701 −0.0568503 0.998383i $$-0.518106\pi$$
−0.0568503 + 0.998383i $$0.518106\pi$$
$$644$$ 0 0
$$645$$ −4.74456 −0.186817
$$646$$ 0 0
$$647$$ −24.6060 −0.967360 −0.483680 0.875245i $$-0.660700\pi$$
−0.483680 + 0.875245i $$0.660700\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ −29.4891 −1.15577
$$652$$ 0 0
$$653$$ −4.51087 −0.176524 −0.0882621 0.996097i $$-0.528131\pi$$
−0.0882621 + 0.996097i $$0.528131\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ −22.9783 −0.895106 −0.447553 0.894258i $$-0.647704\pi$$
−0.447553 + 0.894258i $$0.647704\pi$$
$$660$$ 0 0
$$661$$ 16.5109 0.642199 0.321099 0.947046i $$-0.395948\pi$$
0.321099 + 0.947046i $$0.395948\pi$$
$$662$$ 0 0
$$663$$ −0.372281 −0.0144582
$$664$$ 0 0
$$665$$ −8.74456 −0.339100
$$666$$ 0 0
$$667$$ 6.51087 0.252102
$$668$$ 0 0
$$669$$ −20.2337 −0.782280
$$670$$ 0 0
$$671$$ −39.8614 −1.53883
$$672$$ 0 0
$$673$$ 4.23369 0.163197 0.0815983 0.996665i $$-0.473998\pi$$
0.0815983 + 0.996665i $$0.473998\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 1.86141 0.0715397 0.0357698 0.999360i $$-0.488612\pi$$
0.0357698 + 0.999360i $$0.488612\pi$$
$$678$$ 0 0
$$679$$ −39.8614 −1.52974
$$680$$ 0 0
$$681$$ 14.0000 0.536481
$$682$$ 0 0
$$683$$ 2.00000 0.0765279 0.0382639 0.999268i $$-0.487817\pi$$
0.0382639 + 0.999268i $$0.487817\pi$$
$$684$$ 0 0
$$685$$ 7.48913 0.286145
$$686$$ 0 0
$$687$$ 24.2337 0.924573
$$688$$ 0 0
$$689$$ 7.62772 0.290593
$$690$$ 0 0
$$691$$ −28.9783 −1.10238 −0.551192 0.834378i $$-0.685827\pi$$
−0.551192 + 0.834378i $$0.685827\pi$$
$$692$$ 0 0
$$693$$ −19.1168 −0.726189
$$694$$ 0 0
$$695$$ −3.11684 −0.118229
$$696$$ 0 0
$$697$$ −0.138593 −0.00524960
$$698$$ 0 0
$$699$$ −16.3723 −0.619257
$$700$$ 0 0
$$701$$ −28.2337 −1.06637 −0.533186 0.845998i $$-0.679005\pi$$
−0.533186 + 0.845998i $$0.679005\pi$$
$$702$$ 0 0
$$703$$ −8.74456 −0.329807
$$704$$ 0 0
$$705$$ −2.74456 −0.103366
$$706$$ 0 0
$$707$$ 22.9783 0.864186
$$708$$ 0 0
$$709$$ 35.2119 1.32241 0.661206 0.750204i $$-0.270045\pi$$
0.661206 + 0.750204i $$0.270045\pi$$
$$710$$ 0 0
$$711$$ −9.62772 −0.361068
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 4.37228 0.163514
$$716$$ 0 0
$$717$$ −14.6060 −0.545470
$$718$$ 0 0
$$719$$ 20.4674 0.763304 0.381652 0.924306i $$-0.375355\pi$$
0.381652 + 0.924306i $$0.375355\pi$$
$$720$$ 0 0
$$721$$ 34.9783 1.30266
$$722$$ 0 0
$$723$$ 6.74456 0.250833
$$724$$ 0 0
$$725$$ −2.74456 −0.101930
$$726$$ 0 0
$$727$$ −30.2337 −1.12131 −0.560653 0.828051i $$-0.689450\pi$$
−0.560653 + 0.828051i $$0.689450\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.76631 0.0653294
$$732$$ 0 0
$$733$$ −6.60597 −0.243997 −0.121999 0.992530i $$-0.538930\pi$$
