# Properties

 Label 3240.2.a.o.1.1 Level $3240$ Weight $2$ Character 3240.1 Self dual yes Analytic conductor $25.872$ Analytic rank $0$ 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] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.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: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 3240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -1.44949 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -1.44949 q^{7} +2.00000 q^{17} +2.89898 q^{19} +2.55051 q^{23} +1.00000 q^{25} -7.89898 q^{29} +10.8990 q^{31} -1.44949 q^{35} -6.00000 q^{37} +0.101021 q^{41} -7.79796 q^{43} +4.55051 q^{47} -4.89898 q^{49} +11.7980 q^{53} +10.8990 q^{59} -3.00000 q^{61} +11.2474 q^{67} +9.79796 q^{71} -5.79796 q^{73} +2.89898 q^{79} +0.550510 q^{83} +2.00000 q^{85} +16.7980 q^{89} +2.89898 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{5} + 2 q^{7} + 4 q^{17} - 4 q^{19} + 10 q^{23} + 2 q^{25} - 6 q^{29} + 12 q^{31} + 2 q^{35} - 12 q^{37} + 10 q^{41} + 4 q^{43} + 14 q^{47} + 4 q^{53} + 12 q^{59} - 6 q^{61} - 2 q^{67} + 8 q^{73} - 4 q^{79} + 6 q^{83} + 4 q^{85} + 14 q^{89} - 4 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^17 - 4 * q^19 + 10 * q^23 + 2 * q^25 - 6 * q^29 + 12 * q^31 + 2 * q^35 - 12 * q^37 + 10 * q^41 + 4 * q^43 + 14 * q^47 + 4 * q^53 + 12 * q^59 - 6 * q^61 - 2 * q^67 + 8 * q^73 - 4 * q^79 + 6 * q^83 + 4 * q^85 + 14 * q^89 - 4 * q^95 + 4 * q^97

## 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.44949 −0.547856 −0.273928 0.961750i $$-0.588323\pi$$
−0.273928 + 0.961750i $$0.588323\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 2.89898 0.665072 0.332536 0.943091i $$-0.392096\pi$$
0.332536 + 0.943091i $$0.392096\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.55051 0.531818 0.265909 0.963998i $$-0.414328\pi$$
0.265909 + 0.963998i $$0.414328\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.89898 −1.46680 −0.733402 0.679795i $$-0.762069\pi$$
−0.733402 + 0.679795i $$0.762069\pi$$
$$30$$ 0 0
$$31$$ 10.8990 1.95751 0.978757 0.205023i $$-0.0657268\pi$$
0.978757 + 0.205023i $$0.0657268\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.44949 −0.245008
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.101021 0.0157768 0.00788838 0.999969i $$-0.497489\pi$$
0.00788838 + 0.999969i $$0.497489\pi$$
$$42$$ 0 0
$$43$$ −7.79796 −1.18918 −0.594589 0.804030i $$-0.702685\pi$$
−0.594589 + 0.804030i $$0.702685\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.55051 0.663760 0.331880 0.943322i $$-0.392317\pi$$
0.331880 + 0.943322i $$0.392317\pi$$
$$48$$ 0 0
$$49$$ −4.89898 −0.699854
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 11.7980 1.62057 0.810287 0.586033i $$-0.199311\pi$$
0.810287 + 0.586033i $$0.199311\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.8990 1.41893 0.709463 0.704743i $$-0.248937\pi$$
0.709463 + 0.704743i $$0.248937\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.2474 1.37409 0.687047 0.726613i $$-0.258907\pi$$
0.687047 + 0.726613i $$0.258907\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9.79796 1.16280 0.581402 0.813617i $$-0.302504\pi$$
0.581402 + 0.813617i $$0.302504\pi$$
$$72$$ 0 0
$$73$$ −5.79796 −0.678600 −0.339300 0.940678i $$-0.610190\pi$$
−0.339300 + 0.940678i $$0.610190\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.89898 0.326161 0.163080 0.986613i $$-0.447857\pi$$
0.163080 + 0.986613i $$0.447857\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0.550510 0.0604264 0.0302132 0.999543i $$-0.490381\pi$$
0.0302132 + 0.999543i $$0.490381\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 16.7980 1.78058 0.890290 0.455394i $$-0.150502\pi$$
0.890290 + 0.455394i $$0.150502\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.89898 0.297429
