# Properties

 Label 912.3.o.a.721.1 Level $912$ Weight $3$ Character 912.721 Analytic conductor $24.850$ 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] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, 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("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 912.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: $$24.8502001097$$ 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: $$2$$ 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 $$\chi$$ $$=$$ 912.721 Dual form 912.3.o.a.721.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +4.00000 q^{5} +10.0000 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +4.00000 q^{5} +10.0000 q^{7} -3.00000 q^{9} -10.0000 q^{11} -24.2487i q^{13} -6.92820i q^{15} +10.0000 q^{17} -19.0000 q^{19} -17.3205i q^{21} +20.0000 q^{23} -9.00000 q^{25} +5.19615i q^{27} -34.6410i q^{29} -17.3205i q^{31} +17.3205i q^{33} +40.0000 q^{35} -10.3923i q^{37} -42.0000 q^{39} +34.6410i q^{41} +10.0000 q^{43} -12.0000 q^{45} +80.0000 q^{47} +51.0000 q^{49} -17.3205i q^{51} +41.5692i q^{53} -40.0000 q^{55} +32.9090i q^{57} -34.6410i q^{59} -10.0000 q^{61} -30.0000 q^{63} -96.9948i q^{65} +76.2102i q^{67} -34.6410i q^{69} -103.923i q^{71} -10.0000 q^{73} +15.5885i q^{75} -100.000 q^{77} +17.3205i q^{79} +9.00000 q^{81} -70.0000 q^{83} +40.0000 q^{85} -60.0000 q^{87} -103.923i q^{89} -242.487i q^{91} -30.0000 q^{93} -76.0000 q^{95} +76.2102i q^{97} +30.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{5} + 20 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 8 * q^5 + 20 * q^7 - 6 * q^9 $$2 q + 8 q^{5} + 20 q^{7} - 6 q^{9} - 20 q^{11} + 20 q^{17} - 38 q^{19} + 40 q^{23} - 18 q^{25} + 80 q^{35} - 84 q^{39} + 20 q^{43} - 24 q^{45} + 160 q^{47} + 102 q^{49} - 80 q^{55} - 20 q^{61} - 60 q^{63} - 20 q^{73} - 200 q^{77} + 18 q^{81} - 140 q^{83} + 80 q^{85} - 120 q^{87} - 60 q^{93} - 152 q^{95} + 60 q^{99}+O(q^{100})$$ 2 * q + 8 * q^5 + 20 * q^7 - 6 * q^9 - 20 * q^11 + 20 * q^17 - 38 * q^19 + 40 * q^23 - 18 * q^25 + 80 * q^35 - 84 * q^39 + 20 * q^43 - 24 * q^45 + 160 * q^47 + 102 * q^49 - 80 * q^55 - 20 * q^61 - 60 * q^63 - 20 * q^73 - 200 * q^77 + 18 * q^81 - 140 * q^83 + 80 * q^85 - 120 * q^87 - 60 * q^93 - 152 * q^95 + 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ − 1.73205i − 0.577350i
$$4$$ 0 0
$$5$$ 4.00000 0.800000 0.400000 0.916515i $$-0.369010\pi$$
0.400000 + 0.916515i $$0.369010\pi$$
$$6$$ 0 0
$$7$$ 10.0000 1.42857 0.714286 0.699854i $$-0.246752\pi$$
0.714286 + 0.699854i $$0.246752\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ −10.0000 −0.909091 −0.454545 0.890724i $$-0.650198\pi$$
−0.454545 + 0.890724i $$0.650198\pi$$
$$12$$ 0 0
