# Properties

 Label 2736.2.a.bf.1.3 Level $2736$ Weight $2$ Character 2736.1 Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 171) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.548230$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.15121 q^{5} -3.37228 q^{7} +O(q^{10})$$ $$q+2.15121 q^{5} -3.37228 q^{7} +4.70285 q^{11} +2.00000 q^{13} +2.15121 q^{17} -1.00000 q^{19} +6.85407 q^{23} -0.372281 q^{25} -6.85407 q^{29} +6.74456 q^{31} -7.25450 q^{35} -0.744563 q^{37} -2.55164 q^{41} -6.11684 q^{43} -9.00528 q^{47} +4.37228 q^{49} +11.9574 q^{53} +10.1168 q^{55} -5.10328 q^{59} +12.1168 q^{61} +4.30243 q^{65} +4.00000 q^{67} +13.7081 q^{71} +12.1168 q^{73} -15.8593 q^{77} +4.00000 q^{79} -1.75079 q^{83} +4.62772 q^{85} +11.9574 q^{89} -6.74456 q^{91} -2.15121 q^{95} -15.4891 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^7 $$4 q - 2 q^{7} + 8 q^{13} - 4 q^{19} + 10 q^{25} + 4 q^{31} + 20 q^{37} + 10 q^{43} + 6 q^{49} + 6 q^{55} + 14 q^{61} + 16 q^{67} + 14 q^{73} + 16 q^{79} + 30 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100})$$ 4 * q - 2 * q^7 + 8 * q^13 - 4 * q^19 + 10 * q^25 + 4 * q^31 + 20 * q^37 + 10 * q^43 + 6 * q^49 + 6 * q^55 + 14 * q^61 + 16 * q^67 + 14 * q^73 + 16 * q^79 + 30 * q^85 - 4 * q^91 - 16 * 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
$$5$$ 2.15121 0.962052 0.481026 0.876706i $$-0.340264\pi$$
0.481026 + 0.876706i $$0.340264\pi$$
$$6$$ 0 0
$$7$$ −3.37228 −1.27460 −0.637301 0.770615i $$-0.719949\pi$$
−0.637301 + 0.770615i $$0.719949\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.70285 1.41796 0.708982 0.705227i $$-0.249155\pi$$
0.708982 + 0.705227i $$0.249155\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.15121 0.521746 0.260873 0.965373i $$-0.415990\pi$$
0.260873 + 0.965373i $$0.415990\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.85407 1.42917 0.714586 0.699548i $$-0.246615\pi$$
0.714586 + 0.699548i $$0.246615\pi$$
$$24$$ 0 0
$$25$$ −0.372281 −0.0744563
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.85407 −1.27277 −0.636384 0.771372i $$-0.719571\pi$$
−0.636384 + 0.771372i $$0.719571\pi$$
$$30$$ 0 0
$$31$$ 6.74456 1.21136 0.605680 0.795709i $$-0.292901\pi$$
0.605680 + 0.795709i $$0.292901\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −7.25450 −1.22623
$$36$$ 0 0
$$37$$ −0.744563 −0.122405 −0.0612027 0.998125i $$-0.519494\pi$$
−0.0612027 + 0.998125i $$0.519494\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.55164 −0.398499 −0.199250 0.979949i $$-0.563850\pi$$
−0.199250 + 0.979949i $$0.563850\pi$$
$$42$$ 0 0
$$43$$ −6.11684 −0.932810 −0.466405 0.884571i $$-0.654451\pi$$
−0.466405 + 0.884571i $$0.654451\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.00528 −1.31356 −0.656778 0.754084i $$-0.728081\pi$$
−0.656778 + 0.754084i $$0.728081\pi$$
$$48$$ 0 0
$$49$$ 4.37228 0.624612
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 11.9574 1.64247 0.821234 0.570591i $$-0.193286\pi$$
0.821234 + 0.570591i $$0.193286\pi$$
$$54$$ 0 0
$$55$$ 10.1168 1.36415
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −5.10328 −0.664391 −0.332195 0.943211i $$-0.607789\pi$$
−0.332195 + 0.943211i $$0.607789\pi$$
$$60$$ 0 0
$$61$$ 12.1168 1.55140 0.775701 0.631100i $$-0.217396\pi$$
0.775701 + 0.631100i $$0.217396\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.30243 0.533650
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.7081 1.62686 0.813428 0.581665i $$-0.197599\pi$$
0.813428 + 0.581665i $$0.197599\pi$$
$$72$$ 0 0
$$73$$ 12.1168 1.41817 0.709085 0.705123i $$-0.249108\pi$$
0.709085 + 0.705123i $$0.249108\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −15.8593 −1.80734
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1.75079 −0.192174 −0.0960868 0.995373i $$-0.530633\pi$$