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.34847 −0.227035 −0.113518 0.993536i $$-0.536212\pi$$
−0.113518 + 0.993536i $$0.536212\pi$$
$$108$$ 0 0
$$109$$ 8.79796 0.842692 0.421346 0.906900i $$-0.361558\pi$$
0.421346 + 0.906900i $$0.361558\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.79796 −0.921714 −0.460857 0.887474i $$-0.652458\pi$$
−0.460857 + 0.887474i $$0.652458\pi$$
$$114$$ 0 0
$$115$$ 2.55051 0.237836
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.89898 −0.265749
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −14.3485 −1.27322 −0.636610 0.771186i $$-0.719664\pi$$
−0.636610 + 0.771186i $$0.719664\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.89898 0.602767 0.301383 0.953503i $$-0.402552\pi$$
0.301383 + 0.953503i $$0.402552\pi$$
$$132$$ 0 0
$$133$$ −4.20204 −0.364363
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.5959 1.67419 0.837096 0.547056i $$-0.184251\pi$$
0.837096 + 0.547056i $$0.184251\pi$$
$$138$$ 0 0
$$139$$ −19.5959 −1.66210 −0.831052 0.556195i $$-0.812261\pi$$
−0.831052 + 0.556195i $$0.812261\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −7.89898 −0.655975
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 10.8990 0.875427
$$156$$ 0 0
$$157$$ −4.20204 −0.335359 −0.167680 0.985842i $$-0.553627\pi$$
−0.167680 + 0.985842i $$0.553627\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.69694 −0.291360
$$162$$ 0 0
$$163$$ 11.7980 0.924087 0.462044 0.886857i $$-0.347116\pi$$
0.462044 + 0.886857i $$0.347116\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.34847 0.491259 0.245630 0.969364i $$-0.421005\pi$$
0.245630 + 0.969364i $$0.421005\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ 0 0
$$175$$ −1.44949 −0.109571
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.79796 −0.433360 −0.216680 0.976243i $$-0.569523\pi$$
−0.216680 + 0.976243i $$0.569523\pi$$
$$180$$ 0 0
$$181$$ 19.6969 1.46406 0.732031 0.681271i $$-0.238573\pi$$
0.732031 + 0.681271i $$0.238573\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 5.10102 0.369097 0.184548 0.982823i $$-0.440918\pi$$
0.184548 + 0.982823i $$0.440918\pi$$
$$192$$ 0 0
$$193$$ −21.7980 −1.56905 −0.784526 0.620096i $$-0.787094\pi$$
−0.784526 + 0.620096i $$0.787094\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.5959 −1.68114 −0.840570 0.541703i $$-0.817780\pi$$
−0.840570 + 0.541703i $$0.817780\pi$$
$$198$$ 0 0
$$199$$ 13.1010 0.928707 0.464353 0.885650i $$-0.346287\pi$$
0.464353 + 0.885650i $$0.346287\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 11.4495 0.803597
$$204$$ 0 0
$$205$$ 0.101021 0.00705558
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −7.79796 −0.531816
$$216$$ 0 0
$$217$$ −15.7980 −1.07244
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0.550510 0.0368649 0.0184324 0.999830i $$-0.494132\pi$$
0.0184324 + 0.999830i $$0.494132\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 27.7980 1.84502 0.922508 0.385979i $$-0.126136\pi$$
0.922508 + 0.385979i $$0.126136\pi$$
$$228$$ 0 0
$$229$$ −13.8990 −0.918470 −0.459235 0.888315i $$-0.651876\pi$$
−0.459235 + 0.888315i $$0.651876\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0.202041 0.0132361 0.00661807 0.999978i $$-0.497893\pi$$
0.00661807 + 0.999978i $$0.497893\pi$$
$$234$$ 0 0
$$235$$ 4.55051 0.296843
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −21.7980 −1.40999 −0.704996 0.709211i $$-0.749051\pi$$
−0.704996 + 0.709211i $$0.749051\pi$$
$$240$$ 0 0
$$241$$ 25.6969 1.65529 0.827643 0.561255i $$-0.189681\pi$$
0.827643 + 0.561255i $$0.189681\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.89898 −0.312984
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.89898 0.435460 0.217730 0.976009i $$-0.430135\pi$$
0.217730 + 0.976009i $$0.430135\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −8.20204 −0.511629 −0.255815 0.966726i $$-0.582344\pi$$
