# Properties

 Label 9072.2.a.cd.1.2 Level $9072$ Weight $2$ Character 9072.1 Self dual yes Analytic conductor $72.440$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.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: $$72.4402847137$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.46050$$ of defining polynomial Character $$\chi$$ $$=$$ 9072.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.59358 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q+2.59358 q^{5} +1.00000 q^{7} +4.51459 q^{11} +1.00000 q^{13} +0.945916 q^{17} +4.05408 q^{19} -0.273346 q^{23} +1.72665 q^{25} -2.46050 q^{29} -2.32743 q^{31} +2.59358 q^{35} +1.78074 q^{37} +6.40642 q^{41} +10.4356 q^{43} -12.1623 q^{47} +1.00000 q^{49} +6.27335 q^{53} +11.7089 q^{55} -2.72665 q^{59} -2.27335 q^{61} +2.59358 q^{65} +15.8171 q^{67} +3.27335 q^{71} -1.50739 q^{73} +4.51459 q^{77} -14.7089 q^{79} -0.945916 q^{83} +2.45331 q^{85} +14.3566 q^{89} +1.00000 q^{91} +10.5146 q^{95} -11.4897 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 5 q^{5} + 3 q^{7}+O(q^{10})$$ 3 * q + 5 * q^5 + 3 * q^7 $$3 q + 5 q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 12 q^{17} + 3 q^{19} + 6 q^{25} - q^{29} + 3 q^{31} + 5 q^{35} - 3 q^{37} + 22 q^{41} + 3 q^{43} - 9 q^{47} + 3 q^{49} + 18 q^{53} + 6 q^{55} - 9 q^{59} - 6 q^{61} + 5 q^{65} + 9 q^{71} + 3 q^{73} - 2 q^{77} - 15 q^{79} - 12 q^{83} + 9 q^{85} + 2 q^{89} + 3 q^{91} + 16 q^{95} + 3 q^{97}+O(q^{100})$$ 3 * q + 5 * q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 12 * q^17 + 3 * q^19 + 6 * q^25 - q^29 + 3 * q^31 + 5 * q^35 - 3 * q^37 + 22 * q^41 + 3 * q^43 - 9 * q^47 + 3 * q^49 + 18 * q^53 + 6 * q^55 - 9 * q^59 - 6 * q^61 + 5 * q^65 + 9 * q^71 + 3 * q^73 - 2 * q^77 - 15 * q^79 - 12 * q^83 + 9 * q^85 + 2 * q^89 + 3 * q^91 + 16 * q^95 + 3 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.59358 1.15988 0.579942 0.814658i $$-0.303075\pi$$
0.579942 + 0.814658i $$0.303075\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.51459 1.36120 0.680600 0.732655i $$-0.261719\pi$$
0.680600 + 0.732655i $$0.261719\pi$$
$$12$$ 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
$$17$$ 0.945916 0.229418 0.114709 0.993399i $$-0.463406\pi$$
0.114709 + 0.993399i $$0.463406\pi$$
$$18$$ 0 0
$$19$$ 4.05408 0.930071 0.465035 0.885292i $$-0.346042\pi$$
0.465035 + 0.885292i $$0.346042\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.273346 −0.0569966 −0.0284983 0.999594i $$-0.509073\pi$$
−0.0284983 + 0.999594i $$0.509073\pi$$
$$24$$ 0 0
$$25$$ 1.72665 0.345331
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.46050 −0.456904 −0.228452 0.973555i $$-0.573366\pi$$
−0.228452 + 0.973555i $$0.573366\pi$$
$$30$$ 0 0
$$31$$ −2.32743 −0.418019 −0.209009 0.977914i $$-0.567024\pi$$
−0.209009 + 0.977914i $$0.567024\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.59358 0.438395
$$36$$ 0 0
$$37$$ 1.78074 0.292752 0.146376 0.989229i $$-0.453239\pi$$
0.146376 + 0.989229i $$0.453239\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.40642 1.00051 0.500257 0.865877i $$-0.333239\pi$$
0.500257 + 0.865877i $$0.333239\pi$$
$$42$$ 0 0
$$43$$ 10.4356 1.59141 0.795707 0.605682i $$-0.207100\pi$$
0.795707 + 0.605682i $$0.207100\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.1623 −1.77405 −0.887023 0.461724i $$-0.847231\pi$$
−0.887023 + 0.461724i $$0.847231\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.27335 0.861710 0.430855 0.902421i $$-0.358212\pi$$
0.430855 + 0.902421i $$0.358212\pi$$
$$54$$ 0 0
$$55$$ 11.7089 1.57883
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −2.72665 −0.354980 −0.177490 0.984123i $$-0.556798\pi$$
−0.177490 + 0.984123i $$0.556798\pi$$
$$60$$ 0 0
$$61$$ −2.27335 −0.291072 −0.145536 0.989353i $$-0.546491\pi$$
−0.145536 + 0.989353i $$0.546491\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.59358 0.321694
$$66$$ 0 0
$$67$$ 15.8171 1.93237 0.966184 0.257854i $$-0.0830152\pi$$
0.966184 + 0.257854i $$0.0830152\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 3.27335 0.388475 0.194237 0.980955i $$-0.437777\pi$$
0.194237 + 0.980955i $$0.437777\pi$$
$$72$$ 0 0
$$73$$ −1.50739 −0.176427 −0.0882134 0.996102i $$-0.528116\pi$$