−0.0960868 + 0.995373i $$0.530633\pi$$
$$84$$ 0 0
$$85$$ 4.62772 0.501947
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 11.9574 1.26748 0.633738 0.773547i $$-0.281520\pi$$
0.633738 + 0.773547i $$0.281520\pi$$
$$90$$ 0 0
$$91$$ −6.74456 −0.707022
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.15121 −0.220710
$$96$$ 0 0
$$97$$ −15.4891 −1.57268 −0.786341 0.617792i $$-0.788027\pi$$
−0.786341 + 0.617792i $$0.788027\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.60485 −0.856215 −0.428107 0.903728i $$-0.640820\pi$$
−0.428107 + 0.903728i $$0.640820\pi$$
$$102$$ 0 0
$$103$$ 18.7446 1.84696 0.923478 0.383651i $$-0.125333\pi$$
0.923478 + 0.383651i $$0.125333\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −9.40571 −0.909284 −0.454642 0.890674i $$-0.650233\pi$$
−0.454642 + 0.890674i $$0.650233\pi$$
$$108$$ 0 0
$$109$$ 16.7446 1.60384 0.801919 0.597433i $$-0.203812\pi$$
0.801919 + 0.597433i $$0.203812\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.75079 0.164700 0.0823500 0.996603i $$-0.473757\pi$$
0.0823500 + 0.996603i $$0.473757\pi$$
$$114$$ 0 0
$$115$$ 14.7446 1.37494
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −7.25450 −0.665019
$$120$$ 0 0
$$121$$ 11.1168 1.01062
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −11.5569 −1.03368
$$126$$ 0 0
$$127$$ 1.25544 0.111402 0.0557010 0.998447i $$-0.482261\pi$$
0.0557010 + 0.998447i $$0.482261\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.90200 −0.340919 −0.170460 0.985365i $$-0.554525\pi$$
−0.170460 + 0.985365i $$0.554525\pi$$
$$132$$ 0 0
$$133$$ 3.37228 0.292414
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.6602 1.42338 0.711689 0.702495i $$-0.247931\pi$$
0.711689 + 0.702495i $$0.247931\pi$$
$$138$$ 0 0
$$139$$ 14.1168 1.19738 0.598688 0.800983i $$-0.295689\pi$$
0.598688 + 0.800983i $$0.295689\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 9.40571 0.786545
$$144$$ 0 0
$$145$$ −14.7446 −1.22447
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.35036 −0.110626 −0.0553128 0.998469i $$-0.517616\pi$$
−0.0553128 + 0.998469i $$0.517616\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 14.5090 1.16539
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −23.1138 −1.82163
$$162$$ 0 0
$$163$$ −1.48913 −0.116637 −0.0583186 0.998298i $$-0.518574\pi$$
−0.0583186 + 0.998298i $$0.518574\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4.30243 −0.332932 −0.166466 0.986047i $$-0.553236\pi$$
−0.166466 + 0.986047i $$0.553236\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.85407 0.521105 0.260553 0.965460i $$-0.416095\pi$$
0.260553 + 0.965460i $$0.416095\pi$$
$$174$$ 0 0
$$175$$ 1.25544 0.0949021
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 9.40571 0.703016 0.351508 0.936185i $$-0.385669\pi$$
0.351508 + 0.936185i $$0.385669\pi$$
$$180$$ 0 0
$$181$$ −3.48913 −0.259345 −0.129672 0.991557i $$-0.541393\pi$$
−0.129672 + 0.991557i $$0.541393\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1.60171 −0.117760
$$186$$ 0 0
$$187$$ 10.1168 0.739817
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.20442 −0.593651 −0.296826 0.954932i $$-0.595928\pi$$
−0.296826 + 0.954932i $$0.595928\pi$$
$$192$$ 0 0
$$193$$ −18.2337 −1.31249 −0.656245 0.754548i $$-0.727856\pi$$
−0.656245 + 0.754548i $$0.727856\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.50157 0.249477 0.124738 0.992190i $$-0.460191\pi$$
0.124738 + 0.992190i $$0.460191\pi$$
$$198$$ 0 0
$$199$$ −18.1168 −1.28427 −0.642135 0.766592i $$-0.721951\pi$$
−0.642135 + 0.766592i $$0.721951\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 23.1138 1.62227
$$204$$ 0 0
$$205$$ −5.48913 −0.383377
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.70285 −0.325303
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −13.1586 −0.897412
