# Properties

 Label 225.2.a.f.1.1 Level $225$ Weight $2$ Character 225.1 Self dual yes Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.23607 q^{2} +3.00000 q^{4} -2.23607 q^{8} +O(q^{10})$$ $$q-2.23607 q^{2} +3.00000 q^{4} -2.23607 q^{8} -1.00000 q^{16} +4.47214 q^{17} +4.00000 q^{19} +8.94427 q^{23} +8.00000 q^{31} +6.70820 q^{32} -10.0000 q^{34} -8.94427 q^{38} -20.0000 q^{46} -8.94427 q^{47} -7.00000 q^{49} -4.47214 q^{53} +2.00000 q^{61} -17.8885 q^{62} -13.0000 q^{64} +13.4164 q^{68} +12.0000 q^{76} +16.0000 q^{79} -17.8885 q^{83} +26.8328 q^{92} +20.0000 q^{94} +15.6525 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4}+O(q^{10})$$ 2 * q + 6 * q^4 $$2 q + 6 q^{4} - 2 q^{16} + 8 q^{19} + 16 q^{31} - 20 q^{34} - 40 q^{46} - 14 q^{49} + 4 q^{61} - 26 q^{64} + 24 q^{76} + 32 q^{79} + 40 q^{94}+O(q^{100})$$ 2 * q + 6 * q^4 - 2 * q^16 + 8 * q^19 + 16 * q^31 - 20 * q^34 - 40 * q^46 - 14 * q^49 + 4 * q^61 - 26 * q^64 + 24 * q^76 + 32 * q^79 + 40 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.23607 −1.58114 −0.790569 0.612372i $$-0.790215\pi$$
−0.790569 + 0.612372i $$0.790215\pi$$
$$3$$ 0 0
$$4$$ 3.00000 1.50000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 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
$$23$$ 8.94427 1.86501 0.932505 0.361158i $$-0.117618\pi$$
0.932505 + 0.361158i $$0.117618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 6.70820 1.18585
$$33$$ 0 0
$$34$$ −10.0000 −1.71499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −8.94427 −1.45095
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −20.0000 −2.94884
$$47$$ −8.94427 −1.30466 −0.652328 0.757937i $$-0.726208\pi$$
−0.652328 + 0.757937i $$0.726208\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.47214 −0.614295 −0.307148 0.951662i $$-0.599375\pi$$
−0.307148 + 0.951662i $$0.599375\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −17.8885 −2.27185
$$63$$ 0 0
$$64$$ −13.0000 −1.62500
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 13.4164 1.62698
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 12.0000 1.37649
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −17.8885 −1.96352 −0.981761 0.190117i $$-0.939113\pi$$
−0.981761 + 0.190117i $$0.939113\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 26.8328 2.79751
$$93$$ 0 0
$$94$$ 20.0000 2.06284
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 15.6525 1.58114
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 17.8885 1.72935 0.864675 0.502331i $$-0.167524\pi$$
0.864675 + 0.502331i $$0.167524\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −4.47214 −0.420703 −0.210352 0.977626i $$-0.567461\pi$$
−0.210352 + 0.977626i $$0.567461\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −4.47214 −0.404888
$$123$$ 0 0
$$124$$ 24.0000 2.15526
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 15.6525 1.38350
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −10.0000 −0.857493
$$137$$ −22.3607 −1.91040 −0.955201 0.295958i $$-0.904361\pi$$
−0.955201 + 0.295958i $$0.904361\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −8.94427 −0.725476
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ −35.7771 −2.84627
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 40.0000 3.10460
$$167$$ −8.94427 −0.692129 −0.346064 0.938211i $$-0.612482\pi$$
−0.346064 + 0.938211i $$0.612482\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 22.3607 1.70005 0.850026 0.526742i $$-0.176586\pi$$
0.850026 + 0.526742i $$0.176586\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −20.0000 −1.47442
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −26.8328 −1.95698
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −21.0000 −1.50000
$$197$$ 4.47214 0.318626 0.159313 0.987228i $$-0.449072\pi$$
0.159313 + 0.987228i $$0.449072\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ −13.4164 −0.921443
$$213$$ 0 0
$$214$$ −40.0000 −2.73434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 31.3050 2.12024
