# Properties

 Label 768.2.c.j.767.2 Level $768$ Weight $2$ Character 768.767 Analytic conductor $6.133$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$, degree $$1$$, not 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: $$6.13251087523$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 767.2 Root $$-0.517638i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.767 Dual form 768.2.c.j.767.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} +2.82843i q^{5} +4.89898i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{3} +2.82843i q^{5} +4.89898i q^{7} +3.00000 q^{9} -3.46410 q^{11} -4.89898i q^{15} -8.48528i q^{21} -3.00000 q^{25} -5.19615 q^{27} -2.82843i q^{29} -4.89898i q^{31} +6.00000 q^{33} -13.8564 q^{35} +8.48528i q^{45} -17.0000 q^{49} +14.1421i q^{53} -9.79796i q^{55} -10.3923 q^{59} +14.6969i q^{63} -14.0000 q^{73} +5.19615 q^{75} -16.9706i q^{77} +14.6969i q^{79} +9.00000 q^{81} +17.3205 q^{83} +4.89898i q^{87} +8.48528i q^{93} +2.00000 q^{97} -10.3923 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} - 12 q^{25} + 24 q^{33} - 68 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100})$$ 4 * q + 12 * q^9 - 12 * q^25 + 24 * q^33 - 68 * q^49 - 56 * q^73 + 36 * q^81 + 8 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ −1.73205 −1.00000
$$4$$ 0 0
$$5$$ 2.82843i 1.26491i 0.774597 + 0.632456i $$0.217953\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ 4.89898i 1.85164i 0.377964 + 0.925820i $$0.376624\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ − 4.89898i − 1.26491i
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ − 8.48528i − 1.85164i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ −5.19615 −1.00000
$$28$$ 0 0
$$29$$ − 2.82843i − 0.525226i −0.964901 0.262613i $$-0.915416\pi$$
0.964901 0.262613i $$-0.0845842\pi$$
$$30$$ 0 0
$$31$$ − 4.89898i − 0.879883i −0.898027 0.439941i $$-0.854999\pi$$
0.898027 0.439941i $$-0.145001\pi$$
$$32$$ 0 0
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ −13.8564 −2.34216
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 8.48528i 1.26491i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −17.0000 −2.42857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 14.1421i 1.94257i 0.237915 + 0.971286i $$0.423536\pi$$
−0.237915 + 0.971286i $$0.576464\pi$$
$$54$$ 0 0
$$55$$ − 9.79796i − 1.32116i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −10.3923 −1.35296 −0.676481 0.736460i $$-0.736496\pi$$
−0.676481 + 0.736460i $$0.736496\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 14.6969i 1.85164i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 0 0
$$75$$ 5.19615 0.600000
$$76$$ 0 0
$$77$$ − 16.9706i − 1.93398i
$$78$$ 0 0
$$79$$ 14.6969i 1.65353i 0.562544 + 0.826767i $$0.309823\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 17.3205 1.90117 0.950586 0.310460i $$-0.100483\pi$$
0.950586 + 0.310460i $$0.100483\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.89898i 0.525226i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.48528i 0.879883i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ −10.3923 −1.04447
$$100$$ 0 0
$$101$$ − 19.7990i − 1.97007i −0.172345 0.985037i $$-0.555135\pi$$
0.172345 0.985037i $$-0.444865\pi$$
$$102$$ 0 0
$$103$$ − 14.6969i − 1.44813i −0.689730 0.724066i $$-0.742271\pi$$
0.689730 0.724066i $$-0.257729\pi$$
$$104$$ 0 0
$$105$$ 24.0000 2.34216
$$106$$ 0 0
$$107$$ 17.3205 1.67444 0.837218 0.546869i $$-0.184180\pi$$
0.837218 + 0.546869i $$0.184180\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 5.65685i 0.505964i
$$126$$ 0 0
$$127$$ − 4.89898i − 0.434714i −0.976092 0.217357i $$-0.930256\pi$$
0.976092 0.217357i $$-0.0697436\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.46410 −0.302660 −0.151330 0.988483i $$-0.548356\pi$$
−0.151330 + 0.988483i $$0.548356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ − 14.6969i − 1.26491i