$$216$$ 0 0
$$217$$ −22.7446 −1.54400
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.30243 0.289413
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.9073 0.856686 0.428343 0.903616i $$-0.359097\pi$$
0.428343 + 0.903616i $$0.359097\pi$$
$$228$$ 0 0
$$229$$ 21.3723 1.41232 0.706160 0.708052i $$-0.250426\pi$$
0.706160 + 0.708052i $$0.250426\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7.25450 −0.475258 −0.237629 0.971356i $$-0.576370\pi$$
−0.237629 + 0.971356i $$0.576370\pi$$
$$234$$ 0 0
$$235$$ −19.3723 −1.26371
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −27.0158 −1.74751 −0.873755 0.486367i $$-0.838322\pi$$
−0.873755 + 0.486367i $$0.838322\pi$$
$$240$$ 0 0
$$241$$ 10.2337 0.659210 0.329605 0.944119i $$-0.393084\pi$$
0.329605 + 0.944119i $$0.393084\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 9.40571 0.600909
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.9094 −0.941074 −0.470537 0.882380i $$-0.655940\pi$$
−0.470537 + 0.882380i $$0.655940\pi$$
$$252$$ 0 0
$$253$$ 32.2337 2.02651
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.55164 0.159167 0.0795835 0.996828i $$-0.474641\pi$$
0.0795835 + 0.996828i $$0.474641\pi$$
$$258$$ 0 0
$$259$$ 2.51087 0.156018
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −9.80614 −0.604672 −0.302336 0.953201i $$-0.597767\pi$$
−0.302336 + 0.953201i $$0.597767\pi$$
$$264$$ 0 0
$$265$$ 25.7228 1.58014
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 25.6655 1.56485 0.782426 0.622743i $$-0.213982\pi$$
0.782426 + 0.622743i $$0.213982\pi$$
$$270$$ 0 0
$$271$$ −2.51087 −0.152525 −0.0762624 0.997088i $$-0.524299\pi$$
−0.0762624 + 0.997088i $$0.524299\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.75079 −0.105576
$$276$$ 0 0
$$277$$ −8.11684 −0.487694 −0.243847 0.969814i $$-0.578409\pi$$
−0.243847 + 0.969814i $$0.578409\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.3556 −0.617766 −0.308883 0.951100i $$-0.599955\pi$$
−0.308883 + 0.951100i $$0.599955\pi$$
$$282$$ 0 0
$$283$$ −0.627719 −0.0373140 −0.0186570 0.999826i $$-0.505939\pi$$
−0.0186570 + 0.999826i $$0.505939\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.60485 0.507928
$$288$$ 0 0
$$289$$ −12.3723 −0.727781
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −26.4663 −1.54618 −0.773090 0.634296i $$-0.781290\pi$$
−0.773090 + 0.634296i $$0.781290\pi$$
$$294$$ 0 0
$$295$$ −10.9783 −0.639178
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 13.7081 0.792762
$$300$$ 0 0
$$301$$ 20.6277 1.18896
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 26.0659 1.49253
$$306$$ 0 0
$$307$$ −2.51087 −0.143303 −0.0716516 0.997430i $$-0.522827\pi$$
−0.0716516 + 0.997430i $$0.522827\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 27.0158 1.53193 0.765964 0.642883i $$-0.222262\pi$$
0.765964 + 0.642883i $$0.222262\pi$$
$$312$$ 0 0
$$313$$ −15.4891 −0.875497 −0.437749 0.899097i $$-0.644224\pi$$
−0.437749 + 0.899097i $$0.644224\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −35.0712 −1.96979 −0.984897 0.173139i $$-0.944609\pi$$
−0.984897 + 0.173139i $$0.944609\pi$$
$$318$$ 0 0
$$319$$ −32.2337 −1.80474
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.15121 −0.119697
$$324$$ 0 0
$$325$$ −0.744563 −0.0413009
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 30.3683 1.67426
$$330$$ 0 0
$$331$$ 26.9783 1.48286 0.741429 0.671031i $$-0.234148\pi$$
0.741429 + 0.671031i $$0.234148\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8.60485 0.470133
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 31.7187 1.71766
$$342$$ 0 0
$$343$$ 8.86141 0.478471
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.10114 0.166478 0.0832390 0.996530i $$-0.473474\pi$$
0.0832390 + 0.996530i $$0.473474\pi$$