−0.0882134 + 0.996102i $$0.528116\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.51459 0.514485
$$78$$ 0 0
$$79$$ −14.7089 −1.65489 −0.827443 0.561550i $$-0.810205\pi$$
−0.827443 + 0.561550i $$0.810205\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −0.945916 −0.103828 −0.0519139 0.998652i $$-0.516532\pi$$
−0.0519139 + 0.998652i $$0.516532\pi$$
$$84$$ 0 0
$$85$$ 2.45331 0.266099
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 14.3566 1.52180 0.760899 0.648871i $$-0.224758\pi$$
0.760899 + 0.648871i $$0.224758\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 10.5146 1.07877
$$96$$ 0 0
$$97$$ −11.4897 −1.16660 −0.583300 0.812257i $$-0.698239\pi$$
−0.583300 + 0.812257i $$0.698239\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.67977 0.366150 0.183075 0.983099i $$-0.441395\pi$$
0.183075 + 0.983099i $$0.441395\pi$$
$$102$$ 0 0
$$103$$ 9.72665 0.958396 0.479198 0.877707i $$-0.340928\pi$$
0.479198 + 0.877707i $$0.340928\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1.37432 −0.132860 −0.0664301 0.997791i $$-0.521161\pi$$
−0.0664301 + 0.997791i $$0.521161\pi$$
$$108$$ 0 0
$$109$$ −3.39922 −0.325587 −0.162793 0.986660i $$-0.552050\pi$$
−0.162793 + 0.986660i $$0.552050\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.3887 −0.977288 −0.488644 0.872483i $$-0.662508\pi$$
−0.488644 + 0.872483i $$0.662508\pi$$
$$114$$ 0 0
$$115$$ −0.708945 −0.0661095
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0.945916 0.0867120
$$120$$ 0 0
$$121$$ 9.38151 0.852865
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −8.48968 −0.759340
$$126$$ 0 0
$$127$$ −0.672570 −0.0596809 −0.0298405 0.999555i $$-0.509500\pi$$
−0.0298405 + 0.999555i $$0.509500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 7.91381 0.691433 0.345717 0.938339i $$-0.387636\pi$$
0.345717 + 0.938339i $$0.387636\pi$$
$$132$$ 0 0
$$133$$ 4.05408 0.351534
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.67257 0.313769 0.156884 0.987617i $$-0.449855\pi$$
0.156884 + 0.987617i $$0.449855\pi$$
$$138$$ 0 0
$$139$$ 2.05408 0.174225 0.0871126 0.996198i $$-0.472236\pi$$
0.0871126 + 0.996198i $$0.472236\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.51459 0.377529
$$144$$ 0 0
$$145$$ −6.38151 −0.529956
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 13.5438 1.10955 0.554774 0.832001i $$-0.312805\pi$$
0.554774 + 0.832001i $$0.312805\pi$$
$$150$$ 0 0
$$151$$ −9.92821 −0.807946 −0.403973 0.914771i $$-0.632371\pi$$
−0.403973 + 0.914771i $$0.632371\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.03638 −0.484853
$$156$$ 0 0
$$157$$ 6.05408 0.483169 0.241584 0.970380i $$-0.422333\pi$$
0.241584 + 0.970380i $$0.422333\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.273346 −0.0215427
$$162$$ 0 0
$$163$$ −17.8171 −1.39554 −0.697772 0.716320i $$-0.745825\pi$$
−0.697772 + 0.716320i $$0.745825\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.46770 −0.655250 −0.327625 0.944808i $$-0.606248\pi$$
−0.327625 + 0.944808i $$0.606248\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −17.3566 −1.31960 −0.659799 0.751442i $$-0.729359\pi$$
−0.659799 + 0.751442i $$0.729359\pi$$
$$174$$ 0 0
$$175$$ 1.72665 0.130523
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −11.3494 −0.848295 −0.424147 0.905593i $$-0.639426\pi$$
−0.424147 + 0.905593i $$0.639426\pi$$
$$180$$ 0 0
$$181$$ 21.8889 1.62699 0.813495 0.581572i $$-0.197562\pi$$
0.813495 + 0.581572i $$0.197562\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.61849 0.339558
$$186$$ 0 0
$$187$$ 4.27042 0.312284
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −0.701748 −0.0507767 −0.0253883 0.999678i $$-0.508082\pi$$
−0.0253883 + 0.999678i $$0.508082\pi$$
$$192$$ 0 0
$$193$$ 12.1445 0.874183 0.437092 0.899417i $$-0.356009\pi$$
0.437092 + 0.899417i $$0.356009\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.4107 1.16921 0.584607 0.811317i $$-0.301249\pi$$
0.584607 + 0.811317i $$0.301249\pi$$
$$198$$ 0 0
$$199$$ −22.7060 −1.60959 −0.804794 0.593555i $$-0.797724\pi$$
−0.804794 + 0.593555i $$0.797724\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2.46050 −0.172694
$$204$$ 0 0
$$205$$ 16.6156 1.16048
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 18.3025 1.26601
$$210$$ 0 0
