# Properties

 Label 5184.2.a.bq.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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 81) 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)

 $$f(q)$$ $$=$$ $$q+1.73205 q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q+1.73205 q^{5} -2.00000 q^{7} -3.46410 q^{11} +1.00000 q^{13} +5.19615 q^{17} +2.00000 q^{19} -3.46410 q^{23} -2.00000 q^{25} -1.73205 q^{29} -8.00000 q^{31} -3.46410 q^{35} +7.00000 q^{37} -6.92820 q^{41} +2.00000 q^{43} -6.92820 q^{47} -3.00000 q^{49} -6.00000 q^{55} +13.8564 q^{59} +7.00000 q^{61} +1.73205 q^{65} -10.0000 q^{67} +10.3923 q^{71} -7.00000 q^{73} +6.92820 q^{77} -2.00000 q^{79} -13.8564 q^{83} +9.00000 q^{85} -5.19615 q^{89} -2.00000 q^{91} +3.46410 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^7 $$2 q - 4 q^{7} + 2 q^{13} + 4 q^{19} - 4 q^{25} - 16 q^{31} + 14 q^{37} + 4 q^{43} - 6 q^{49} - 12 q^{55} + 14 q^{61} - 20 q^{67} - 14 q^{73} - 4 q^{79} + 18 q^{85} - 4 q^{91} + 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 + 2 * q^13 + 4 * q^19 - 4 * q^25 - 16 * q^31 + 14 * q^37 + 4 * q^43 - 6 * q^49 - 12 * q^55 + 14 * q^61 - 20 * q^67 - 14 * q^73 - 4 * q^79 + 18 * q^85 - 4 * q^91 + 4 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 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
$$17$$ 5.19615 1.26025 0.630126 0.776493i $$-0.283003\pi$$
0.630126 + 0.776493i $$0.283003\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ −2.00000 −0.400000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.73205 −0.321634 −0.160817 0.986984i $$-0.551413\pi$$
−0.160817 + 0.986984i $$0.551413\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.46410 −0.585540
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.92820 −1.08200 −0.541002 0.841021i $$-0.681955\pi$$
−0.541002 + 0.841021i $$0.681955\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −6.00000 −0.809040
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 13.8564 1.80395 0.901975 0.431788i $$-0.142117\pi$$
0.901975 + 0.431788i $$0.142117\pi$$
$$60$$ 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
$$65$$ 1.73205 0.214834
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.92820 0.789542
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −13.8564 −1.52094 −0.760469 0.649374i $$-0.775031\pi$$
−0.760469 + 0.649374i $$0.775031\pi$$
$$84$$ 0 0
$$85$$ 9.00000 0.976187
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −5.19615 −0.550791 −0.275396 0.961331i $$-0.588809\pi$$
−0.275396 + 0.961331i $$0.588809\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 3.46410 0.355409
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.92820 −0.689382 −0.344691 0.938716i $$-0.612016\pi$$
−0.344691 + 0.938716i $$0.612016\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.73205 −0.162938 −0.0814688 0.996676i $$-0.525961\pi$$
−0.0814688 + 0.996676i $$0.525961\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −10.3923 −0.952661
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −12.1244 −1.08444
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.46410 0.302660 0.151330 0.988483i $$-0.451644\pi$$
0.151330 + 0.988483i $$0.451644\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.73205 0.147979 0.0739895 0.997259i $$-0.476427\pi$$
0.0739895 + 0.997259i $$0.476427\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.46410 −0.289683
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −8.66025 −0.709476 −0.354738 0.934966i $$-0.615430\pi$$
−0.354738 + 0.934966i $$0.615430\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −13.8564 −1.11297
$$156$$ 0 0
$$157$$ −17.0000 −1.35675 −0.678374 0.734717i $$-0.737315\pi$$
−0.678374 + 0.734717i $$0.737315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.92820 0.546019
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 19.0526 1.44854 0.724270 0.689517i $$-0.242177\pi$$