$$13$$ − 24.2487i − 1.86529i −0.360801 0.932643i $$-0.617497\pi$$
0.360801 0.932643i $$-0.382503\pi$$
$$14$$ 0 0
$$15$$ − 6.92820i − 0.461880i
$$16$$ 0 0
$$17$$ 10.0000 0.588235 0.294118 0.955769i $$-0.404974\pi$$
0.294118 + 0.955769i $$0.404974\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ − 17.3205i − 0.824786i
$$22$$ 0 0
$$23$$ 20.0000 0.869565 0.434783 0.900535i $$-0.356825\pi$$
0.434783 + 0.900535i $$0.356825\pi$$
$$24$$ 0 0
$$25$$ −9.00000 −0.360000
$$26$$ 0 0
$$27$$ 5.19615i 0.192450i
$$28$$ 0 0
$$29$$ − 34.6410i − 1.19452i −0.802049 0.597259i $$-0.796256\pi$$
0.802049 0.597259i $$-0.203744\pi$$
$$30$$ 0 0
$$31$$ − 17.3205i − 0.558726i −0.960186 0.279363i $$-0.909877\pi$$
0.960186 0.279363i $$-0.0901233\pi$$
$$32$$ 0 0
$$33$$ 17.3205i 0.524864i
$$34$$ 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
$$39$$ −42.0000 −1.07692
$$40$$ 0 0
$$41$$ 34.6410i 0.844903i 0.906386 + 0.422451i $$0.138830\pi$$
−0.906386 + 0.422451i $$0.861170\pi$$
$$42$$ 0 0
$$43$$ 10.0000 0.232558 0.116279 0.993217i $$-0.462903\pi$$
0.116279 + 0.993217i $$0.462903\pi$$
$$44$$ 0 0
$$45$$ −12.0000 −0.266667
$$46$$ 0 0
$$47$$ 80.0000 1.70213 0.851064 0.525062i $$-0.175958\pi$$
0.851064 + 0.525062i $$0.175958\pi$$
$$48$$ 0 0
$$49$$ 51.0000 1.04082
$$50$$ 0 0
$$51$$ − 17.3205i − 0.339618i
$$52$$ 0 0
$$53$$ 41.5692i 0.784325i 0.919896 + 0.392162i $$0.128273\pi$$
−0.919896 + 0.392162i $$0.871727\pi$$
$$54$$ 0 0
$$55$$ −40.0000 −0.727273
$$56$$ 0 0
$$57$$ 32.9090i 0.577350i
$$58$$ 0 0
$$59$$ − 34.6410i − 0.587136i −0.955938 0.293568i $$-0.905157\pi$$
0.955938 0.293568i $$-0.0948427\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −0.163934 −0.0819672 0.996635i $$-0.526120\pi$$
−0.0819672 + 0.996635i $$0.526120\pi$$
$$62$$ 0 0
$$63$$ −30.0000 −0.476190
$$64$$ 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
$$69$$ − 34.6410i − 0.502044i
$$70$$ 0 0
$$71$$ − 103.923i − 1.46370i −0.681463 0.731852i $$-0.738656\pi$$
0.681463 0.731852i $$-0.261344\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −0.136986 −0.0684932 0.997652i $$-0.521819\pi$$
−0.0684932 + 0.997652i $$0.521819\pi$$
$$74$$ 0 0
$$75$$ 15.5885i 0.207846i
$$76$$ 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
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ −70.0000 −0.843373 −0.421687 0.906742i $$-0.638562\pi$$
−0.421687 + 0.906742i $$0.638562\pi$$
$$84$$ 0 0
$$85$$ 40.0000 0.470588
$$86$$ 0 0
$$87$$ −60.0000 −0.689655
$$88$$ 0 0
$$89$$ − 103.923i − 1.16767i −0.811871 0.583837i $$-0.801551\pi$$
0.811871 0.583837i $$-0.198449\pi$$
$$90$$ 0 0
$$91$$ − 242.487i − 2.66469i
$$92$$ 0 0
$$93$$ −30.0000 −0.322581
$$94$$ 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
$$99$$ 30.0000 0.303030
$$100$$ 0 0
