# Properties

 Label 48.4.c.a.47.1 Level $48$ Weight $4$ Character 48.47 Analytic conductor $2.832$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(47,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("48.47");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.c (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: $$2.83209168028$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 47.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.47 Dual form 48.4.c.a.47.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.19615i q^{3} -31.1769i q^{7} -27.0000 q^{9} +O(q^{10})$$ $$q-5.19615i q^{3} -31.1769i q^{7} -27.0000 q^{9} +70.0000 q^{13} +155.885i q^{19} -162.000 q^{21} +125.000 q^{25} +140.296i q^{27} -155.885i q^{31} +110.000 q^{37} -363.731i q^{39} -218.238i q^{43} -629.000 q^{49} +810.000 q^{57} +182.000 q^{61} +841.777i q^{63} +654.715i q^{67} -1190.00 q^{73} -649.519i q^{75} +1091.19i q^{79} +729.000 q^{81} -2182.38i q^{91} -810.000 q^{93} +1330.00 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{9}+O(q^{10})$$ 2 * q - 54 * q^9 $$2 q - 54 q^{9} + 140 q^{13} - 324 q^{21} + 250 q^{25} + 220 q^{37} - 1258 q^{49} + 1620 q^{57} + 364 q^{61} - 2380 q^{73} + 1458 q^{81} - 1620 q^{93} + 2660 q^{97}+O(q^{100})$$ 2 * q - 54 * q^9 + 140 * q^13 - 324 * q^21 + 250 * q^25 + 220 * q^37 - 1258 * q^49 + 1620 * q^57 + 364 * q^61 - 2380 * q^73 + 1458 * q^81 - 1620 * q^93 + 2660 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ − 5.19615i − 1.00000i
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ − 31.1769i − 1.68340i −0.539949 0.841698i $$-0.681557\pi$$
0.539949 0.841698i $$-0.318443\pi$$
$$8$$ 0 0
$$9$$ −27.0000 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 70.0000 1.49342 0.746712 0.665148i $$-0.231631\pi$$
0.746712 + 0.665148i $$0.231631\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 155.885i 1.88223i 0.338086 + 0.941115i $$0.390220\pi$$
−0.338086 + 0.941115i $$0.609780\pi$$
$$20$$ 0 0
$$21$$ −162.000 −1.68340
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 125.000 1.00000
$$26$$ 0 0
$$27$$ 140.296i 1.00000i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ − 155.885i − 0.903151i −0.892233 0.451576i $$-0.850862\pi$$
0.892233 0.451576i $$-0.149138\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 110.000 0.488754 0.244377 0.969680i $$-0.421417\pi$$
0.244377 + 0.969680i $$0.421417\pi$$
$$38$$ 0 0
$$39$$ − 363.731i − 1.49342i
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ − 218.238i − 0.773978i −0.922084 0.386989i $$-0.873515\pi$$
0.922084 0.386989i $$-0.126485\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −629.000 −1.83382
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 810.000 1.88223
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 182.000 0.382012 0.191006 0.981589i $$-0.438825\pi$$
0.191006 + 0.981589i $$0.438825\pi$$
$$62$$ 0 0
$$63$$ 841.777i 1.68340i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 654.715i 1.19382i 0.802307 + 0.596912i $$0.203606\pi$$
−0.802307 + 0.596912i $$0.796394\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −1190.00 −1.90793 −0.953966 0.299916i $$-0.903041\pi$$
−0.953966 + 0.299916i $$0.903041\pi$$
$$74$$ 0 0
$$75$$ − 649.519i − 1.00000i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1091.19i 1.55403i 0.629480 + 0.777017i $$0.283268\pi$$
−0.629480 + 0.777017i $$0.716732\pi$$
$$80$$ 0 0
$$81$$ 729.000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ − 2182.38i − 2.51402i
$$92$$ 0 0
$$93$$ −810.000 −0.903151
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1330.00 1.39218 0.696088 0.717957i $$-0.254922\pi$$
0.696088 + 0.717957i $$0.254922\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ − 1028.84i − 0.984218i −0.870534 0.492109i $$-0.836226\pi$$
0.870534 0.492109i $$-0.163774\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 646.000 0.567666 0.283833 0.958874i $$-0.408394\pi$$
0.283833 + 0.958874i $$0.408394\pi$$
$$110$$ 0 0
$$111$$ − 571.577i − 0.488754i
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1890.00 −1.49342
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1331.00 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2837.10i 1.98230i 0.132754 + 0.991149i $$0.457618\pi$$