0.724270 + 0.689517i $$0.242177\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 20.7846 1.55351 0.776757 0.629800i $$-0.216863\pi$$
0.776757 + 0.629800i $$0.216863\pi$$
$$180$$ 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
$$185$$ 12.1244 0.891400
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.3205 −1.25327 −0.626634 0.779314i $$-0.715568\pi$$
−0.626634 + 0.779314i $$0.715568\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.19615 0.370211 0.185105 0.982719i $$-0.440737\pi$$
0.185105 + 0.982719i $$0.440737\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3.46410 0.243132
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.92820 −0.479234
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 3.46410 0.236250
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.19615 0.349531
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −3.46410 −0.229920 −0.114960 0.993370i $$-0.536674\pi$$
−0.114960 + 0.993370i $$0.536674\pi$$
$$228$$ 0 0
$$229$$ 1.00000 0.0660819 0.0330409 0.999454i $$-0.489481\pi$$
0.0330409 + 0.999454i $$0.489481\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −25.9808 −1.70206 −0.851028 0.525120i $$-0.824020\pi$$
−0.851028 + 0.525120i $$0.824020\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 27.7128 1.79259 0.896296 0.443455i $$-0.146248\pi$$
0.896296 + 0.443455i $$0.146248\pi$$
$$240$$ 0 0
$$241$$ 29.0000 1.86805 0.934027 0.357202i $$-0.116269\pi$$
0.934027 + 0.357202i $$0.116269\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −5.19615 −0.331970
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.3923 0.655956 0.327978 0.944685i $$-0.393633\pi$$
0.327978 + 0.944685i $$0.393633\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 8.66025 0.540212 0.270106 0.962831i $$-0.412941\pi$$
0.270106 + 0.962831i $$0.412941\pi$$
$$258$$ 0 0
$$259$$ −14.0000 −0.869918
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6.92820 −0.427211 −0.213606 0.976920i $$-0.568521\pi$$
−0.213606 + 0.976920i $$0.568521\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 15.5885 0.950445 0.475223 0.879866i $$-0.342368\pi$$
0.475223 + 0.879866i $$0.342368\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.92820 0.417786
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.1244 0.723278 0.361639 0.932318i $$-0.382217\pi$$
0.361639 + 0.932318i $$0.382217\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 13.8564 0.817918
$$288$$ 0 0
$$289$$ 10.0000 0.588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −19.0526 −1.11306 −0.556531 0.830827i $$-0.687868\pi$$
−0.556531 + 0.830827i $$0.687868\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −3.46410 −0.200334
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12.1244 0.694239
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.92820 0.392862 0.196431 0.980518i $$-0.437065\pi$$
0.196431 + 0.980518i $$0.437065\pi$$
$$312$$ 0 0
$$313$$ −25.0000 −1.41308 −0.706542 0.707671i $$-0.749746\pi$$
−0.706542 + 0.707671i $$0.749746\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.66025 0.486408 0.243204 0.969975i $$-0.421801\pi$$
0.243204 + 0.969975i $$0.421801\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 10.3923 0.578243
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 13.8564 0.763928
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −17.3205 −0.946320
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 27.7128 1.50073
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.46410 0.185963 0.0929814 0.995668i $$-0.470360\pi$$
0.0929814 + 0.995668i $$0.470360\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −13.8564 −0.737502 −0.368751 0.929528i $$-0.620215\pi$$
−0.368751 + 0.929528i $$0.620215\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −10.3923 −0.548485 −0.274242 0.961661i $$-0.588427\pi$$
−0.274242 + 0.961661i $$0.588427\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −12.1244 −0.634618
