# Properties

 Label 5184.2.a.bl.1.2 Level $5184$ Weight $2$ Character 5184.1 Self dual yes Analytic conductor $41.394$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(1,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.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: $$41.3944484078$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5184.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +1.73205 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +1.73205 q^{7} -1.73205 q^{11} +3.00000 q^{13} -4.00000 q^{17} +6.92820 q^{19} -8.66025 q^{23} -4.00000 q^{25} +1.00000 q^{29} -5.19615 q^{31} -1.73205 q^{35} +8.00000 q^{37} -5.00000 q^{41} -8.66025 q^{43} +12.1244 q^{47} -4.00000 q^{49} -8.00000 q^{53} +1.73205 q^{55} -1.73205 q^{59} +7.00000 q^{61} -3.00000 q^{65} -8.66025 q^{67} +3.46410 q^{71} -12.0000 q^{73} -3.00000 q^{77} +5.19615 q^{79} +8.66025 q^{83} +4.00000 q^{85} +4.00000 q^{89} +5.19615 q^{91} -6.92820 q^{95} -3.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} + 6 q^{13} - 8 q^{17} - 8 q^{25} + 2 q^{29} + 16 q^{37} - 10 q^{41} - 8 q^{49} - 16 q^{53} + 14 q^{61} - 6 q^{65} - 24 q^{73} - 6 q^{77} + 8 q^{85} + 8 q^{89} - 6 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^13 - 8 * q^17 - 8 * q^25 + 2 * q^29 + 16 * q^37 - 10 * q^41 - 8 * q^49 - 16 * q^53 + 14 * q^61 - 6 * q^65 - 24 * q^73 - 6 * q^77 + 8 * q^85 + 8 * q^89 - 6 * 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 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ 1.73205 0.654654 0.327327 0.944911i $$-0.393852\pi$$
0.327327 + 0.944911i $$0.393852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.73205 −0.522233 −0.261116 0.965307i $$-0.584091\pi$$
−0.261116 + 0.965307i $$0.584091\pi$$
$$12$$ 0 0
$$13$$ 3.00000 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 6.92820 1.58944 0.794719 0.606977i $$-0.207618\pi$$
0.794719 + 0.606977i $$0.207618\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.66025 −1.80579 −0.902894 0.429863i $$-0.858562\pi$$
−0.902894 + 0.429863i $$0.858562\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ −5.19615 −0.933257 −0.466628 0.884454i $$-0.654531\pi$$
−0.466628 + 0.884454i $$0.654531\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.73205 −0.292770
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −8.66025 −1.32068 −0.660338 0.750968i $$-0.729587\pi$$
−0.660338 + 0.750968i $$0.729587\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.1244 1.76852 0.884260 0.466996i $$-0.154664\pi$$
0.884260 + 0.466996i $$0.154664\pi$$
$$48$$ 0 0
$$49$$ −4.00000 −0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.00000 −1.09888 −0.549442 0.835532i $$-0.685160\pi$$
−0.549442 + 0.835532i $$0.685160\pi$$
$$54$$ 0 0
$$55$$ 1.73205 0.233550
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −1.73205 −0.225494 −0.112747 0.993624i $$-0.535965\pi$$
−0.112747 + 0.993624i $$0.535965\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ −8.66025 −1.05802 −0.529009 0.848616i $$-0.677436\pi$$
−0.529009 + 0.848616i $$0.677436\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 3.46410 0.411113 0.205557 0.978645i $$-0.434100\pi$$
0.205557 + 0.978645i $$0.434100\pi$$
$$72$$ 0 0
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 5.19615 0.584613 0.292306 0.956325i $$-0.405577\pi$$
0.292306 + 0.956325i $$0.405577\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 8.66025 0.950586 0.475293 0.879827i $$-0.342342\pi$$
0.475293 + 0.879827i $$0.342342\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 5.19615 0.544705
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.92820 −0.710819