−0.132754 + 0.991149i $$0.542382\pi$$
$$128$$ 0 0
$$129$$ −1134.00 −0.773978
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 4860.00 3.16854
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 2026.50i 1.23659i 0.785948 + 0.618293i $$0.212175\pi$$
−0.785948 + 0.618293i $$0.787825\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3268.38i 1.83382i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ − 3273.58i − 1.76424i −0.471027 0.882119i $$-0.656117\pi$$
0.471027 0.882119i $$-0.343883\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3850.00 −1.95709 −0.978546 0.206028i $$-0.933946\pi$$
−0.978546 + 0.206028i $$0.933946\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2400.62i 1.15357i 0.816897 + 0.576783i $$0.195692\pi$$
−0.816897 + 0.576783i $$0.804308\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 2703.00 1.23031
$$170$$ 0 0
$$171$$ − 4208.88i − 1.88223i
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ − 3897.11i − 1.68340i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −3458.00 −1.42006 −0.710031 0.704171i $$-0.751319\pi$$
−0.710031 + 0.704171i $$0.751319\pi$$
$$182$$ 0 0
$$183$$ − 945.700i − 0.382012i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4374.00 1.68340
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −1150.00 −0.428906 −0.214453 0.976734i $$-0.568797\pi$$
−0.214453 + 0.976734i $$0.568797\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ − 2026.50i − 0.721883i −0.932588 0.360942i $$-0.882455\pi$$
0.932588 0.360942i $$-0.117545\pi$$
$$200$$ 0 0
$$201$$ 3402.00 1.19382
$$202$$ 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
$$211$$ − 1091.19i − 0.356023i −0.984028 0.178011i $$-0.943034\pi$$
0.984028 0.178011i $$-0.0569664\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4860.00 −1.52036
$$218$$ 0 0
$$219$$ 6183.42i 1.90793i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5830.08i 1.75072i 0.483469 + 0.875362i $$0.339377\pi$$
−0.483469 + 0.875362i $$0.660623\pi$$
$$224$$ 0 0
$$225$$ −3375.00 −1.00000
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −4466.00 −1.28874 −0.644370 0.764714i $$-0.722880\pi$$
−0.644370 + 0.764714i $$0.722880\pi$$
$$230$$ 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
$$237$$ 5670.00 1.55403
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 7378.00 1.97203 0.986014 0.166662i $$-0.0532990\pi$$
0.986014 + 0.166662i $$0.0532990\pi$$
$$242$$ 0 0
$$243$$ − 3788.00i − 1.00000i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 10911.9i 2.81097i
$$248$$ 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
$$259$$ − 3429.46i − 0.822766i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ − 8885.42i − 1.99170i −0.0910064 0.995850i $$-0.529008\pi$$
0.0910064 0.995850i $$-0.470992\pi$$
$$272$$ 0 0
$$273$$ −11340.0 −2.51402
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4030.00 0.874149 0.437074 0.899425i $$-0.356015\pi$$
0.437074 + 0.899425i $$0.356015\pi$$
$$278$$ 0 0
$$279$$ 4208.88i 0.903151i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ − 7700.70i − 1.61752i −0.588137 0.808761i $$-0.700138\pi$$
0.588137 0.808761i $$-0.299862\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 4913.00 1.00000
$$290$$ 0 0
$$291$$ − 6910.88i − 1.39218i
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −6804.00 −1.30291
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1590.02i − 0.295594i −0.989018 0.147797i $$-0.952782\pi$$
0.989018 0.147797i $$-0.0472182\pi$$
$$308$$ 0 0
$$309$$ −5346.00 −0.984218
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 10010.0 1.80766 0.903832 0.427888i $$-0.140742\pi$$
0.903832 + 0.427888i $$0.140742\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$325$$ 8750.00 1.49342
$$326$$ 0 0
$$327$$ − 3356.71i − 0.567666i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12003.1i 1.99320i 0.0823644 + 0.996602i $$0.473753\pi$$
−0.0823644 + 0.996602i $$0.526247\pi$$
$$332$$ 0 0
$$333$$ −2970.00 −0.488754
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4930.00 0.796897 0.398448 0.917191i $$-0.369549\pi$$
0.398448 + 0.917191i $$0.369549\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8916.60i 1.40365i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ −11914.0 −1.82734 −0.913670 0.406456i $$-0.866764\pi$$
