# Properties

 Label 768.2.c.i.767.1 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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 192) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 767.1 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.767 Dual form 768.2.c.i.767.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} -3.46410i q^{5} +2.00000i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{3} -3.46410i q^{5} +2.00000i q^{7} +3.00000 q^{9} +3.46410 q^{11} +6.00000i q^{15} -3.46410i q^{21} -7.00000 q^{25} -5.19615 q^{27} -10.3923i q^{29} -10.0000i q^{31} -6.00000 q^{33} +6.92820 q^{35} -10.3923i q^{45} +3.00000 q^{49} -3.46410i q^{53} -12.0000i q^{55} -10.3923 q^{59} +6.00000i q^{63} +14.0000 q^{73} +12.1244 q^{75} +6.92820i q^{77} -10.0000i q^{79} +9.00000 q^{81} -17.3205 q^{83} +18.0000i q^{87} +17.3205i q^{93} +2.00000 q^{97} +10.3923 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} - 28 q^{25} - 24 q^{33} + 12 q^{49} + 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100})$$ 4 * q + 12 * q^9 - 28 * q^25 - 24 * q^33 + 12 * 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
$$5$$ − 3.46410i − 1.54919i −0.632456 0.774597i $$-0.717953\pi$$
0.632456 0.774597i $$-0.282047\pi$$
$$6$$ 0 0
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 6.00000i 1.54919i
$$16$$ 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
$$21$$ − 3.46410i − 0.755929i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −7.00000 −1.40000
$$26$$ 0 0
$$27$$ −5.19615 −1.00000
$$28$$ 0 0
$$29$$ − 10.3923i − 1.92980i −0.262613 0.964901i $$-0.584584\pi$$
0.262613 0.964901i $$-0.415416\pi$$
$$30$$ 0 0
$$31$$ − 10.0000i − 1.79605i −0.439941 0.898027i $$-0.645001\pi$$
0.439941 0.898027i $$-0.354999\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 6.92820 1.17108
$$36$$ 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
$$45$$ − 10.3923i − 1.54919i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 3.46410i − 0.475831i −0.971286 0.237915i $$-0.923536\pi$$
0.971286 0.237915i $$-0.0764641\pi$$
$$54$$ 0 0
$$55$$ − 12.0000i − 1.61808i
$$56$$ 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
$$63$$ 6.00000i 0.755929i
$$64$$ 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 $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ 12.1244 1.40000
$$76$$ 0 0
$$77$$ 6.92820i 0.789542i
$$78$$ 0 0
$$79$$ − 10.0000i − 1.12509i −0.826767 0.562544i $$-0.809823\pi$$
0.826767 0.562544i $$-0.190177\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −17.3205 −1.90117 −0.950586 0.310460i $$-0.899517\pi$$
−0.950586 + 0.310460i $$0.899517\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 18.0000i 1.92980i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 17.3205i 1.79605i
$$94$$ 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
$$101$$ − 3.46410i − 0.344691i −0.985037 0.172345i $$-0.944865\pi$$
0.985037 0.172345i $$-0.0551346\pi$$
$$102$$ 0 0
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 0 0
$$105$$ −12.0000 −1.17108
$$106$$ 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
$$125$$ 6.92820i 0.619677i
$$126$$ 0 0
$$127$$ 22.0000i 1.95218i 0.217357 + 0.976092i $$0.430256\pi$$
−0.217357 + 0.976092i $$0.569744\pi$$
$$128$$ 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
$$135$$ 18.0000i 1.54919i
$$136$$ 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
$$145$$ −36.0000 −2.98964
$$146$$ 0 0
$$147$$ −5.19615 −0.428571
$$148$$ 0 0
$$149$$ 24.2487i 1.98653i 0.115857 + 0.993266i $$0.463039\pi$$
−0.115857 + 0.993266i $$0.536961\pi$$
$$150$$ 0 0
$$151$$ 2.00000i 0.162758i 0.996683 + 0.0813788i $$0.0259324\pi$$
−0.996683 + 0.0813788i $$0.974068\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −34.6410 −2.78243
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ 6.00000i 0.475831i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 20.7846i 1.61808i
$$166$$ 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
$$173$$ 17.3205i 1.31685i 0.752645 + 0.658427i $$0.228778\pi$$
−0.752645 + 0.658427i $$0.771222\pi$$
$$174$$ 0 0
$$175$$ − 14.0000i − 1.05830i
$$176$$ 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
$$189$$ − 10.3923i − 0.755929i
$$190$$ 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 $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.2487i 1.72765i 0.503793 + 0.863825i $$0.331938\pi$$
−0.503793 + 0.863825i $$0.668062\pi$$
$$198$$ 0 0
