# Properties

 Label 2736.3.o.e.721.1 Level $2736$ Weight $3$ Character 2736.721 Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (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: $$74.5506003290$$ 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: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 721.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.721 Dual form 2736.3.o.e.721.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{5} +10.0000 q^{7} +O(q^{10})$$ $$q-4.00000 q^{5} +10.0000 q^{7} +10.0000 q^{11} -24.2487i q^{13} -10.0000 q^{17} -19.0000 q^{19} -20.0000 q^{23} -9.00000 q^{25} +34.6410i q^{29} -17.3205i q^{31} -40.0000 q^{35} -10.3923i q^{37} -34.6410i q^{41} +10.0000 q^{43} -80.0000 q^{47} +51.0000 q^{49} -41.5692i q^{53} -40.0000 q^{55} +34.6410i q^{59} -10.0000 q^{61} +96.9948i q^{65} +76.2102i q^{67} +103.923i q^{71} -10.0000 q^{73} +100.000 q^{77} +17.3205i q^{79} +70.0000 q^{83} +40.0000 q^{85} +103.923i q^{89} -242.487i q^{91} +76.0000 q^{95} +76.2102i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{5} + 20 q^{7}+O(q^{10})$$ 2 * q - 8 * q^5 + 20 * q^7 $$2 q - 8 q^{5} + 20 q^{7} + 20 q^{11} - 20 q^{17} - 38 q^{19} - 40 q^{23} - 18 q^{25} - 80 q^{35} + 20 q^{43} - 160 q^{47} + 102 q^{49} - 80 q^{55} - 20 q^{61} - 20 q^{73} + 200 q^{77} + 140 q^{83} + 80 q^{85} + 152 q^{95}+O(q^{100})$$ 2 * q - 8 * q^5 + 20 * q^7 + 20 * q^11 - 20 * q^17 - 38 * q^19 - 40 * q^23 - 18 * q^25 - 80 * q^35 + 20 * q^43 - 160 * q^47 + 102 * q^49 - 80 * q^55 - 20 * q^61 - 20 * q^73 + 200 * q^77 + 140 * q^83 + 80 * q^85 + 152 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$4$$ 0 0
$$5$$ −4.00000 −0.800000 −0.400000 0.916515i $$-0.630990\pi$$
−0.400000 + 0.916515i $$0.630990\pi$$
$$6$$ 0 0
$$7$$ 10.0000 1.42857 0.714286 0.699854i $$-0.246752\pi$$
0.714286 + 0.699854i $$0.246752\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 10.0000 0.909091 0.454545 0.890724i $$-0.349802\pi$$
0.454545 + 0.890724i $$0.349802\pi$$
$$12$$ 0 0
$$13$$ − 24.2487i − 1.86529i −0.360801 0.932643i $$-0.617497\pi$$
0.360801 0.932643i $$-0.382503\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −10.0000 −0.588235 −0.294118 0.955769i $$-0.595026\pi$$
−0.294118 + 0.955769i $$0.595026\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −20.0000 −0.869565 −0.434783 0.900535i $$-0.643175\pi$$
−0.434783 + 0.900535i $$0.643175\pi$$
$$24$$ 0 0
$$25$$ −9.00000 −0.360000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 34.6410i 1.19452i 0.802049 + 0.597259i $$0.203744\pi$$
−0.802049 + 0.597259i $$0.796256\pi$$
$$30$$ 0 0
$$31$$ − 17.3205i − 0.558726i −0.960186 0.279363i $$-0.909877\pi$$
0.960186 0.279363i $$-0.0901233\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −40.0000 −1.14286
$$36$$ 0 0
$$37$$ − 10.3923i − 0.280873i −0.990090 0.140437i $$-0.955149\pi$$
0.990090 0.140437i $$-0.0448506\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 34.6410i − 0.844903i −0.906386 0.422451i $$-0.861170\pi$$
0.906386 0.422451i $$-0.138830\pi$$
$$42$$ 0 0
$$43$$ 10.0000 0.232558 0.116279 0.993217i $$-0.462903\pi$$
0.116279 + 0.993217i $$0.462903\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −80.0000 −1.70213 −0.851064 0.525062i $$-0.824042\pi$$
−0.851064 + 0.525062i $$0.824042\pi$$
$$48$$ 0 0
$$49$$ 51.0000 1.04082
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 41.5692i − 0.784325i −0.919896 0.392162i $$-0.871727\pi$$
0.919896 0.392162i $$-0.128273\pi$$
$$54$$ 0 0
$$55$$ −40.0000 −0.727273
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 34.6410i 0.587136i 0.955938 + 0.293568i $$0.0948427\pi$$
−0.955938 + 0.293568i $$0.905157\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −0.163934 −0.0819672 0.996635i $$-0.526120\pi$$
−0.0819672 + 0.996635i $$0.526120\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 96.9948i 1.49223i
$$66$$ 0 0
$$67$$ 76.2102i 1.13747i 0.822522 + 0.568733i $$0.192566\pi$$
−0.822522 + 0.568733i $$0.807434\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 103.923i 1.46370i 0.681463 + 0.731852i $$0.261344\pi$$