$$101$$ 100.000 0.990099 0.495050 0.868865i $$-0.335150\pi$$
0.495050 + 0.868865i $$0.335150\pi$$
$$102$$ 0 0
$$103$$ − 183.597i − 1.78250i −0.453513 0.891249i $$-0.649830\pi$$
0.453513 0.891249i $$-0.350170\pi$$
$$104$$ 0 0
$$105$$ − 69.2820i − 0.659829i
$$106$$ 0 0
$$107$$ − 62.3538i − 0.582746i −0.956610 0.291373i $$-0.905888\pi$$
0.956610 0.291373i $$-0.0941121\pi$$
$$108$$ 0 0
$$109$$ − 155.885i − 1.43013i −0.699056 0.715067i $$-0.746396\pi$$
0.699056 0.715067i $$-0.253604\pi$$
$$110$$ 0 0
$$111$$ −18.0000 −0.162162
$$112$$ 0 0
$$113$$ − 6.92820i − 0.0613115i −0.999530 0.0306558i $$-0.990240\pi$$
0.999530 0.0306558i $$-0.00975956\pi$$
$$114$$ 0 0
$$115$$ 80.0000 0.695652
$$116$$ 0 0
$$117$$ 72.7461i 0.621762i
$$118$$ 0 0
$$119$$ 100.000 0.840336
$$120$$ 0 0
$$121$$ −21.0000 −0.173554
$$122$$ 0 0
$$123$$ 60.0000 0.487805
$$124$$ 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
$$129$$ − 17.3205i − 0.134268i
$$130$$ 0 0
$$131$$ 38.0000 0.290076 0.145038 0.989426i $$-0.453670\pi$$
0.145038 + 0.989426i $$0.453670\pi$$
$$132$$ 0 0
$$133$$ −190.000 −1.42857
$$134$$ 0 0
$$135$$ 20.7846i 0.153960i
$$136$$ 0 0
$$137$$ 190.000 1.38686 0.693431 0.720523i $$-0.256098\pi$$
0.693431 + 0.720523i $$0.256098\pi$$
$$138$$ 0 0
$$139$$ −50.0000 −0.359712 −0.179856 0.983693i $$-0.557563\pi$$
−0.179856 + 0.983693i $$0.557563\pi$$
$$140$$ 0 0
$$141$$ − 138.564i − 0.982724i
$$142$$ 0 0
$$143$$ 242.487i 1.69571i
$$144$$ 0 0
$$145$$ − 138.564i − 0.955614i
$$146$$ 0 0
$$147$$ − 88.3346i − 0.600916i
$$148$$ 0 0
$$149$$ −20.0000 −0.134228 −0.0671141 0.997745i $$-0.521379\pi$$
−0.0671141 + 0.997745i $$0.521379\pi$$
$$150$$ 0 0
$$151$$ 225.167i 1.49117i 0.666411 + 0.745585i $$0.267830\pi$$
−0.666411 + 0.745585i $$0.732170\pi$$
$$152$$ 0 0
$$153$$ −30.0000 −0.196078
$$154$$ 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
$$159$$ 72.0000 0.452830
$$160$$ 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
$$165$$ 69.2820i 0.419891i
$$166$$ 0 0
$$167$$ 131.636i 0.788239i 0.919059 + 0.394119i $$0.128950\pi$$
−0.919059 + 0.394119i $$0.871050\pi$$
$$168$$ 0 0
$$169$$ −419.000 −2.47929
$$170$$ 0 0
$$171$$ 57.0000 0.333333
$$172$$ 0 0
$$173$$ 235.559i 1.36161i 0.732464 + 0.680806i $$0.238370\pi$$
−0.732464 + 0.680806i $$0.761630\pi$$
$$174$$ 0 0
$$175$$ −90.0000 −0.514286
$$176$$ 0 0
$$177$$ −60.0000 −0.338983
$$178$$ 0 0
$$179$$ 103.923i 0.580576i 0.956939 + 0.290288i $$0.0937511\pi$$
−0.956939 + 0.290288i $$0.906249\pi$$
$$180$$ 0 0
$$181$$ 259.808i 1.43540i 0.696352 + 0.717701i $$0.254805\pi$$
−0.696352 + 0.717701i $$0.745195\pi$$
$$182$$ 0 0
$$183$$ 17.3205i 0.0946476i
$$184$$ 0 0
$$185$$ − 41.5692i − 0.224698i
$$186$$ 0 0
$$187$$ −100.000 −0.534759
$$188$$ 0 0
$$189$$ 51.9615i 0.274929i
$$190$$ 0 0