$$348$$ 0 0
$$349$$ −10.8614 −0.581398 −0.290699 0.956815i $$-0.593888\pi$$
−0.290699 + 0.956815i $$0.593888\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −22.3130 −1.18760 −0.593800 0.804612i $$-0.702373\pi$$
−0.593800 + 0.804612i $$0.702373\pi$$
$$354$$ 0 0
$$355$$ 29.4891 1.56512
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 27.8167 1.46811 0.734055 0.679090i $$-0.237626\pi$$
0.734055 + 0.679090i $$0.237626\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 26.0659 1.36435
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −40.3236 −2.09349
$$372$$ 0 0
$$373$$ 10.2337 0.529880 0.264940 0.964265i $$-0.414648\pi$$
0.264940 + 0.964265i $$0.414648\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −13.7081 −0.706005
$$378$$ 0 0
$$379$$ −17.2554 −0.886352 −0.443176 0.896435i $$-0.646148\pi$$
−0.443176 + 0.896435i $$0.646148\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0.800857 0.0409219 0.0204609 0.999791i $$-0.493487\pi$$
0.0204609 + 0.999791i $$0.493487\pi$$
$$384$$ 0 0
$$385$$ −34.1168 −1.73876
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 16.6602 0.844706 0.422353 0.906431i $$-0.361204\pi$$
0.422353 + 0.906431i $$0.361204\pi$$
$$390$$ 0 0
$$391$$ 14.7446 0.745665
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.60485 0.432957
$$396$$ 0 0
$$397$$ 0.116844 0.00586423 0.00293212 0.999996i $$-0.499067\pi$$
0.00293212 + 0.999996i $$0.499067\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 16.2598 0.811975 0.405987 0.913879i $$-0.366928\pi$$
0.405987 + 0.913879i $$0.366928\pi$$
$$402$$ 0 0
$$403$$ 13.4891 0.671941
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.50157 −0.173566
$$408$$ 0 0
$$409$$ −15.4891 −0.765888 −0.382944 0.923772i $$-0.625090\pi$$
−0.382944 + 0.923772i $$0.625090\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 17.2097 0.846834
$$414$$ 0 0
$$415$$ −3.76631 −0.184881
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 1.75079 0.0855314 0.0427657 0.999085i $$-0.486383\pi$$
0.0427657 + 0.999085i $$0.486383\pi$$
$$420$$ 0 0
$$421$$ 36.9783 1.80221 0.901105 0.433601i $$-0.142757\pi$$
0.901105 + 0.433601i $$0.142757\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −0.800857 −0.0388472
$$426$$ 0 0
$$427$$ −40.8614 −1.97742
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7.80400 −0.375905 −0.187953 0.982178i $$-0.560185\pi$$
−0.187953 + 0.982178i $$0.560185\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.85407 −0.327875
$$438$$ 0 0
$$439$$ −16.2337 −0.774792 −0.387396 0.921913i $$-0.626625\pi$$
−0.387396 + 0.921913i $$0.626625\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.5069 −0.594218 −0.297109 0.954843i $$-0.596023\pi$$
−0.297109 + 0.954843i $$0.596023\pi$$
$$444$$ 0 0
$$445$$ 25.7228 1.21938
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −33.4695 −1.57952 −0.789761 0.613414i $$-0.789796\pi$$
−0.789761 + 0.613414i $$0.789796\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −14.5090 −0.680192
$$456$$ 0 0
$$457$$ −2.62772 −0.122919 −0.0614597 0.998110i $$-0.519576\pi$$
−0.0614597 + 0.998110i $$0.519576\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5.65278 0.263276 0.131638 0.991298i $$-0.457976\pi$$
0.131638 + 0.991298i $$0.457976\pi$$
$$462$$ 0 0
$$463$$ −3.37228 −0.156723 −0.0783616 0.996925i $$-0.524969\pi$$
−0.0783616 + 0.996925i $$0.524969\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −30.5174 −1.41218 −0.706089 0.708123i $$-0.749542\pi$$
−0.706089 + 0.708123i $$0.749542\pi$$
$$468$$ 0 0
$$469$$ −13.4891 −0.622870
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −28.7666 −1.32269
$$474$$ 0 0
$$475$$ 0.372281 0.0170814
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −8.45578 −0.386355 −0.193177 0.981164i $$-0.561879\pi$$