−0.681463 + 0.731852i $$0.738656\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −0.136986 −0.0684932 0.997652i $$-0.521819\pi$$
−0.0684932 + 0.997652i $$0.521819\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 100.000 1.29870
$$78$$ 0 0
$$79$$ 17.3205i 0.219247i 0.993973 + 0.109623i $$0.0349645\pi$$
−0.993973 + 0.109623i $$0.965035\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 70.0000 0.843373 0.421687 0.906742i $$-0.361438\pi$$
0.421687 + 0.906742i $$0.361438\pi$$
$$84$$ 0 0
$$85$$ 40.0000 0.470588
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 103.923i 1.16767i 0.811871 + 0.583837i $$0.198449\pi$$
−0.811871 + 0.583837i $$0.801551\pi$$
$$90$$ 0 0
$$91$$ − 242.487i − 2.66469i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 76.0000 0.800000
$$96$$ 0 0
$$97$$ 76.2102i 0.785673i 0.919608 + 0.392836i $$0.128506\pi$$
−0.919608 + 0.392836i $$0.871494\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −100.000 −0.990099 −0.495050 0.868865i $$-0.664850\pi$$
−0.495050 + 0.868865i $$0.664850\pi$$
$$102$$ 0 0
$$103$$ − 183.597i − 1.78250i −0.453513 0.891249i $$-0.649830\pi$$
0.453513 0.891249i $$-0.350170\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 62.3538i 0.582746i 0.956610 + 0.291373i $$0.0941121\pi$$
−0.956610 + 0.291373i $$0.905888\pi$$
$$108$$ 0 0
$$109$$ − 155.885i − 1.43013i −0.699056 0.715067i $$-0.746396\pi$$
0.699056 0.715067i $$-0.253604\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.92820i 0.0613115i 0.999530 + 0.0306558i $$0.00975956\pi$$
−0.999530 + 0.0306558i $$0.990240\pi$$
$$114$$ 0 0
$$115$$ 80.0000 0.695652
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −100.000 −0.840336
$$120$$ 0 0
$$121$$ −21.0000 −0.173554
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 136.000 1.08800
$$126$$ 0 0
$$127$$ 114.315i 0.900121i 0.892998 + 0.450060i $$0.148598\pi$$
−0.892998 + 0.450060i $$0.851402\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −38.0000 −0.290076 −0.145038 0.989426i $$-0.546330\pi$$
−0.145038 + 0.989426i $$0.546330\pi$$
$$132$$ 0 0
$$133$$ −190.000 −1.42857
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −190.000 −1.38686 −0.693431 0.720523i $$-0.743902\pi$$
−0.693431 + 0.720523i $$0.743902\pi$$
$$138$$ 0 0
$$139$$ −50.0000 −0.359712 −0.179856 0.983693i $$-0.557563\pi$$
−0.179856 + 0.983693i $$0.557563\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 242.487i − 1.69571i
$$144$$ 0 0
$$145$$ − 138.564i − 0.955614i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 20.0000 0.134228 0.0671141 0.997745i $$-0.478621\pi$$
0.0671141 + 0.997745i $$0.478621\pi$$
$$150$$ 0 0
$$151$$ 225.167i 1.49117i 0.666411 + 0.745585i $$0.267830\pi$$
−0.666411 + 0.745585i $$0.732170\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 69.2820i 0.446981i
$$156$$ 0 0
$$157$$ 230.000 1.46497 0.732484 0.680784i $$-0.238361\pi$$
0.732484 + 0.680784i $$0.238361\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −200.000 −1.24224
$$162$$ 0 0
$$163$$ −170.000 −1.04294 −0.521472 0.853268i $$-0.674617\pi$$
−0.521472 + 0.853268i $$0.674617\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 131.636i − 0.788239i −0.919059 0.394119i $$-0.871050\pi$$
0.919059 0.394119i $$-0.128950\pi$$
$$168$$ 0 0
$$169$$ −419.000 −2.47929
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 235.559i − 1.36161i −0.732464 0.680806i $$-0.761630\pi$$
0.732464 0.680806i $$-0.238370\pi$$
$$174$$ 0 0
$$175$$ −90.0000 −0.514286
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 103.923i − 0.580576i −0.956939 0.290288i $$-0.906249\pi$$
0.956939 0.290288i $$-0.0937511\pi$$
$$180$$ 0 0
$$181$$ 259.808i 1.43540i 0.696352 + 0.717701i $$0.254805\pi$$
−0.696352 + 0.717701i $$0.745195\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 41.5692i 0.224698i
$$186$$ 0 0
$$187$$ −100.000 −0.534759
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −332.000 −1.73822 −0.869110 0.494619i $$-0.835308\pi$$
−0.869110 + 0.494619i $$0.835308\pi$$
$$192$$ 0 0
$$193$$ − 96.9948i − 0.502564i −0.967914 0.251282i $$-0.919148\pi$$