$$96$$ 0 0
$$97$$ −3.00000 −0.304604 −0.152302 0.988334i $$-0.548669\pi$$
−0.152302 + 0.988334i $$0.548669\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −13.0000 −1.29355 −0.646774 0.762682i $$-0.723882\pi$$
−0.646774 + 0.762682i $$0.723882\pi$$
$$102$$ 0 0
$$103$$ −1.73205 −0.170664 −0.0853320 0.996353i $$-0.527195\pi$$
−0.0853320 + 0.996353i $$0.527195\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.92820 0.669775 0.334887 0.942258i $$-0.391302\pi$$
0.334887 + 0.942258i $$0.391302\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.00000 0.0940721 0.0470360 0.998893i $$-0.485022\pi$$
0.0470360 + 0.998893i $$0.485022\pi$$
$$114$$ 0 0
$$115$$ 8.66025 0.807573
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.92820 −0.635107
$$120$$ 0 0
$$121$$ −8.00000 −0.727273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ 13.8564 1.22956 0.614779 0.788700i $$-0.289245\pi$$
0.614779 + 0.788700i $$0.289245\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −15.5885 −1.36197 −0.680985 0.732297i $$-0.738448\pi$$
−0.680985 + 0.732297i $$0.738448\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.0000 −1.11066 −0.555332 0.831628i $$-0.687409\pi$$
−0.555332 + 0.831628i $$0.687409\pi$$
$$138$$ 0 0
$$139$$ 1.73205 0.146911 0.0734553 0.997299i $$-0.476597\pi$$
0.0734553 + 0.997299i $$0.476597\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −5.19615 −0.434524
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −11.0000 −0.901155 −0.450578 0.892737i $$-0.648782\pi$$
−0.450578 + 0.892737i $$0.648782\pi$$
$$150$$ 0 0
$$151$$ −22.5167 −1.83238 −0.916190 0.400744i $$-0.868752\pi$$
−0.916190 + 0.400744i $$0.868752\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 5.19615 0.417365
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −15.0000 −1.18217
$$162$$ 0 0
$$163$$ −6.92820 −0.542659 −0.271329 0.962487i $$-0.587463\pi$$
−0.271329 + 0.962487i $$0.587463\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 15.5885 1.20627 0.603136 0.797639i $$-0.293918\pi$$
0.603136 + 0.797639i $$0.293918\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ 0 0
$$175$$ −6.92820 −0.523723
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −6.92820 −0.517838 −0.258919 0.965899i $$-0.583366\pi$$
−0.258919 + 0.965899i $$0.583366\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 6.92820 0.506640
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.73205 −0.125327 −0.0626634 0.998035i $$-0.519959\pi$$
−0.0626634 + 0.998035i $$0.519959\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ 3.46410 0.245564 0.122782 0.992434i $$-0.460818\pi$$
0.122782 + 0.992434i $$0.460818\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.73205 0.121566
$$204$$ 0 0
$$205$$ 5.00000 0.349215
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ −8.66025 −0.596196 −0.298098 0.954535i $$-0.596352\pi$$
−0.298098 + 0.954535i $$0.596352\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.66025 0.590624
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 1.73205 0.115987 0.0579934 0.998317i $$-0.481530\pi$$
0.0579934 + 0.998317i $$0.481530\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.1244 0.804722 0.402361 0.915481i $$-0.368190\pi$$
0.402361 + 0.915481i $$0.368190\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ −12.1244 −0.790906
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 19.0526 1.23241 0.616204 0.787587i $$-0.288670\pi$$
0.616204 + 0.787587i $$0.288670\pi$$
$$240$$ 0 0
$$241$$ −9.00000 −0.579741 −0.289870 0.957066i $$-0.593612\pi$$