−0.913670 + 0.406456i $$0.866764\pi$$
$$350$$ 0 0
$$351$$ 9820.73i 1.49342i
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 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
$$361$$ −17441.0 −2.54279
$$362$$ 0 0
$$363$$ 6916.08i 1.00000i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 13374.9i − 1.90235i −0.308646 0.951177i $$-0.599876\pi$$
0.308646 0.951177i $$-0.400124\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12350.0 1.71437 0.857183 0.515011i $$-0.172212\pi$$
0.857183 + 0.515011i $$0.172212\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12003.1i 1.62680i 0.581702 + 0.813402i $$0.302387\pi$$
−0.581702 + 0.813402i $$0.697613\pi$$
$$380$$ 0 0
$$381$$ 14742.0 1.98230
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 5892.44i 0.773978i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1190.00 0.150439 0.0752196 0.997167i $$-0.476034\pi$$
0.0752196 + 0.997167i $$0.476034\pi$$
$$398$$ 0 0
$$399$$ − 25253.3i − 3.16854i
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ − 10911.9i − 1.34879i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −8246.00 −0.996916 −0.498458 0.866914i $$-0.666100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 10530.0 1.23659
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −17138.0 −1.98398 −0.991989 0.126322i $$-0.959683\pi$$
−0.991989 + 0.126322i $$0.959683\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 5674.20i − 0.643077i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −2590.00 −0.287454 −0.143727 0.989617i $$-0.545909\pi$$
−0.143727 + 0.989617i $$0.545909\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 10756.0i − 1.16938i −0.811257 0.584690i $$-0.801216\pi$$
0.811257 0.584690i $$-0.198784\pi$$
$$440$$ 0 0
$$441$$ 16983.0 1.83382
$$442$$ 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
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −17010.0 −1.76424
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −12710.0 −1.30098 −0.650491 0.759514i $$-0.725437\pi$$
−0.650491 + 0.759514i $$0.725437\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ − 2400.62i − 0.240964i −0.992716 0.120482i $$-0.961556\pi$$
0.992716 0.120482i $$-0.0384440\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 20412.0 2.00968
$$470$$ 0 0
$$471$$ 20005.2i 1.95709i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 19485.6i 1.88223i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 7700.00 0.729916
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 5019.48i − 0.467052i −0.972351 0.233526i $$-0.924974\pi$$
0.972351 0.233526i $$-0.0750265\pi$$
$$488$$ 0 0
$$489$$ 12474.0 1.15357
$$490$$ 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
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 16367.9i 1.46839i 0.678938 + 0.734195i $$0.262440\pi$$
−0.678938 + 0.734195i $$0.737560\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 14045.2i − 1.23031i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 37100.5i 3.21180i
$$512$$ 0 0
$$513$$ −21870.0 −1.88223
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 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
$$523$$ − 20670.3i − 1.72820i −0.503320 0.864100i $$-0.667888\pi$$
0.503320 0.864100i $$-0.332112\pi$$
$$524$$ 0 0
$$525$$ −20250.0 −1.68340
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 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
$$541$$ 22678.0 1.80222 0.901112 0.433586i $$-0.142752\pi$$
0.901112 + 0.433586i $$0.142752\pi$$
$$542$$ 0 0
$$543$$ 17968.3i 1.42006i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 25533.9i − 1.99589i −0.0640963 0.997944i $$-0.520416\pi$$
0.0640963 0.997944i $$-0.479584\pi$$
$$548$$ 0 0
$$549$$ −4914.00 −0.382012
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 34020.0 2.61605
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ − 15276.7i − 1.15588i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 22728.0i − 1.68340i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ − 14185.5i − 1.03966i −0.854270 0.519829i $$-0.825996\pi$$
0.854270 0.519829i $$-0.174004\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −17710.0 −1.27778 −0.638888 0.769300i $$-0.720605\pi$$