$$366$$ 0 0
$$367$$ −20.0000 −1.04399 −0.521996 0.852948i $$-0.674812\pi$$
−0.521996 + 0.852948i $$0.674812\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.73205 −0.0892052
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 17.3205 0.885037 0.442518 0.896759i $$-0.354085\pi$$
0.442518 + 0.896759i $$0.354085\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −27.7128 −1.40510 −0.702548 0.711637i $$-0.747954\pi$$
−0.702548 + 0.711637i $$0.747954\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3.46410 −0.174298
$$396$$ 0 0
$$397$$ −29.0000 −1.45547 −0.727734 0.685859i $$-0.759427\pi$$
−0.727734 + 0.685859i $$0.759427\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.1244 −0.605461 −0.302731 0.953076i $$-0.597898\pi$$
−0.302731 + 0.953076i $$0.597898\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.2487 −1.20196
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −27.7128 −1.36366
$$414$$ 0 0
$$415$$ −24.0000 −1.17811
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6.92820 −0.338465 −0.169232 0.985576i $$-0.554129\pi$$
−0.169232 + 0.985576i $$0.554129\pi$$
$$420$$ 0 0
$$421$$ 25.0000 1.21843 0.609213 0.793007i $$-0.291486\pi$$
0.609213 + 0.793007i $$0.291486\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −10.3923 −0.504101
$$426$$ 0 0
$$427$$ −14.0000 −0.677507
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.92820 −0.331421
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −34.6410 −1.64584 −0.822922 0.568154i $$-0.807658\pi$$
−0.822922 + 0.568154i $$0.807658\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 20.7846 0.980886 0.490443 0.871473i $$-0.336835\pi$$
0.490443 + 0.871473i $$0.336835\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.46410 −0.162400
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.8564 0.645357 0.322679 0.946509i $$-0.395417\pi$$
0.322679 + 0.946509i $$0.395417\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −20.7846 −0.961797 −0.480899 0.876776i $$-0.659689\pi$$
−0.480899 + 0.876776i $$0.659689\pi$$
$$468$$ 0 0
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −6.92820 −0.318559
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.2487 1.10795 0.553976 0.832533i $$-0.313110\pi$$
0.553976 + 0.832533i $$0.313110\pi$$
$$480$$ 0 0
$$481$$ 7.00000 0.319173
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.46410 0.157297
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17.3205 −0.781664 −0.390832 0.920462i $$-0.627813\pi$$
−0.390832 + 0.920462i $$0.627813\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −20.7846 −0.932317
$$498$$ 0 0
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −20.7846 −0.926740 −0.463370 0.886165i $$-0.653360\pi$$
−0.463370 + 0.886165i $$0.653360\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 27.7128 1.22835 0.614174 0.789170i $$-0.289489\pi$$
0.614174 + 0.789170i $$0.289489\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −13.8564 −0.610586
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20.7846 −0.910590 −0.455295 0.890341i $$-0.650466\pi$$
−0.455295 + 0.890341i $$0.650466\pi$$
$$522$$ 0 0
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −41.5692 −1.81078
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6.92820 −0.300094
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 10.3923 0.447628
$$540$$ 0 0
$$541$$ −11.0000 −0.472927 −0.236463 0.971640i $$-0.575988\pi$$
−0.236463 + 0.971640i $$0.575988\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −19.0526 −0.816122
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.46410 −0.147576
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −36.3731 −1.54118 −0.770588 0.637333i $$-0.780037\pi$$
−0.770588 + 0.637333i $$0.780037\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 34.6410 1.45994 0.729972 0.683477i $$-0.239533\pi$$
0.729972 + 0.683477i $$0.239533\pi$$
$$564$$ 0 0