−0.289870 + 0.957066i $$0.593612\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.00000 0.255551
$$246$$ 0 0
$$247$$ 20.7846 1.32249
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.7846 1.31191 0.655956 0.754799i $$-0.272265\pi$$
0.655956 + 0.754799i $$0.272265\pi$$
$$252$$ 0 0
$$253$$ 15.0000 0.943042
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −31.0000 −1.93373 −0.966863 0.255294i $$-0.917828\pi$$
−0.966863 + 0.255294i $$0.917828\pi$$
$$258$$ 0 0
$$259$$ 13.8564 0.860995
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −19.0526 −1.17483 −0.587416 0.809285i $$-0.699855\pi$$
−0.587416 + 0.809285i $$0.699855\pi$$
$$264$$ 0 0
$$265$$ 8.00000 0.491436
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −16.0000 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 0 0
$$271$$ −13.8564 −0.841717 −0.420858 0.907126i $$-0.638271\pi$$
−0.420858 + 0.907126i $$0.638271\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.92820 0.417786
$$276$$ 0 0
$$277$$ −27.0000 −1.62227 −0.811136 0.584857i $$-0.801151\pi$$
−0.811136 + 0.584857i $$0.801151\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 19.0000 1.13344 0.566722 0.823909i $$-0.308211\pi$$
0.566722 + 0.823909i $$0.308211\pi$$
$$282$$ 0 0
$$283$$ 29.4449 1.75032 0.875158 0.483838i $$-0.160758\pi$$
0.875158 + 0.483838i $$0.160758\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.66025 −0.511199
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 7.00000 0.408944 0.204472 0.978872i $$-0.434452\pi$$
0.204472 + 0.978872i $$0.434452\pi$$
$$294$$ 0 0
$$295$$ 1.73205 0.100844
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −25.9808 −1.50251
$$300$$ 0 0
$$301$$ −15.0000 −0.864586
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.00000 −0.400819
$$306$$ 0 0
$$307$$ −6.92820 −0.395413 −0.197707 0.980261i $$-0.563349\pi$$
−0.197707 + 0.980261i $$0.563349\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.19615 0.294647 0.147323 0.989088i $$-0.452934\pi$$
0.147323 + 0.989088i $$0.452934\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.0000 0.954815 0.477408 0.878682i $$-0.341577\pi$$
0.477408 + 0.878682i $$0.341577\pi$$
$$318$$ 0 0
$$319$$ −1.73205 −0.0969762
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −27.7128 −1.54198
$$324$$ 0 0
$$325$$ −12.0000 −0.665640
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 21.0000 1.15777
$$330$$ 0 0
$$331$$ −12.1244 −0.666415 −0.333207 0.942854i $$-0.608131\pi$$
−0.333207 + 0.942854i $$0.608131\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 8.66025 0.473160
$$336$$ 0 0
$$337$$ 15.0000 0.817102 0.408551 0.912735i $$-0.366034\pi$$
0.408551 + 0.912735i $$0.366034\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 9.00000 0.487377
$$342$$ 0 0
$$343$$ −19.0526 −1.02874
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 22.5167 1.20876 0.604379 0.796697i $$-0.293421\pi$$
0.604379 + 0.796697i $$0.293421\pi$$
$$348$$ 0 0
$$349$$ −11.0000 −0.588817 −0.294408 0.955680i $$-0.595123\pi$$
−0.294408 + 0.955680i $$0.595123\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −29.0000 −1.54351 −0.771757 0.635917i $$-0.780622\pi$$
−0.771757 + 0.635917i $$0.780622\pi$$
$$354$$ 0 0
$$355$$ −3.46410 −0.183855
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 29.0000 1.52632
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ 0 0
$$367$$ 32.9090 1.71783 0.858917 0.512115i $$-0.171138\pi$$
0.858917 + 0.512115i $$0.171138\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −13.8564 −0.719389
$$372$$ 0 0
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ −10.3923 −0.533817 −0.266908 0.963722i $$-0.586002\pi$$