−0.121999 + 0.992530i $$0.538930\pi$$
$$734$$ 0 0
$$735$$ −12.1168 −0.446937
$$736$$ 0 0
$$737$$ 5.48913 0.202195
$$738$$ 0 0
$$739$$ 47.9565 1.76411 0.882054 0.471148i $$-0.156160\pi$$
0.882054 + 0.471148i $$0.156160\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ −38.0000 −1.39408 −0.697042 0.717030i $$-0.745501\pi$$
−0.697042 + 0.717030i $$0.745501\pi$$
$$744$$ 0 0
$$745$$ 13.1168 0.480564
$$746$$ 0 0
$$747$$ −14.7446 −0.539475
$$748$$ 0 0
$$749$$ −7.11684 −0.260044
$$750$$ 0 0
$$751$$ −38.8397 −1.41728 −0.708640 0.705571i $$-0.750691\pi$$
−0.708640 + 0.705571i $$0.750691\pi$$
$$752$$ 0 0
$$753$$ 18.9783 0.691606
$$754$$ 0 0
$$755$$ −20.2337 −0.736379
$$756$$ 0 0
$$757$$ 2.74456 0.0997528 0.0498764 0.998755i $$-0.484117\pi$$
0.0498764 + 0.998755i $$0.484117\pi$$
$$758$$ 0 0
$$759$$ −10.3723 −0.376490
$$760$$ 0 0
$$761$$ 8.51087 0.308519 0.154259 0.988030i $$-0.450701\pi$$
0.154259 + 0.988030i $$0.450701\pi$$
$$762$$ 0 0
$$763$$ 40.4674 1.46502
$$764$$ 0 0
$$765$$ −0.372281 −0.0134599
$$766$$ 0 0
$$767$$ 2.74456 0.0991004
$$768$$ 0 0
$$769$$ 39.9565 1.44087 0.720434 0.693523i $$-0.243943\pi$$
0.720434 + 0.693523i $$0.243943\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ 0 0
$$773$$ 49.7228 1.78841 0.894203 0.447662i $$-0.147743\pi$$
0.894203 + 0.447662i $$0.147743\pi$$
$$774$$ 0 0
$$775$$ 6.74456 0.242272
$$776$$ 0 0
$$777$$ −19.1168 −0.685813
$$778$$ 0 0
$$779$$ 0.744563 0.0266767
$$780$$ 0 0
$$781$$ 19.1168 0.684054
$$782$$ 0 0
$$783$$ −2.74456 −0.0980827
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 22.7446 0.810756 0.405378 0.914149i $$-0.367140\pi$$
0.405378 + 0.914149i $$0.367140\pi$$
$$788$$ 0 0
$$789$$ −2.51087 −0.0893895
$$790$$ 0 0
$$791$$ 8.74456 0.310921
$$792$$ 0 0
$$793$$ 9.11684 0.323749
$$794$$ 0 0
$$795$$ 7.62772 0.270527
$$796$$ 0 0
$$797$$ 36.0951 1.27855 0.639277 0.768977i $$-0.279234\pi$$
0.639277 + 0.768977i $$0.279234\pi$$
$$798$$ 0 0
$$799$$ 1.02175 0.0361469
$$800$$ 0 0
$$801$$ −13.1168 −0.463461
$$802$$ 0 0
$$803$$ −26.2337 −0.925767
$$804$$ 0 0
$$805$$ −10.3723 −0.365575
$$806$$ 0 0
$$807$$ 25.7228 0.905486
$$808$$ 0 0
$$809$$ 40.9783 1.44072 0.720359 0.693601i $$-0.243977\pi$$
0.720359 + 0.693601i $$0.243977\pi$$
$$810$$ 0 0
$$811$$ −44.9783 −1.57940 −0.789700 0.613493i $$-0.789764\pi$$
−0.789700 + 0.613493i $$0.789764\pi$$
$$812$$ 0 0
$$813$$ −3.48913 −0.122369
$$814$$ 0 0
$$815$$ 17.8614 0.625658
$$816$$ 0 0
$$817$$ −9.48913 −0.331982
$$818$$ 0 0
$$819$$ 4.37228 0.152780
$$820$$ 0 0
$$821$$ 15.6277 0.545411 0.272706 0.962098i $$-0.412082\pi$$
0.272706 + 0.962098i $$0.412082\pi$$
$$822$$ 0 0
$$823$$ 10.9783 0.382678 0.191339 0.981524i $$-0.438717\pi$$