$$565$$ −3.00000 −0.126211
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 32.9090 1.37962 0.689808 0.723993i $$-0.257695\pi$$
0.689808 + 0.723993i $$0.257695\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.92820 0.288926
$$576$$ 0 0
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 27.7128 1.14972
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 38.1051 1.57277 0.786383 0.617739i $$-0.211951\pi$$
0.786383 + 0.617739i $$0.211951\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 15.5885 0.640141 0.320071 0.947394i $$-0.396293\pi$$
0.320071 + 0.947394i $$0.396293\pi$$
$$594$$ 0 0
$$595$$ −18.0000 −0.737928
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −13.8564 −0.566157 −0.283079 0.959097i $$-0.591356\pi$$
−0.283079 + 0.959097i $$0.591356\pi$$
$$600$$ 0 0
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1.73205 0.0704179
$$606$$ 0 0
$$607$$ −26.0000 −1.05531 −0.527654 0.849460i $$-0.676928\pi$$
−0.527654 + 0.849460i $$0.676928\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.92820 −0.280285
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.1244 −0.488108 −0.244054 0.969762i $$-0.578477\pi$$
−0.244054 + 0.969762i $$0.578477\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 36.3731 1.45029
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −3.46410 −0.137469
$$636$$ 0 0
$$637$$ −3.00000 −0.118864
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 22.5167 0.889355 0.444677 0.895691i $$-0.353318\pi$$
0.444677 + 0.895691i $$0.353318\pi$$
$$642$$ 0 0
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 31.1769 1.22569 0.612845 0.790203i $$-0.290025\pi$$
0.612845 + 0.790203i $$0.290025\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13.8564 −0.542243 −0.271122 0.962545i $$-0.587395\pi$$
−0.271122 + 0.962545i $$0.587395\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.46410 −0.134942 −0.0674711 0.997721i $$-0.521493\pi$$
−0.0674711 + 0.997721i $$0.521493\pi$$
$$660$$ 0 0
$$661$$ −17.0000 −0.661223 −0.330612 0.943767i $$-0.607255\pi$$
−0.330612 + 0.943767i $$0.607255\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −6.92820 −0.268664
$$666$$ 0 0
$$667$$ 6.00000 0.232321
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24.2487 −0.936111
$$672$$ 0 0
$$673$$ −25.0000 −0.963679 −0.481840 0.876259i $$-0.660031\pi$$
−0.481840 + 0.876259i $$0.660031\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13.8564 0.532545 0.266272 0.963898i $$-0.414208\pi$$
0.266272 + 0.963898i $$0.414208\pi$$
$$678$$ 0 0
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −20.7846 −0.795301 −0.397650 0.917537i $$-0.630174\pi$$
−0.397650 + 0.917537i $$0.630174\pi$$
$$684$$ 0 0
$$685$$ 3.00000 0.114624
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 13.8564 0.525603
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −46.7654 −1.76630 −0.883152 0.469087i $$-0.844583\pi$$
−0.883152 + 0.469087i $$0.844583\pi$$
$$702$$ 0 0
$$703$$ 14.0000 0.528020
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 13.8564 0.521124
$$708$$ 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
$$713$$ 27.7128 1.03785
$$714$$ 0 0
$$715$$ −6.00000 −0.224387
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 10.3923 0.387568 0.193784 0.981044i $$-0.437924\pi$$
0.193784 + 0.981044i $$0.437924\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 3.46410 0.128654
$$726$$ 0 0
$$727$$ 34.0000 1.26099 0.630495 0.776193i $$-0.282852\pi$$
0.630495 + 0.776193i $$0.282852\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10.3923 0.384373
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.6410 1.27602
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.92820 0.254171 0.127086 0.991892i $$-0.459438\pi$$
0.127086 + 0.991892i $$0.459438\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −34.6410 −1.26072
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.4449 1.06738 0.533688 0.845682i $$-0.320806\pi$$