0.967914 0.251282i $$-0.0808521\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −160.000 −0.812183 −0.406091 0.913832i $$-0.633109\pi$$
−0.406091 + 0.913832i $$0.633109\pi$$
$$198$$ 0 0
$$199$$ −98.0000 −0.492462 −0.246231 0.969211i $$-0.579192\pi$$
−0.246231 + 0.969211i $$0.579192\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 346.410i 1.70645i
$$204$$ 0 0
$$205$$ 138.564i 0.675922i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −190.000 −0.909091
$$210$$ 0 0
$$211$$ − 173.205i − 0.820877i −0.911888 0.410439i $$-0.865376\pi$$
0.911888 0.410439i $$-0.134624\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −40.0000 −0.186047
$$216$$ 0 0
$$217$$ − 173.205i − 0.798180i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 242.487i 1.09723i
$$222$$ 0 0
$$223$$ 79.6743i 0.357284i 0.983914 + 0.178642i $$0.0571704\pi$$
−0.983914 + 0.178642i $$0.942830\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 76.2102i − 0.335728i −0.985810 0.167864i $$-0.946313\pi$$
0.985810 0.167864i $$-0.0536869\pi$$
$$228$$ 0 0
$$229$$ 110.000 0.480349 0.240175 0.970730i $$-0.422795\pi$$
0.240175 + 0.970730i $$0.422795\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −190.000 −0.815451 −0.407725 0.913105i $$-0.633678\pi$$
−0.407725 + 0.913105i $$0.633678\pi$$
$$234$$ 0 0
$$235$$ 320.000 1.36170
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −128.000 −0.535565 −0.267782 0.963479i $$-0.586291\pi$$
−0.267782 + 0.963479i $$0.586291\pi$$
$$240$$ 0 0
$$241$$ 138.564i 0.574955i 0.957787 + 0.287477i $$0.0928166\pi$$
−0.957787 + 0.287477i $$0.907183\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −204.000 −0.832653
$$246$$ 0 0
$$247$$ 460.726i 1.86529i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2.00000 −0.00796813 −0.00398406 0.999992i $$-0.501268\pi$$
−0.00398406 + 0.999992i $$0.501268\pi$$
$$252$$ 0 0
$$253$$ −200.000 −0.790514
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 491.902i − 1.91402i −0.290059 0.957009i $$-0.593675\pi$$
0.290059 0.957009i $$-0.406325\pi$$
$$258$$ 0 0
$$259$$ − 103.923i − 0.401247i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −200.000 −0.760456 −0.380228 0.924893i $$-0.624155\pi$$
−0.380228 + 0.924893i $$0.624155\pi$$
$$264$$ 0 0
$$265$$ 166.277i 0.627460i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 415.692i 1.54532i 0.634818 + 0.772662i $$0.281075\pi$$
−0.634818 + 0.772662i $$0.718925\pi$$
$$270$$ 0 0
$$271$$ −170.000 −0.627306 −0.313653 0.949538i $$-0.601553\pi$$
−0.313653 + 0.949538i $$0.601553\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −90.0000 −0.327273
$$276$$ 0 0
$$277$$ −10.0000 −0.0361011 −0.0180505 0.999837i $$-0.505746\pi$$
−0.0180505 + 0.999837i $$0.505746\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 381.051i − 1.35605i −0.735037 0.678027i $$-0.762835\pi$$
0.735037 0.678027i $$-0.237165\pi$$
$$282$$ 0 0
$$283$$ 70.0000 0.247350 0.123675 0.992323i $$-0.460532\pi$$
0.123675 + 0.992323i $$0.460532\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 346.410i − 1.20700i
$$288$$ 0 0
$$289$$ −189.000 −0.653979
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 180.133i − 0.614789i −0.951582 0.307395i $$-0.900543\pi$$
0.951582 0.307395i $$-0.0994572\pi$$
$$294$$ 0 0
$$295$$ − 138.564i − 0.469709i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 484.974i 1.62199i
$$300$$ 0 0
$$301$$ 100.000 0.332226
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 40.0000 0.131148
$$306$$ 0 0
$$307$$ 145.492i 0.473916i 0.971520 + 0.236958i $$0.0761504\pi$$
−0.971520 + 0.236958i $$0.923850\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 580.000 1.86495 0.932476 0.361232i $$-0.117644\pi$$
0.932476 + 0.361232i $$0.117644\pi$$
$$312$$ 0 0
$$313$$ −370.000 −1.18211 −0.591054 0.806632i $$-0.701288\pi$$
−0.591054 + 0.806632i $$0.701288\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 27.7128i − 0.0874221i −0.999044 0.0437111i $$-0.986082\pi$$
0.999044 0.0437111i $$-0.0139181\pi$$
$$318$$ 0 0
$$319$$ 346.410i 1.08593i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 190.000 0.588235
$$324$$ 0 0
