# Properties

 Label 1840.2.m.b.1839.1 Level $1840$ Weight $2$ Character 1840.1839 Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ CM discriminant -115 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.m (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{-23})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 49$$ x^4 + 9*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 1839.1 Root $$1.11803 + 2.39792i$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1839 Dual form 1840.2.m.b.1839.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.23607 q^{5} -4.79583i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-2.23607 q^{5} -4.79583i q^{7} -3.00000 q^{9} -6.70820 q^{17} +4.79583i q^{23} +5.00000 q^{25} -1.00000 q^{29} +10.7238i q^{31} +10.7238i q^{35} +11.1803 q^{37} +7.00000 q^{41} -9.59166i q^{43} +6.70820 q^{45} -16.0000 q^{49} +2.23607 q^{53} +10.7238i q^{59} +14.3875i q^{63} -4.79583i q^{67} +10.7238i q^{71} +9.00000 q^{81} +14.3875i q^{83} +15.0000 q^{85} +4.47214 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^9 $$4 q - 12 q^{9} + 20 q^{25} - 4 q^{29} + 28 q^{41} - 64 q^{49} + 36 q^{81} + 60 q^{85}+O(q^{100})$$ 4 * q - 12 * q^9 + 20 * q^25 - 4 * q^29 + 28 * q^41 - 64 * q^49 + 36 * q^81 + 60 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ −2.23607 −1.00000
$$6$$ 0 0
$$7$$ − 4.79583i − 1.81265i −0.422577 0.906327i $$-0.638874\pi$$
0.422577 0.906327i $$-0.361126\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.70820 −1.62698 −0.813489 0.581580i $$-0.802435\pi$$
−0.813489 + 0.581580i $$0.802435\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.79583i 1.00000i
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 10.7238i 1.92605i 0.269408 + 0.963026i $$0.413172\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 10.7238i 1.81265i
$$36$$ 0 0
$$37$$ 11.1803 1.83804 0.919018 0.394215i $$-0.128983\pi$$
0.919018 + 0.394215i $$0.128983\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ − 9.59166i − 1.46271i −0.681994 0.731357i $$-0.738887\pi$$
0.681994 0.731357i $$-0.261113\pi$$
$$44$$ 0 0
$$45$$ 6.70820 1.00000
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −16.0000 −2.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.23607 0.307148 0.153574 0.988137i $$-0.450922\pi$$
0.153574 + 0.988137i $$0.450922\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.7238i 1.39612i 0.716039 + 0.698060i $$0.245953\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 14.3875i 1.81265i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.79583i − 0.585904i −0.956127 0.292952i $$-0.905362\pi$$
0.956127 0.292952i $$-0.0946376\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.7238i 1.27268i 0.771408 + 0.636341i $$0.219553\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 14.3875i 1.57923i 0.613601 + 0.789616i $$0.289720\pi$$
−0.613601 + 0.789616i $$0.710280\pi$$
$$84$$ 0 0
$$85$$ 15.0000 1.62698
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.47214 0.454077 0.227038 0.973886i $$-0.427096\pi$$
0.227038 + 0.973886i $$0.427096\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −17.0000 −1.69156 −0.845782 0.533529i $$-0.820865\pi$$
−0.845782 + 0.533529i $$0.820865\pi$$
$$102$$ 0 0
$$103$$ 9.59166i 0.945095i 0.881305 + 0.472547i $$0.156665\pi$$
−0.881305 + 0.472547i $$0.843335\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.79583i − 0.463631i −0.972760 0.231815i $$-0.925534\pi$$
0.972760 0.231815i $$-0.0744665\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −15.6525 −1.47246 −0.736231 0.676731i $$-0.763396\pi$$
−0.736231 + 0.676731i $$0.763396\pi$$
$$114$$ 0 0
$$115$$ − 10.7238i − 1.00000i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 32.1714i 2.94915i
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 21.4476i − 1.87389i −0.349482 0.936943i $$-0.613642\pi$$
0.349482 0.936943i $$-0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −13.4164 −1.14624 −0.573121 0.819471i $$-0.694267\pi$$
−0.573121 + 0.819471i $$0.694267\pi$$
$$138$$ 0 0
$$139$$ 10.7238i 0.909581i 0.890598 + 0.454791i $$0.150286\pi$$
−0.890598 + 0.454791i $$0.849714\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 2.23607 0.185695
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 21.4476i 1.74538i 0.488273 + 0.872691i $$0.337627\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 20.1246 1.62698
$$154$$ 0 0
$$155$$ − 23.9792i − 1.92605i
$$156$$ 0 0
$$157$$ −24.5967 −1.96303 −0.981517 0.191375i $$-0.938705\pi$$
−0.981517 + 0.191375i $$0.938705\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 23.0000 1.81265