$$211$$ −4.56148 −0.314025 −0.157012 0.987597i $$-0.550186\pi$$
−0.157012 + 0.987597i $$0.550186\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 27.0656 1.84586
$$216$$ 0 0
$$217$$ −2.32743 −0.157996
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.945916 0.0636292
$$222$$ 0 0
$$223$$ −13.3245 −0.892275 −0.446137 0.894964i $$-0.647201\pi$$
−0.446137 + 0.894964i $$0.647201\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.38151 0.0916943 0.0458472 0.998948i $$-0.485401\pi$$
0.0458472 + 0.998948i $$0.485401\pi$$
$$228$$ 0 0
$$229$$ −17.9794 −1.18811 −0.594055 0.804424i $$-0.702474\pi$$
−0.594055 + 0.804424i $$0.702474\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.9823 1.24357 0.621786 0.783187i $$-0.286408\pi$$
0.621786 + 0.783187i $$0.286408\pi$$
$$234$$ 0 0
$$235$$ −31.5438 −2.05769
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4.89183 0.316426 0.158213 0.987405i $$-0.449427\pi$$
0.158213 + 0.987405i $$0.449427\pi$$
$$240$$ 0 0
$$241$$ −26.1593 −1.68507 −0.842535 0.538641i $$-0.818938\pi$$
−0.842535 + 0.538641i $$0.818938\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.59358 0.165698
$$246$$ 0 0
$$247$$ 4.05408 0.257955
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.4576 1.16503 0.582516 0.812819i $$-0.302068\pi$$
0.582516 + 0.812819i $$0.302068\pi$$
$$252$$ 0 0
$$253$$ −1.23405 −0.0775838
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 11.7339 0.731938 0.365969 0.930627i $$-0.380738\pi$$
0.365969 + 0.930627i $$0.380738\pi$$
$$258$$ 0 0
$$259$$ 1.78074 0.110650
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7.52179 −0.463813 −0.231907 0.972738i $$-0.574496\pi$$
−0.231907 + 0.972738i $$0.574496\pi$$
$$264$$ 0 0
$$265$$ 16.2704 0.999484
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 18.8348 1.14838 0.574190 0.818722i $$-0.305317\pi$$
0.574190 + 0.818722i $$0.305317\pi$$
$$270$$ 0 0
$$271$$ 23.9823 1.45682 0.728410 0.685141i $$-0.240260\pi$$
0.728410 + 0.685141i $$0.240260\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 7.79513 0.470064
$$276$$ 0 0
$$277$$ 7.16225 0.430338 0.215169 0.976577i $$-0.430970\pi$$
0.215169 + 0.976577i $$0.430970\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −14.8817 −0.887768 −0.443884 0.896084i $$-0.646400\pi$$
−0.443884 + 0.896084i $$0.646400\pi$$
$$282$$ 0 0
$$283$$ −19.9971 −1.18870 −0.594351 0.804205i $$-0.702591\pi$$
−0.594351 + 0.804205i $$0.702591\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.40642 0.378159
$$288$$ 0 0
$$289$$ −16.1052 −0.947367
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −15.0656 −0.880139 −0.440070 0.897964i $$-0.645046\pi$$
−0.440070 + 0.897964i $$0.645046\pi$$
$$294$$ 0 0
$$295$$ −7.07179 −0.411736
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −0.273346 −0.0158080
$$300$$ 0 0
$$301$$ 10.4356 0.601498
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −5.89610 −0.337610
$$306$$ 0 0
$$307$$ 27.2704 1.55641 0.778203 0.628013i $$-0.216132\pi$$
0.778203 + 0.628013i $$0.216132\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −15.9823 −0.906273 −0.453136 0.891441i $$-0.649695\pi$$
−0.453136 + 0.891441i $$0.649695\pi$$
$$312$$ 0 0
$$313$$ 11.5979 0.655549 0.327775 0.944756i $$-0.393701\pi$$
0.327775 + 0.944756i $$0.393701\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.01771 0.113326 0.0566629 0.998393i $$-0.481954\pi$$
0.0566629 + 0.998393i $$0.481954\pi$$
$$318$$ 0 0
$$319$$ −11.1082 −0.621938
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.83482 0.213375
$$324$$ 0 0
$$325$$ 1.72665 0.0957775
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −12.1623 −0.670527
$$330$$ 0 0
$$331$$ 19.7089 1.08330 0.541651 0.840604i $$-0.317800\pi$$
0.541651 + 0.840604i $$0.317800\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 41.0229 2.24132
$$336$$ 0 0
$$337$$ −29.0512 −1.58252 −0.791259 0.611481i $$-0.790574\pi$$
−0.791259 + 0.611481i $$0.790574\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.5074 −0.569007
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 29.0833 1.56127 0.780636 0.624986i $$-0.214895\pi$$
0.780636 + 0.624986i $$0.214895\pi$$
$$348$$ 0 0
$$349$$ 24.7630 1.32553 0.662767 0.748825i $$-0.269382\pi$$