$$191$$ 332.000 1.73822 0.869110 0.494619i $$-0.164692\pi$$
0.869110 + 0.494619i $$0.164692\pi$$
$$192$$ 0 0
$$193$$ − 96.9948i − 0.502564i −0.967914 0.251282i $$-0.919148\pi$$
0.967914 0.251282i $$-0.0808521\pi$$
$$194$$ 0 0
$$195$$ −168.000 −0.861538
$$196$$ 0 0
$$197$$ 160.000 0.812183 0.406091 0.913832i $$-0.366891\pi$$
0.406091 + 0.913832i $$0.366891\pi$$
$$198$$ 0 0
$$199$$ −98.0000 −0.492462 −0.246231 0.969211i $$-0.579192\pi$$
−0.246231 + 0.969211i $$0.579192\pi$$
$$200$$ 0 0
$$201$$ 132.000 0.656716
$$202$$ 0 0
$$203$$ − 346.410i − 1.70645i
$$204$$ 0 0
$$205$$ 138.564i 0.675922i
$$206$$ 0 0
$$207$$ −60.0000 −0.289855
$$208$$ 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
$$213$$ −180.000 −0.845070
$$214$$ 0 0
$$215$$ 40.0000 0.186047
$$216$$ 0 0
$$217$$ − 173.205i − 0.798180i
$$218$$ 0 0
$$219$$ 17.3205i 0.0790891i
$$220$$ 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
$$225$$ 27.0000 0.120000
$$226$$ 0 0
$$227$$ 76.2102i 0.335728i 0.985810 + 0.167864i $$0.0536869\pi$$
−0.985810 + 0.167864i $$0.946313\pi$$
$$228$$ 0 0
$$229$$ 110.000 0.480349 0.240175 0.970730i $$-0.422795\pi$$
0.240175 + 0.970730i $$0.422795\pi$$
$$230$$ 0 0
$$231$$ 173.205i 0.749806i
$$232$$ 0 0
$$233$$ 190.000 0.815451 0.407725 0.913105i $$-0.366322\pi$$
0.407725 + 0.913105i $$0.366322\pi$$
$$234$$ 0 0
$$235$$ 320.000 1.36170
$$236$$ 0 0
$$237$$ 30.0000 0.126582
$$238$$ 0 0
$$239$$ 128.000 0.535565 0.267782 0.963479i $$-0.413709\pi$$
0.267782 + 0.963479i $$0.413709\pi$$
$$240$$ 0 0
$$241$$ 138.564i 0.574955i 0.957787 + 0.287477i $$0.0928166\pi$$
−0.957787 + 0.287477i $$0.907183\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 0.0641500i
$$244$$ 0 0
$$245$$ 204.000 0.832653
$$246$$ 0 0
$$247$$ 460.726i 1.86529i
$$248$$ 0 0
$$249$$ 121.244i 0.486922i
$$250$$ 0 0
$$251$$ 2.00000 0.00796813 0.00398406 0.999992i $$-0.498732\pi$$
0.00398406 + 0.999992i $$0.498732\pi$$
$$252$$ 0 0
$$253$$ −200.000 −0.790514
$$254$$ 0 0
$$255$$ − 69.2820i − 0.271694i
$$256$$ 0 0
$$257$$ 491.902i 1.91402i 0.290059 + 0.957009i $$0.406325\pi$$
−0.290059 + 0.957009i $$0.593675\pi$$
$$258$$ 0 0
$$259$$ − 103.923i − 0.401247i
$$260$$ 0 0
$$261$$ 103.923i 0.398173i
$$262$$ 0 0
$$263$$ 200.000 0.760456 0.380228 0.924893i $$-0.375845\pi$$
0.380228 + 0.924893i $$0.375845\pi$$
$$264$$ 0 0
$$265$$ 166.277i 0.627460i
$$266$$ 0 0
$$267$$ −180.000 −0.674157
$$268$$ 0 0
$$269$$ − 415.692i − 1.54532i −0.634818 0.772662i $$-0.718925\pi$$
0.634818 0.772662i $$-0.281075\pi$$
$$270$$ 0 0
$$271$$ −170.000 −0.627306 −0.313653 0.949538i $$-0.601553\pi$$
−0.313653 + 0.949538i $$0.601553\pi$$
$$272$$ 0 0
$$273$$ −420.000 −1.53846
$$274$$ 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
$$279$$ 51.9615i 0.186242i
$$280$$ 0 0