−0.266908 + 0.963722i $$0.586002\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −25.9808 −1.32755 −0.663777 0.747930i $$-0.731048\pi$$
−0.663777 + 0.747930i $$0.731048\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −29.0000 −1.47036 −0.735179 0.677873i $$-0.762902\pi$$
−0.735179 + 0.677873i $$0.762902\pi$$
$$390$$ 0 0
$$391$$ 34.6410 1.75187
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −5.19615 −0.261447
$$396$$ 0 0
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −13.0000 −0.649189 −0.324595 0.945853i $$-0.605228\pi$$
−0.324595 + 0.945853i $$0.605228\pi$$
$$402$$ 0 0
$$403$$ −15.5885 −0.776516
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −13.8564 −0.686837
$$408$$ 0 0
$$409$$ −9.00000 −0.445021 −0.222511 0.974930i $$-0.571425\pi$$
−0.222511 + 0.974930i $$0.571425\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −3.00000 −0.147620
$$414$$ 0 0
$$415$$ −8.66025 −0.425115
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 22.5167 1.10001 0.550005 0.835161i $$-0.314626\pi$$
0.550005 + 0.835161i $$0.314626\pi$$
$$420$$ 0 0
$$421$$ 15.0000 0.731055 0.365528 0.930800i $$-0.380889\pi$$
0.365528 + 0.930800i $$0.380889\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 16.0000 0.776114
$$426$$ 0 0
$$427$$ 12.1244 0.586739
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −27.7128 −1.33488 −0.667440 0.744664i $$-0.732610\pi$$
−0.667440 + 0.744664i $$0.732610\pi$$
$$432$$ 0 0
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −60.0000 −2.87019
$$438$$ 0 0
$$439$$ 5.19615 0.247999 0.123999 0.992282i $$-0.460428\pi$$
0.123999 + 0.992282i $$0.460428\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −32.9090 −1.56355 −0.781776 0.623559i $$-0.785686\pi$$
−0.781776 + 0.623559i $$0.785686\pi$$
$$444$$ 0 0
$$445$$ −4.00000 −0.189618
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 8.66025 0.407795
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −5.19615 −0.243599
$$456$$ 0 0
$$457$$ 21.0000 0.982339 0.491169 0.871064i $$-0.336570\pi$$
0.491169 + 0.871064i $$0.336570\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1.00000 0.0465746 0.0232873 0.999729i $$-0.492587\pi$$
0.0232873 + 0.999729i $$0.492587\pi$$
$$462$$ 0 0
$$463$$ −32.9090 −1.52941 −0.764705 0.644381i $$-0.777115\pi$$
−0.764705 + 0.644381i $$0.777115\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.92820 −0.320599 −0.160300 0.987068i $$-0.551246\pi$$
−0.160300 + 0.987068i $$0.551246\pi$$
$$468$$ 0 0
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 15.0000 0.689701
$$474$$ 0 0
$$475$$ −27.7128 −1.27155
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 8.66025 0.395697 0.197849 0.980233i $$-0.436605\pi$$
0.197849 + 0.980233i $$0.436605\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.00000 0.136223
$$486$$ 0 0
$$487$$ −27.7128 −1.25579 −0.627894 0.778299i $$-0.716083\pi$$
−0.627894 + 0.778299i $$0.716083\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.73205 −0.0781664 −0.0390832 0.999236i $$-0.512444\pi$$
−0.0390832 + 0.999236i $$0.512444\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ 1.73205 0.0775372 0.0387686 0.999248i $$-0.487656\pi$$
0.0387686 + 0.999248i $$0.487656\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 24.2487 1.08120 0.540598 0.841281i $$-0.318198\pi$$
0.540598 + 0.841281i $$0.318198\pi$$
$$504$$ 0 0
$$505$$ 13.0000 0.578492
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 37.0000 1.64000 0.819998 0.572366i $$-0.193974\pi$$
0.819998 + 0.572366i $$0.193974\pi$$
$$510$$ 0 0