0.662767 + 0.748825i $$0.269382\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 33.3025 1.77251 0.886257 0.463193i $$-0.153296\pi$$
0.886257 + 0.463193i $$0.153296\pi$$
$$354$$ 0 0
$$355$$ 8.48968 0.450586
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 25.5366 1.34777 0.673884 0.738837i $$-0.264625\pi$$
0.673884 + 0.738837i $$0.264625\pi$$
$$360$$ 0 0
$$361$$ −2.56440 −0.134968
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3.90954 −0.204635
$$366$$ 0 0
$$367$$ −27.4504 −1.43290 −0.716449 0.697639i $$-0.754234\pi$$
−0.716449 + 0.697639i $$0.754234\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.27335 0.325696
$$372$$ 0 0
$$373$$ 16.3274 0.845402 0.422701 0.906269i $$-0.361082\pi$$
0.422701 + 0.906269i $$0.361082\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.46050 −0.126722
$$378$$ 0 0
$$379$$ −12.0364 −0.618267 −0.309134 0.951019i $$-0.600039\pi$$
−0.309134 + 0.951019i $$0.600039\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12.4356 −0.635429 −0.317715 0.948186i $$-0.602915\pi$$
−0.317715 + 0.948186i $$0.602915\pi$$
$$384$$ 0 0
$$385$$ 11.7089 0.596743
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −20.6008 −1.04450 −0.522250 0.852792i $$-0.674907\pi$$
−0.522250 + 0.852792i $$0.674907\pi$$
$$390$$ 0 0
$$391$$ −0.258562 −0.0130761
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −38.1488 −1.91948
$$396$$ 0 0
$$397$$ −23.6372 −1.18631 −0.593157 0.805087i $$-0.702119\pi$$
−0.593157 + 0.805087i $$0.702119\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.56440 0.128060 0.0640300 0.997948i $$-0.479605\pi$$
0.0640300 + 0.997948i $$0.479605\pi$$
$$402$$ 0 0
$$403$$ −2.32743 −0.115938
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.03930 0.398493
$$408$$ 0 0
$$409$$ −34.3245 −1.69724 −0.848619 0.529005i $$-0.822565\pi$$
−0.848619 + 0.529005i $$0.822565\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2.72665 −0.134170
$$414$$ 0 0
$$415$$ −2.45331 −0.120428
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4.05701 0.198198 0.0990989 0.995078i $$-0.468404\pi$$
0.0990989 + 0.995078i $$0.468404\pi$$
$$420$$ 0 0
$$421$$ −21.0689 −1.02683 −0.513417 0.858139i $$-0.671621\pi$$
−0.513417 + 0.858139i $$0.671621\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1.63327 0.0792252
$$426$$ 0 0
$$427$$ −2.27335 −0.110015
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.6185 1.08949 0.544747 0.838600i $$-0.316626\pi$$
0.544747 + 0.838600i $$0.316626\pi$$
$$432$$ 0 0
$$433$$ 2.41789 0.116196 0.0580982 0.998311i $$-0.481496\pi$$
0.0580982 + 0.998311i $$0.481496\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.10817 −0.0530109
$$438$$ 0 0
$$439$$ 23.4897 1.12110 0.560551 0.828120i $$-0.310589\pi$$
0.560551 + 0.828120i $$0.310589\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −13.4179 −0.637503 −0.318752 0.947838i $$-0.603264\pi$$
−0.318752 + 0.947838i $$0.603264\pi$$
$$444$$ 0 0
$$445$$ 37.2350 1.76511
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.16225 0.432393 0.216197 0.976350i $$-0.430635\pi$$
0.216197 + 0.976350i $$0.430635\pi$$
$$450$$ 0 0
$$451$$ 28.9224 1.36190
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.59358 0.121589
$$456$$ 0 0
$$457$$ 8.81711 0.412447 0.206224 0.978505i $$-0.433883\pi$$
0.206224 + 0.978505i $$0.433883\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5.65913 0.263572 0.131786 0.991278i $$-0.457929\pi$$
0.131786 + 0.991278i $$0.457929\pi$$
$$462$$ 0 0
$$463$$ −15.7267 −0.730880 −0.365440 0.930835i $$-0.619081\pi$$
−0.365440 + 0.930835i $$0.619081\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −21.9971 −1.01790 −0.508952 0.860795i $$-0.669967\pi$$
−0.508952 + 0.860795i $$0.669967\pi$$
$$468$$ 0 0
$$469$$ 15.8171 0.730366
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 47.1124 2.16623
$$474$$ 0 0
$$475$$ 7.00000 0.321182
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.9751 1.14114 0.570571 0.821249i $$-0.306722\pi$$
0.570571 + 0.821249i $$0.306722\pi$$
$$480$$ 0 0
$$481$$ 1.78074 0.0811947
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −29.7994 −1.35312
$$486$$ 0 0
$$487$$ 17.5979 0.797435 0.398717 0.917074i $$-0.369455\pi$$
0.398717 + 0.917074i $$0.369455\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 13.7951 0.622566 0.311283 0.950317i $$-0.399241\pi$$