$$281$$ 381.051i 1.35605i 0.735037 + 0.678027i $$0.237165\pi$$
−0.735037 + 0.678027i $$0.762835\pi$$
$$282$$ 0 0
$$283$$ 70.0000 0.247350 0.123675 0.992323i $$-0.460532\pi$$
0.123675 + 0.992323i $$0.460532\pi$$
$$284$$ 0 0
$$285$$ 131.636i 0.461880i
$$286$$ 0 0
$$287$$ 346.410i 1.20700i
$$288$$ 0 0
$$289$$ −189.000 −0.653979
$$290$$ 0 0
$$291$$ 132.000 0.453608
$$292$$ 0 0
$$293$$ 180.133i 0.614789i 0.951582 + 0.307395i $$0.0994572\pi$$
−0.951582 + 0.307395i $$0.900543\pi$$
$$294$$ 0 0
$$295$$ − 138.564i − 0.469709i
$$296$$ 0 0
$$297$$ − 51.9615i − 0.174955i
$$298$$ 0 0
$$299$$ − 484.974i − 1.62199i
$$300$$ 0 0
$$301$$ 100.000 0.332226
$$302$$ 0 0
$$303$$ − 173.205i − 0.571634i
$$304$$ 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
$$309$$ −318.000 −1.02913
$$310$$ 0 0
$$311$$ −580.000 −1.86495 −0.932476 0.361232i $$-0.882356\pi$$
−0.932476 + 0.361232i $$0.882356\pi$$
$$312$$ 0 0
$$313$$ −370.000 −1.18211 −0.591054 0.806632i $$-0.701288\pi$$
−0.591054 + 0.806632i $$0.701288\pi$$
$$314$$ 0 0
$$315$$ −120.000 −0.380952
$$316$$ 0 0
$$317$$ 27.7128i 0.0874221i 0.999044 + 0.0437111i $$0.0139181\pi$$
−0.999044 + 0.0437111i $$0.986082\pi$$
$$318$$ 0 0
$$319$$ 346.410i 1.08593i
$$320$$ 0 0
$$321$$ −108.000 −0.336449
$$322$$ 0 0
$$323$$ −190.000 −0.588235
$$324$$ 0 0
$$325$$ 218.238i 0.671503i
$$326$$ 0 0
$$327$$ −270.000 −0.825688
$$328$$ 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
$$333$$ 31.1769i 0.0936244i
$$334$$ 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
$$339$$ −12.0000 −0.0353982
$$340$$ 0 0
$$341$$ 173.205i 0.507933i
$$342$$ 0 0
$$343$$ 20.0000 0.0583090
$$344$$ 0 0
$$345$$ − 138.564i − 0.401635i
$$346$$ 0 0
$$347$$ 590.000 1.70029 0.850144 0.526550i $$-0.176515\pi$$
0.850144 + 0.526550i $$0.176515\pi$$
$$348$$ 0 0
$$349$$ 98.0000 0.280802 0.140401 0.990095i $$-0.455161\pi$$
0.140401 + 0.990095i $$0.455161\pi$$
$$350$$ 0 0
$$351$$ 126.000 0.358974
$$352$$ 0 0
$$353$$ 190.000 0.538244 0.269122 0.963106i $$-0.413267\pi$$
0.269122 + 0.963106i $$0.413267\pi$$
$$354$$ 0 0
$$355$$ − 415.692i − 1.17096i
$$356$$ 0 0
$$357$$ − 173.205i − 0.485168i
$$358$$ 0 0
$$359$$ 200.000 0.557103 0.278552 0.960421i $$-0.410146\pi$$
0.278552 + 0.960421i $$0.410146\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 36.3731i 0.100201i
$$364$$ 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
$$369$$ − 103.923i − 0.281634i
$$370$$ 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
$$375$$ 235.559i 0.628157i
$$376$$ 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
$$381$$ 198.000 0.519685
$$382$$ 0 0
$$383$$ 630.466i 1.64613i 0.567950 + 0.823063i $$0.307737\pi$$
−0.567950 + 0.823063i $$0.692263\pi$$
$$384$$ 0 0
$$385$$ −400.000 −1.03896
$$386$$ 0 0