−0.638888 + 0.769300i $$0.720605\pi$$
$$578$$ 0 0
$$579$$ 5975.58i 0.428906i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 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
$$589$$ 24300.0 1.69994
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10530.0 −0.721883
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −29302.0 −1.98877 −0.994387 0.105801i $$-0.966259\pi$$
−0.994387 + 0.105801i $$0.966259\pi$$
$$602$$ 0 0
$$603$$ − 17677.3i − 1.19382i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 9321.90i 0.623335i 0.950191 + 0.311667i $$0.100887\pi$$
−0.950191 + 0.311667i $$0.899113\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 17390.0 1.14580 0.572900 0.819625i $$-0.305818\pi$$
0.572900 + 0.819625i $$0.305818\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ − 15432.6i − 1.00208i −0.865424 0.501040i $$-0.832951\pi$$
0.865424 0.501040i $$-0.167049\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15625.0 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 31644.6i 1.99643i 0.0596825 + 0.998217i $$0.480991\pi$$
−0.0596825 + 0.998217i $$0.519009\pi$$
$$632$$ 0 0
$$633$$ −5670.00 −0.356023
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −44030.0 −2.73867
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 29836.3i 1.82991i 0.403561 + 0.914953i $$0.367772\pi$$
−0.403561 + 0.914953i $$0.632228\pi$$
$$644$$ 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
$$651$$ 25253.3i 1.52036i
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 32130.0 1.90793
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −20482.0 −1.20523 −0.602615 0.798032i $$-0.705875\pi$$
−0.602615 + 0.798032i $$0.705875\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 30294.0 1.75072
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 24050.0 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$674$$ 0 0
$$675$$ 17537.0i 1.00000i
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ − 41465.3i − 2.34358i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 23206.0i 1.28874i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32579.9i 1.79363i 0.442408 + 0.896814i $$0.354124\pi$$
−0.442408 + 0.896814i $$0.645876\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 17147.3i 0.919947i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −36146.0 −1.91466 −0.957328 0.289003i $$-0.906676\pi$$
−0.957328 + 0.289003i $$0.906676\pi$$
$$710$$ 0 0
$$711$$ − 29462.2i − 1.55403i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 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
$$721$$ −32076.0 −1.65683
$$722$$ 0 0
$$723$$ − 38337.2i − 1.97203i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 37692.9i − 1.92290i −0.274971 0.961452i $$-0.588668\pi$$
0.274971 0.961452i $$-0.411332\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −15050.0 −0.758369 −0.379184 0.925321i $$-0.623795\pi$$
−0.379184 + 0.925321i $$0.623795\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 25097.4i 1.24929i 0.780910 + 0.624644i $$0.214756\pi$$
−0.780910 + 0.624644i $$0.785244\pi$$
$$740$$ 0 0
$$741$$ 56700.0 2.81097
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 33827.0i − 1.64363i −0.569757 0.821813i $$-0.692963\pi$$
0.569757 0.821813i $$-0.307037\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 41470.0 1.99109 0.995543 0.0943039i $$-0.0300625\pi$$
0.995543 + 0.0943039i $$0.0300625\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ − 20140.3i − 0.955606i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −4606.00 −0.215990 −0.107995 0.994151i $$-0.534443\pi$$
−0.107995 + 0.994151i $$0.534443\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ − 19485.6i − 0.903151i
$$776$$ 0 0
$$777$$ −17820.0 −0.822766
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 8137.17i 0.368563i 0.982874 + 0.184281i $$0.0589958\pi$$
−0.982874 + 0.184281i $$0.941004\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 12740.0 0.570505
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ − 24162.1i − 1.04617i −0.852280 0.523087i $$-0.824780\pi$$
0.852280 0.523087i $$-0.175220\pi$$
$$812$$ 0 0
$$813$$ −46170.0 −1.99170
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 34020.0 1.45680
$$818$$ 0 0
$$819$$ 58924.4i 2.51402i