$$325$$ 218.238i 0.671503i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −800.000 −2.43161
$$330$$ 0 0
$$331$$ 173.205i 0.523278i 0.965166 + 0.261639i $$0.0842630\pi$$
−0.965166 + 0.261639i $$0.915737\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 304.841i − 0.909973i
$$336$$ 0 0
$$337$$ 339.482i 1.00736i 0.863889 + 0.503682i $$0.168022\pi$$
−0.863889 + 0.503682i $$0.831978\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 173.205i − 0.507933i
$$342$$ 0 0
$$343$$ 20.0000 0.0583090
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −590.000 −1.70029 −0.850144 0.526550i $$-0.823485\pi$$
−0.850144 + 0.526550i $$0.823485\pi$$
$$348$$ 0 0
$$349$$ 98.0000 0.280802 0.140401 0.990095i $$-0.455161\pi$$
0.140401 + 0.990095i $$0.455161\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −190.000 −0.538244 −0.269122 0.963106i $$-0.586733\pi$$
−0.269122 + 0.963106i $$0.586733\pi$$
$$354$$ 0 0
$$355$$ − 415.692i − 1.17096i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −200.000 −0.557103 −0.278552 0.960421i $$-0.589854\pi$$
−0.278552 + 0.960421i $$0.589854\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 40.0000 0.109589
$$366$$ 0 0
$$367$$ −170.000 −0.463215 −0.231608 0.972809i $$-0.574399\pi$$
−0.231608 + 0.972809i $$0.574399\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 415.692i − 1.12046i
$$372$$ 0 0
$$373$$ − 356.802i − 0.956575i −0.878203 0.478287i $$-0.841258\pi$$
0.878203 0.478287i $$-0.158742\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 840.000 2.22812
$$378$$ 0 0
$$379$$ − 207.846i − 0.548407i −0.961672 0.274203i $$-0.911586\pi$$
0.961672 0.274203i $$-0.0884141\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 630.466i − 1.64613i −0.567950 0.823063i $$-0.692263\pi$$
0.567950 0.823063i $$-0.307737\pi$$
$$384$$ 0 0
$$385$$ −400.000 −1.03896
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 128.000 0.329049 0.164524 0.986373i $$-0.447391\pi$$
0.164524 + 0.986373i $$0.447391\pi$$
$$390$$ 0 0
$$391$$ 200.000 0.511509
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 69.2820i − 0.175398i
$$396$$ 0 0
$$397$$ 650.000 1.63728 0.818640 0.574307i $$-0.194729\pi$$
0.818640 + 0.574307i $$0.194729\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 173.205i 0.431933i 0.976401 + 0.215966i $$0.0692902\pi$$
−0.976401 + 0.215966i $$0.930710\pi$$
$$402$$ 0 0
$$403$$ −420.000 −1.04218
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 103.923i − 0.255339i
$$408$$ 0 0
$$409$$ 173.205i 0.423484i 0.977326 + 0.211742i $$0.0679137\pi$$
−0.977326 + 0.211742i $$0.932086\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 346.410i 0.838766i
$$414$$ 0 0
$$415$$ −280.000 −0.674699
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −38.0000 −0.0906921 −0.0453461 0.998971i $$-0.514439\pi$$
−0.0453461 + 0.998971i $$0.514439\pi$$
$$420$$ 0 0
$$421$$ − 17.3205i − 0.0411413i −0.999788 0.0205707i $$-0.993452\pi$$
0.999788 0.0205707i $$-0.00654831\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 90.0000 0.211765
$$426$$ 0 0
$$427$$ −100.000 −0.234192
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ − 353.338i − 0.816024i −0.912977 0.408012i $$-0.866222\pi$$
0.912977 0.408012i $$-0.133778\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 380.000 0.869565
$$438$$ 0 0
$$439$$ 121.244i 0.276181i 0.990420 + 0.138091i $$0.0440965\pi$$
−0.990420 + 0.138091i $$0.955903\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −110.000 −0.248307 −0.124153 0.992263i $$-0.539622\pi$$
−0.124153 + 0.992263i $$0.539622\pi$$
$$444$$ 0 0
$$445$$ − 415.692i − 0.934140i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 311.769i − 0.694363i −0.937798 0.347182i $$-0.887139\pi$$
0.937798 0.347182i $$-0.112861\pi$$
$$450$$ 0 0
$$451$$ − 346.410i − 0.768093i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 969.948i 2.13175i
$$456$$ 0 0
$$457$$ 290.000 0.634573 0.317287 0.948330i $$-0.397228\pi$$
0.317287 + 0.948330i $$0.397228\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 728.000 1.57918 0.789588 0.613638i $$-0.210294\pi$$
0.789588 + 0.613638i $$0.210294\pi$$
$$462$$ 0 0