$$199$$ − 14.0000i − 0.992434i −0.868199 0.496217i $$-0.834722\pi$$
0.868199 0.496217i $$-0.165278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.7846 1.45879
$$204$$ 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
$$217$$ 20.0000 1.35769
$$218$$ 0 0
$$219$$ −24.2487 −1.63858
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 26.0000i − 1.74109i −0.492090 0.870544i $$-0.663767\pi$$
0.492090 0.870544i $$-0.336233\pi$$
$$224$$ 0 0
$$225$$ −21.0000 −1.40000
$$226$$ 0 0
$$227$$ 10.3923 0.689761 0.344881 0.938647i $$-0.387919\pi$$
0.344881 + 0.938647i $$0.387919\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ − 12.0000i − 0.789542i
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 17.3205i 1.12509i
$$238$$ 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 $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ −15.5885 −1.00000
$$244$$ 0 0
$$245$$ − 10.3923i − 0.663940i
$$246$$ 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 $$-0.0571337\pi$$
0.983935 + 0.178529i $$0.0571337\pi$$
$$252$$ 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
$$261$$ − 31.1769i − 1.92980i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 10.3923i − 0.633630i −0.948487 0.316815i $$-0.897387\pi$$
0.948487 0.316815i $$-0.102613\pi$$
$$270$$ 0 0
$$271$$ 22.0000i 1.33640i 0.743980 + 0.668202i $$0.232936\pi$$
−0.743980 + 0.668202i $$0.767064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −24.2487 −1.46225
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ − 30.0000i − 1.79605i
$$280$$ 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
$$293$$ − 31.1769i − 1.82137i −0.413096 0.910687i $$-0.635553\pi$$
0.413096 0.910687i $$-0.364447\pi$$
$$294$$ 0 0
$$295$$ 36.0000i 2.09600i
$$296$$ 0 0
$$297$$ −18.0000 −1.04447
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 24.2487i 1.37946i
$$310$$ 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 $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 0 0
$$315$$ 20.7846 1.17108
$$316$$ 0 0
$$317$$ 17.3205i 0.972817i 0.873732 + 0.486408i $$0.161693\pi$$
−0.873732 + 0.486408i $$0.838307\pi$$
$$318$$ 0 0
$$319$$ − 36.0000i − 2.01561i
$$320$$ 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 $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 34.6410i − 1.87592i
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −24.2487 −1.30174 −0.650870 0.759190i $$-0.725596\pi$$
−0.650870 + 0.759190i $$0.725596\pi$$
$$348$$ 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
$$365$$ − 48.4974i − 2.53847i
$$366$$ 0 0
$$367$$ 38.0000i 1.98358i 0.127862 + 0.991792i $$0.459188\pi$$
−0.127862 + 0.991792i $$0.540812\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.92820 0.359694
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ − 12.0000i − 0.619677i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ − 38.1051i − 1.95218i
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 24.0000 1.22315
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 24.2487i 1.22946i 0.788738 + 0.614729i $$0.210735\pi$$
−0.788738 + 0.614729i $$0.789265\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ −34.6410 −1.74298
$$396$$ 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
$$405$$ − 31.1769i − 1.54919i
$$406$$ 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
$$413$$ − 20.7846i − 1.02274i
$$414$$ 0 0
$$415$$ 60.0000i 2.94528i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 10.3923 0.507697 0.253849 0.967244i $$-0.418303\pi$$
0.253849 + 0.967244i $$0.418303\pi$$
$$420$$ 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
$$435$$ 62.3538 2.98964
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 34.0000i 1.62273i 0.584539 + 0.811366i $$0.301275\pi$$
−0.584539 + 0.811366i $$0.698725\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 31.1769 1.48126 0.740630 0.671913i $$-0.234527\pi$$
0.740630 + 0.671913i $$0.234527\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 42.0000i − 1.98653i
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 3.46410i − 0.162758i
$$454$$ 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
$$461$$ − 38.1051i − 1.77473i −0.461065 0.887366i $$-0.652533\pi$$
0.461065 0.887366i $$-0.347467\pi$$
$$462$$ 0 0
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ 0 0
$$465$$ 60.0000 2.78243
$$466$$ 0 0
$$467$$ −17.3205 −0.801498 −0.400749 0.916188i $$-0.631250\pi$$
−0.400749 + 0.916188i $$0.631250\pi$$