$$511$$ −20.7846 −0.919457
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1.73205 0.0763233
$$516$$ 0 0
$$517$$ −21.0000 −0.923579
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ 10.3923 0.454424 0.227212 0.973845i $$-0.427039\pi$$
0.227212 + 0.973845i $$0.427039\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.7846 0.905392
$$528$$ 0 0
$$529$$ 52.0000 2.26087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −15.0000 −0.649722
$$534$$ 0 0
$$535$$ −6.92820 −0.299532
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.92820 0.298419
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 15.5885 0.666514 0.333257 0.942836i $$-0.391852\pi$$
0.333257 + 0.942836i $$0.391852\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.92820 0.295151
$$552$$ 0 0
$$553$$ 9.00000 0.382719
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.0000 0.677942 0.338971 0.940797i $$-0.389921\pi$$
0.338971 + 0.940797i $$0.389921\pi$$
$$558$$ 0 0
$$559$$ −25.9808 −1.09887
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.5167 0.948964 0.474482 0.880265i $$-0.342635\pi$$
0.474482 + 0.880265i $$0.342635\pi$$
$$564$$ 0 0
$$565$$ −1.00000 −0.0420703
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −31.0000 −1.29959 −0.649794 0.760111i $$-0.725145\pi$$
−0.649794 + 0.760111i $$0.725145\pi$$
$$570$$ 0 0
$$571$$ −8.66025 −0.362420 −0.181210 0.983444i $$-0.558001\pi$$
−0.181210 + 0.983444i $$0.558001\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 34.6410 1.44463
$$576$$ 0 0
$$577$$ −4.00000 −0.166522 −0.0832611 0.996528i $$-0.526534\pi$$
−0.0832611 + 0.996528i $$0.526534\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 15.0000 0.622305
$$582$$ 0 0
$$583$$ 13.8564 0.573874
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5.19615 −0.214468 −0.107234 0.994234i $$-0.534199\pi$$
−0.107234 + 0.994234i $$0.534199\pi$$
$$588$$ 0 0
$$589$$ −36.0000 −1.48335
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 20.0000 0.821302 0.410651 0.911793i $$-0.365302\pi$$
0.410651 + 0.911793i $$0.365302\pi$$
$$594$$ 0 0
$$595$$ 6.92820 0.284029
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.73205 0.0707697 0.0353848 0.999374i $$-0.488734\pi$$
0.0353848 + 0.999374i $$0.488734\pi$$
$$600$$ 0 0
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 8.00000 0.325246
$$606$$ 0 0
$$607$$ 12.1244 0.492112 0.246056 0.969256i $$-0.420865\pi$$
0.246056 + 0.969256i $$0.420865\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 36.3731 1.47150
$$612$$ 0 0
$$613$$ −8.00000 −0.323117 −0.161558 0.986863i $$-0.551652\pi$$
−0.161558 + 0.986863i $$0.551652\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −5.00000 −0.201292 −0.100646 0.994922i $$-0.532091\pi$$
−0.100646 + 0.994922i $$0.532091\pi$$
$$618$$ 0 0
$$619$$ 15.5885 0.626553 0.313276 0.949662i $$-0.398573\pi$$
0.313276 + 0.949662i $$0.398573\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 6.92820 0.277573
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ 41.5692 1.65484 0.827422 0.561580i $$-0.189806\pi$$
0.827422 + 0.561580i $$0.189806\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −13.8564 −0.549875
$$636$$ 0 0
$$637$$ −12.0000 −0.475457
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −7.00000 −0.276483 −0.138242 0.990399i $$-0.544145\pi$$
−0.138242 + 0.990399i $$0.544145\pi$$
$$642$$ 0 0
$$643$$ 5.19615 0.204916 0.102458 0.994737i $$-0.467329\pi$$
0.102458 + 0.994737i $$0.467329\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −13.8564 −0.544752 −0.272376 0.962191i $$-0.587809\pi$$
−0.272376 + 0.962191i $$0.587809\pi$$