−0.255815 + 0.966726i $$0.582344\pi$$
$$258$$ 0 0
$$259$$ 8.69694 0.540401
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 11.7980 0.724743
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 16.5959 1.01187 0.505935 0.862571i $$-0.331147\pi$$
0.505935 + 0.862571i $$0.331147\pi$$
$$270$$ 0 0
$$271$$ 21.1010 1.28180 0.640898 0.767626i $$-0.278562\pi$$
0.640898 + 0.767626i $$0.278562\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 17.8990 1.06776 0.533882 0.845559i $$-0.320733\pi$$
0.533882 + 0.845559i $$0.320733\pi$$
$$282$$ 0 0
$$283$$ −18.3485 −1.09070 −0.545352 0.838207i $$-0.683604\pi$$
−0.545352 + 0.838207i $$0.683604\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.146428 −0.00864338
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 21.5959 1.26165 0.630823 0.775926i $$-0.282717\pi$$
0.630823 + 0.775926i $$0.282717\pi$$
$$294$$ 0 0
$$295$$ 10.8990 0.634563
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 11.3031 0.651498
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −29.2474 −1.66924 −0.834620 0.550826i $$-0.814313\pi$$
−0.834620 + 0.550826i $$0.814313\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.89898 −0.164386 −0.0821930 0.996616i $$-0.526192\pi$$
−0.0821930 + 0.996616i $$0.526192\pi$$
$$312$$ 0 0
$$313$$ 23.5959 1.33372 0.666860 0.745183i $$-0.267638\pi$$
0.666860 + 0.745183i $$0.267638\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.20204 0.348341 0.174171 0.984715i $$-0.444276\pi$$
0.174171 + 0.984715i $$0.444276\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.79796 0.322607
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −6.59592 −0.363645
$$330$$ 0 0
$$331$$ 2.89898 0.159342 0.0796712 0.996821i $$-0.474613\pi$$
0.0796712 + 0.996821i $$0.474613\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 11.2474 0.614514
$$336$$ 0 0
$$337$$ −10.2020 −0.555741 −0.277870 0.960619i $$-0.589629\pi$$
−0.277870 + 0.960619i $$0.589629\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 17.2474 0.931275
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −13.5959 −0.729867 −0.364934 0.931034i $$-0.618908\pi$$
−0.364934 + 0.931034i $$0.618908\pi$$
$$348$$ 0 0
$$349$$ −31.6969 −1.69670 −0.848349 0.529437i $$-0.822403\pi$$
−0.848349 + 0.529437i $$0.822403\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 9.79796 0.520022
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0.696938 0.0367830 0.0183915 0.999831i $$-0.494145\pi$$
0.0183915 + 0.999831i $$0.494145\pi$$
$$360$$ 0 0
$$361$$ −10.5959 −0.557680
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −5.79796 −0.303479
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −17.1010 −0.887841
$$372$$ 0 0
$$373$$ −17.5959 −0.911082 −0.455541 0.890215i $$-0.650554\pi$$
−0.455541 + 0.890215i $$0.650554\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 17.1010 0.878420 0.439210 0.898384i $$-0.355258\pi$$
0.439210 + 0.898384i $$0.355258\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.00000 −0.102195 −0.0510976 0.998694i $$-0.516272\pi$$
−0.0510976 + 0.998694i $$0.516272\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −22.5959 −1.14566 −0.572829 0.819675i $$-0.694154\pi$$
−0.572829 + 0.819675i $$0.694154\pi$$
$$390$$ 0 0
$$391$$ 5.10102 0.257970
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2.89898 0.145863
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 13.5959 0.678948 0.339474 0.940615i $$-0.389751\pi$$
0.339474 + 0.940615i $$0.389751\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 29.5959 1.46342 0.731712 0.681614i $$-0.238722\pi$$
0.731712 + 0.681614i $$0.238722\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −15.7980 −0.777367
$$414$$ 0 0
$$415$$ 0.550510 0.0270235
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −13.5959 −0.662624 −0.331312 0.943521i $$-0.607491\pi$$
