Properties

 Label 5184.2.a.ba.1.1 Level $5184$ Weight $2$ Character 5184.1 Self dual yes Analytic conductor $41.394$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 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)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+3.00000 q^{5} -1.00000 q^{7} +3.00000 q^{11} +1.00000 q^{13} -6.00000 q^{17} +4.00000 q^{19} +3.00000 q^{23} +4.00000 q^{25} +3.00000 q^{29} +5.00000 q^{31} -3.00000 q^{35} -2.00000 q^{37} -3.00000 q^{41} +1.00000 q^{43} +9.00000 q^{47} -6.00000 q^{49} -6.00000 q^{53} +9.00000 q^{55} -3.00000 q^{59} +13.0000 q^{61} +3.00000 q^{65} +7.00000 q^{67} +12.0000 q^{71} -10.0000 q^{73} -3.00000 q^{77} +11.0000 q^{79} -9.00000 q^{83} -18.0000 q^{85} -6.00000 q^{89} -1.00000 q^{91} +12.0000 q^{95} +11.0000 q^{97} +O(q^{100})$$

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
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 9.00000 1.21356
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ −18.0000 −1.95237
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 12.0000 1.23117
$$96$$ 0 0
$$97$$ 11.0000 1.11688 0.558440 0.829545i $$-0.311400\pi$$
0.558440 + 0.829545i $$0.311400\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 9.00000 0.839254
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 21.0000 1.83478 0.917389 0.397991i $$-0.130293\pi$$
0.917389 + 0.397991i $$0.130293\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 3.00000 0.250873
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −13.0000 −1.05792 −0.528962 0.848645i $$-0.677419\pi$$
−0.528962 + 0.848645i $$0.677419\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 15.0000 1.20483
$$156$$ 0 0
$$157$$ 13.0000 1.03751 0.518756 0.854922i $$-0.326395\pi$$
0.518756 + 0.854922i $$0.326395\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −17.0000 −1.17033 −0.585164 0.810915i $$-0.698970\pi$$
−0.585164 + 0.810915i $$0.698970\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 3.00000 0.204598
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 27.0000 1.79205 0.896026 0.444001i $$-0.146441\pi$$
0.896026 + 0.444001i $$0.146441\pi$$
$$228$$ 0 0
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 27.0000 1.76129
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 27.0000 1.74648 0.873242 0.487286i $$-0.162013\pi$$
0.873242 + 0.487286i $$0.162013\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157 −0.0322078 0.999481i $$-0.510254\pi$$
−0.0322078 + 0.999481i $$0.510254\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 9.00000 0.561405 0.280702 0.959795i $$-0.409433\pi$$
0.280702 + 0.959795i $$0.409433\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.00000 −0.178965 −0.0894825 0.995988i $$-0.528521\pi$$
−0.0894825 + 0.995988i $$0.528521\pi$$
$$282$$ 0 0
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3.00000 0.177084
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 0 0
$$295$$ −9.00000 −0.524000
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 39.0000 2.23313
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 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
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ −11.0000 −0.604615 −0.302307 0.953211i $$-0.597757\pi$$
−0.302307 + 0.953211i $$0.597757\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 21.0000 1.14735
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 9.00000 0.483145 0.241573 0.970383i $$-0.422337\pi$$
0.241573 + 0.970383i $$0.422337\pi$$
$$348$$ 0 0
$$349$$ 1.00000 0.0535288 0.0267644 0.999642i $$-0.491480\pi$$
0.0267644 + 0.999642i $$0.491480\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 36.0000 1.91068