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 29.4449 2.42857
$$148$$ 0 0
$$149$$ 2.82843i 0.231714i 0.993266 + 0.115857i $$0.0369614\pi$$
−0.993266 + 0.115857i $$0.963039\pi$$
$$150$$ 0 0
$$151$$ 24.4949i 1.99337i 0.0813788 + 0.996683i $$0.474068\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 13.8564 1.11297
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ − 24.4949i − 1.94257i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 16.9706i 1.32116i
$$166$$ 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
$$173$$ 19.7990i 1.50529i 0.658427 + 0.752645i $$0.271222\pi$$
−0.658427 + 0.752645i $$0.728778\pi$$
$$174$$ 0 0
$$175$$ − 14.6969i − 1.11098i
$$176$$ 0 0
$$177$$ 18.0000 1.35296
$$178$$ 0 0
$$179$$ 24.2487 1.81243 0.906217 0.422813i $$-0.138957\pi$$
0.906217 + 0.422813i $$0.138957\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 25.4558i − 1.85164i
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.1421i 1.00759i 0.863825 + 0.503793i $$0.168062\pi$$
−0.863825 + 0.503793i $$0.831938\pi$$
$$198$$ 0 0
$$199$$ 24.4949i 1.73640i 0.496217 + 0.868199i $$0.334722\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 13.8564 0.972529
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 24.0000 1.62923
$$218$$ 0 0
$$219$$ 24.2487 1.63858
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.6969i 0.984180i 0.870544 + 0.492090i $$0.163767\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 0 0
$$225$$ −9.00000 −0.600000
$$226$$ 0 0
$$227$$ −10.3923 −0.689761 −0.344881 0.938647i $$-0.612081\pi$$
−0.344881 + 0.938647i $$0.612081\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 29.3939i 1.93398i
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 25.4558i − 1.65353i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ −15.5885 −1.00000
$$244$$ 0 0
$$245$$ − 48.0833i − 3.07193i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −30.0000 −1.90117
$$250$$ 0 0
$$251$$ −31.1769 −1.96787 −0.983935 0.178529i $$-0.942866\pi$$
−0.983935 + 0.178529i $$0.942866\pi$$
$$252$$ 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
$$259$$ 0 0
$$260$$ 0 0
$$261$$ − 8.48528i − 0.525226i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −40.0000 −2.45718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 31.1127i 1.89697i 0.316815 + 0.948487i $$0.397387\pi$$
−0.316815 + 0.948487i $$0.602613\pi$$
$$270$$ 0 0
$$271$$ − 24.4949i − 1.48796i −0.668202 0.743980i $$-0.732936\pi$$
0.668202 0.743980i $$-0.267064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 10.3923 0.626680
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ − 14.6969i − 0.879883i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −3.46410 −0.203069
$$292$$ 0 0
$$293$$ 14.1421i 0.826192i 0.910687 + 0.413096i $$0.135553\pi$$
−0.910687 + 0.413096i $$0.864447\pi$$
$$294$$ 0 0
$$295$$ − 29.3939i − 1.71138i
$$296$$ 0 0
$$297$$ 18.0000 1.04447
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 34.2929i 1.97007i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 25.4558i 1.44813i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 34.0000 1.92179 0.960897 0.276907i $$-0.0893093\pi$$
0.960897 + 0.276907i $$0.0893093\pi$$
$$314$$ 0 0
$$315$$ −41.5692 −2.34216
$$316$$ 0 0
$$317$$ 31.1127i 1.74746i 0.486408 + 0.873732i $$0.338307\pi$$
−0.486408 + 0.873732i $$0.661693\pi$$
$$318$$ 0 0
$$319$$ 9.79796i 0.548580i
$$320$$ 0 0
$$321$$ −30.0000 −1.67444
$$322$$ 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
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.9706i 0.919007i
$$342$$ 0 0
$$343$$ − 48.9898i − 2.64520i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.2487 1.30174 0.650870 0.759190i $$-0.274404\pi$$
0.650870 + 0.759190i $$0.274404\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 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