−0.193177 + 0.981164i $$0.561879\pi$$
$$480$$ 0 0
$$481$$ −1.48913 −0.0678983
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −33.3204 −1.51300
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6.85407 −0.309320 −0.154660 0.987968i $$-0.549428\pi$$
−0.154660 + 0.987968i $$0.549428\pi$$
$$492$$ 0 0
$$493$$ −14.7446 −0.664062
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −46.2277 −2.07360
$$498$$ 0 0
$$499$$ −6.11684 −0.273828 −0.136914 0.990583i $$-0.543718\pi$$
−0.136914 + 0.990583i $$0.543718\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 22.1639 0.988240 0.494120 0.869394i $$-0.335490\pi$$
0.494120 + 0.869394i $$0.335490\pi$$
$$504$$ 0 0
$$505$$ −18.5109 −0.823723
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −10.3556 −0.459006 −0.229503 0.973308i $$-0.573710\pi$$
−0.229503 + 0.973308i $$0.573710\pi$$
$$510$$ 0 0
$$511$$ −40.8614 −1.80760
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 40.3236 1.77687
$$516$$ 0 0
$$517$$ −42.3505 −1.86257
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 7.65492 0.335368 0.167684 0.985841i $$-0.446371\pi$$
0.167684 + 0.985841i $$0.446371\pi$$
$$522$$ 0 0
$$523$$ −16.2337 −0.709850 −0.354925 0.934895i $$-0.615494\pi$$
−0.354925 + 0.934895i $$0.615494\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.5090 0.632022
$$528$$ 0 0
$$529$$ 23.9783 1.04253
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −5.10328 −0.221048
$$534$$ 0 0
$$535$$ −20.2337 −0.874779
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 20.5622 0.885677
$$540$$ 0 0
$$541$$ 3.88316 0.166950 0.0834750 0.996510i $$-0.473398\pi$$
0.0834750 + 0.996510i $$0.473398\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 36.0211 1.54298
$$546$$ 0 0
$$547$$ 12.2337 0.523075 0.261537 0.965193i $$-0.415771\pi$$
0.261537 + 0.965193i $$0.415771\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.85407 0.291993
$$552$$ 0 0
$$553$$ −13.4891 −0.573616
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.35036 0.0572165 0.0286082 0.999591i $$-0.490892\pi$$
0.0286082 + 0.999591i $$0.490892\pi$$
$$558$$ 0 0
$$559$$ −12.2337 −0.517430
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −25.8146 −1.08795 −0.543977 0.839100i $$-0.683082\pi$$
−0.543977 + 0.839100i $$0.683082\pi$$
$$564$$ 0 0
$$565$$ 3.76631 0.158450
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −20.5622 −0.862012 −0.431006 0.902349i $$-0.641841\pi$$
−0.431006 + 0.902349i $$0.641841\pi$$
$$570$$ 0 0
$$571$$ −25.4891 −1.06669 −0.533343 0.845899i $$-0.679065\pi$$
−0.533343 + 0.845899i $$0.679065\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.55164 −0.106411
$$576$$ 0 0
$$577$$ −23.8832 −0.994269 −0.497134 0.867674i $$-0.665614\pi$$
−0.497134 + 0.867674i $$0.665614\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5.90414 0.244945
$$582$$ 0 0
$$583$$ 56.2337 2.32896
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −32.1191 −1.32570 −0.662849 0.748753i $$-0.730653\pi$$
−0.662849 + 0.748753i $$0.730653\pi$$
$$588$$ 0 0
$$589$$ −6.74456 −0.277905
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −27.4163 −1.12585 −0.562926 0.826508i $$-0.690324\pi$$
−0.562926 + 0.826508i $$0.690324\pi$$
$$594$$ 0 0
$$595$$ −15.6060 −0.639782
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.60485 0.351585 0.175792 0.984427i $$-0.443751\pi$$
0.175792 + 0.984427i $$0.443751\pi$$
$$600$$ 0 0
$$601$$ −36.7446 −1.49884 −0.749421 0.662094i $$-0.769668\pi$$
−0.749421 + 0.662094i $$0.769668\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 23.9147 0.972271
$$606$$ 0 0
$$607$$ −17.2554 −0.700377 −0.350188 0.936679i $$-0.613882\pi$$
−0.350188 + 0.936679i $$0.613882\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −18.0106 −0.728629
$$612$$ 0 0
$$613$$ −37.6060 −1.51889 −0.759445 0.650571i $$-0.774530\pi$$