$$648$$ 0 0
$$649$$ 3.00000 0.117760
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.00000 −0.0391330 −0.0195665 0.999809i $$-0.506229\pi$$
−0.0195665 + 0.999809i $$0.506229\pi$$
$$654$$ 0 0
$$655$$ 15.5885 0.609091
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 39.8372 1.55184 0.775918 0.630834i $$-0.217287\pi$$
0.775918 + 0.630834i $$0.217287\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ −8.66025 −0.335326
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −12.1244 −0.468056
$$672$$ 0 0
$$673$$ 5.00000 0.192736 0.0963679 0.995346i $$-0.469277\pi$$
0.0963679 + 0.995346i $$0.469277\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 11.0000 0.422764 0.211382 0.977403i $$-0.432204\pi$$
0.211382 + 0.977403i $$0.432204\pi$$
$$678$$ 0 0
$$679$$ −5.19615 −0.199410
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −34.6410 −1.32550 −0.662751 0.748840i $$-0.730611\pi$$
−0.662751 + 0.748840i $$0.730611\pi$$
$$684$$ 0 0
$$685$$ 13.0000 0.496704
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 19.0526 0.724793 0.362397 0.932024i $$-0.381959\pi$$
0.362397 + 0.932024i $$0.381959\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.73205 −0.0657004
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16.0000 0.604312 0.302156 0.953259i $$-0.402294\pi$$
0.302156 + 0.953259i $$0.402294\pi$$
$$702$$ 0 0
$$703$$ 55.4256 2.09042
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −22.5167 −0.846826
$$708$$ 0 0
$$709$$ −27.0000 −1.01401 −0.507003 0.861944i $$-0.669247\pi$$
−0.507003 + 0.861944i $$0.669247\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 45.0000 1.68526
$$714$$ 0 0
$$715$$ 5.19615 0.194325
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −41.5692 −1.55027 −0.775135 0.631795i $$-0.782318\pi$$
−0.775135 + 0.631795i $$0.782318\pi$$
$$720$$ 0 0
$$721$$ −3.00000 −0.111726
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4.00000 −0.148556
$$726$$ 0 0
$$727$$ −22.5167 −0.835097 −0.417548 0.908655i $$-0.637111\pi$$
−0.417548 + 0.908655i $$0.637111\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 34.6410 1.28124
$$732$$ 0 0
$$733$$ −21.0000 −0.775653 −0.387826 0.921732i $$-0.626774\pi$$
−0.387826 + 0.921732i $$0.626774\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15.0000 0.552532
$$738$$ 0 0
$$739$$ −3.46410 −0.127429 −0.0637145 0.997968i $$-0.520295\pi$$
−0.0637145 + 0.997968i $$0.520295\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.73205 0.0635428 0.0317714 0.999495i $$-0.489885\pi$$
0.0317714 + 0.999495i $$0.489885\pi$$
$$744$$ 0 0
$$745$$ 11.0000 0.403009
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 25.9808 0.948051 0.474026 0.880511i $$-0.342800\pi$$
0.474026 + 0.880511i $$0.342800\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 22.5167 0.819465
$$756$$ 0 0
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 43.0000 1.55875 0.779374 0.626559i $$-0.215537\pi$$
0.779374 + 0.626559i $$0.215537\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.19615 −0.187622
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 40.0000 1.43870 0.719350 0.694648i $$-0.244440\pi$$
0.719350 + 0.694648i $$0.244440\pi$$
$$774$$ 0 0
$$775$$ 20.7846 0.746605
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −34.6410 −1.24114
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −17.0000 −0.606756
$$786$$ 0 0
$$787$$ −25.9808 −0.926114 −0.463057 0.886328i $$-0.653248\pi$$
−0.463057 + 0.886328i $$0.653248\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.73205 0.0615846
$$792$$ 0 0
$$793$$ 21.0000 0.745732