0.533688 + 0.845682i $$0.320806\pi$$
$$762$$ 0 0
$$763$$ 22.0000 0.796453
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13.8564 0.500326
$$768$$ 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
$$773$$ 25.9808 0.934463 0.467232 0.884135i $$-0.345251\pi$$
0.467232 + 0.884135i $$0.345251\pi$$
$$774$$ 0 0
$$775$$ 16.0000 0.574737
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −13.8564 −0.496457
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −29.4449 −1.05093
$$786$$ 0 0
$$787$$ 26.0000 0.926800 0.463400 0.886149i $$-0.346629\pi$$
0.463400 + 0.886149i $$0.346629\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3.46410 0.123169
$$792$$ 0 0
$$793$$ 7.00000 0.248577
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 53.6936 1.90192 0.950962 0.309308i $$-0.100097\pi$$
0.950962 + 0.309308i $$0.100097\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 24.2487 0.855718
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −46.7654 −1.64418 −0.822091 0.569355i $$-0.807193\pi$$
−0.822091 + 0.569355i $$0.807193\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −27.7128 −0.970737
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −12.1244 −0.423143 −0.211571 0.977363i $$-0.567858\pi$$
−0.211571 + 0.977363i $$0.567858\pi$$
$$822$$ 0 0
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −10.3923 −0.361376 −0.180688 0.983540i $$-0.557832\pi$$
−0.180688 + 0.983540i $$0.557832\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −15.5885 −0.540108
$$834$$ 0 0
$$835$$ 30.0000 1.03819
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 45.0333 1.55472 0.777361 0.629054i $$-0.216558\pi$$
0.777361 + 0.629054i $$0.216558\pi$$
$$840$$ 0 0
$$841$$ −26.0000 −0.896552
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −20.7846 −0.715012
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −24.2487 −0.831235
$$852$$ 0 0
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 22.5167 0.769154 0.384577 0.923093i $$-0.374347\pi$$
0.384577 + 0.923093i $$0.374347\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −31.1769 −1.06127 −0.530637 0.847599i $$-0.678047\pi$$
−0.530637 + 0.847599i $$0.678047\pi$$
$$864$$ 0 0
$$865$$ 33.0000 1.12203
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 6.92820 0.235023
$$870$$ 0 0
$$871$$ −10.0000 −0.338837
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 24.2487 0.819756
$$876$$ 0 0
$$877$$ −53.0000 −1.78968 −0.894841 0.446384i $$-0.852711\pi$$
−0.894841 + 0.446384i $$0.852711\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 20.7846 0.700251 0.350126 0.936703i $$-0.386139\pi$$
0.350126 + 0.936703i $$0.386139\pi$$
$$882$$ 0 0
$$883$$ 56.0000 1.88455 0.942275 0.334840i $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −3.46410 −0.116313 −0.0581566 0.998307i $$-0.518522\pi$$
−0.0581566 + 0.998307i $$0.518522\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −13.8564 −0.463687
$$894$$ 0 0
$$895$$ 36.0000 1.20335
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 13.8564 0.462137
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −3.46410 −0.115151
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.2487 0.803396 0.401698 0.915772i $$-0.368420\pi$$
0.401698 + 0.915772i $$0.368420\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −6.92820 −0.228789
$$918$$ 0 0
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.3923 0.342067
$$924$$ 0 0
$$925$$ −14.0000 −0.460317
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −50.2295 −1.64798 −0.823988 0.566608i $$-0.808256\pi$$
−0.823988 + 0.566608i $$0.808256\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −31.1769 −1.01959
$$936$$ 0 0
$$937$$ −25.0000 −0.816714 −0.408357 0.912822i $$-0.633898\pi$$
−0.408357 + 0.912822i $$0.633898\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −50.2295 −1.63743 −0.818717 0.574197i $$-0.805314\pi$$