−0.331312 + 0.943521i $$0.607491\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 4.34847 0.210437
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5.10102 −0.245708 −0.122854 0.992425i $$-0.539205\pi$$
−0.122854 + 0.992425i $$0.539205\pi$$
$$432$$ 0 0
$$433$$ 7.79796 0.374746 0.187373 0.982289i $$-0.440003\pi$$
0.187373 + 0.982289i $$0.440003\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.39388 0.353697
$$438$$ 0 0
$$439$$ 1.79796 0.0858119 0.0429059 0.999079i $$-0.486338\pi$$
0.0429059 + 0.999079i $$0.486338\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −26.1464 −1.24225 −0.621127 0.783710i $$-0.713325\pi$$
−0.621127 + 0.783710i $$0.713325\pi$$
$$444$$ 0 0
$$445$$ 16.7980 0.796300
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 17.5959 0.830403 0.415201 0.909730i $$-0.363711\pi$$
0.415201 + 0.909730i $$0.363711\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −21.5959 −1.01021 −0.505107 0.863057i $$-0.668547\pi$$
−0.505107 + 0.863057i $$0.668547\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −21.8990 −1.01994 −0.509969 0.860193i $$-0.670343\pi$$
−0.509969 + 0.860193i $$0.670343\pi$$
$$462$$ 0 0
$$463$$ 31.7980 1.47778 0.738888 0.673828i $$-0.235351\pi$$
0.738888 + 0.673828i $$0.235351\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 11.7980 0.545944 0.272972 0.962022i $$-0.411993\pi$$
0.272972 + 0.962022i $$0.411993\pi$$
$$468$$ 0 0
$$469$$ −16.3031 −0.752805
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 2.89898 0.133014
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 18.4949 0.845053 0.422527 0.906350i $$-0.361143\pi$$
0.422527 + 0.906350i $$0.361143\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 25.5959 1.15986 0.579931 0.814666i $$-0.303080\pi$$
0.579931 + 0.814666i $$0.303080\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −5.79796 −0.261658 −0.130829 0.991405i $$-0.541764\pi$$
−0.130829 + 0.991405i $$0.541764\pi$$
$$492$$ 0 0
$$493$$ −15.7980 −0.711504
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −14.2020 −0.637049
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 19.0454 0.849193 0.424596 0.905383i $$-0.360416\pi$$
0.424596 + 0.905383i $$0.360416\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.8990 0.616061 0.308031 0.951376i $$-0.400330\pi$$
0.308031 + 0.951376i $$0.400330\pi$$
$$510$$ 0 0
$$511$$ 8.40408 0.371775
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 10.0000 0.440653
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −28.7980 −1.26166 −0.630831 0.775920i $$-0.717286\pi$$
−0.630831 + 0.775920i $$0.717286\pi$$
$$522$$ 0 0
$$523$$ −19.6515 −0.859301 −0.429651 0.902995i $$-0.641363\pi$$
−0.429651 + 0.902995i $$0.641363\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 21.7980 0.949534
$$528$$ 0 0
$$529$$ −16.4949 −0.717169
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −2.34847 −0.101533
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 21.8990 0.941511 0.470755 0.882264i $$-0.343981\pi$$
0.470755 + 0.882264i $$0.343981\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 8.79796 0.376863
$$546$$ 0 0
$$547$$ −41.2474 −1.76361 −0.881807 0.471611i $$-0.843673\pi$$
−0.881807 + 0.471611i $$0.843673\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −22.8990 −0.975529
$$552$$ 0 0
$$553$$ −4.20204 −0.178689
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −24.2020 −1.02547 −0.512737 0.858546i $$-0.671368\pi$$
−0.512737 + 0.858546i $$0.671368\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −4.34847 −0.183266 −0.0916331 0.995793i $$-0.529209\pi$$
−0.0916331 + 0.995793i $$0.529209\pi$$
$$564$$ 0 0
$$565$$ −9.79796 −0.412203
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 6.20204 0.259547 0.129774 0.991544i $$-0.458575\pi$$
0.129774 + 0.991544i $$0.458575\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.55051 0.106364