0.191339 + 0.981524i $$0.438717\pi$$
$$824$$ 0 0
$$825$$ 4.37228 0.152223
$$826$$ 0 0
$$827$$ 37.2554 1.29550 0.647749 0.761854i $$-0.275710\pi$$
0.647749 + 0.761854i $$0.275710\pi$$
$$828$$ 0 0
$$829$$ 4.51087 0.156669 0.0783346 0.996927i $$-0.475040\pi$$
0.0783346 + 0.996927i $$0.475040\pi$$
$$830$$ 0 0
$$831$$ −21.2554 −0.737343
$$832$$ 0 0
$$833$$ 4.51087 0.156293
$$834$$ 0 0
$$835$$ −2.74456 −0.0949795
$$836$$ 0 0
$$837$$ 6.74456 0.233126
$$838$$ 0 0
$$839$$ −53.5842 −1.84993 −0.924966 0.380049i $$-0.875907\pi$$
−0.924966 + 0.380049i $$0.875907\pi$$
$$840$$ 0 0
$$841$$ −21.4674 −0.740254
$$842$$ 0 0
$$843$$ 3.48913 0.120172
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ −35.4891 −1.21942
$$848$$ 0 0
$$849$$ −25.4891 −0.874785
$$850$$ 0 0
$$851$$ −10.3723 −0.355557
$$852$$ 0 0
$$853$$ 49.1168 1.68173 0.840864 0.541246i $$-0.182047\pi$$
0.840864 + 0.541246i $$0.182047\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ −28.8397 −0.985144 −0.492572 0.870272i $$-0.663943\pi$$
−0.492572 + 0.870272i $$0.663943\pi$$
$$858$$ 0 0
$$859$$ −1.35053 −0.0460796 −0.0230398 0.999735i $$-0.507334\pi$$
−0.0230398 + 0.999735i $$0.507334\pi$$
$$860$$ 0 0
$$861$$ 1.62772 0.0554725
$$862$$ 0 0
$$863$$ 11.4891 0.391094 0.195547 0.980694i $$-0.437352\pi$$
0.195547 + 0.980694i $$0.437352\pi$$
$$864$$ 0 0
$$865$$ 11.4891 0.390642
$$866$$ 0 0
$$867$$ −16.8614 −0.572643
$$868$$ 0 0
$$869$$ −42.0951 −1.42798
$$870$$ 0 0
$$871$$ −1.25544 −0.0425389
$$872$$ 0 0
$$873$$ 9.11684 0.308558
$$874$$ 0 0
$$875$$ 4.37228 0.147810
$$876$$ 0 0
$$877$$ −15.4891 −0.523031 −0.261515 0.965199i $$-0.584222\pi$$
−0.261515 + 0.965199i $$0.584222\pi$$
$$878$$ 0 0
$$879$$ −2.74456 −0.0925718
$$880$$ 0 0
$$881$$ −21.7228 −0.731860 −0.365930 0.930642i $$-0.619249\pi$$
−0.365930 + 0.930642i $$0.619249\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 2.74456 0.0922575
$$886$$ 0 0
$$887$$ −52.6060 −1.76634 −0.883168 0.469057i $$-0.844594\pi$$
−0.883168 + 0.469057i $$0.844594\pi$$
$$888$$ 0 0
$$889$$ 69.9565 2.34627
$$890$$ 0 0
$$891$$ 4.37228 0.146477
$$892$$ 0 0
$$893$$ −5.48913 −0.183687
$$894$$ 0 0
$$895$$ −3.25544 −0.108817
$$896$$ 0 0
$$897$$ 2.37228 0.0792082
$$898$$ 0 0
$$899$$ −18.5109 −0.617372
$$900$$ 0 0
$$901$$ −2.83966 −0.0946027
$$902$$ 0 0
$$903$$ −20.7446 −0.690336
$$904$$ 0 0
$$905$$ 5.11684 0.170090
$$906$$ 0 0
$$907$$ 11.2554 0.373731 0.186865 0.982386i $$-0.440167\pi$$
0.186865 + 0.982386i $$0.440167\pi$$
$$908$$ 0 0
$$909$$ −5.25544 −0.174312
$$910$$ 0 0
$$911$$ −27.7228 −0.918498 −0.459249 0.888308i $$-0.651881\pi$$
−0.459249 + 0.888308i $$0.651881\pi$$
$$912$$ 0 0
$$913$$ −64.4674 −2.13356