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ 17.8885 1.18730 0.593652 0.804722i $$-0.297686\pi$$
0.593652 + 0.804722i $$0.297686\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.3607 1.46490 0.732448 0.680823i $$-0.238378\pi$$
0.732448 + 0.680823i $$0.238378\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 24.5967 1.58114
$$243$$ 0 0
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −17.8885 −1.13592
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −9.00000 −0.562500
$$257$$ 31.3050 1.95275 0.976375 0.216085i $$-0.0693287\pi$$
0.976375 + 0.216085i $$0.0693287\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 8.94427 0.551527 0.275764 0.961225i $$-0.411069\pi$$
0.275764 + 0.961225i $$0.411069\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ −4.47214 −0.271163
$$273$$ 0 0
$$274$$ 50.0000 3.02061
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ −8.94427 −0.536442
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −31.3050 −1.82885 −0.914427 0.404750i $$-0.867359\pi$$
−0.914427 + 0.404750i $$0.867359\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −17.8885 −1.02937
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 48.0000 2.70021
$$317$$ −22.3607 −1.25590 −0.627950 0.778253i $$-0.716106\pi$$
−0.627950 + 0.778253i $$0.716106\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 17.8885 0.995345
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −53.6656 −2.94528
$$333$$ 0 0
$$334$$ 20.0000 1.09435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 29.0689 1.58114
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −50.0000 −2.68802
$$347$$ −35.7771 −1.92061 −0.960307 0.278944i $$-0.910016\pi$$
−0.960307 + 0.278944i $$0.910016\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −31.3050 −1.66619 −0.833097 0.553127i $$-0.813435\pi$$
−0.833097 + 0.553127i $$0.813435\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$362$$ 49.1935 2.58555
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ −8.94427 −0.466252
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 20.0000 1.03142
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 8.94427 0.457031 0.228515 0.973540i $$-0.426613\pi$$
0.228515 + 0.973540i $$0.426613\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 40.0000 2.02289
$$392$$ 15.6525 0.790569
$$393$$ 0 0
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ −35.7771 −1.79334
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 62.6099 3.04780
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 53.6656 2.59403
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −42.0000 −2.01144
$$437$$ 35.7771 1.71145
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −17.8885 −0.849910 −0.424955 0.905214i $$-0.639710\pi$$
−0.424955 + 0.905214i $$0.639710\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −13.4164 −0.631055
$$453$$ 0 0
$$454$$ −40.0000 −1.87729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 58.1378 2.71660
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −50.0000 −2.31621
$$467$$ −35.7771 −1.65557 −0.827783 0.561048i $$-0.810398\pi$$
−0.827783 + 0.561048i $$0.810398\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −4.47214 −0.203700
$$483$$ 0 0
$$484$$ −33.0000 −1.50000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ −4.47214 −0.202444
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −44.0000 −1.96971 −0.984855 0.173379i $$-0.944532\pi$$
−0.984855 + 0.173379i $$0.944532\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −44.7214 −1.99403 −0.997013 0.0772283i $$-0.975393\pi$$
−0.997013 + 0.0772283i $$0.975393\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11.1803 −0.494106
$$513$$ 0 0
$$514$$ −70.0000 −3.08757
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −20.0000 −0.872041
$$527$$ 35.7771 1.55847
$$528$$ 0 0
$$529$$ 57.0000 2.47826
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −71.5542 −3.07352