$$576$$ 0 0
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.797959 −0.0331049
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −22.3485 −0.922420 −0.461210 0.887291i $$-0.652585\pi$$
−0.461210 + 0.887291i $$0.652585\pi$$
$$588$$ 0 0
$$589$$ 31.5959 1.30189
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −27.3939 −1.12493 −0.562466 0.826821i $$-0.690147\pi$$
−0.562466 + 0.826821i $$0.690147\pi$$
$$594$$ 0 0
$$595$$ −2.89898 −0.118847
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −4.69694 −0.191912 −0.0959559 0.995386i $$-0.530591\pi$$
−0.0959559 + 0.995386i $$0.530591\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ −32.1464 −1.30478 −0.652392 0.757882i $$-0.726234\pi$$
−0.652392 + 0.757882i $$0.726234\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 9.59592 0.387575 0.193788 0.981043i $$-0.437923\pi$$
0.193788 + 0.981043i $$0.437923\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −43.1918 −1.73884 −0.869419 0.494076i $$-0.835507\pi$$
−0.869419 + 0.494076i $$0.835507\pi$$
$$618$$ 0 0
$$619$$ 4.69694 0.188786 0.0943929 0.995535i $$-0.469909\pi$$
0.0943929 + 0.995535i $$0.469909\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.3485 −0.975501
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 23.5959 0.939339 0.469669 0.882842i $$-0.344373\pi$$
0.469669 + 0.882842i $$0.344373\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −14.3485 −0.569402
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4.10102 0.161981 0.0809903 0.996715i $$-0.474192\pi$$
0.0809903 + 0.996715i $$0.474192\pi$$
$$642$$ 0 0
$$643$$ 14.1464 0.557881 0.278940 0.960308i $$-0.410017\pi$$
0.278940 + 0.960308i $$0.410017\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −11.4495 −0.450126 −0.225063 0.974344i $$-0.572259\pi$$
−0.225063 + 0.974344i $$0.572259\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8.20204 −0.320971 −0.160485 0.987038i $$-0.551306\pi$$
−0.160485 + 0.987038i $$0.551306\pi$$
$$654$$ 0 0
$$655$$ 6.89898 0.269565
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −29.7980 −1.16076 −0.580382 0.814344i $$-0.697097\pi$$
−0.580382 + 0.814344i $$0.697097\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.20204 −0.162948
$$666$$ 0 0
$$667$$ −20.1464 −0.780073
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −16.0000 −0.616755 −0.308377 0.951264i $$-0.599786\pi$$
−0.308377 + 0.951264i $$0.599786\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −33.7980 −1.29896 −0.649481 0.760378i $$-0.725014\pi$$
−0.649481 + 0.760378i $$0.725014\pi$$
$$678$$ 0 0
$$679$$ −2.89898 −0.111253
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 35.3939 1.35431 0.677155 0.735841i $$-0.263213\pi$$
0.677155 + 0.735841i $$0.263213\pi$$
$$684$$ 0 0
$$685$$ 19.5959 0.748722
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 29.1010 1.10705 0.553527 0.832831i $$-0.313281\pi$$
0.553527 + 0.832831i $$0.313281\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −19.5959 −0.743316
$$696$$ 0 0
$$697$$ 0.202041 0.00765285
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 28.3939 1.07242 0.536211 0.844084i $$-0.319855\pi$$
0.536211 + 0.844084i $$0.319855\pi$$
$$702$$ 0 0
$$703$$ −17.3939 −0.656022
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.89898 −0.109027
$$708$$ 0 0
$$709$$ −19.6969 −0.739734 −0.369867 0.929085i $$-0.620597\pi$$
−0.369867 + 0.929085i $$0.620597\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 27.7980 1.04104
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −40.2929 −1.50267 −0.751335 0.659921i $$-0.770590\pi$$
−0.751335 + 0.659921i $$0.770590\pi$$
$$720$$ 0 0
$$721$$ −14.4949 −0.539818
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.89898 −0.293361
$$726$$ 0 0
$$727$$ −6.75255 −0.250438 −0.125219 0.992129i $$-0.539963\pi$$