0.311283 + 0.950317i $$0.399241\pi$$
$$492$$ 0 0
$$493$$ −2.32743 −0.104822
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.27335 0.146830
$$498$$ 0 0
$$499$$ −13.0875 −0.585879 −0.292939 0.956131i $$-0.594633\pi$$
−0.292939 + 0.956131i $$0.594633\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −22.3068 −0.994611 −0.497305 0.867576i $$-0.665677\pi$$
−0.497305 + 0.867576i $$0.665677\pi$$
$$504$$ 0 0
$$505$$ 9.54377 0.424692
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 15.8932 0.704453 0.352226 0.935915i $$-0.385425\pi$$
0.352226 + 0.935915i $$0.385425\pi$$
$$510$$ 0 0
$$511$$ −1.50739 −0.0666831
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 25.2268 1.11163
$$516$$ 0 0
$$517$$ −54.9076 −2.41483
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.41789 −0.193551 −0.0967756 0.995306i $$-0.530853\pi$$
−0.0967756 + 0.995306i $$0.530853\pi$$
$$522$$ 0 0
$$523$$ −25.2733 −1.10513 −0.552563 0.833471i $$-0.686350\pi$$
−0.552563 + 0.833471i $$0.686350\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.20155 −0.0959012
$$528$$ 0 0
$$529$$ −22.9253 −0.996751
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 6.40642 0.277493
$$534$$ 0 0
$$535$$ −3.56440 −0.154103
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4.51459 0.194457
$$540$$ 0 0
$$541$$ −3.43852 −0.147834 −0.0739168 0.997264i $$-0.523550\pi$$
−0.0739168 + 0.997264i $$0.523550\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −8.81616 −0.377643
$$546$$ 0 0
$$547$$ 6.92821 0.296229 0.148114 0.988970i $$-0.452680\pi$$
0.148114 + 0.988970i $$0.452680\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −9.97509 −0.424953
$$552$$ 0 0
$$553$$ −14.7089 −0.625488
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −33.5835 −1.42298 −0.711488 0.702698i $$-0.751979\pi$$
−0.711488 + 0.702698i $$0.751979\pi$$
$$558$$ 0 0
$$559$$ 10.4356 0.441379
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 42.4792 1.79028 0.895142 0.445781i $$-0.147074\pi$$
0.895142 + 0.445781i $$0.147074\pi$$
$$564$$ 0 0
$$565$$ −26.9439 −1.13354
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.4035 −0.436137 −0.218069 0.975933i $$-0.569976\pi$$
−0.218069 + 0.975933i $$0.569976\pi$$
$$570$$ 0 0
$$571$$ −17.8496 −0.746983 −0.373491 0.927634i $$-0.621839\pi$$
−0.373491 + 0.927634i $$0.621839\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −0.471974 −0.0196827
$$576$$ 0 0
$$577$$ 11.9430 0.497193 0.248597 0.968607i $$-0.420031\pi$$
0.248597 + 0.968607i $$0.420031\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.945916 −0.0392432
$$582$$ 0 0
$$583$$ 28.3216 1.17296
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 23.8597 0.984796 0.492398 0.870370i $$-0.336120\pi$$
0.492398 + 0.870370i $$0.336120\pi$$
$$588$$ 0 0
$$589$$ −9.43560 −0.388787
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −19.5801 −0.804060 −0.402030 0.915626i $$-0.631695\pi$$
−0.402030 + 0.915626i $$0.631695\pi$$
$$594$$ 0 0
$$595$$ 2.45331 0.100576
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 18.5467 0.757797 0.378899 0.925438i $$-0.376303\pi$$
0.378899 + 0.925438i $$0.376303\pi$$
$$600$$ 0 0
$$601$$ −18.1986 −0.742338 −0.371169 0.928565i $$-0.621043\pi$$
−0.371169 + 0.928565i $$0.621043\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 24.3317 0.989224
$$606$$ 0 0
$$607$$ 22.3097 0.905524 0.452762 0.891631i $$-0.350439\pi$$
0.452762 + 0.891631i $$0.350439\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.1623 −0.492032
$$612$$ 0 0
$$613$$ 10.2370 0.413467 0.206734 0.978397i $$-0.433717\pi$$
0.206734 + 0.978397i $$0.433717\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11.3274 0.456025 0.228013 0.973658i $$-0.426777\pi$$
0.228013 + 0.973658i $$0.426777\pi$$
$$618$$ 0 0
$$619$$ −8.63327 −0.347000 −0.173500 0.984834i $$-0.555508\pi$$
−0.173500 + 0.984834i $$0.555508\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 14.3566 0.575185
$$624$$ 0 0
$$625$$ −30.6519 −1.22608
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.68443 0.0671626
$$630$$ 0 0
$$631$$ 14.8535 0.591308 0.295654 0.955295i $$-0.404462\pi$$
0.295654 + 0.955295i $$0.404462\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1.74436 −0.0692229