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ −67.0820 −2.86560
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ −22.3607 −0.947452 −0.473726 0.880672i $$-0.657091\pi$$
−0.473726 + 0.880672i $$0.657091\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 35.7771 1.50782 0.753912 0.656975i $$-0.228164\pi$$
0.753912 + 0.656975i $$0.228164\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −6.70820 −0.279024
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 70.0000 2.89167
$$587$$ 17.8885 0.738339 0.369170 0.929362i $$-0.379642\pi$$
0.369170 + 0.929362i $$0.379642\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −4.47214 −0.183649 −0.0918243 0.995775i $$-0.529270\pi$$
−0.0918243 + 0.995775i $$0.529270\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 26.8328 1.08821
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −49.1935 −1.98046 −0.990228 0.139459i $$-0.955464\pi$$
−0.990228 + 0.139459i $$0.955464\pi$$
$$618$$ 0 0
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −35.7771 −1.42314
$$633$$ 0 0
$$634$$ 50.0000 1.98575
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −40.0000 −1.57378
$$647$$ 44.7214 1.75818 0.879089 0.476658i $$-0.158152\pi$$
0.879089 + 0.476658i $$0.158152\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 49.1935 1.92509 0.962545 0.271122i $$-0.0873945\pi$$
0.962545 + 0.271122i $$0.0873945\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 62.6099 2.43340
$$663$$ 0 0
$$664$$ 40.0000 1.55230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −26.8328 −1.03819
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −39.0000 −1.50000
$$677$$ 31.3050 1.20315 0.601574 0.798817i $$-0.294541\pi$$
0.601574 + 0.798817i $$0.294541\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 35.7771 1.36897 0.684486 0.729026i $$-0.260027\pi$$
0.684486 + 0.729026i $$0.260027\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −52.0000 −1.97817 −0.989087 0.147335i $$-0.952930\pi$$
−0.989087 + 0.147335i $$0.952930\pi$$
$$692$$ 67.0820 2.55008
$$693$$ 0 0
$$694$$ 80.0000 3.03676
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −76.0263 −2.87764
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 70.0000 2.63448
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 71.5542 2.67972
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 6.70820 0.249653
$$723$$ 0 0
$$724$$ −66.0000 −2.45287
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 60.0000 2.21163
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −44.7214 −1.64067 −0.820334 0.571885i $$-0.806212\pi$$
−0.820334 + 0.571885i $$0.806212\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 8.94427 0.326164
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ −8.94427 −0.324871
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −4.47214 −0.160852 −0.0804258 0.996761i $$-0.525628\pi$$
−0.0804258 + 0.996761i $$0.525628\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −89.4427 −3.19847
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 13.4164 0.477940
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 48.0000 1.70131
$$797$$ −49.1935 −1.74252 −0.871262 0.490819i $$-0.836698\pi$$
−0.871262 + 0.490819i $$0.836698\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 58.1378 2.03274
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −35.7771 −1.24409 −0.622046 0.782981i $$-0.713698\pi$$
−0.622046 + 0.782981i $$0.713698\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −31.3050 −1.08465
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −84.9706 −2.92828
$$843$$ 0 0
$$844$$ −84.0000 −2.89140
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 4.47214 0.153574
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −40.0000 −1.36717
$$857$$ 58.1378 1.98595 0.992974 0.118331i $$-0.0377545\pi$$
0.992974 + 0.118331i $$0.0377545\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −44.7214 −1.52233 −0.761166 0.648557i $$-0.775373\pi$$