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ − 23.9792i − 1.81265i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 21.4476i − 1.60307i −0.597948 0.801535i $$-0.704017\pi$$
0.597948 0.801535i $$-0.295983\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −25.0000 −1.83804
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4.79583i 0.336601i
$$204$$ 0 0
$$205$$ −15.6525 −1.09322
$$206$$ 0 0
$$207$$ − 14.3875i − 1.00000i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 10.7238i 0.738257i 0.929378 + 0.369129i $$0.120344\pi$$
−0.929378 + 0.369129i $$0.879656\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 21.4476i 1.46271i
$$216$$ 0 0
$$217$$ 51.4296 3.49127
$$218$$ 0 0
$$219$$ 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
$$225$$ −15.0000 −1.00000
$$226$$ 0 0
$$227$$ 28.7750i 1.90986i 0.296826 + 0.954932i $$0.404072\pi$$
−0.296826 + 0.954932i $$0.595928\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 10.7238i 0.693665i 0.937927 + 0.346833i $$0.112743\pi$$
−0.937927 + 0.346833i $$0.887257\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 35.7771 2.28571
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 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
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ − 53.6190i − 3.33172i
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 0 0
$$263$$ 14.3875i 0.887171i 0.896232 + 0.443585i $$0.146294\pi$$
−0.896232 + 0.443585i $$0.853706\pi$$
$$264$$ 0 0
$$265$$ −5.00000 −0.307148
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 31.0000 1.89010 0.945052 0.326921i $$-0.106011\pi$$
0.945052 + 0.326921i $$0.106011\pi$$
$$270$$ 0 0
$$271$$ − 32.1714i − 1.95427i −0.212610 0.977137i $$-0.568196\pi$$
0.212610 0.977137i $$-0.431804\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ − 32.1714i − 1.92605i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 33.5708i 1.99558i 0.0664602 + 0.997789i $$0.478829\pi$$
−0.0664602 + 0.997789i $$0.521171\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 33.5708i − 1.98162i
$$288$$ 0 0
$$289$$ 28.0000 1.64706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.70820 −0.391897 −0.195949 0.980614i $$-0.562779\pi$$
−0.195949 + 0.980614i $$0.562779\pi$$
$$294$$ 0 0
$$295$$ − 23.9792i − 1.39612i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −46.0000 −2.65140
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 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
$$311$$ 21.4476i 1.21618i 0.793867 + 0.608091i $$0.208065\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 0 0
$$313$$ 11.1803 0.631950 0.315975 0.948767i $$-0.397668\pi$$
0.315975 + 0.948767i $$0.397668\pi$$
$$314$$ 0 0
$$315$$ − 32.1714i − 1.81265i
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 32.1714i − 1.76830i −0.467202 0.884150i $$-0.654738\pi$$
0.467202 0.884150i $$-0.345262\pi$$
$$332$$ 0 0
$$333$$ −33.5410 −1.83804
$$334$$ 0 0
$$335$$ 10.7238i 0.585904i
$$336$$ 0 0
$$337$$ −31.3050 −1.70529 −0.852645 0.522491i $$-0.825003\pi$$
−0.852645 + 0.522491i $$0.825003\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 43.1625i 2.33056i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ − 23.9792i − 1.27268i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.79583i − 0.250340i −0.992135 0.125170i $$-0.960052\pi$$
0.992135 0.125170i $$-0.0399477\pi$$
$$368$$ 0 0
$$369$$ −21.0000 −1.09322
$$370$$ 0 0
$$371$$ − 10.7238i − 0.556752i
$$372$$ 0 0
$$373$$ −4.47214 −0.231558 −0.115779 0.993275i $$-0.536937\pi$$
−0.115779 + 0.993275i $$0.536937\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 33.5708i 1.71539i 0.514160 + 0.857694i $$0.328104\pi$$
−0.514160 + 0.857694i $$0.671896\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 28.7750i 1.46271i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ − 32.1714i − 1.62698i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −20.1246 −1.00000
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 39.0000 1.92843 0.964213 0.265129i $$-0.0854146\pi$$
0.964213 + 0.265129i $$0.0854146\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 51.4296 2.53068
$$414$$ 0 0
$$415$$ − 32.1714i − 1.57923i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −33.5410 −1.62698
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −24.5967 −1.18204 −0.591022 0.806655i $$-0.701275\pi$$
−0.591022 + 0.806655i $$0.701275\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 21.4476i 1.02364i 0.859093 + 0.511819i $$0.171028\pi$$
−0.859093 + 0.511819i $$0.828972\pi$$
$$440$$ 0 0
$$441$$ 48.0000 2.28571