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 34.1593 1.34921 0.674606 0.738178i $$-0.264313\pi$$
0.674606 + 0.738178i $$0.264313\pi$$
$$642$$ 0 0
$$643$$ 10.8348 0.427284 0.213642 0.976912i $$-0.431467\pi$$
0.213642 + 0.976912i $$0.431467\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32.9692 1.29615 0.648077 0.761575i $$-0.275573\pi$$
0.648077 + 0.761575i $$0.275573\pi$$
$$648$$ 0 0
$$649$$ −12.3097 −0.483199
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3.93113 0.153837 0.0769185 0.997037i $$-0.475492\pi$$
0.0769185 + 0.997037i $$0.475492\pi$$
$$654$$ 0 0
$$655$$ 20.5251 0.801982
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 16.8171 0.655102 0.327551 0.944834i $$-0.393777\pi$$
0.327551 + 0.944834i $$0.393777\pi$$
$$660$$ 0 0
$$661$$ −17.0216 −0.662063 −0.331032 0.943620i $$-0.607397\pi$$
−0.331032 + 0.943620i $$0.607397\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10.5146 0.407738
$$666$$ 0 0
$$667$$ 0.672570 0.0260420
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.2632 −0.396207
$$672$$ 0 0
$$673$$ 28.7453 1.10805 0.554025 0.832500i $$-0.313091\pi$$
0.554025 + 0.832500i $$0.313091\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.03638 0.231997 0.115998 0.993249i $$-0.462993\pi$$
0.115998 + 0.993249i $$0.462993\pi$$
$$678$$ 0 0
$$679$$ −11.4897 −0.440934
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 20.5113 0.784842 0.392421 0.919786i $$-0.371638\pi$$
0.392421 + 0.919786i $$0.371638\pi$$
$$684$$ 0 0
$$685$$ 9.52510 0.363935
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 6.27335 0.238995
$$690$$ 0 0
$$691$$ 15.0029 0.570738 0.285369 0.958418i $$-0.407884\pi$$
0.285369 + 0.958418i $$0.407884\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 5.32743 0.202081
$$696$$ 0 0
$$697$$ 6.05993 0.229536
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −38.5113 −1.45455 −0.727275 0.686346i $$-0.759214\pi$$
−0.727275 + 0.686346i $$0.759214\pi$$
$$702$$ 0 0
$$703$$ 7.21926 0.272280
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 3.67977 0.138392
$$708$$ 0 0
$$709$$ 7.64008 0.286929 0.143465 0.989655i $$-0.454176\pi$$
0.143465 + 0.989655i $$0.454176\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0.636194 0.0238257
$$714$$ 0 0
$$715$$ 11.7089 0.437890
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −30.0364 −1.12017 −0.560084 0.828436i $$-0.689231\pi$$
−0.560084 + 0.828436i $$0.689231\pi$$
$$720$$ 0 0
$$721$$ 9.72665 0.362240
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4.24844 −0.157783
$$726$$ 0 0
$$727$$ −3.45623 −0.128185 −0.0640923 0.997944i $$-0.520415\pi$$
−0.0640923 + 0.997944i $$0.520415\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 9.87120 0.365099
$$732$$ 0 0
$$733$$ 38.5261 1.42299 0.711496 0.702690i $$-0.248018\pi$$
0.711496 + 0.702690i $$0.248018\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 71.4078 2.63034
$$738$$ 0 0
$$739$$ −45.1239 −1.65991 −0.829955 0.557830i $$-0.811634\pi$$
−0.829955 + 0.557830i $$0.811634\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.48676 0.348035 0.174018 0.984743i $$-0.444325\pi$$
0.174018 + 0.984743i $$0.444325\pi$$
$$744$$ 0 0
$$745$$ 35.1268 1.28695
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −1.37432 −0.0502165
$$750$$ 0 0
$$751$$ 9.83190 0.358771 0.179386 0.983779i $$-0.442589\pi$$
0.179386 + 0.983779i $$0.442589\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −25.7496 −0.937124
$$756$$ 0 0
$$757$$ −41.8171 −1.51987 −0.759934 0.650000i $$-0.774769\pi$$
−0.759934 + 0.650000i $$0.774769\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.9794 −0.833001 −0.416501 0.909135i $$-0.636744\pi$$
−0.416501 + 0.909135i $$0.636744\pi$$
$$762$$ 0 0
$$763$$ −3.39922 −0.123060
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.72665 −0.0984538
$$768$$ 0 0
$$769$$ 6.08658 0.219488 0.109744 0.993960i $$-0.464997\pi$$
0.109744 + 0.993960i $$0.464997\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 41.8214 1.50421 0.752105 0.659043i $$-0.229038\pi$$
0.752105 + 0.659043i $$0.229038\pi$$
$$774$$ 0 0
$$775$$ −4.01867 −0.144355
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 25.9722 0.930550
$$780$$ 0 0
$$781$$ 14.7778 0.528792