−0.761166 + 0.648557i $$0.775373\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 31.3050 1.06012
$$873$$ 0 0
$$874$$ −80.0000 −2.70604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ −35.7771 −1.20742
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 40.0000 1.34383
$$887$$ −8.94427 −0.300319 −0.150160 0.988662i $$-0.547979\pi$$
−0.150160 + 0.988662i $$0.547979\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −35.7771 −1.19723
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 10.0000 0.332595
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 53.6656 1.78096
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −78.0000 −2.57719
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 67.0820 2.19735
$$933$$ 0 0
$$934$$ 80.0000 2.61768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.8885 0.581300 0.290650 0.956830i $$-0.406129\pi$$
0.290650 + 0.956830i $$0.406129\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −58.1378 −1.88327 −0.941634 0.336640i $$-0.890710\pi$$
−0.941634 + 0.336640i $$0.890710\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 6.00000 0.193247
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 24.5967 0.790569
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 4.47214 0.143076 0.0715382 0.997438i $$-0.477209\pi$$
0.0715382 + 0.997438i $$0.477209\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 62.6099 1.99695 0.998473 0.0552438i $$-0.0175936\pi$$
0.998473 + 0.0552438i $$0.0175936\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 53.6656 1.70389
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 98.3870 3.11439
$$999$$ 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 225.2.a.f.1.1 2
3.2 odd 2 inner 225.2.a.f.1.2 2
4.3 odd 2 3600.2.a.bs.1.2 2
5.2 odd 4 45.2.b.a.19.1 2
5.3 odd 4 45.2.b.a.19.2 yes 2
5.4 even 2 inner 225.2.a.f.1.2 2
12.11 even 2 3600.2.a.bs.1.1 2
15.2 even 4 45.2.b.a.19.2 yes 2
15.8 even 4 45.2.b.a.19.1 2
15.14 odd 2 CM 225.2.a.f.1.1 2
20.3 even 4 720.2.f.d.289.1 2
20.7 even 4 720.2.f.d.289.2 2
20.19 odd 2 3600.2.a.bs.1.1 2
35.13 even 4 2205.2.d.a.1324.2 2
35.27 even 4 2205.2.d.a.1324.1 2
40.3 even 4 2880.2.f.j.1729.2 2
40.13 odd 4 2880.2.f.k.1729.2 2
40.27 even 4 2880.2.f.j.1729.1 2
40.37 odd 4 2880.2.f.k.1729.1 2
45.2 even 12 405.2.j.c.109.2 4
45.7 odd 12 405.2.j.c.109.1 4
45.13 odd 12 405.2.j.c.379.1 4
45.22 odd 12 405.2.j.c.379.2 4
45.23 even 12 405.2.j.c.379.2 4
45.32 even 12 405.2.j.c.379.1 4
45.38 even 12 405.2.j.c.109.1 4
45.43 odd 12 405.2.j.c.109.2 4
60.23 odd 4 720.2.f.d.289.2 2
60.47 odd 4 720.2.f.d.289.1 2
60.59 even 2 3600.2.a.bs.1.2 2
105.62 odd 4 2205.2.d.a.1324.2 2
105.83 odd 4 2205.2.d.a.1324.1 2
120.53 even 4 2880.2.f.k.1729.1 2
120.77 even 4 2880.2.f.k.1729.2 2
120.83 odd 4 2880.2.f.j.1729.1 2
120.107 odd 4 2880.2.f.j.1729.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.b.a.19.1 2 5.2 odd 4
45.2.b.a.19.1 2 15.8 even 4
45.2.b.a.19.2 yes 2 5.3 odd 4
45.2.b.a.19.2 yes 2 15.2 even 4
225.2.a.f.1.1 2 1.1 even 1 trivial
225.2.a.f.1.1 2 15.14 odd 2 CM
225.2.a.f.1.2 2 3.2 odd 2 inner
225.2.a.f.1.2 2 5.4 even 2 inner
405.2.j.c.109.1 4 45.7 odd 12
405.2.j.c.109.1 4 45.38 even 12
405.2.j.c.109.2 4 45.2 even 12
405.2.j.c.109.2 4 45.43 odd 12
405.2.j.c.379.1 4 45.13 odd 12
405.2.j.c.379.1 4 45.32 even 12
405.2.j.c.379.2 4 45.22 odd 12
405.2.j.c.379.2 4 45.23 even 12
720.2.f.d.289.1 2 20.3 even 4
720.2.f.d.289.1 2 60.47 odd 4
720.2.f.d.289.2 2 20.7 even 4
720.2.f.d.289.2 2 60.23 odd 4
2205.2.d.a.1324.1 2 35.27 even 4
2205.2.d.a.1324.1 2 105.83 odd 4
2205.2.d.a.1324.2 2 35.13 even 4
2205.2.d.a.1324.2 2 105.62 odd 4
2880.2.f.j.1729.1 2 40.27 even 4
2880.2.f.j.1729.1 2 120.83 odd 4
2880.2.f.j.1729.2 2 40.3 even 4
2880.2.f.j.1729.2 2 120.107 odd 4
2880.2.f.k.1729.1 2 40.37 odd 4
2880.2.f.k.1729.1 2 120.53 even 4
2880.2.f.k.1729.2 2 40.13 odd 4
2880.2.f.k.1729.2 2 120.77 even 4
3600.2.a.bs.1.1 2 12.11 even 2
3600.2.a.bs.1.1 2 20.19 odd 2
3600.2.a.bs.1.2 2 4.3 odd 2
3600.2.a.bs.1.2 2 60.59 even 2