$$463$$ 790.000 1.70626 0.853132 0.521696i $$-0.174700\pi$$
0.853132 + 0.521696i $$0.174700\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −530.000 −1.13490 −0.567452 0.823407i $$-0.692071\pi$$
−0.567452 + 0.823407i $$0.692071\pi$$
$$468$$ 0 0
$$469$$ 762.102i 1.62495i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 100.000 0.211416
$$474$$ 0 0
$$475$$ 171.000 0.360000
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −80.0000 −0.167015 −0.0835073 0.996507i $$-0.526612\pi$$
−0.0835073 + 0.996507i $$0.526612\pi$$
$$480$$ 0 0
$$481$$ −252.000 −0.523909
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 304.841i − 0.628538i
$$486$$ 0 0
$$487$$ 509.223i 1.04563i 0.852445 + 0.522816i $$0.175119\pi$$
−0.852445 + 0.522816i $$0.824881\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 418.000 0.851324 0.425662 0.904882i $$-0.360041\pi$$
0.425662 + 0.904882i $$0.360041\pi$$
$$492$$ 0 0
$$493$$ − 346.410i − 0.702658i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1039.23i 2.09101i
$$498$$ 0 0
$$499$$ −470.000 −0.941884 −0.470942 0.882164i $$-0.656086\pi$$
−0.470942 + 0.882164i $$0.656086\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 100.000 0.198807 0.0994036 0.995047i $$-0.468307\pi$$
0.0994036 + 0.995047i $$0.468307\pi$$
$$504$$ 0 0
$$505$$ 400.000 0.792079
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 450.333i − 0.884741i −0.896832 0.442371i $$-0.854138\pi$$
0.896832 0.442371i $$-0.145862\pi$$
$$510$$ 0 0
$$511$$ −100.000 −0.195695
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 734.390i 1.42600i
$$516$$ 0 0
$$517$$ −800.000 −1.54739
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 311.769i − 0.598405i −0.954190 0.299203i $$-0.903279\pi$$
0.954190 0.299203i $$-0.0967207\pi$$
$$522$$ 0 0
$$523$$ 789.815i 1.51016i 0.655631 + 0.755081i $$0.272403\pi$$
−0.655631 + 0.755081i $$0.727597\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 173.205i 0.328662i
$$528$$ 0 0
$$529$$ −129.000 −0.243856
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −840.000 −1.57598
$$534$$ 0 0
$$535$$ − 249.415i − 0.466197i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 510.000 0.946197
$$540$$ 0 0
$$541$$ 650.000 1.20148 0.600739 0.799445i $$-0.294873\pi$$
0.600739 + 0.799445i $$0.294873\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 623.538i 1.14411i
$$546$$ 0 0
$$547$$ − 595.825i − 1.08926i −0.838676 0.544630i $$-0.816670\pi$$
0.838676 0.544630i $$-0.183330\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 658.179i − 1.19452i
$$552$$ 0 0
$$553$$ 173.205i 0.313210i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 80.0000 0.143627 0.0718133 0.997418i $$-0.477121\pi$$
0.0718133 + 0.997418i $$0.477121\pi$$
$$558$$ 0 0
$$559$$ − 242.487i − 0.433787i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 339.482i − 0.602987i −0.953468 0.301494i $$-0.902515\pi$$
0.953468 0.301494i $$-0.0974852\pi$$
$$564$$ 0 0
$$565$$ − 27.7128i − 0.0490492i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 658.179i 1.15673i 0.815778 + 0.578365i $$0.196309\pi$$
−0.815778 + 0.578365i $$0.803691\pi$$
$$570$$ 0 0
$$571$$ 610.000 1.06830 0.534151 0.845389i $$-0.320632\pi$$
0.534151 + 0.845389i $$0.320632\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 180.000 0.313043
$$576$$ 0 0
$$577$$ 170.000 0.294627 0.147314 0.989090i $$-0.452937\pi$$
0.147314 + 0.989090i $$0.452937\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 700.000 1.20482
$$582$$ 0 0
$$583$$ − 415.692i − 0.713023i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −650.000 −1.10733 −0.553663 0.832741i $$-0.686770\pi$$
−0.553663 + 0.832741i $$0.686770\pi$$
$$588$$ 0 0
$$589$$ 329.090i 0.558726i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −910.000 −1.53457 −0.767285 0.641306i $$-0.778393\pi$$
−0.767285 + 0.641306i $$0.778393\pi$$
$$594$$ 0 0
$$595$$ 400.000 0.672269
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 34.6410i 0.0578314i 0.999582 + 0.0289157i $$0.00920544\pi$$
−0.999582 + 0.0289157i $$0.990795\pi$$
$$600$$ 0 0
$$601$$ − 173.205i − 0.288195i −0.989564 0.144097i $$-0.953972\pi$$