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 15.7017 0.560419
$$786$$ 0 0
$$787$$ −32.2920 −1.15109 −0.575543 0.817772i $$-0.695209\pi$$
−0.575543 + 0.817772i $$0.695209\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −10.3887 −0.369380
$$792$$ 0 0
$$793$$ −2.27335 −0.0807289
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −46.5657 −1.64944 −0.824722 0.565539i $$-0.808668\pi$$
−0.824722 + 0.565539i $$0.808668\pi$$
$$798$$ 0 0
$$799$$ −11.5045 −0.406999
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6.80525 −0.240152
$$804$$ 0 0
$$805$$ −0.708945 −0.0249870
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 10.8023 0.379790 0.189895 0.981804i $$-0.439185\pi$$
0.189895 + 0.981804i $$0.439185\pi$$
$$810$$ 0 0
$$811$$ −5.58307 −0.196048 −0.0980240 0.995184i $$-0.531252\pi$$
−0.0980240 + 0.995184i $$0.531252\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −46.2101 −1.61867
$$816$$ 0 0
$$817$$ 42.3068 1.48013
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.7879 1.10941 0.554703 0.832048i $$-0.312832\pi$$
0.554703 + 0.832048i $$0.312832\pi$$
$$822$$ 0 0
$$823$$ 36.0000 1.25488 0.627441 0.778664i $$-0.284103\pi$$
0.627441 + 0.778664i $$0.284103\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 15.9224 0.553675 0.276837 0.960917i $$-0.410714\pi$$
0.276837 + 0.960917i $$0.410714\pi$$
$$828$$ 0 0
$$829$$ 35.4720 1.23199 0.615996 0.787749i $$-0.288754\pi$$
0.615996 + 0.787749i $$0.288754\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0.945916 0.0327740
$$834$$ 0 0
$$835$$ −21.9617 −0.760014
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −54.6782 −1.88770 −0.943850 0.330373i $$-0.892825\pi$$
−0.943850 + 0.330373i $$0.892825\pi$$
$$840$$ 0 0
$$841$$ −22.9459 −0.791238
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −31.1230 −1.07066
$$846$$ 0 0
$$847$$ 9.38151 0.322353
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −0.486758 −0.0166858
$$852$$ 0 0
$$853$$ −2.19767 −0.0752468 −0.0376234 0.999292i $$-0.511979\pi$$
−0.0376234 + 0.999292i $$0.511979\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.7765 0.538914 0.269457 0.963012i $$-0.413156\pi$$
0.269457 + 0.963012i $$0.413156\pi$$
$$858$$ 0 0
$$859$$ −5.57626 −0.190260 −0.0951298 0.995465i $$-0.530327\pi$$
−0.0951298 + 0.995465i $$0.530327\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 23.1268 0.787247 0.393623 0.919272i $$-0.371221\pi$$
0.393623 + 0.919272i $$0.371221\pi$$
$$864$$ 0 0
$$865$$ −45.0157 −1.53058
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −66.4048 −2.25263
$$870$$ 0 0
$$871$$ 15.8171 0.535942
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −8.48968 −0.287004
$$876$$ 0 0
$$877$$ 3.92528 0.132547 0.0662737 0.997801i $$-0.478889\pi$$
0.0662737 + 0.997801i $$0.478889\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −27.1986 −0.916345 −0.458173 0.888863i $$-0.651496\pi$$
−0.458173 + 0.888863i $$0.651496\pi$$
$$882$$ 0 0
$$883$$ −8.21341 −0.276403 −0.138202 0.990404i $$-0.544132\pi$$
−0.138202 + 0.990404i $$0.544132\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −6.48114 −0.217615 −0.108808 0.994063i $$-0.534703\pi$$
−0.108808 + 0.994063i $$0.534703\pi$$
$$888$$ 0 0
$$889$$ −0.672570 −0.0225573
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −49.3068 −1.64999
$$894$$ 0 0
$$895$$ −29.4356 −0.983924
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 5.72665 0.190995
$$900$$ 0 0
$$901$$ 5.93406 0.197692
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 56.7706 1.88712
$$906$$ 0 0
$$907$$ −10.1288 −0.336321 −0.168161 0.985760i $$-0.553783\pi$$
−0.168161 + 0.985760i $$0.553783\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45.9224 1.52148 0.760738 0.649059i $$-0.224837\pi$$
0.760738 + 0.649059i $$0.224837\pi$$
$$912$$ 0 0
$$913$$ −4.27042 −0.141330
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 7.91381 0.261337
$$918$$ 0 0
$$919$$ 4.92432 0.162438 0.0812192 0.996696i $$-0.474119\pi$$
0.0812192 + 0.996696i $$0.474119\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 3.27335 0.107744
$$924$$ 0 0
$$925$$ 3.07472 0.101096
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −0.00758649 −0.000248905 0 −0.000124452 1.00000i $$-0.500040\pi$$
−0.000124452 1.00000i $$0.500040\pi$$