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −30.0000 −1.57027
$$366$$ 0 0
$$367$$ −13.0000 −0.678594 −0.339297 0.940679i $$-0.610189\pi$$
−0.339297 + 0.940679i $$0.610189\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ 1.00000 0.0517780 0.0258890 0.999665i $$-0.491758\pi$$
0.0258890 + 0.999665i $$0.491758\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 0 0
$$385$$ −9.00000 −0.458682
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 33.0000 1.66041
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 0 0
$$403$$ 5.00000 0.249068
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.00000 −0.297409
$$408$$ 0 0
$$409$$ 23.0000 1.13728 0.568638 0.822588i $$-0.307470\pi$$
0.568638 + 0.822588i $$0.307470\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3.00000 0.147620
$$414$$ 0 0
$$415$$ −27.0000 −1.32538
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9.00000 0.439679 0.219839 0.975536i $$-0.429447\pi$$
0.219839 + 0.975536i $$0.429447\pi$$
$$420$$ 0 0
$$421$$ −35.0000 −1.70580 −0.852898 0.522078i $$-0.825157\pi$$
−0.852898 + 0.522078i $$0.825157\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −24.0000 −1.16417
$$426$$ 0 0
$$427$$ −13.0000 −0.629114
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ 35.0000 1.67046 0.835229 0.549902i $$-0.185335\pi$$
0.835229 + 0.549902i $$0.185335\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9.00000 −0.427603 −0.213801 0.976877i $$-0.568585\pi$$
−0.213801 + 0.976877i $$0.568585\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −37.0000 −1.73079 −0.865393 0.501093i $$-0.832931\pi$$
−0.865393 + 0.501093i $$0.832931\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 0 0
$$463$$ −19.0000 −0.883005 −0.441502 0.897260i $$-0.645554\pi$$
−0.441502 + 0.897260i $$0.645554\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ −7.00000 −0.323230
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ 16.0000 0.734130
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −27.0000 −1.23366 −0.616831 0.787096i $$-0.711584\pi$$
−0.616831 + 0.787096i $$0.711584\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 33.0000 1.49845
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.00000 −0.135388 −0.0676941 0.997706i $$-0.521564\pi$$
−0.0676941 + 0.997706i $$0.521564\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ 45.0000 2.00247
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 39.0000 1.72864 0.864322 0.502938i $$-0.167748\pi$$
0.864322 + 0.502938i $$0.167748\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −21.0000 −0.925371
$$516$$ 0 0
$$517$$ 27.0000 1.18746
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 0 0
$$523$$ −8.00000 −0.349816 −0.174908 0.984585i $$-0.555963\pi$$
−0.174908 + 0.984585i $$0.555963\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −30.0000 −1.30682
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.00000 −0.129944
$$534$$ 0 0
$$535$$ 36.0000 1.55642
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ 13.0000 0.555840 0.277920 0.960604i $$-0.410355\pi$$
0.277920 + 0.960604i $$0.410355\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ −11.0000 −0.467768
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ 0 0
$$565$$ 27.0000 1.13590
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 31.0000 1.29731 0.648655 0.761083i $$-0.275332\pi$$
0.648655 + 0.761083i $$0.275332\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 0 0
$$589$$ 20.0000 0.824086
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 18.0000 0.737928
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 39.0000 1.59350 0.796748 0.604311i $$-0.206552\pi$$