$$914$$ 0 0
$$915$$ 9.11684 0.301394
$$916$$ 0 0
$$917$$ −52.4674 −1.73263
$$918$$ 0 0
$$919$$ −3.11684 −0.102815 −0.0514076 0.998678i $$-0.516371\pi$$
−0.0514076 + 0.998678i $$0.516371\pi$$
$$920$$ 0 0
$$921$$ 9.11684 0.300410
$$922$$ 0 0
$$923$$ −4.37228 −0.143915
$$924$$ 0 0
$$925$$ 4.37228 0.143760
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 47.3505 1.55352 0.776760 0.629796i $$-0.216862\pi$$
0.776760 + 0.629796i $$0.216862\pi$$
$$930$$ 0 0
$$931$$ −24.2337 −0.794227
$$932$$ 0 0
$$933$$ −31.7228 −1.03856
$$934$$ 0 0
$$935$$ −1.62772 −0.0532321
$$936$$ 0 0
$$937$$ −17.2554 −0.563711 −0.281855 0.959457i $$-0.590950\pi$$
−0.281855 + 0.959457i $$0.590950\pi$$
$$938$$ 0 0
$$939$$ −28.9783 −0.945669
$$940$$ 0 0
$$941$$ 9.39403 0.306237 0.153118 0.988208i $$-0.451068\pi$$
0.153118 + 0.988208i $$0.451068\pi$$
$$942$$ 0 0
$$943$$ 0.883156 0.0287595
$$944$$ 0 0
$$945$$ 4.37228 0.142230
$$946$$ 0 0
$$947$$ −53.7228 −1.74576 −0.872878 0.487938i $$-0.837749\pi$$
−0.872878 + 0.487938i $$0.837749\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 58.3288 1.88945 0.944727 0.327857i $$-0.106326\pi$$
0.944727 + 0.327857i $$0.106326\pi$$
$$954$$ 0 0
$$955$$ −8.74456 −0.282967
$$956$$ 0 0
$$957$$ −12.0000 −0.387905
$$958$$ 0 0
$$959$$ 32.7446 1.05738
$$960$$ 0 0
$$961$$ 14.4891 0.467391
$$962$$ 0 0
$$963$$ 1.62772 0.0524525
$$964$$ 0 0
$$965$$ 21.1168 0.679775
$$966$$ 0 0
$$967$$ −17.2554 −0.554897 −0.277449 0.960740i $$-0.589489\pi$$
−0.277449 + 0.960740i $$0.589489\pi$$
$$968$$ 0 0
$$969$$ −0.744563 −0.0239188
$$970$$ 0 0
$$971$$ −29.2119 −0.937456 −0.468728 0.883343i $$-0.655288\pi$$
−0.468728 + 0.883343i $$0.655288\pi$$
$$972$$ 0 0
$$973$$ −13.6277 −0.436885
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ −22.0000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$978$$ 0 0
$$979$$ −57.3505 −1.83293
$$980$$ 0 0
$$981$$ −9.25544 −0.295503
$$982$$ 0 0
$$983$$ 52.2337 1.66600 0.832998 0.553276i $$-0.186623\pi$$
0.832998 + 0.553276i $$0.186623\pi$$
$$984$$ 0 0
$$985$$ −3.48913 −0.111173
$$986$$ 0 0
$$987$$ −12.0000 −0.381964
$$988$$ 0 0
$$989$$ −11.2554 −0.357902
$$990$$ 0 0
$$991$$ −39.8614 −1.26624 −0.633120 0.774054i $$-0.718226\pi$$
−0.633120 + 0.774054i $$0.718226\pi$$
$$992$$ 0 0
$$993$$ 22.0000 0.698149
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ 53.7228 1.70142 0.850709 0.525636i $$-0.176173\pi$$
0.850709 + 0.525636i $$0.176173\pi$$
$$998$$ 0 0
$$999$$ 4.37228 0.138333
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 6240.2.a.bo.1.1 yes 2
4.3 odd 2 6240.2.a.bk.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bk.1.2 2 4.3 odd 2
6240.2.a.bo.1.1 yes 2 1.1 even 1 trivial