$$468$$ 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
$$477$$ − 10.3923i − 0.475831i
$$478$$ 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
$$485$$ − 6.92820i − 0.314594i
$$486$$ 0 0
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 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
$$495$$ − 36.0000i − 1.61808i
$$496$$ 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
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 22.5167 1.00000
$$508$$ 0 0
$$509$$ 45.0333i 1.99607i 0.0626839 + 0.998033i $$0.480034\pi$$
−0.0626839 + 0.998033i $$0.519966\pi$$
$$510$$ 0 0
$$511$$ 28.0000i 1.23865i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −48.4974 −2.13705
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ − 30.0000i − 1.31685i
$$520$$ 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
$$525$$ 24.2487i 1.05830i
$$526$$ 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
$$535$$ − 60.0000i − 2.59403i
$$536$$ 0 0
$$537$$ −42.0000 −1.81243
$$538$$ 0 0
$$539$$ 10.3923 0.447628
$$540$$ 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
$$553$$ 20.0000 0.850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 45.0333i 1.90812i 0.299611 + 0.954062i $$0.403143\pi$$
−0.299611 + 0.954062i $$0.596857\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 38.1051 1.60594 0.802970 0.596020i $$-0.203252\pi$$
0.802970 + 0.596020i $$0.203252\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 18.0000i 0.755929i
$$568$$ 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 $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 0 0
$$579$$ −45.0333 −1.87152
$$580$$ 0 0
$$581$$ − 34.6410i − 1.43715i
$$582$$ 0 0
$$583$$ − 12.0000i − 0.496989i
$$584$$ 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
$$591$$ − 42.0000i − 1.72765i
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 24.2487i 0.992434i
$$598$$ 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 $$-0.512988\pi$$
−0.0407909 + 0.999168i $$0.512988\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 3.46410i − 0.140836i
$$606$$ 0 0
$$607$$ 22.0000i 0.892952i 0.894795 + 0.446476i $$0.147321\pi$$
−0.894795 + 0.446476i $$0.852679\pi$$
$$608$$ 0 0
$$609$$ −36.0000 −1.45879
$$610$$ 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
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 50.0000i 1.99047i 0.0975126 + 0.995234i $$0.468911\pi$$
−0.0975126 + 0.995234i $$0.531089\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 76.2102 3.02431
$$636$$ 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
$$651$$ −34.6410 −1.35769
$$652$$ 0 0
$$653$$ 17.3205i 0.677804i 0.940822 + 0.338902i $$0.110055\pi$$
−0.940822 + 0.338902i $$0.889945\pi$$
$$654$$ 0 0
$$655$$ 12.0000i 0.468879i
$$656$$ 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
$$669$$ 45.0333i 1.74109i
$$670$$ 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
$$675$$ 36.3731 1.40000
$$676$$ 0 0
$$677$$ 51.9615i 1.99704i 0.0543526 + 0.998522i $$0.482690\pi$$
−0.0543526 + 0.998522i $$0.517310\pi$$
$$678$$ 0 0
$$679$$ 4.00000i 0.153506i
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ −51.9615 −1.98825 −0.994126 0.108227i $$-0.965483\pi$$
−0.994126 + 0.108227i $$0.965483\pi$$
$$684$$ 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
$$693$$ 20.7846i 0.789542i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 38.1051i − 1.43921i −0.694383 0.719605i $$-0.744323\pi$$
0.694383 0.719605i $$-0.255677\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.92820 0.260562
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 0 0
$$711$$ − 30.0000i − 1.12509i
$$712$$ 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
$$721$$ 28.0000 1.04277
$$722$$ 0 0
$$723$$ −17.3205 −0.644157
$$724$$ 0 0
$$725$$ 72.7461i 2.70172i
$$726$$ 0 0
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ 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
$$735$$ 18.0000i 0.663940i
$$736$$ 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
$$745$$ 84.0000 3.07752
$$746$$ 0 0
$$747$$ −51.9615 −1.90117
$$748$$ 0 0
$$749$$ 34.6410i 1.26576i
$$750$$ 0 0
$$751$$ − 10.0000i − 0.364905i −0.983215 0.182453i $$-0.941596\pi$$
0.983215 0.182453i $$-0.0584036\pi$$
$$752$$ 0 0
$$753$$ −54.0000 −1.96787
$$754$$ 0 0
$$755$$ 6.92820 0.252143
$$756$$ 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 $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 51.9615i 1.86893i 0.356060 + 0.934463i $$0.384120\pi$$