$$930$$ 0 0
$$931$$ 4.05408 0.132867
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 11.0757 0.362213
$$936$$ 0 0
$$937$$ 21.1623 0.691341 0.345670 0.938356i $$-0.387652\pi$$
0.345670 + 0.938356i $$0.387652\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −4.55816 −0.148592 −0.0742959 0.997236i $$-0.523671\pi$$
−0.0742959 + 0.997236i $$0.523671\pi$$
$$942$$ 0 0
$$943$$ −1.75117 −0.0570260
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13.7352 0.446334 0.223167 0.974780i $$-0.428360\pi$$
0.223167 + 0.974780i $$0.428360\pi$$
$$948$$ 0 0
$$949$$ −1.50739 −0.0489320
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.80699 −0.285286 −0.142643 0.989774i $$-0.545560\pi$$
−0.142643 + 0.989774i $$0.545560\pi$$
$$954$$ 0 0
$$955$$ −1.82004 −0.0588951
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.67257 0.118593
$$960$$ 0 0
$$961$$ −25.5831 −0.825260
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 31.4978 1.01395
$$966$$ 0 0
$$967$$ −38.3284 −1.23256 −0.616279 0.787528i $$-0.711361\pi$$
−0.616279 + 0.787528i $$0.711361\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −31.0187 −0.995436 −0.497718 0.867339i $$-0.665829\pi$$
−0.497718 + 0.867339i $$0.665829\pi$$
$$972$$ 0 0
$$973$$ 2.05408 0.0658509
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.7424 1.68738 0.843689 0.536832i $$-0.180379\pi$$
0.843689 + 0.536832i $$0.180379\pi$$
$$978$$ 0 0
$$979$$ 64.8142 2.07147
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −18.3029 −0.583772 −0.291886 0.956453i $$-0.594283\pi$$
−0.291886 + 0.956453i $$0.594283\pi$$
$$984$$ 0 0
$$985$$ 42.5624 1.35615
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2.85253 −0.0907052
$$990$$ 0 0
$$991$$ 12.6008 0.400277 0.200138 0.979768i $$-0.435861\pi$$
0.200138 + 0.979768i $$0.435861\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −58.8899 −1.86693
$$996$$ 0 0
$$997$$ 11.7424 0.371885 0.185943 0.982561i $$-0.440466\pi$$
0.185943 + 0.982561i $$0.440466\pi$$
$$998$$ 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 9072.2.a.cd.1.2 3
3.2 odd 2 9072.2.a.bq.1.2 3
4.3 odd 2 567.2.a.g.1.3 3
9.2 odd 6 1008.2.r.k.337.3 6
9.4 even 3 3024.2.r.g.2017.2 6
9.5 odd 6 1008.2.r.k.673.3 6
9.7 even 3 3024.2.r.g.1009.2 6
12.11 even 2 567.2.a.d.1.1 3
28.27 even 2 3969.2.a.p.1.3 3
36.7 odd 6 189.2.f.a.64.1 6
36.11 even 6 63.2.f.b.22.3 6
36.23 even 6 63.2.f.b.43.3 yes 6
36.31 odd 6 189.2.f.a.127.1 6
84.83 odd 2 3969.2.a.m.1.1 3
252.11 even 6 441.2.h.c.373.1 6
252.23 even 6 441.2.h.c.214.1 6
252.31 even 6 1323.2.g.b.667.1 6
252.47 odd 6 441.2.g.d.67.3 6
252.59 odd 6 441.2.g.d.79.3 6
252.67 odd 6 1323.2.g.c.667.1 6
252.79 odd 6 1323.2.g.c.361.1 6
252.83 odd 6 441.2.f.d.148.3 6
252.95 even 6 441.2.g.e.79.3 6
252.103 even 6 1323.2.h.e.802.3 6
252.115 even 6 1323.2.h.e.226.3 6
252.131 odd 6 441.2.h.b.214.1 6
252.139 even 6 1323.2.f.c.883.1 6
252.151 odd 6 1323.2.h.d.226.3 6
252.167 odd 6 441.2.f.d.295.3 6
252.187 even 6 1323.2.g.b.361.1 6
252.191 even 6 441.2.g.e.67.3 6
252.223 even 6 1323.2.f.c.442.1 6
252.227 odd 6 441.2.h.b.373.1 6
252.247 odd 6 1323.2.h.d.802.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 6 36.11 even 6
63.2.f.b.43.3 yes 6 36.23 even 6
189.2.f.a.64.1 6 36.7 odd 6
189.2.f.a.127.1 6 36.31 odd 6
441.2.f.d.148.3 6 252.83 odd 6
441.2.f.d.295.3 6 252.167 odd 6
441.2.g.d.67.3 6 252.47 odd 6
441.2.g.d.79.3 6 252.59 odd 6
441.2.g.e.67.3 6 252.191 even 6
441.2.g.e.79.3 6 252.95 even 6
441.2.h.b.214.1 6 252.131 odd 6
441.2.h.b.373.1 6 252.227 odd 6
441.2.h.c.214.1 6 252.23 even 6
441.2.h.c.373.1 6 252.11 even 6
567.2.a.d.1.1 3 12.11 even 2
567.2.a.g.1.3 3 4.3 odd 2
1008.2.r.k.337.3 6 9.2 odd 6
1008.2.r.k.673.3 6 9.5 odd 6
1323.2.f.c.442.1 6 252.223 even 6
1323.2.f.c.883.1 6 252.139 even 6
1323.2.g.b.361.1 6 252.187 even 6
1323.2.g.b.667.1 6 252.31 even 6
1323.2.g.c.361.1 6 252.79 odd 6
1323.2.g.c.667.1 6 252.67 odd 6
1323.2.h.d.226.3 6 252.151 odd 6
1323.2.h.d.802.3 6 252.247 odd 6
1323.2.h.e.226.3 6 252.115 even 6
1323.2.h.e.802.3 6 252.103 even 6
3024.2.r.g.1009.2 6 9.7 even 3
3024.2.r.g.2017.2 6 9.4 even 3
3969.2.a.m.1.1 3 84.83 odd 2
3969.2.a.p.1.3 3 28.27 even 2
9072.2.a.bq.1.2 3 3.2 odd 2
9072.2.a.cd.1.2 3 1.1 even 1 trivial