0.796748 + 0.604311i $$0.206552\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −6.00000 −0.243935
$$606$$ 0 0
$$607$$ 41.0000 1.66414 0.832069 0.554672i $$-0.187156\pi$$
0.832069 + 0.554672i $$0.187156\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −3.00000 −0.120775 −0.0603877 0.998175i $$-0.519234\pi$$
−0.0603877 + 0.998175i $$0.519234\pi$$
$$618$$ 0 0
$$619$$ 13.0000 0.522514 0.261257 0.965269i $$-0.415863\pi$$
0.261257 + 0.965269i $$0.415863\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −48.0000 −1.90482
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ 0 0
$$643$$ −41.0000 −1.61688 −0.808441 0.588577i $$-0.799688\pi$$
−0.808441 + 0.588577i $$0.799688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −9.00000 −0.353281
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −21.0000 −0.821794 −0.410897 0.911682i $$-0.634784\pi$$
−0.410897 + 0.911682i $$0.634784\pi$$
$$654$$ 0 0
$$655$$ 63.0000 2.46161
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −21.0000 −0.818044 −0.409022 0.912525i $$-0.634130\pi$$
−0.409022 + 0.912525i $$0.634130\pi$$
$$660$$ 0 0
$$661$$ −11.0000 −0.427850 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 39.0000 1.50558
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15.0000 0.576497 0.288248 0.957556i $$-0.406927\pi$$
0.288248 + 0.957556i $$0.406927\pi$$
$$678$$ 0 0
$$679$$ −11.0000 −0.422141
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ −9.00000 −0.343872
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 1.00000 0.0380418 0.0190209 0.999819i $$-0.493945\pi$$
0.0190209 + 0.999819i $$0.493945\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −15.0000 −0.568982
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15.0000 −0.564133
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 15.0000 0.561754
$$714$$ 0 0
$$715$$ 9.00000 0.336581
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 7.00000 0.260694
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 12.0000 0.445669
$$726$$ 0 0
$$727$$ −37.0000 −1.37225 −0.686127 0.727482i $$-0.740691\pi$$
−0.686127 + 0.727482i $$0.740691\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −6.00000 −0.221918
$$732$$ 0 0
$$733$$ −23.0000 −0.849524 −0.424762 0.905305i $$-0.639642\pi$$
−0.424762 + 0.905305i $$0.639642\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0000 0.773545
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 0 0
$$745$$ 45.0000 1.64867
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −31.0000 −1.13121 −0.565603 0.824678i $$-0.691357\pi$$
−0.565603 + 0.824678i $$0.691357\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −39.0000 −1.41936
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3.00000 −0.108324
$$768$$ 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
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 39.0000 1.39197
$$786$$ 0 0
$$787$$ 43.0000 1.53278 0.766392 0.642373i $$-0.222050\pi$$
0.766392 + 0.642373i $$0.222050\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9.00000 −0.320003
$$792$$ 0 0
$$793$$ 13.0000 0.461644
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −9.00000 −0.318796 −0.159398 0.987214i $$-0.550955\pi$$
−0.159398 + 0.987214i $$0.550955\pi$$
$$798$$ 0 0
$$799$$ −54.0000 −1.91038
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −30.0000 −1.05868
$$804$$ 0 0
$$805$$ −9.00000 −0.317208
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −60.0000 −2.10171
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 51.0000 1.77991 0.889956 0.456046i $$-0.150735\pi$$
0.889956 + 0.456046i $$0.150735\pi$$