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −41.0000 −1.93491 −0.967455 0.253044i $$-0.918568\pi$$
−0.967455 + 0.253044i $$0.918568\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −42.4853 −1.98738 −0.993689 0.112170i $$-0.964220\pi$$
−0.993689 + 0.112170i $$0.964220\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 43.1625i − 1.99732i −0.0517364 0.998661i $$-0.516476\pi$$
0.0517364 0.998661i $$-0.483524\pi$$
$$468$$ 0 0
$$469$$ −23.0000 −1.06204
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.70820 −0.307148
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.0000 −0.454077
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10.7238i 0.483959i 0.970281 + 0.241979i $$0.0777966\pi$$
−0.970281 + 0.241979i $$0.922203\pi$$
$$492$$ 0 0
$$493$$ 6.70820 0.302122
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 51.4296 2.30693
$$498$$ 0 0
$$499$$ − 32.1714i − 1.44019i −0.693875 0.720095i $$-0.744098\pi$$
0.693875 0.720095i $$-0.255902\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14.3875i 0.641507i 0.947163 + 0.320753i $$0.103936\pi$$
−0.947163 + 0.320753i $$0.896064\pi$$
$$504$$ 0 0
$$505$$ 38.0132 1.69156
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 21.4476i − 0.945095i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ − 9.59166i − 0.419414i −0.977764 0.209707i $$-0.932749\pi$$
0.977764 0.209707i $$-0.0672510\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 71.9375i − 3.13365i
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ − 32.1714i − 1.39612i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 10.7238i 0.463631i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 46.9574 1.98965 0.994825 0.101603i $$-0.0323971\pi$$
0.994825 + 0.101603i $$0.0323971\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 33.5708i 1.41484i 0.706793 + 0.707421i $$0.250141\pi$$
−0.706793 + 0.707421i $$0.749859\pi$$
$$564$$ 0 0
$$565$$ 35.0000 1.47246
$$566$$ 0 0
$$567$$ − 43.1625i − 1.81265i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 23.9792i 1.00000i
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 69.0000 2.86260
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ − 71.9375i − 2.94915i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 21.4476i 0.876326i 0.898896 + 0.438163i $$0.144371\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 0 0
$$603$$ 14.3875i 0.585904i
$$604$$ 0 0
$$605$$ 24.5967 1.00000
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 31.3050 1.26440 0.632198 0.774807i $$-0.282153\pi$$
0.632198 + 0.774807i $$0.282153\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −24.5967 −0.990228 −0.495114 0.868828i $$-0.664874\pi$$
−0.495114 + 0.868828i $$0.664874\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −75.0000 −2.99045
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 32.1714i − 1.27268i
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 33.5708i 1.32390i 0.749546 + 0.661952i $$0.230272\pi$$
−0.749546 + 0.661952i $$0.769728\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 47.9583i 1.87389i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 4.79583i − 0.185695i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 29.0689 1.11721 0.558604 0.829435i $$-0.311337\pi$$
0.558604 + 0.829435i $$0.311337\pi$$
$$678$$ 0 0
$$679$$ − 21.4476i − 0.823084i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 30.0000 1.14624
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 21.4476i − 0.815906i −0.913003 0.407953i $$-0.866243\pi$$
0.913003 0.407953i $$-0.133757\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 23.9792i − 0.909581i
$$696$$ 0 0
$$697$$ −46.9574 −1.77864
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 81.5291i 3.06622i
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −51.4296 −1.92605
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 53.6190i 1.99965i 0.0186469 + 0.999826i $$0.494064\pi$$
−0.0186469 + 0.999826i $$0.505936\pi$$
$$720$$ 0 0
$$721$$ 46.0000 1.71313
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5.00000 −0.185695
$$726$$ 0 0
$$727$$ 52.7541i 1.95654i 0.207328 + 0.978272i $$0.433523\pi$$
−0.207328 + 0.978272i $$0.566477\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 64.3428i 2.37981i
$$732$$ 0 0
$$733$$ −42.4853 −1.56923 −0.784615 0.619983i $$-0.787139\pi$$
−0.784615 + 0.619983i $$0.787139\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 53.6190i 1.97241i 0.165535 + 0.986204i $$0.447065\pi$$
−0.165535 + 0.986204i $$0.552935\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.59166i 0.351884i 0.984401 + 0.175942i $$0.0562971\pi$$