0.989564 0.144097i $$-0.0460279\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 84.0000 0.138843
$$606$$ 0 0
$$607$$ − 703.213i − 1.15851i −0.815148 0.579253i $$-0.803344\pi$$
0.815148 0.579253i $$-0.196656\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1939.90i 3.17495i
$$612$$ 0 0
$$613$$ 350.000 0.570962 0.285481 0.958384i $$-0.407847\pi$$
0.285481 + 0.958384i $$0.407847\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −610.000 −0.988655 −0.494327 0.869276i $$-0.664586\pi$$
−0.494327 + 0.869276i $$0.664586\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.0161551 0.00807754 0.999967i $$-0.497429\pi$$
0.00807754 + 0.999967i $$0.497429\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1039.23i 1.66811i
$$624$$ 0 0
$$625$$ −319.000 −0.510400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 103.923i 0.165219i
$$630$$ 0 0
$$631$$ −350.000 −0.554675 −0.277338 0.960773i $$-0.589452\pi$$
−0.277338 + 0.960773i $$0.589452\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 457.261i − 0.720097i
$$636$$ 0 0
$$637$$ − 1236.68i − 1.94142i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 588.897i − 0.918716i −0.888251 0.459358i $$-0.848079\pi$$
0.888251 0.459358i $$-0.151921\pi$$
$$642$$ 0 0
$$643$$ −650.000 −1.01089 −0.505443 0.862860i $$-0.668671\pi$$
−0.505443 + 0.862860i $$0.668671\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 820.000 1.26739 0.633694 0.773584i $$-0.281538\pi$$
0.633694 + 0.773584i $$0.281538\pi$$
$$648$$ 0 0
$$649$$ 346.410i 0.533760i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 560.000 0.857580 0.428790 0.903404i $$-0.358940\pi$$
0.428790 + 0.903404i $$0.358940\pi$$
$$654$$ 0 0
$$655$$ 152.000 0.232061
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 450.333i − 0.683358i −0.939817 0.341679i $$-0.889004\pi$$
0.939817 0.341679i $$-0.110996\pi$$
$$660$$ 0 0
$$661$$ − 398.372i − 0.602680i −0.953517 0.301340i $$-0.902566\pi$$
0.953517 0.301340i $$-0.0974340\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 760.000 1.14286
$$666$$ 0 0
$$667$$ − 692.820i − 1.03871i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −100.000 −0.149031
$$672$$ 0 0
$$673$$ − 630.466i − 0.936800i −0.883516 0.468400i $$-0.844831\pi$$
0.883516 0.468400i $$-0.155169\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 526.543i − 0.777760i −0.921288 0.388880i $$-0.872862\pi$$
0.921288 0.388880i $$-0.127138\pi$$
$$678$$ 0 0
$$679$$ 762.102i 1.12239i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 478.046i − 0.699921i −0.936764 0.349960i $$-0.886195\pi$$
0.936764 0.349960i $$-0.113805\pi$$
$$684$$ 0 0
$$685$$ 760.000 1.10949
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1008.00 −1.46299
$$690$$ 0 0
$$691$$ −470.000 −0.680174 −0.340087 0.940394i $$-0.610456\pi$$
−0.340087 + 0.940394i $$0.610456\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 200.000 0.287770
$$696$$ 0 0
$$697$$ 346.410i 0.497002i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 560.000 0.798859 0.399429 0.916764i $$-0.369208\pi$$
0.399429 + 0.916764i $$0.369208\pi$$
$$702$$ 0 0
$$703$$ 197.454i 0.280873i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1000.00 −1.41443
$$708$$ 0 0
$$709$$ −982.000 −1.38505 −0.692525 0.721394i $$-0.743502\pi$$
−0.692525 + 0.721394i $$0.743502\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 346.410i 0.485849i
$$714$$ 0 0
$$715$$ 969.948i 1.35657i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 520.000 0.723227 0.361613 0.932328i $$-0.382226\pi$$
0.361613 + 0.932328i $$0.382226\pi$$
$$720$$ 0 0
$$721$$ − 1835.97i − 2.54643i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 311.769i − 0.430026i
$$726$$ 0 0
$$727$$ 790.000 1.08666 0.543329 0.839520i $$-0.317164\pi$$
0.543329 + 0.839520i $$0.317164\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −100.000 −0.136799
$$732$$ 0 0
$$733$$ −1150.00 −1.56889 −0.784447 0.620195i $$-0.787053\pi$$
−0.784447 + 0.620195i $$0.787053\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 762.102i 1.03406i
$$738$$ 0 0