$$822$$ 0 0
$$823$$ −19.0000 −0.662298 −0.331149 0.943578i $$-0.607436\pi$$
−0.331149 + 0.943578i $$0.607436\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 36.0000 1.24733
$$834$$ 0 0
$$835$$ −27.0000 −0.934374
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 9.00000 0.310715 0.155357 0.987858i $$-0.450347\pi$$
0.155357 + 0.987858i $$0.450347\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −36.0000 −1.23844
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ 13.0000 0.445112 0.222556 0.974920i $$-0.428560\pi$$
0.222556 + 0.974920i $$0.428560\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −27.0000 −0.922302 −0.461151 0.887322i $$-0.652563\pi$$
−0.461151 + 0.887322i $$0.652563\pi$$
$$858$$ 0 0
$$859$$ −41.0000 −1.39890 −0.699451 0.714681i $$-0.746572\pi$$
−0.699451 + 0.714681i $$0.746572\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −27.0000 −0.918028
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 33.0000 1.11945
$$870$$ 0 0
$$871$$ 7.00000 0.237186
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ −23.0000 −0.776655 −0.388327 0.921521i $$-0.626947\pi$$
−0.388327 + 0.921521i $$0.626947\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.0000 −0.705111 −0.352555 0.935791i $$-0.614687\pi$$
−0.352555 + 0.935791i $$0.614687\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 36.0000 1.20469
$$894$$ 0 0
$$895$$ 36.0000 1.20335
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 15.0000 0.500278
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −47.0000 −1.56061 −0.780305 0.625400i $$-0.784936\pi$$
−0.780305 + 0.625400i $$0.784936\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45.0000 1.49092 0.745458 0.666552i $$-0.232231\pi$$
0.745458 + 0.666552i $$0.232231\pi$$
$$912$$ 0 0
$$913$$ −27.0000 −0.893570
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −21.0000 −0.693481
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −27.0000 −0.885841 −0.442921 0.896561i $$-0.646058\pi$$
−0.442921 + 0.896561i $$0.646058\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −54.0000 −1.76599
$$936$$ 0 0
$$937$$ −34.0000 −1.11073 −0.555366 0.831606i $$-0.687422\pi$$
−0.555366 + 0.831606i $$0.687422\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −21.0000 −0.684580 −0.342290 0.939594i $$-0.611203\pi$$
−0.342290 + 0.939594i $$0.611203\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ 0 0
$$955$$ −45.0000 −1.45617
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 33.0000 1.06231
$$966$$ 0 0
$$967$$ −43.0000 −1.38279 −0.691393 0.722478i $$-0.743003\pi$$
−0.691393 + 0.722478i $$0.743003\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 5.00000 0.160293
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 57.0000 1.82359 0.911796 0.410644i $$-0.134696\pi$$
0.911796 + 0.410644i $$0.134696\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −51.0000 −1.62665 −0.813324 0.581811i $$-0.802344\pi$$
−0.813324 + 0.581811i $$0.802344\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 3.00000 0.0953945
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ 0 0
$$997$$ 1.00000 0.0316703 0.0158352 0.999875i $$-0.494959\pi$$
0.0158352 + 0.999875i $$0.494959\pi$$
$$998$$ 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.ba.1.1 1
3.2 odd 2 5184.2.a.e.1.1 1
4.3 odd 2 5184.2.a.bb.1.1 1
8.3 odd 2 1296.2.a.b.1.1 1
8.5 even 2 324.2.a.a.1.1 1
9.2 odd 6 576.2.i.f.193.1 2
9.4 even 3 1728.2.i.d.1153.1 2
9.5 odd 6 576.2.i.f.385.1 2
9.7 even 3 1728.2.i.d.577.1 2
12.11 even 2 5184.2.a.f.1.1 1
24.5 odd 2 324.2.a.c.1.1 1
24.11 even 2 1296.2.a.k.1.1 1
36.7 odd 6 1728.2.i.c.577.1 2
36.11 even 6 576.2.i.e.193.1 2