−0.125219 + 0.992129i $$0.539963\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −15.5959 −0.576836
$$732$$ 0 0
$$733$$ 9.59592 0.354433 0.177217 0.984172i $$-0.443291\pi$$
0.177217 + 0.984172i $$0.443291\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −42.8990 −1.57806 −0.789032 0.614352i $$-0.789418\pi$$
−0.789032 + 0.614352i $$0.789418\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −37.0454 −1.35906 −0.679532 0.733646i $$-0.737817\pi$$
−0.679532 + 0.733646i $$0.737817\pi$$
$$744$$ 0 0
$$745$$ 21.0000 0.769380
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3.40408 0.124382
$$750$$ 0 0
$$751$$ 45.7980 1.67119 0.835596 0.549345i $$-0.185123\pi$$
0.835596 + 0.549345i $$0.185123\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ −7.59592 −0.276078 −0.138039 0.990427i $$-0.544080\pi$$
−0.138039 + 0.990427i $$0.544080\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −35.0000 −1.26875 −0.634375 0.773026i $$-0.718742\pi$$
−0.634375 + 0.773026i $$0.718742\pi$$
$$762$$ 0 0
$$763$$ −12.7526 −0.461673
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −24.7980 −0.894237 −0.447119 0.894475i $$-0.647550\pi$$
−0.447119 + 0.894475i $$0.647550\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −25.7980 −0.927888 −0.463944 0.885865i $$-0.653566\pi$$
−0.463944 + 0.885865i $$0.653566\pi$$
$$774$$ 0 0
$$775$$ 10.8990 0.391503
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0.292856 0.0104927
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4.20204 −0.149977
$$786$$ 0 0
$$787$$ −10.4041 −0.370865 −0.185433 0.982657i $$-0.559369\pi$$
−0.185433 + 0.982657i $$0.559369\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.2020 0.504966
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ 0 0
$$799$$ 9.10102 0.321971
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −3.69694 −0.130300
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 22.8990 0.804092 0.402046 0.915619i $$-0.368299\pi$$
0.402046 + 0.915619i $$0.368299\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 11.7980 0.413264
$$816$$ 0 0
$$817$$ −22.6061 −0.790888
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16.7980 0.586253 0.293126 0.956074i $$-0.405304\pi$$
0.293126 + 0.956074i $$0.405304\pi$$
$$822$$ 0 0
$$823$$ 6.34847 0.221294 0.110647 0.993860i $$-0.464708\pi$$
0.110647 + 0.993860i $$0.464708\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.5403 1.37495 0.687476 0.726208i $$-0.258719\pi$$
0.687476 + 0.726208i $$0.258719\pi$$
$$828$$ 0 0
$$829$$ −28.1918 −0.979143 −0.489571 0.871963i $$-0.662847\pi$$
−0.489571 + 0.871963i $$0.662847\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −9.79796 −0.339479
$$834$$ 0 0
$$835$$ 6.34847 0.219698
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 24.6969 0.852633 0.426317 0.904574i $$-0.359811\pi$$
0.426317 + 0.904574i $$0.359811\pi$$
$$840$$ 0 0
$$841$$ 33.3939 1.15151
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 15.9444 0.547856
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −15.3031 −0.524582
$$852$$ 0 0
$$853$$ 37.7980 1.29418 0.647089 0.762415i $$-0.275986\pi$$
0.647089 + 0.762415i $$0.275986\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −39.5959 −1.35257 −0.676285 0.736640i $$-0.736411\pi$$
−0.676285 + 0.736640i $$0.736411\pi$$
$$858$$ 0 0
$$859$$ −39.1918 −1.33721 −0.668604 0.743619i $$-0.733108\pi$$
−0.668604 + 0.743619i $$0.733108\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −52.5505 −1.78884 −0.894420 0.447228i $$-0.852411\pi$$
−0.894420 + 0.447228i $$0.852411\pi$$
$$864$$ 0 0
$$865$$ 12.0000 0.408012
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1.44949 −0.0490017
$$876$$ 0 0
$$877$$ −39.5959 −1.33706 −0.668530 0.743686i $$-0.733076\pi$$
−0.668530 + 0.743686i $$0.733076\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −15.8990 −0.535650 −0.267825 0.963468i $$-0.586305\pi$$