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ −1.73205 −0.0909091
$$364$$ 0 0
$$365$$ − 39.5980i − 2.07265i
$$366$$ 0 0
$$367$$ − 4.89898i − 0.255725i −0.991792 0.127862i $$-0.959188\pi$$
0.991792 0.127862i $$-0.0408116\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −69.2820 −3.59694
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ − 9.79796i − 0.505964i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 8.48528i 0.434714i
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 48.0000 2.44631
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 31.1127i − 1.57748i −0.614729 0.788738i $$-0.710735\pi$$
0.614729 0.788738i $$-0.289265\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ −41.5692 −2.09157
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 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
$$405$$ 25.4558i 1.26491i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 50.9117i − 2.50520i
$$414$$ 0 0
$$415$$ 48.9898i 2.40481i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −10.3923 −0.507697 −0.253849 0.967244i $$-0.581697\pi$$
−0.253849 + 0.967244i $$0.581697\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 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
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −13.8564 −0.664364
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 24.4949i 1.16908i 0.811366 + 0.584539i $$0.198725\pi$$
−0.811366 + 0.584539i $$0.801275\pi$$
$$440$$ 0 0
$$441$$ −51.0000 −2.42857
$$442$$ 0 0
$$443$$ −31.1769 −1.48126 −0.740630 0.671913i $$-0.765473\pi$$
−0.740630 + 0.671913i $$0.765473\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 4.89898i − 0.231714i
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 42.4264i − 1.99337i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 19.7990i 0.922131i 0.887366 + 0.461065i $$0.152533\pi$$
−0.887366 + 0.461065i $$0.847467\pi$$
$$462$$ 0 0
$$463$$ 34.2929i 1.59372i 0.604161 + 0.796862i $$0.293508\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 17.3205 0.801498 0.400749 0.916188i $$-0.368750\pi$$
0.400749 + 0.916188i $$0.368750\pi$$
$$468$$ 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
$$477$$ 42.4264i 1.94257i
$$478$$ 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
$$485$$ 5.65685i 0.256865i
$$486$$ 0 0
$$487$$ 44.0908i 1.99795i 0.0453143 + 0.998973i $$0.485571\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −38.1051 −1.71966 −0.859830 0.510581i $$-0.829431\pi$$
−0.859830 + 0.510581i $$0.829431\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ − 29.3939i − 1.32116i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 56.0000 2.49197
$$506$$ 0 0
$$507$$ 22.5167 1.00000
$$508$$ 0 0
$$509$$ − 2.82843i − 0.125368i −0.998033 0.0626839i $$-0.980034\pi$$
0.998033 0.0626839i $$-0.0199660\pi$$
$$510$$ 0 0
$$511$$ − 68.5857i − 3.03405i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 41.5692 1.83176
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ − 34.2929i − 1.50529i
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 25.4558i 1.11098i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −31.1769 −1.35296
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 48.9898i 2.11801i
$$536$$ 0 0
$$537$$ −42.0000 −1.81243
$$538$$ 0 0
$$539$$ 58.8897 2.53656
$$540$$ 0 0
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −72.0000 −3.06175
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 14.1421i − 0.599222i −0.954062 0.299611i $$-0.903143\pi$$
0.954062 0.299611i $$-0.0968568\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −38.1051 −1.60594 −0.802970 0.596020i $$-0.796748\pi$$
−0.802970 + 0.596020i $$0.796748\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 44.0908i 1.85164i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 0 0
$$579$$ 45.0333 1.87152
$$580$$ 0 0