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 29.0000 1.02723 0.513616 0.858020i $$-0.328305\pi$$
0.513616 + 0.858020i $$0.328305\pi$$
$$798$$ 0 0
$$799$$ −48.4974 −1.71572
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 20.7846 0.733473
$$804$$ 0 0
$$805$$ 15.0000 0.528681
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 4.00000 0.140633 0.0703163 0.997525i $$-0.477599\pi$$
0.0703163 + 0.997525i $$0.477599\pi$$
$$810$$ 0 0
$$811$$ 20.7846 0.729846 0.364923 0.931038i $$-0.381095\pi$$
0.364923 + 0.931038i $$0.381095\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 6.92820 0.242684
$$816$$ 0 0
$$817$$ −60.0000 −2.09913
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −7.00000 −0.244302 −0.122151 0.992512i $$-0.538979\pi$$
−0.122151 + 0.992512i $$0.538979\pi$$
$$822$$ 0 0
$$823$$ −46.7654 −1.63014 −0.815069 0.579364i $$-0.803301\pi$$
−0.815069 + 0.579364i $$0.803301\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.92820 −0.240917 −0.120459 0.992718i $$-0.538437\pi$$
−0.120459 + 0.992718i $$0.538437\pi$$
$$828$$ 0 0
$$829$$ −24.0000 −0.833554 −0.416777 0.909009i $$-0.636840\pi$$
−0.416777 + 0.909009i $$0.636840\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 16.0000 0.554367
$$834$$ 0 0
$$835$$ −15.5885 −0.539461
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 25.9808 0.896956 0.448478 0.893794i $$-0.351966\pi$$
0.448478 + 0.893794i $$0.351966\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4.00000 0.137604
$$846$$ 0 0
$$847$$ −13.8564 −0.476112
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −69.2820 −2.37496
$$852$$ 0 0
$$853$$ 23.0000 0.787505 0.393753 0.919216i $$-0.371177\pi$$
0.393753 + 0.919216i $$0.371177\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 43.0000 1.46885 0.734426 0.678689i $$-0.237451\pi$$
0.734426 + 0.678689i $$0.237451\pi$$
$$858$$ 0 0
$$859$$ −36.3731 −1.24103 −0.620517 0.784193i $$-0.713077\pi$$
−0.620517 + 0.784193i $$0.713077\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 55.4256 1.88671 0.943355 0.331785i $$-0.107651\pi$$
0.943355 + 0.331785i $$0.107651\pi$$
$$864$$ 0 0
$$865$$ 13.0000 0.442013
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −9.00000 −0.305304
$$870$$ 0 0
$$871$$ −25.9808 −0.880325
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 15.5885 0.526986
$$876$$ 0 0
$$877$$ 43.0000 1.45201 0.726003 0.687691i $$-0.241376\pi$$
0.726003 + 0.687691i $$0.241376\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −52.0000 −1.75192 −0.875962 0.482380i $$-0.839773\pi$$
−0.875962 + 0.482380i $$0.839773\pi$$
$$882$$ 0 0
$$883$$ −20.7846 −0.699458 −0.349729 0.936851i $$-0.613726\pi$$
−0.349729 + 0.936851i $$0.613726\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 5.19615 0.174470 0.0872349 0.996188i $$-0.472197\pi$$
0.0872349 + 0.996188i $$0.472197\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 84.0000 2.81095
$$894$$ 0 0
$$895$$ 6.92820 0.231584
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −5.19615 −0.173301
$$900$$ 0 0
$$901$$ 32.0000 1.06607
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.9090 1.09272 0.546362 0.837549i $$-0.316012\pi$$
0.546362 + 0.837549i $$0.316012\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32.9090 −1.09032 −0.545161 0.838331i $$-0.683532\pi$$
−0.545161 + 0.838331i $$0.683532\pi$$
$$912$$ 0 0
$$913$$ −15.0000 −0.496428
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −27.0000 −0.891619
$$918$$ 0 0
$$919$$ 13.8564 0.457081 0.228540 0.973534i $$-0.426605\pi$$
0.228540 + 0.973534i $$0.426605\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.3923 0.342067