−0.759445 + 0.650571i $$0.774530\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.15121 −0.0866046 −0.0433023 0.999062i $$-0.513788\pi$$
−0.0433023 + 0.999062i $$0.513788\pi$$
$$618$$ 0 0
$$619$$ 38.9783 1.56667 0.783334 0.621601i $$-0.213517\pi$$
0.783334 + 0.621601i $$0.213517\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −40.3236 −1.61553
$$624$$ 0 0
$$625$$ −23.0000 −0.920000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −1.60171 −0.0638645
$$630$$ 0 0
$$631$$ 2.11684 0.0842702 0.0421351 0.999112i $$-0.486584\pi$$
0.0421351 + 0.999112i $$0.486584\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2.70071 0.107175
$$636$$ 0 0
$$637$$ 8.74456 0.346472
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10.3556 0.409023 0.204512 0.978864i $$-0.434439\pi$$
0.204512 + 0.978864i $$0.434439\pi$$
$$642$$ 0 0
$$643$$ 17.8832 0.705243 0.352621 0.935766i $$-0.385290\pi$$
0.352621 + 0.935766i $$0.385290\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −17.6101 −0.692326 −0.346163 0.938174i $$-0.612516\pi$$
−0.346163 + 0.938174i $$0.612516\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.95207 0.115523 0.0577617 0.998330i $$-0.481604\pi$$
0.0577617 + 0.998330i $$0.481604\pi$$
$$654$$ 0 0
$$655$$ −8.39403 −0.327982
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −41.9253 −1.63318 −0.816588 0.577221i $$-0.804137\pi$$
−0.816588 + 0.577221i $$0.804137\pi$$
$$660$$ 0 0
$$661$$ 4.74456 0.184542 0.0922710 0.995734i $$-0.470587\pi$$
0.0922710 + 0.995734i $$0.470587\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 7.25450 0.281317
$$666$$ 0 0
$$667$$ −46.9783 −1.81901
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 56.9838 2.19983
$$672$$ 0 0
$$673$$ −36.7446 −1.41640 −0.708199 0.706012i $$-0.750492\pi$$
−0.708199 + 0.706012i $$0.750492\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13.5591 0.521117 0.260559 0.965458i $$-0.416093\pi$$
0.260559 + 0.965458i $$0.416093\pi$$
$$678$$ 0 0
$$679$$ 52.2337 2.00454
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 35.2203 1.34767 0.673833 0.738884i $$-0.264647\pi$$
0.673833 + 0.738884i $$0.264647\pi$$
$$684$$ 0 0
$$685$$ 35.8397 1.36936
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 23.9147 0.911078
$$690$$ 0 0
$$691$$ −9.88316 −0.375973 −0.187986 0.982172i $$-0.560196\pi$$
−0.187986 + 0.982172i $$0.560196\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 30.3683 1.15194
$$696$$ 0 0
$$697$$ −5.48913 −0.207915
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −29.0180 −1.09599 −0.547997 0.836480i $$-0.684610\pi$$
−0.547997 + 0.836480i $$0.684610\pi$$
$$702$$ 0 0
$$703$$ 0.744563 0.0280817
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 29.0180 1.09133
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 46.2277 1.73124
$$714$$ 0 0
$$715$$ 20.2337 0.756697
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −7.40357 −0.276107 −0.138053 0.990425i $$-0.544085\pi$$
−0.138053 + 0.990425i $$0.544085\pi$$
$$720$$ 0 0
$$721$$ −63.2119 −2.35414
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 2.55164 0.0947656
$$726$$ 0 0
$$727$$ −14.3505 −0.532232 −0.266116 0.963941i $$-0.585740\pi$$
−0.266116 + 0.963941i $$0.585740\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −13.1586 −0.486690
$$732$$ 0 0
$$733$$ −50.4674 −1.86406 −0.932028 0.362387i $$-0.881962\pi$$
−0.932028 + 0.362387i $$0.881962\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.8114 0.692928
$$738$$ 0 0
$$739$$ −9.88316 −0.363558 −0.181779 0.983339i $$-0.558186\pi$$
−0.181779 + 0.983339i $$0.558186\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −42.7261 −1.56747 −0.783735 0.621096i $$-0.786688\pi$$
−0.783735 + 0.621096i $$0.786688\pi$$
$$744$$ 0 0
$$745$$ −2.90491 −0.106428