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 45611.8i 1.93187i 0.258786 + 0.965935i $$0.416677\pi$$
−0.258786 + 0.965935i $$0.583323\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −17066.0 −0.714990 −0.357495 0.933915i $$-0.616369\pi$$
−0.357495 + 0.933915i $$0.616369\pi$$
$$830$$ 0 0
$$831$$ − 20940.5i − 0.874149i
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 21870.0 0.903151
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 24389.0 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 41496.5i 1.68340i
$$848$$ 0 0
$$849$$ −40014.0 −1.61752
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −46690.0 −1.87413 −0.937066 0.349151i $$-0.886470\pi$$
−0.937066 + 0.349151i $$0.886470\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 39438.8i 1.56651i 0.621699 + 0.783256i $$0.286443\pi$$
−0.621699 + 0.783256i $$0.713557\pi$$
$$860$$ 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
$$867$$ − 25528.7i − 1.00000i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 45830.1i 1.78288i
$$872$$ 0 0
$$873$$ −35910.0 −1.39218
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50150.0 1.93095 0.965476 0.260491i $$-0.0838846\pi$$
0.965476 + 0.260491i $$0.0838846\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ − 48230.7i − 1.83816i −0.394076 0.919078i $$-0.628935\pi$$
0.394076 0.919078i $$-0.371065\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 88452.0 3.33699
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 35354.6i 1.30291i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 31208.1i 1.14250i 0.820776 + 0.571250i $$0.193541\pi$$
−0.820776 + 0.571250i $$0.806459\pi$$
$$908$$ 0 0
$$909$$ 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
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 55650.8i − 1.99755i −0.0494625 0.998776i $$-0.515751\pi$$
0.0494625 0.998776i $$-0.484249\pi$$
$$920$$ 0 0
$$921$$ −8262.00 −0.295594
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 13750.0 0.488754
$$926$$ 0 0
$$927$$ 27778.6i 0.984218i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ − 98051.4i − 3.45167i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −55510.0 −1.93536 −0.967680 0.252181i $$-0.918852\pi$$
−0.967680 + 0.252181i $$0.918852\pi$$
$$938$$ 0 0
$$939$$ − 52013.5i − 1.80766i
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ −83300.0 −2.84935
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5491.00 0.184317
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 33390.5i 1.11041i 0.831714 + 0.555204i $$0.187360\pi$$
−0.831714 + 0.555204i $$0.812640\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 63180.0 2.08166
$$974$$ 0 0
$$975$$ − 45466.3i − 1.49342i
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −17442.0 −0.567666
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 42556.5i − 1.36413i −0.731292 0.682064i $$-0.761082\pi$$
0.731292 0.682064i $$-0.238918\pi$$
$$992$$ 0 0
$$993$$ 62370.0 1.99320
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 28910.0 0.918344 0.459172 0.888347i $$-0.348146\pi$$
0.459172 + 0.888347i $$0.348146\pi$$
$$998$$ 0 0
$$999$$ 15432.6i 0.488754i
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 48.4.c.a.47.1 2
3.2 odd 2 CM 48.4.c.a.47.1 2
4.3 odd 2 inner 48.4.c.a.47.2 yes 2
8.3 odd 2 192.4.c.a.191.1 2
8.5 even 2 192.4.c.a.191.2 2
12.11 even 2 inner 48.4.c.a.47.2 yes 2
16.3 odd 4 768.4.f.a.383.1 4
16.5 even 4 768.4.f.a.383.2 4
16.11 odd 4 768.4.f.a.383.3 4
16.13 even 4 768.4.f.a.383.4 4
24.5 odd 2 192.4.c.a.191.2 2
24.11 even 2 192.4.c.a.191.1 2
48.5 odd 4 768.4.f.a.383.2 4
48.11 even 4 768.4.f.a.383.3 4
48.29 odd 4 768.4.f.a.383.4 4
48.35 even 4 768.4.f.a.383.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
48.4.c.a.47.1 2 1.1 even 1 trivial
48.4.c.a.47.1 2 3.2 odd 2 CM
48.4.c.a.47.2 yes 2 4.3 odd 2 inner
48.4.c.a.47.2 yes 2 12.11 even 2 inner
192.4.c.a.191.1 2 8.3 odd 2
192.4.c.a.191.1 2 24.11 even 2
192.4.c.a.191.2 2 8.5 even 2
192.4.c.a.191.2 2 24.5 odd 2
768.4.f.a.383.1 4 16.3 odd 4
768.4.f.a.383.1 4 48.35 even 4
768.4.f.a.383.2 4 16.5 even 4
768.4.f.a.383.2 4 48.5 odd 4
768.4.f.a.383.3 4 16.11 odd 4
768.4.f.a.383.3 4 48.11 even 4
768.4.f.a.383.4 4 16.13 even 4
768.4.f.a.383.4 4 48.29 odd 4