$$924$$ 0 0
$$925$$ −32.0000 −1.05215
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 35.0000 1.14831 0.574156 0.818746i $$-0.305330\pi$$
0.574156 + 0.818746i $$0.305330\pi$$
$$930$$ 0 0
$$931$$ −27.7128 −0.908251
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −6.92820 −0.226576
$$936$$ 0 0
$$937$$ 4.00000 0.130674 0.0653372 0.997863i $$-0.479188\pi$$
0.0653372 + 0.997863i $$0.479188\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 7.00000 0.228193 0.114097 0.993470i $$-0.463603\pi$$
0.114097 + 0.993470i $$0.463603\pi$$
$$942$$ 0 0
$$943$$ 43.3013 1.41008
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −15.5885 −0.506557 −0.253278 0.967393i $$-0.581509\pi$$
−0.253278 + 0.967393i $$0.581509\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ 1.73205 0.0560478
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −22.5167 −0.727101
$$960$$ 0 0
$$961$$ −4.00000 −0.129032
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 11.0000 0.354103
$$966$$ 0 0
$$967$$ −32.9090 −1.05828 −0.529140 0.848534i $$-0.677486\pi$$
−0.529140 + 0.848534i $$0.677486\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 38.1051 1.22285 0.611426 0.791302i $$-0.290596\pi$$
0.611426 + 0.791302i $$0.290596\pi$$
$$972$$ 0 0
$$973$$ 3.00000 0.0961756
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 25.0000 0.799821 0.399910 0.916554i $$-0.369041\pi$$
0.399910 + 0.916554i $$0.369041\pi$$
$$978$$ 0 0
$$979$$ −6.92820 −0.221426
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 8.66025 0.276219 0.138110 0.990417i $$-0.455897\pi$$
0.138110 + 0.990417i $$0.455897\pi$$
$$984$$ 0 0
$$985$$ 22.0000 0.700978
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 75.0000 2.38486
$$990$$ 0 0
$$991$$ 55.4256 1.76065 0.880327 0.474368i $$-0.157323\pi$$
0.880327 + 0.474368i $$0.157323\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −3.46410 −0.109819
$$996$$ 0 0
$$997$$ 13.0000 0.411714 0.205857 0.978582i $$-0.434002\pi$$
0.205857 + 0.978582i $$0.434002\pi$$
$$998$$ 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 5184.2.a.bl.1.2 2
3.2 odd 2 5184.2.a.bx.1.2 2
4.3 odd 2 inner 5184.2.a.bl.1.1 2
8.3 odd 2 2592.2.a.p.1.1 2
8.5 even 2 2592.2.a.p.1.2 2
9.2 odd 6 576.2.i.k.193.1 4
9.4 even 3 1728.2.i.l.1153.1 4
9.5 odd 6 576.2.i.k.385.1 4
9.7 even 3 1728.2.i.l.577.1 4
12.11 even 2 5184.2.a.bx.1.1 2
24.5 odd 2 2592.2.a.l.1.2 2
24.11 even 2 2592.2.a.l.1.1 2
36.7 odd 6 1728.2.i.l.577.2 4
36.11 even 6 576.2.i.k.193.2 4
36.23 even 6 576.2.i.k.385.2 4
36.31 odd 6 1728.2.i.l.1153.2 4
72.5 odd 6 288.2.i.d.97.2 yes 4
72.11 even 6 288.2.i.d.193.1 yes 4
72.13 even 6 864.2.i.d.289.1 4
72.29 odd 6 288.2.i.d.193.2 yes 4
72.43 odd 6 864.2.i.d.577.2 4
72.59 even 6 288.2.i.d.97.1 4
72.61 even 6 864.2.i.d.577.1 4
72.67 odd 6 864.2.i.d.289.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.d.97.1 4 72.59 even 6
288.2.i.d.97.2 yes 4 72.5 odd 6
288.2.i.d.193.1 yes 4 72.11 even 6
288.2.i.d.193.2 yes 4 72.29 odd 6
576.2.i.k.193.1 4 9.2 odd 6
576.2.i.k.193.2 4 36.11 even 6
576.2.i.k.385.1 4 9.5 odd 6
576.2.i.k.385.2 4 36.23 even 6
864.2.i.d.289.1 4 72.13 even 6
864.2.i.d.289.2 4 72.67 odd 6
864.2.i.d.577.1 4 72.61 even 6
864.2.i.d.577.2 4 72.43 odd 6
1728.2.i.l.577.1 4 9.7 even 3
1728.2.i.l.577.2 4 36.7 odd 6
1728.2.i.l.1153.1 4 9.4 even 3
1728.2.i.l.1153.2 4 36.31 odd 6
2592.2.a.l.1.1 2 24.11 even 2
2592.2.a.l.1.2 2 24.5 odd 2
2592.2.a.p.1.1 2 8.3 odd 2
2592.2.a.p.1.2 2 8.5 even 2
5184.2.a.bl.1.1 2 4.3 odd 2 inner
5184.2.a.bl.1.2 2 1.1 even 1 trivial
5184.2.a.bx.1.1 2 12.11 even 2
5184.2.a.bx.1.2 2 3.2 odd 2