36.23 even 6 576.2.i.e.385.1 2
36.31 odd 6 1728.2.i.c.1153.1 2
40.13 odd 4 8100.2.d.c.649.2 2
40.29 even 2 8100.2.a.g.1.1 1
40.37 odd 4 8100.2.d.c.649.1 2
72.5 odd 6 36.2.e.a.25.1 yes 2
72.11 even 6 144.2.i.a.49.1 2
72.13 even 6 108.2.e.a.73.1 2
72.29 odd 6 36.2.e.a.13.1 2
72.43 odd 6 432.2.i.c.145.1 2
72.59 even 6 144.2.i.a.97.1 2
72.61 even 6 108.2.e.a.37.1 2
72.67 odd 6 432.2.i.c.289.1 2
120.29 odd 2 8100.2.a.j.1.1 1
120.53 even 4 8100.2.d.h.649.2 2
120.77 even 4 8100.2.d.h.649.1 2
360.13 odd 12 2700.2.s.b.2449.1 4
360.29 odd 6 900.2.i.b.301.1 2
360.77 even 12 900.2.s.b.349.2 4
360.133 odd 12 2700.2.s.b.1549.2 4
360.149 odd 6 900.2.i.b.601.1 2
360.157 odd 12 2700.2.s.b.2449.2 4
360.173 even 12 900.2.s.b.49.2 4
360.229 even 6 2700.2.i.b.1801.1 2
360.277 odd 12 2700.2.s.b.1549.1 4
360.293 even 12 900.2.s.b.349.1 4
360.317 even 12 900.2.s.b.49.1 4
360.349 even 6 2700.2.i.b.901.1 2
504.5 even 6 1764.2.i.c.1537.1 2
504.13 odd 6 5292.2.j.a.3529.1 2
504.61 odd 6 5292.2.l.c.361.1 2
504.101 even 6 1764.2.i.c.373.1 2
504.149 odd 6 1764.2.i.a.1537.1 2
504.157 odd 6 5292.2.l.c.3313.1 2
504.173 even 6 1764.2.l.a.949.1 2
504.205 even 6 5292.2.l.a.361.1 2
504.221 odd 6 1764.2.l.c.961.1 2
504.229 odd 6 5292.2.i.a.2125.1 2
504.277 even 6 5292.2.i.c.1549.1 2
504.293 even 6 1764.2.j.b.1177.1 2
504.317 odd 6 1764.2.l.c.949.1 2
504.349 odd 6 5292.2.j.a.1765.1 2
504.373 even 6 5292.2.i.c.2125.1 2
504.389 odd 6 1764.2.i.a.373.1 2
504.437 even 6 1764.2.l.a.961.1 2
504.445 even 6 5292.2.l.a.3313.1 2
504.461 even 6 1764.2.j.b.589.1 2
504.493 odd 6 5292.2.i.a.1549.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.e.a.13.1 2 72.29 odd 6
36.2.e.a.25.1 yes 2 72.5 odd 6
108.2.e.a.37.1 2 72.61 even 6
108.2.e.a.73.1 2 72.13 even 6
144.2.i.a.49.1 2 72.11 even 6
144.2.i.a.97.1 2 72.59 even 6
324.2.a.a.1.1 1 8.5 even 2
324.2.a.c.1.1 1 24.5 odd 2
432.2.i.c.145.1 2 72.43 odd 6
432.2.i.c.289.1 2 72.67 odd 6
576.2.i.e.193.1 2 36.11 even 6
576.2.i.e.385.1 2 36.23 even 6
576.2.i.f.193.1 2 9.2 odd 6
576.2.i.f.385.1 2 9.5 odd 6
900.2.i.b.301.1 2 360.29 odd 6
900.2.i.b.601.1 2 360.149 odd 6
900.2.s.b.49.1 4 360.317 even 12
900.2.s.b.49.2 4 360.173 even 12
900.2.s.b.349.1 4 360.293 even 12
900.2.s.b.349.2 4 360.77 even 12
1296.2.a.b.1.1 1 8.3 odd 2
1296.2.a.k.1.1 1 24.11 even 2
1728.2.i.c.577.1 2 36.7 odd 6
1728.2.i.c.1153.1 2 36.31 odd 6
1728.2.i.d.577.1 2 9.7 even 3
1728.2.i.d.1153.1 2 9.4 even 3
1764.2.i.a.373.1 2 504.389 odd 6
1764.2.i.a.1537.1 2 504.149 odd 6
1764.2.i.c.373.1 2 504.101 even 6
1764.2.i.c.1537.1 2 504.5 even 6
1764.2.j.b.589.1 2 504.461 even 6
1764.2.j.b.1177.1 2 504.293 even 6
1764.2.l.a.949.1 2 504.173 even 6
1764.2.l.a.961.1 2 504.437 even 6
1764.2.l.c.949.1 2 504.317 odd 6
1764.2.l.c.961.1 2 504.221 odd 6
2700.2.i.b.901.1 2 360.349 even 6
2700.2.i.b.1801.1 2 360.229 even 6
2700.2.s.b.1549.1 4 360.277 odd 12
2700.2.s.b.1549.2 4 360.133 odd 12
2700.2.s.b.2449.1 4 360.13 odd 12
2700.2.s.b.2449.2 4 360.157 odd 12
5184.2.a.e.1.1 1 3.2 odd 2
5184.2.a.f.1.1 1 12.11 even 2
5184.2.a.ba.1.1 1 1.1 even 1 trivial
5184.2.a.bb.1.1 1 4.3 odd 2
5292.2.i.a.1549.1 2 504.493 odd 6
5292.2.i.a.2125.1 2 504.229 odd 6
5292.2.i.c.1549.1 2 504.277 even 6
5292.2.i.c.2125.1 2 504.373 even 6
5292.2.j.a.1765.1 2 504.349 odd 6
5292.2.j.a.3529.1 2 504.13 odd 6
5292.2.l.a.361.1 2 504.205 even 6
5292.2.l.a.3313.1 2 504.445 even 6
5292.2.l.c.361.1 2 504.61 odd 6
5292.2.l.c.3313.1 2 504.157 odd 6
8100.2.a.g.1.1 1 40.29 even 2
8100.2.a.j.1.1 1 120.29 odd 2
8100.2.d.c.649.1 2 40.37 odd 4
8100.2.d.c.649.2 2 40.13 odd 4
8100.2.d.h.649.1 2 120.77 even 4
8100.2.d.h.649.2 2 120.53 even 4