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 31.7187 1.15898
$$750$$ 0 0
$$751$$ 44.4674 1.62264 0.811319 0.584604i $$-0.198750\pi$$
0.811319 + 0.584604i $$0.198750\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 8.60485 0.313163
$$756$$ 0 0
$$757$$ 12.1168 0.440394 0.220197 0.975455i $$-0.429330\pi$$
0.220197 + 0.975455i $$0.429330\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −25.2651 −0.915858 −0.457929 0.888989i $$-0.651409\pi$$
−0.457929 + 0.888989i $$0.651409\pi$$
$$762$$ 0 0
$$763$$ −56.4674 −2.04426
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −10.2066 −0.368538
$$768$$ 0 0
$$769$$ 24.1168 0.869676 0.434838 0.900509i $$-0.356806\pi$$
0.434838 + 0.900509i $$0.356806\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 2.55164 0.0917762 0.0458881 0.998947i $$-0.485388\pi$$
0.0458881 + 0.998947i $$0.485388\pi$$
$$774$$ 0 0
$$775$$ −2.51087 −0.0901933
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.55164 0.0914220
$$780$$ 0 0
$$781$$ 64.4674 2.30682
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.30243 0.153560
$$786$$ 0 0
$$787$$ −41.9565 −1.49559 −0.747794 0.663931i $$-0.768887\pi$$
−0.747794 + 0.663931i $$0.768887\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5.90414 −0.209927
$$792$$ 0 0
$$793$$ 24.2337 0.860563
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11.1565 0.395183 0.197592 0.980284i $$-0.436688\pi$$
0.197592 + 0.980284i $$0.436688\pi$$
$$798$$ 0 0
$$799$$ −19.3723 −0.685342
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 56.9838 2.01091
$$804$$ 0 0
$$805$$ −49.7228 −1.75250
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −34.6708 −1.21896 −0.609480 0.792802i $$-0.708622\pi$$
−0.609480 + 0.792802i $$0.708622\pi$$
$$810$$ 0 0
$$811$$ 25.2554 0.886838 0.443419 0.896314i $$-0.353765\pi$$
0.443419 + 0.896314i $$0.353765\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3.20343 −0.112211
$$816$$ 0 0
$$817$$ 6.11684 0.214001
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −17.4611 −0.609395 −0.304698 0.952449i $$-0.598555\pi$$
−0.304698 + 0.952449i $$0.598555\pi$$
$$822$$ 0 0
$$823$$ 31.6060 1.10171 0.550857 0.834599i $$-0.314301\pi$$
0.550857 + 0.834599i $$0.314301\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 42.7261 1.48573 0.742866 0.669440i $$-0.233466\pi$$
0.742866 + 0.669440i $$0.233466\pi$$
$$828$$ 0 0
$$829$$ −12.7446 −0.442637 −0.221318 0.975202i $$-0.571036\pi$$
−0.221318 + 0.975202i $$0.571036\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 9.40571 0.325889
$$834$$ 0 0
$$835$$ −9.25544 −0.320298
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 7.80400 0.269424 0.134712 0.990885i $$-0.456989\pi$$
0.134712 + 0.990885i $$0.456989\pi$$
$$840$$ 0 0
$$841$$ 17.9783 0.619940
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −19.3609 −0.666036
$$846$$ 0 0
$$847$$ −37.4891 −1.28814
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5.10328 −0.174938
$$852$$ 0 0
$$853$$ 30.4674 1.04318 0.521592 0.853195i $$-0.325339\pi$$
0.521592 + 0.853195i $$0.325339\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42.0743 1.43723 0.718616 0.695407i $$-0.244776\pi$$
0.718616 + 0.695407i $$0.244776\pi$$
$$858$$ 0 0
$$859$$ 14.1168 0.481661 0.240830 0.970567i $$-0.422580\pi$$
0.240830 + 0.970567i $$0.422580\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −22.3130 −0.759543 −0.379771 0.925080i $$-0.623997\pi$$
−0.379771 + 0.925080i $$0.623997\pi$$
$$864$$ 0 0
$$865$$ 14.7446 0.501330
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 18.8114 0.638134
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 38.9732 1.31753
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25.2651 0.851201 0.425601 0.904911i $$-0.360063\pi$$
0.425601 + 0.904911i $$0.360063\pi$$
$$882$$ 0 0