$$739$$ −578.000 −0.782138 −0.391069 0.920361i $$-0.627895\pi$$
−0.391069 + 0.920361i $$0.627895\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 235.559i − 0.317038i −0.987356 0.158519i $$-0.949328\pi$$
0.987356 0.158519i $$-0.0506718\pi$$
$$744$$ 0 0
$$745$$ −80.0000 −0.107383
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 623.538i 0.832494i
$$750$$ 0 0
$$751$$ − 952.628i − 1.26848i −0.773137 0.634240i $$-0.781313\pi$$
0.773137 0.634240i $$-0.218687\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 900.666i − 1.19294i
$$756$$ 0 0
$$757$$ −250.000 −0.330251 −0.165125 0.986273i $$-0.552803\pi$$
−0.165125 + 0.986273i $$0.552803\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 770.000 1.01183 0.505913 0.862584i $$-0.331156\pi$$
0.505913 + 0.862584i $$0.331156\pi$$
$$762$$ 0 0
$$763$$ − 1558.85i − 2.04305i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 840.000 1.09518
$$768$$ 0 0
$$769$$ 110.000 0.143043 0.0715215 0.997439i $$-0.477215\pi$$
0.0715215 + 0.997439i $$0.477215\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 145.492i 0.188218i 0.995562 + 0.0941088i $$0.0300002\pi$$
−0.995562 + 0.0941088i $$0.970000\pi$$
$$774$$ 0 0
$$775$$ 155.885i 0.201141i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 658.179i 0.844903i
$$780$$ 0 0
$$781$$ 1039.23i 1.33064i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −920.000 −1.17197
$$786$$ 0 0
$$787$$ 96.9948i 0.123246i 0.998099 + 0.0616232i $$0.0196277\pi$$
−0.998099 + 0.0616232i $$0.980372\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 69.2820i 0.0875879i
$$792$$ 0 0
$$793$$ 242.487i 0.305785i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 339.482i 0.425950i 0.977058 + 0.212975i $$0.0683153\pi$$
−0.977058 + 0.212975i $$0.931685\pi$$
$$798$$ 0 0
$$799$$ 800.000 1.00125
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −100.000 −0.124533
$$804$$ 0 0
$$805$$ 800.000 0.993789
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 182.000 0.224969 0.112485 0.993653i $$-0.464119\pi$$
0.112485 + 0.993653i $$0.464119\pi$$
$$810$$ 0 0
$$811$$ 831.384i 1.02513i 0.858647 + 0.512567i $$0.171306\pi$$
−0.858647 + 0.512567i $$0.828694\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 680.000 0.834356
$$816$$ 0 0
$$817$$ −190.000 −0.232558
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.00000 0.00974421 0.00487211 0.999988i $$-0.498449\pi$$
0.00487211 + 0.999988i $$0.498449\pi$$
$$822$$ 0 0
$$823$$ −950.000 −1.15431 −0.577157 0.816633i $$-0.695838\pi$$
−0.577157 + 0.816633i $$0.695838\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 478.046i − 0.578048i −0.957322 0.289024i $$-0.906669\pi$$
0.957322 0.289024i $$-0.0933308\pi$$
$$828$$ 0 0
$$829$$ 1195.12i 1.44163i 0.693125 + 0.720817i $$0.256233\pi$$
−0.693125 + 0.720817i $$0.743767\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −510.000 −0.612245
$$834$$ 0 0
$$835$$ 526.543i 0.630591i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1177.79i − 1.40381i −0.712272 0.701904i $$-0.752334\pi$$
0.712272 0.701904i $$-0.247666\pi$$
$$840$$ 0 0
$$841$$ −359.000 −0.426873
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1676.00 1.98343
$$846$$ 0 0
$$847$$ −210.000 −0.247934
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 207.846i 0.244237i
$$852$$ 0 0
$$853$$ 890.000 1.04338 0.521688 0.853136i $$-0.325302\pi$$
0.521688 + 0.853136i $$0.325302\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 1254.00i − 1.46325i −0.681708 0.731625i $$-0.738762\pi$$
0.681708 0.731625i $$-0.261238\pi$$
$$858$$ 0 0
$$859$$ −182.000 −0.211874 −0.105937 0.994373i $$-0.533784\pi$$
−0.105937 + 0.994373i $$0.533784\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1080.80i 1.25238i 0.779672 + 0.626188i $$0.215386\pi$$
−0.779672 + 0.626188i $$0.784614\pi$$
$$864$$ 0 0
$$865$$ 942.236i 1.08929i
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 173.205i 0.199315i
$$870$$ 0 0
$$871$$ 1848.00 2.12170
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1360.00 1.55429
$$876$$ 0 0
$$877$$ − 1188.19i − 1.35483i −0.735601 0.677416i $$-0.763100\pi$$