−0.267825 + 0.963468i $$0.586305\pi$$
$$882$$ 0 0
$$883$$ −45.0454 −1.51590 −0.757949 0.652313i $$-0.773799\pi$$
−0.757949 + 0.652313i $$0.773799\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 37.5959 1.26235 0.631174 0.775642i $$-0.282574\pi$$
0.631174 + 0.775642i $$0.282574\pi$$
$$888$$ 0 0
$$889$$ 20.7980 0.697541
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 13.1918 0.441448
$$894$$ 0 0
$$895$$ −5.79796 −0.193804
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −86.0908 −2.87129
$$900$$ 0 0
$$901$$ 23.5959 0.786094
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.6969 0.654748
$$906$$ 0 0
$$907$$ −36.3485 −1.20693 −0.603466 0.797389i $$-0.706214\pi$$
−0.603466 + 0.797389i $$0.706214\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 38.4949 1.27539 0.637696 0.770288i $$-0.279887\pi$$
0.637696 + 0.770288i $$0.279887\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −10.0000 −0.330229
$$918$$ 0 0
$$919$$ −4.69694 −0.154938 −0.0774689 0.996995i $$-0.524684\pi$$
−0.0774689 + 0.996995i $$0.524684\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.59592 0.0523604 0.0261802 0.999657i $$-0.491666\pi$$
0.0261802 + 0.999657i $$0.491666\pi$$
$$930$$ 0 0
$$931$$ −14.2020 −0.465453
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 56.9898 1.86178 0.930888 0.365305i $$-0.119035\pi$$
0.930888 + 0.365305i $$0.119035\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −27.6969 −0.902894 −0.451447 0.892298i $$-0.649092\pi$$
−0.451447 + 0.892298i $$0.649092\pi$$
$$942$$ 0 0
$$943$$ 0.257654 0.00839036
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 46.1464 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 33.7980 1.09482 0.547412 0.836863i $$-0.315613\pi$$
0.547412 + 0.836863i $$0.315613\pi$$
$$954$$ 0 0
$$955$$ 5.10102 0.165065
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −28.4041 −0.917216
$$960$$ 0 0
$$961$$ 87.7878 2.83186
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −21.7980 −0.701701
$$966$$ 0 0
$$967$$ −14.8434 −0.477330 −0.238665 0.971102i $$-0.576710\pi$$
−0.238665 + 0.971102i $$0.576710\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −60.6969 −1.94786 −0.973929 0.226854i $$-0.927156\pi$$
−0.973929 + 0.226854i $$0.927156\pi$$
$$972$$ 0 0
$$973$$ 28.4041 0.910593
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −45.3939 −1.45228 −0.726139 0.687548i $$-0.758687\pi$$
−0.726139 + 0.687548i $$0.758687\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6.55051 0.208929 0.104464 0.994529i $$-0.466687\pi$$
0.104464 + 0.994529i $$0.466687\pi$$
$$984$$ 0 0
$$985$$ −23.5959 −0.751828
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −19.8888 −0.632426
$$990$$ 0 0
$$991$$ −0.696938 −0.0221390 −0.0110695 0.999939i $$-0.503524\pi$$
−0.0110695 + 0.999939i $$0.503524\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 13.1010 0.415330
$$996$$ 0 0
$$997$$ 35.3939 1.12094 0.560468 0.828176i $$-0.310621\pi$$
0.560468 + 0.828176i $$0.310621\pi$$
$$998$$ 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 3240.2.a.o.1.1 2
3.2 odd 2 3240.2.a.j.1.1 2
4.3 odd 2 6480.2.a.bl.1.2 2
9.2 odd 6 360.2.q.c.121.2 4
9.4 even 3 1080.2.q.c.721.2 4
9.5 odd 6 360.2.q.c.241.2 yes 4
9.7 even 3 1080.2.q.c.361.2 4
12.11 even 2 6480.2.a.bc.1.2 2
36.7 odd 6 2160.2.q.g.1441.1 4
36.11 even 6 720.2.q.g.481.1 4
36.23 even 6 720.2.q.g.241.1 4
36.31 odd 6 2160.2.q.g.721.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.c.121.2 4 9.2 odd 6
360.2.q.c.241.2 yes 4 9.5 odd 6
720.2.q.g.241.1 4 36.23 even 6
720.2.q.g.481.1 4 36.11 even 6
1080.2.q.c.361.2 4 9.7 even 3
1080.2.q.c.721.2 4 9.4 even 3
2160.2.q.g.721.1 4 36.31 odd 6
2160.2.q.g.1441.1 4 36.7 odd 6
3240.2.a.j.1.1 2 3.2 odd 2
3240.2.a.o.1.1 2 1.1 even 1 trivial
6480.2.a.bc.1.2 2 12.11 even 2
6480.2.a.bl.1.2 2 4.3 odd 2