$$581$$ 84.8528i 3.52029i
$$582$$ 0 0
$$583$$ − 48.9898i − 2.02895i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 17.3205 0.714894 0.357447 0.933933i $$-0.383647\pi$$
0.357447 + 0.933933i $$0.383647\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ − 24.4949i − 1.00759i
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 42.4264i − 1.73640i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 2.82843i 0.114992i
$$606$$ 0 0
$$607$$ − 44.0908i − 1.78959i −0.446476 0.894795i $$-0.647321\pi$$
0.446476 0.894795i $$-0.352679\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 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
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 4.89898i 0.195025i 0.995234 + 0.0975126i $$0.0310886\pi$$
−0.995234 + 0.0975126i $$0.968911\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 13.8564 0.549875
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ −41.5692 −1.62923
$$652$$ 0 0
$$653$$ − 48.0833i − 1.88164i −0.338902 0.940822i $$-0.610055\pi$$
0.338902 0.940822i $$-0.389945\pi$$
$$654$$ 0 0
$$655$$ − 9.79796i − 0.382838i
$$656$$ 0 0
$$657$$ −42.0000 −1.63858
$$658$$ 0 0
$$659$$ 24.2487 0.944596 0.472298 0.881439i $$-0.343425\pi$$
0.472298 + 0.881439i $$0.343425\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ − 25.4558i − 0.984180i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 15.5885 0.600000
$$676$$ 0 0
$$677$$ 2.82843i 0.108705i 0.998522 + 0.0543526i $$0.0173095\pi$$
−0.998522 + 0.0543526i $$0.982690\pi$$
$$678$$ 0 0
$$679$$ 9.79796i 0.376011i
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 0 0
$$683$$ 51.9615 1.98825 0.994126 0.108227i $$-0.0345173\pi$$
0.994126 + 0.108227i $$0.0345173\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ − 50.9117i − 1.93398i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 36.7696i − 1.38877i −0.719605 0.694383i $$-0.755677\pi$$
0.719605 0.694383i $$-0.244323\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 96.9948 3.64787
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 0 0
$$711$$ 44.0908i 1.65353i
$$712$$ 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
$$721$$ 72.0000 2.68142
$$722$$ 0 0
$$723$$ 17.3205 0.644157
$$724$$ 0 0
$$725$$ 8.48528i 0.315135i
$$726$$ 0 0
$$727$$ − 53.8888i − 1.99862i −0.0370879 0.999312i $$-0.511808\pi$$
0.0370879 0.999312i $$-0.488192\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 83.2827i 3.07193i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 0 0
$$747$$ 51.9615 1.90117
$$748$$ 0 0
$$749$$ 84.8528i 3.10045i
$$750$$ 0 0
$$751$$ 53.8888i 1.96643i 0.182453 + 0.983215i $$0.441596\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 0 0
$$753$$ 54.0000 1.96787
$$754$$ 0 0
$$755$$ −69.2820 −2.52143
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 19.7990i − 0.712120i −0.934463 0.356060i $$-0.884120\pi$$
0.934463 0.356060i $$-0.115880\pi$$
$$774$$ 0 0
$$775$$ 14.6969i 0.527930i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 14.6969i 0.525226i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 69.2820 2.45718
$$796$$ 0 0
$$797$$ 53.7401i 1.90357i 0.306762 + 0.951786i $$0.400754\pi$$
−0.306762 + 0.951786i $$0.599246\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 48.4974 1.71144
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 53.8888i − 1.89697i
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 42.4264i 1.48796i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 48.0833i 1.67812i 0.544041 + 0.839059i $$0.316894\pi$$
−0.544041 + 0.839059i $$0.683106\pi$$
$$822$$ 0 0
$$823$$ − 34.2929i − 1.19537i −0.801730 0.597687i $$-0.796087\pi$$
0.801730 0.597687i $$-0.203913\pi$$
$$824$$ 0 0
$$825$$ −18.0000 −0.626680
$$826$$ 0 0
$$827$$ −10.3923 −0.361376 −0.180688 0.983540i $$-0.557832\pi$$