0.735601 0.677416i $$-0.236900\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −550.000 −0.624291 −0.312145 0.950034i $$-0.601048\pi$$
−0.312145 + 0.950034i $$0.601048\pi$$
$$882$$ 0 0
$$883$$ 1450.00 1.64213 0.821065 0.570835i $$-0.193381\pi$$
0.821065 + 0.570835i $$0.193381\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1254.00i − 1.41376i −0.707334 0.706880i $$-0.750102\pi$$
0.707334 0.706880i $$-0.249898\pi$$
$$888$$ 0 0
$$889$$ 1143.15i 1.28589i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1520.00 1.70213
$$894$$ 0 0
$$895$$ 415.692i 0.464461i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 600.000 0.667408
$$900$$ 0 0
$$901$$ 415.692i 0.461368i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 1039.23i − 1.14832i
$$906$$ 0 0
$$907$$ 110.851i 0.122217i 0.998131 + 0.0611087i $$0.0194636\pi$$
−0.998131 + 0.0611087i $$0.980536\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 796.743i 0.874581i 0.899320 + 0.437291i $$0.144062\pi$$
−0.899320 + 0.437291i $$0.855938\pi$$
$$912$$ 0 0
$$913$$ 700.000 0.766703
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −380.000 −0.414395
$$918$$ 0 0
$$919$$ −62.0000 −0.0674646 −0.0337323 0.999431i $$-0.510739\pi$$
−0.0337323 + 0.999431i $$0.510739\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 2520.00 2.73023
$$924$$ 0 0
$$925$$ 93.5307i 0.101114i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 242.000 0.260495 0.130248 0.991482i $$-0.458423\pi$$
0.130248 + 0.991482i $$0.458423\pi$$
$$930$$ 0 0
$$931$$ −969.000 −1.04082
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 400.000 0.427807
$$936$$ 0 0
$$937$$ 110.000 0.117396 0.0586980 0.998276i $$-0.481305\pi$$
0.0586980 + 0.998276i $$0.481305\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 796.743i 0.846699i 0.905967 + 0.423349i $$0.139146\pi$$
−0.905967 + 0.423349i $$0.860854\pi$$
$$942$$ 0 0
$$943$$ 692.820i 0.734698i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1450.00 1.53115 0.765576 0.643346i $$-0.222454\pi$$
0.765576 + 0.643346i $$0.222454\pi$$
$$948$$ 0 0
$$949$$ 242.487i 0.255519i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 353.338i − 0.370764i −0.982667 0.185382i $$-0.940648\pi$$
0.982667 0.185382i $$-0.0593523\pi$$
$$954$$ 0 0
$$955$$ 1328.00 1.39058
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1900.00 −1.98123
$$960$$ 0 0
$$961$$ 661.000 0.687825
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 387.979i 0.402051i
$$966$$ 0 0
$$967$$ −470.000 −0.486039 −0.243020 0.970021i $$-0.578138\pi$$
−0.243020 + 0.970021i $$0.578138\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 519.615i − 0.535134i −0.963539 0.267567i $$-0.913780\pi$$
0.963539 0.267567i $$-0.0862197\pi$$
$$972$$ 0 0
$$973$$ −500.000 −0.513875
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1669.70i 1.70900i 0.519448 + 0.854502i $$0.326138\pi$$
−0.519448 + 0.854502i $$0.673862\pi$$
$$978$$ 0 0
$$979$$ 1039.23i 1.06152i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1690.48i 1.71972i 0.510533 + 0.859858i $$0.329448\pi$$
−0.510533 + 0.859858i $$0.670552\pi$$
$$984$$ 0 0
$$985$$ 640.000 0.649746
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −200.000 −0.202224
$$990$$ 0 0
$$991$$ − 571.577i − 0.576768i −0.957515 0.288384i $$-0.906882\pi$$
0.957515 0.288384i $$-0.0931179\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 392.000 0.393970
$$996$$ 0 0
$$997$$ 1550.00 1.55466 0.777332 0.629091i $$-0.216573\pi$$
0.777332 + 0.629091i $$0.216573\pi$$
$$998$$ 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 2736.3.o.e.721.1 2
3.2 odd 2 912.3.o.a.721.1 2
4.3 odd 2 171.3.c.c.37.1 2
12.11 even 2 57.3.c.a.37.2 yes 2
19.18 odd 2 inner 2736.3.o.e.721.2 2
57.56 even 2 912.3.o.a.721.2 2
76.75 even 2 171.3.c.c.37.2 2
228.227 odd 2 57.3.c.a.37.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
57.3.c.a.37.1 2 228.227 odd 2
57.3.c.a.37.2 yes 2 12.11 even 2
171.3.c.c.37.1 2 4.3 odd 2
171.3.c.c.37.2 2 76.75 even 2
912.3.o.a.721.1 2 3.2 odd 2
912.3.o.a.721.2 2 57.56 even 2
2736.3.o.e.721.1 2 1.1 even 1 trivial
2736.3.o.e.721.2 2 19.18 odd 2 inner