−0.984401 + 0.175942i $$0.943703\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 43.1625i − 1.57923i
$$748$$ 0 0
$$749$$ −23.0000 −0.840402
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 47.9583i − 1.74538i
$$756$$ 0 0
$$757$$ −15.6525 −0.568899 −0.284449 0.958691i $$-0.591811\pi$$
−0.284449 + 0.958691i $$0.591811\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −45.0000 −1.62698
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −40.2492 −1.44766 −0.723832 0.689976i $$-0.757621\pi$$
−0.723832 + 0.689976i $$0.757621\pi$$
$$774$$ 0 0
$$775$$ 53.6190i 1.92605i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 55.0000 1.96303
$$786$$ 0 0
$$787$$ − 4.79583i − 0.170953i −0.996340 0.0854765i $$-0.972759\pi$$
0.996340 0.0854765i $$-0.0272412\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 75.0666i 2.66906i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 20.1246 0.712850 0.356425 0.934324i $$-0.383995\pi$$
0.356425 + 0.934324i $$0.383995\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −51.4296 −1.81265
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 19.0000 0.668004 0.334002 0.942572i $$-0.391601\pi$$
0.334002 + 0.942572i $$0.391601\pi$$
$$810$$ 0 0
$$811$$ − 32.1714i − 1.12969i −0.825197 0.564846i $$-0.808936\pi$$
0.825197 0.564846i $$-0.191064\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 43.1625i − 1.50091i −0.660923 0.750453i $$-0.729835\pi$$
0.660923 0.750453i $$-0.270165\pi$$
$$828$$ 0 0
$$829$$ −21.0000 −0.729360 −0.364680 0.931133i $$-0.618822\pi$$
−0.364680 + 0.931133i $$0.618822\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 107.331 3.71881
$$834$$ 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
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −29.0689 −1.00000
$$846$$ 0 0
$$847$$ 52.7541i 1.81265i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 53.6190i 1.83804i
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ − 32.1714i − 1.09767i −0.835929 0.548837i $$-0.815071\pi$$
0.835929 0.548837i $$-0.184929\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −13.4164 −0.454077
$$874$$ 0 0
$$875$$ 53.6190i 1.81265i
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 47.9583i 1.60307i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 10.7238i − 0.357659i
$$900$$ 0 0
$$901$$ −15.0000 −0.499722
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.7541i 1.75167i 0.482608 + 0.875836i $$0.339690\pi$$
−0.482608 + 0.875836i $$0.660310\pi$$
$$908$$ 0 0
$$909$$ 51.0000 1.69156
$$910$$ 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
$$916$$ 0 0
$$917$$ −102.859 −3.39671
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 55.9017 1.83804
$$926$$ 0 0
$$927$$ − 28.7750i − 0.945095i
$$928$$ 0 0
$$929$$ −29.0000 −0.951459 −0.475730 0.879592i $$-0.657816\pi$$
−0.475730 + 0.879592i $$0.657816\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 58.1378 1.89928 0.949639 0.313346i $$-0.101450\pi$$
0.949639 + 0.313346i $$0.101450\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 33.5708i 1.09322i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22.3607 0.724333 0.362167 0.932113i $$-0.382037\pi$$
0.362167 + 0.932113i $$0.382037\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 64.3428i 2.07774i
$$960$$ 0 0
$$961$$ −84.0000 −2.70968
$$962$$ 0 0
$$963$$ 14.3875i 0.463631i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 51.4296 1.64876
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.5410 −1.07307 −0.536536 0.843877i $$-0.680267\pi$$
−0.536536 + 0.843877i $$0.680267\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 62.3458i − 1.98852i −0.106979 0.994261i $$-0.534118\pi$$
0.106979 0.994261i $$-0.465882\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 46.0000 1.46271
$$990$$ 0 0
$$991$$ 53.6190i 1.70326i 0.524140 + 0.851632i $$0.324387\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 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 1840.2.m.b.1839.1 4
4.3 odd 2 inner 1840.2.m.b.1839.2 yes 4
5.4 even 2 inner 1840.2.m.b.1839.4 yes 4
20.19 odd 2 inner 1840.2.m.b.1839.3 yes 4
23.22 odd 2 inner 1840.2.m.b.1839.4 yes 4
92.91 even 2 inner 1840.2.m.b.1839.3 yes 4
115.114 odd 2 CM 1840.2.m.b.1839.1 4
460.459 even 2 inner 1840.2.m.b.1839.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.b.1839.1 4 1.1 even 1 trivial
1840.2.m.b.1839.1 4 115.114 odd 2 CM
1840.2.m.b.1839.2 yes 4 4.3 odd 2 inner
1840.2.m.b.1839.2 yes 4 460.459 even 2 inner
1840.2.m.b.1839.3 yes 4 20.19 odd 2 inner
1840.2.m.b.1839.3 yes 4 92.91 even 2 inner
1840.2.m.b.1839.4 yes 4 5.4 even 2 inner
1840.2.m.b.1839.4 yes 4 23.22 odd 2 inner