−0.180688 + 0.983540i $$0.557832\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 25.4558i 0.879883i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 21.0000 0.724138
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 36.7696i − 1.26491i
$$846$$ 0 0
$$847$$ 4.89898i 0.168331i
$$848$$ 0 0
$$849$$ 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
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −56.0000 −1.90406
$$866$$ 0 0
$$867$$ −29.4449 −1.00000
$$868$$ 0 0
$$869$$ − 50.9117i − 1.72706i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ −27.7128 −0.936864
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 0 0
$$879$$ − 24.4949i − 0.826192i
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 50.9117i 1.71138i
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −31.1769 −1.04447
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 68.5857i 2.29257i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −13.8564 −0.462137
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ − 59.3970i − 1.97007i
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −60.0000 −1.98571
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 16.9706i − 0.560417i
$$918$$ 0 0
$$919$$ − 34.2929i − 1.13122i −0.824674 0.565608i $$-0.808641\pi$$
0.824674 0.565608i $$-0.191359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 44.0908i − 1.44813i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −58.0000 −1.89478 −0.947389 0.320085i $$-0.896288\pi$$
−0.947389 + 0.320085i $$0.896288\pi$$
$$938$$ 0 0
$$939$$ −58.8897 −1.92179
$$940$$ 0 0
$$941$$ − 48.0833i − 1.56747i −0.621096 0.783735i $$-0.713312\pi$$
0.621096 0.783735i $$-0.286688\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 72.0000 2.34216
$$946$$ 0 0
$$947$$ 24.2487 0.787977 0.393989 0.919115i $$-0.371095\pi$$
0.393989 + 0.919115i $$0.371095\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 53.8888i − 1.74746i
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 16.9706i − 0.548580i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 7.00000 0.225806
$$962$$ 0 0
$$963$$ 51.9615 1.67444
$$964$$ 0 0
$$965$$ − 73.5391i − 2.36731i
$$966$$ 0 0
$$967$$ 4.89898i 0.157541i 0.996893 + 0.0787703i $$0.0250994\pi$$
−0.996893 + 0.0787703i $$0.974901\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3.46410 −0.111168 −0.0555842 0.998454i $$-0.517702\pi$$
−0.0555842 + 0.998454i $$0.517702\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ −40.0000 −1.27451
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 24.4949i − 0.778106i −0.921215 0.389053i $$-0.872802\pi$$
0.921215 0.389053i $$-0.127198\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −69.2820 −2.19639
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 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 768.2.c.j.767.2 4
3.2 odd 2 inner 768.2.c.j.767.3 4
4.3 odd 2 inner 768.2.c.j.767.4 4
8.3 odd 2 inner 768.2.c.j.767.1 4
8.5 even 2 inner 768.2.c.j.767.3 4
12.11 even 2 inner 768.2.c.j.767.1 4
16.3 odd 4 384.2.f.a.191.4 yes 4
16.5 even 4 384.2.f.a.191.3 yes 4
16.11 odd 4 384.2.f.a.191.1 4
16.13 even 4 384.2.f.a.191.2 yes 4
24.5 odd 2 CM 768.2.c.j.767.2 4
24.11 even 2 inner 768.2.c.j.767.4 4
48.5 odd 4 384.2.f.a.191.2 yes 4
48.11 even 4 384.2.f.a.191.4 yes 4
48.29 odd 4 384.2.f.a.191.3 yes 4
48.35 even 4 384.2.f.a.191.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.f.a.191.1 4 16.11 odd 4
384.2.f.a.191.1 4 48.35 even 4
384.2.f.a.191.2 yes 4 16.13 even 4
384.2.f.a.191.2 yes 4 48.5 odd 4
384.2.f.a.191.3 yes 4 16.5 even 4
384.2.f.a.191.3 yes 4 48.29 odd 4
384.2.f.a.191.4 yes 4 16.3 odd 4
384.2.f.a.191.4 yes 4 48.11 even 4
768.2.c.j.767.1 4 8.3 odd 2 inner
768.2.c.j.767.1 4 12.11 even 2 inner
768.2.c.j.767.2 4 1.1 even 1 trivial
768.2.c.j.767.2 4 24.5 odd 2 CM
768.2.c.j.767.3 4 3.2 odd 2 inner
768.2.c.j.767.3 4 8.5 even 2 inner
768.2.c.j.767.4 4 4.3 odd 2 inner
768.2.c.j.767.4 4 24.11 even 2 inner