$$883$$ 14.1168 0.475070 0.237535 0.971379i $$-0.423661\pi$$
0.237535 + 0.971379i $$0.423661\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −10.2066 −0.342703 −0.171351 0.985210i $$-0.554813\pi$$
−0.171351 + 0.985210i $$0.554813\pi$$
$$888$$ 0 0
$$889$$ −4.23369 −0.141993
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 9.00528 0.301350
$$894$$ 0 0
$$895$$ 20.2337 0.676338
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −46.2277 −1.54178
$$900$$ 0 0
$$901$$ 25.7228 0.856951
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −7.50585 −0.249503
$$906$$ 0 0
$$907$$ −34.7446 −1.15367 −0.576837 0.816859i $$-0.695713\pi$$
−0.576837 + 0.816859i $$0.695713\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.7156 0.818863 0.409432 0.912341i $$-0.365727\pi$$
0.409432 + 0.912341i $$0.365727\pi$$
$$912$$ 0 0
$$913$$ −8.23369 −0.272495
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 13.1586 0.434536
$$918$$ 0 0
$$919$$ 26.9783 0.889930 0.444965 0.895548i $$-0.353216\pi$$
0.444965 + 0.895548i $$0.353216\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 27.4163 0.902418
$$924$$ 0 0
$$925$$ 0.277187 0.00911384
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 46.2277 1.51668 0.758341 0.651858i $$-0.226010\pi$$
0.758341 + 0.651858i $$0.226010\pi$$
$$930$$ 0 0
$$931$$ −4.37228 −0.143296
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 21.7635 0.711742
$$936$$ 0 0
$$937$$ 12.1168 0.395840 0.197920 0.980218i $$-0.436581\pi$$
0.197920 + 0.980218i $$0.436581\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −10.3556 −0.337584 −0.168792 0.985652i $$-0.553987\pi$$
−0.168792 + 0.985652i $$0.553987\pi$$
$$942$$ 0 0
$$943$$ −17.4891 −0.569524
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 11.9574 0.388562 0.194281 0.980946i $$-0.437763\pi$$
0.194281 + 0.980946i $$0.437763\pi$$
$$948$$ 0 0
$$949$$ 24.2337 0.786659
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −24.8646 −0.805444 −0.402722 0.915322i $$-0.631936\pi$$
−0.402722 + 0.915322i $$0.631936\pi$$
$$954$$ 0 0
$$955$$ −17.6495 −0.571123
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −56.1829 −1.81424
$$960$$ 0 0
$$961$$ 14.4891 0.467391
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −39.2246 −1.26268
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.4194 −1.10457 −0.552286 0.833655i $$-0.686244\pi$$
−0.552286 + 0.833655i $$0.686244\pi$$
$$972$$ 0 0
$$973$$ −47.6060 −1.52618
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −17.0606 −0.545818 −0.272909 0.962040i $$-0.587986\pi$$
−0.272909 + 0.962040i $$0.587986\pi$$
$$978$$ 0 0
$$979$$ 56.2337 1.79724
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −39.2246 −1.25107 −0.625534 0.780197i $$-0.715119\pi$$
−0.625534 + 0.780197i $$0.715119\pi$$
$$984$$ 0 0
$$985$$ 7.53262 0.240009
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −41.9253 −1.33315
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −38.9732 −1.23553
$$996$$ 0 0
$$997$$ 32.3505 1.02455 0.512276 0.858821i $$-0.328803\pi$$
0.512276 + 0.858821i $$0.328803\pi$$
$$998$$ 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 2736.2.a.bf.1.3 4
3.2 odd 2 inner 2736.2.a.bf.1.2 4
4.3 odd 2 171.2.a.e.1.3 yes 4
12.11 even 2 171.2.a.e.1.2 4
20.19 odd 2 4275.2.a.bp.1.2 4
28.27 even 2 8379.2.a.bw.1.3 4
60.59 even 2 4275.2.a.bp.1.3 4
76.75 even 2 3249.2.a.bf.1.2 4
84.83 odd 2 8379.2.a.bw.1.2 4
228.227 odd 2 3249.2.a.bf.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 12.11 even 2
171.2.a.e.1.3 yes 4 4.3 odd 2
2736.2.a.bf.1.2 4 3.2 odd 2 inner
2736.2.a.bf.1.3 4 1.1 even 1 trivial
3249.2.a.bf.1.2 4 76.75 even 2
3249.2.a.bf.1.3 4 228.227 odd 2
4275.2.a.bp.1.2 4 20.19 odd 2
4275.2.a.bp.1.3 4 60.59 even 2
8379.2.a.bw.1.2 4 84.83 odd 2
8379.2.a.bw.1.3 4 28.27 even 2