−0.356060 + 0.934463i $$0.615880\pi$$
$$774$$ 0 0
$$775$$ 70.0000i 2.51447i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 54.0000i 1.92980i
$$784$$ 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
$$795$$ 20.7846 0.737154
$$796$$ 0 0
$$797$$ 17.3205i 0.613524i 0.951786 + 0.306762i $$0.0992455\pi$$
−0.951786 + 0.306762i $$0.900754\pi$$
$$798$$ 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
$$807$$ 18.0000i 0.633630i
$$808$$ 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
$$813$$ − 38.1051i − 1.33640i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 31.1769i − 1.08808i −0.839059 0.544041i $$-0.816894\pi$$
0.839059 0.544041i $$-0.183106\pi$$
$$822$$ 0 0
$$823$$ − 46.0000i − 1.60346i −0.597687 0.801730i $$-0.703913\pi$$
0.597687 0.801730i $$-0.296087\pi$$
$$824$$ 0 0
$$825$$ 42.0000 1.46225
$$826$$ 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
$$837$$ 51.9615i 1.79605i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −79.0000 −2.72414
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 45.0333i 1.54919i
$$846$$ 0 0
$$847$$ 2.00000i 0.0687208i
$$848$$ 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
$$865$$ 60.0000 2.04006
$$866$$ 0 0
$$867$$ −29.4449 −1.00000
$$868$$ 0 0
$$869$$ − 34.6410i − 1.17512i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ −13.8564 −0.468432
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 0 0
$$879$$ 54.0000i 1.82137i
$$880$$ 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
$$885$$ − 62.3538i − 2.09600i
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −44.0000 −1.47571
$$890$$ 0 0
$$891$$ 31.1769 1.04447
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ − 84.0000i − 2.80781i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −103.923 −3.46603
$$900$$ 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
$$909$$ − 10.3923i − 0.344691i
$$910$$ 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
$$917$$ − 6.92820i − 0.228789i
$$918$$ 0 0
$$919$$ 50.0000i 1.64935i 0.565608 + 0.824674i $$0.308641\pi$$
−0.565608 + 0.824674i $$0.691359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 42.0000i − 1.37946i
$$928$$ 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
$$941$$ − 38.1051i − 1.24219i −0.783735 0.621096i $$-0.786688\pi$$
0.783735 0.621096i $$-0.213312\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −36.0000 −1.17108
$$946$$ 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
$$951$$ − 30.0000i − 0.972817i
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 62.3538i 2.01561i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −69.0000 −2.22581
$$962$$ 0 0
$$963$$ 51.9615 1.67444
$$964$$ 0 0
$$965$$ − 90.0666i − 2.89935i
$$966$$ 0 0
$$967$$ − 62.0000i − 1.99379i −0.0787703 0.996893i $$-0.525099\pi$$
0.0787703 0.996893i $$-0.474901\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3.46410 0.111168 0.0555842 0.998454i $$-0.482298\pi$$
0.0555842 + 0.998454i $$0.482298\pi$$
$$972$$ 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
$$985$$ 84.0000 2.67646
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 58.0000i − 1.84243i −0.389053 0.921215i $$-0.627198\pi$$
0.389053 0.921215i $$-0.372802\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −48.4974 −1.53747
$$996$$ 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.i.767.1 4
3.2 odd 2 inner 768.2.c.i.767.4 4
4.3 odd 2 inner 768.2.c.i.767.3 4
8.3 odd 2 inner 768.2.c.i.767.2 4
8.5 even 2 inner 768.2.c.i.767.4 4
12.11 even 2 inner 768.2.c.i.767.2 4
16.3 odd 4 192.2.f.a.95.3 yes 4
16.5 even 4 192.2.f.a.95.4 yes 4
16.11 odd 4 192.2.f.a.95.2 yes 4
16.13 even 4 192.2.f.a.95.1 4
24.5 odd 2 CM 768.2.c.i.767.1 4
24.11 even 2 inner 768.2.c.i.767.3 4
48.5 odd 4 192.2.f.a.95.1 4
48.11 even 4 192.2.f.a.95.3 yes 4
48.29 odd 4 192.2.f.a.95.4 yes 4
48.35 even 4 192.2.f.a.95.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.f.a.95.1 4 16.13 even 4
192.2.f.a.95.1 4 48.5 odd 4
192.2.f.a.95.2 yes 4 16.11 odd 4
192.2.f.a.95.2 yes 4 48.35 even 4
192.2.f.a.95.3 yes 4 16.3 odd 4
192.2.f.a.95.3 yes 4 48.11 even 4
192.2.f.a.95.4 yes 4 16.5 even 4
192.2.f.a.95.4 yes 4 48.29 odd 4
768.2.c.i.767.1 4 1.1 even 1 trivial
768.2.c.i.767.1 4 24.5 odd 2 CM
768.2.c.i.767.2 4 8.3 odd 2 inner
768.2.c.i.767.2 4 12.11 even 2 inner
768.2.c.i.767.3 4 4.3 odd 2 inner
768.2.c.i.767.3 4 24.11 even 2 inner
768.2.c.i.767.4 4 3.2 odd 2 inner
768.2.c.i.767.4 4 8.5 even 2 inner