−0.818717 + 0.574197i $$0.805314\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.3205 0.562841 0.281420 0.959585i $$-0.409194\pi$$
0.281420 + 0.959585i $$0.409194\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −5.19615 −0.168320 −0.0841599 0.996452i $$-0.526821\pi$$
−0.0841599 + 0.996452i $$0.526821\pi$$
$$954$$ 0 0
$$955$$ −30.0000 −0.970777
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −3.46410 −0.111862
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −1.73205 −0.0557567
$$966$$ 0 0
$$967$$ 46.0000 1.47926 0.739630 0.673014i $$-0.235000\pi$$
0.739630 + 0.673014i $$0.235000\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31.1769 1.00051 0.500257 0.865877i $$-0.333239\pi$$
0.500257 + 0.865877i $$0.333239\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −48.4974 −1.55157 −0.775785 0.630997i $$-0.782646\pi$$
−0.775785 + 0.630997i $$0.782646\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 34.6410 1.10488 0.552438 0.833554i $$-0.313697\pi$$
0.552438 + 0.833554i $$0.313697\pi$$
$$984$$ 0 0
$$985$$ 9.00000 0.286764
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.92820 −0.220304
$$990$$ 0 0
$$991$$ 34.0000 1.08005 0.540023 0.841650i $$-0.318416\pi$$
0.540023 + 0.841650i $$0.318416\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −34.6410 −1.09819
$$996$$ 0 0
$$997$$ 7.00000 0.221692 0.110846 0.993838i $$-0.464644\pi$$
0.110846 + 0.993838i $$0.464644\pi$$
$$998$$ 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.bq.1.2 2
3.2 odd 2 inner 5184.2.a.bq.1.1 2
4.3 odd 2 5184.2.a.br.1.2 2
8.3 odd 2 81.2.a.a.1.2 yes 2
8.5 even 2 1296.2.a.o.1.1 2
12.11 even 2 5184.2.a.br.1.1 2
24.5 odd 2 1296.2.a.o.1.2 2
24.11 even 2 81.2.a.a.1.1 2
40.3 even 4 2025.2.b.k.649.1 4
40.19 odd 2 2025.2.a.j.1.1 2
40.27 even 4 2025.2.b.k.649.4 4
56.27 even 2 3969.2.a.i.1.2 2
72.5 odd 6 1296.2.i.s.865.1 4
72.11 even 6 81.2.c.b.28.2 4
72.13 even 6 1296.2.i.s.865.2 4
72.29 odd 6 1296.2.i.s.433.1 4
72.43 odd 6 81.2.c.b.28.1 4
72.59 even 6 81.2.c.b.55.2 4
72.61 even 6 1296.2.i.s.433.2 4
72.67 odd 6 81.2.c.b.55.1 4
88.43 even 2 9801.2.a.v.1.1 2
120.59 even 2 2025.2.a.j.1.2 2
120.83 odd 4 2025.2.b.k.649.3 4
120.107 odd 4 2025.2.b.k.649.2 4
168.83 odd 2 3969.2.a.i.1.1 2
216.11 even 18 729.2.e.o.82.1 12
216.43 odd 18 729.2.e.o.82.2 12
216.59 even 18 729.2.e.o.649.1 12
216.67 odd 18 729.2.e.o.163.1 12
216.83 even 18 729.2.e.o.568.2 12
216.115 odd 18 729.2.e.o.325.2 12
216.131 even 18 729.2.e.o.406.1 12
216.139 odd 18 729.2.e.o.406.2 12
216.155 even 18 729.2.e.o.325.1 12
216.187 odd 18 729.2.e.o.568.1 12
216.203 even 18 729.2.e.o.163.2 12
216.211 odd 18 729.2.e.o.649.2 12
264.131 odd 2 9801.2.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 24.11 even 2
81.2.a.a.1.2 yes 2 8.3 odd 2
81.2.c.b.28.1 4 72.43 odd 6
81.2.c.b.28.2 4 72.11 even 6
81.2.c.b.55.1 4 72.67 odd 6
81.2.c.b.55.2 4 72.59 even 6
729.2.e.o.82.1 12 216.11 even 18
729.2.e.o.82.2 12 216.43 odd 18
729.2.e.o.163.1 12 216.67 odd 18
729.2.e.o.163.2 12 216.203 even 18
729.2.e.o.325.1 12 216.155 even 18
729.2.e.o.325.2 12 216.115 odd 18
729.2.e.o.406.1 12 216.131 even 18
729.2.e.o.406.2 12 216.139 odd 18
729.2.e.o.568.1 12 216.187 odd 18
729.2.e.o.568.2 12 216.83 even 18
729.2.e.o.649.1 12 216.59 even 18
729.2.e.o.649.2 12 216.211 odd 18
1296.2.a.o.1.1 2 8.5 even 2
1296.2.a.o.1.2 2 24.5 odd 2
1296.2.i.s.433.1 4 72.29 odd 6
1296.2.i.s.433.2 4 72.61 even 6
1296.2.i.s.865.1 4 72.5 odd 6
1296.2.i.s.865.2 4 72.13 even 6
2025.2.a.j.1.1 2 40.19 odd 2
2025.2.a.j.1.2 2 120.59 even 2
2025.2.b.k.649.1 4 40.3 even 4
2025.2.b.k.649.2 4 120.107 odd 4
2025.2.b.k.649.3 4 120.83 odd 4
2025.2.b.k.649.4 4 40.27 even 4
3969.2.a.i.1.1 2 168.83 odd 2
3969.2.a.i.1.2 2 56.27 even 2
5184.2.a.bq.1.1 2 3.2 odd 2 inner
5184.2.a.bq.1.2 2 1.1 even 1 trivial
5184.2.a.br.1.1 2 12.11 even 2
5184.2.a.br.1.2 2 4.3 odd 2
9801.2.a.v.1.1 2 88.43 even 2
9801.2.a.v.1.2 2 264.131 odd 2