$$387$$ −30.0000 −0.0775194
$$388$$ 0 0
$$389$$ −128.000 −0.329049 −0.164524 0.986373i $$-0.552609\pi$$
−0.164524 + 0.986373i $$0.552609\pi$$
$$390$$ 0 0
$$391$$ 200.000 0.511509
$$392$$ 0 0
$$393$$ − 65.8179i − 0.167476i
$$394$$ 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
$$399$$ 329.090i 0.824786i
$$400$$ 0 0
$$401$$ − 173.205i − 0.431933i −0.976401 0.215966i $$-0.930710\pi$$
0.976401 0.215966i $$-0.0692902\pi$$
$$402$$ 0 0
$$403$$ −420.000 −1.04218
$$404$$ 0 0
$$405$$ 36.0000 0.0888889
$$406$$ 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
$$411$$ − 329.090i − 0.800705i
$$412$$ 0 0
$$413$$ − 346.410i − 0.838766i
$$414$$ 0 0
$$415$$ −280.000 −0.674699
$$416$$ 0 0
$$417$$ 86.6025i 0.207680i
$$418$$ 0 0
$$419$$ 38.0000 0.0906921 0.0453461 0.998971i $$-0.485561\pi$$
0.0453461 + 0.998971i $$0.485561\pi$$
$$420$$ 0 0
$$421$$ − 17.3205i − 0.0411413i −0.999788 0.0205707i $$-0.993452\pi$$
0.999788 0.0205707i $$-0.00654831\pi$$
$$422$$ 0 0
$$423$$ −240.000 −0.567376
$$424$$ 0 0
$$425$$ −90.0000 −0.211765
$$426$$ 0 0
$$427$$ −100.000 −0.234192
$$428$$ 0 0
$$429$$ 420.000 0.979021
$$430$$ 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
$$435$$ −240.000 −0.551724
$$436$$ 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
$$441$$ −153.000 −0.346939
$$442$$ 0 0
$$443$$ 110.000 0.248307 0.124153 0.992263i $$-0.460378\pi$$
0.124153 + 0.992263i $$0.460378\pi$$
$$444$$ 0 0
$$445$$ − 415.692i − 0.934140i
$$446$$ 0 0
$$447$$ 34.6410i 0.0774967i
$$448$$ 0 0
$$449$$ 311.769i 0.694363i 0.937798 + 0.347182i $$0.112861\pi$$
−0.937798 + 0.347182i $$0.887139\pi$$
$$450$$ 0 0
$$451$$ − 346.410i − 0.768093i
$$452$$ 0 0
$$453$$ 390.000 0.860927
$$454$$ 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
$$459$$ 51.9615i 0.113206i
$$460$$ 0 0
$$461$$ −728.000 −1.57918 −0.789588 0.613638i $$-0.789706\pi$$
−0.789588 + 0.613638i $$0.789706\pi$$
$$462$$ 0 0
$$463$$ 790.000 1.70626 0.853132 0.521696i $$-0.174700\pi$$
0.853132 + 0.521696i $$0.174700\pi$$
$$464$$ 0 0
$$465$$ −120.000 −0.258065
$$466$$ 0 0
$$467$$ 530.000 1.13490 0.567452 0.823407i $$-0.307929\pi$$
0.567452 + 0.823407i $$0.307929\pi$$
$$468$$ 0 0
$$469$$ 762.102i 1.62495i
$$470$$ 0 0
$$471$$ − 398.372i − 0.845800i
$$472$$ 0 0
$$473$$ −100.000 −0.211416
$$474$$ 0 0
$$475$$ 171.000 0.360000
$$476$$ 0 0
$$477$$ − 124.708i − 0.261442i
$$478$$ 0 0
$$479$$ 80.0000 0.167015 0.0835073 0.996507i $$-0.473388\pi$$
0.0835073 + 0.996507i $$0.473388\pi$$
$$480$$ 0 0
$$481$$ −252.000 −0.523909
$$482$$ 0 0
$$483$$ − 346.410i − 0.717205i
$$484$$ 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
$$489$$ 294.449i 0.602144i
$$490$$ 0 0
$$491$$ −418.000 −0.851324 −0.425662 0.904882i $$-0.639959\pi$$
−0.425662 + 0.904882i $$0.639959\pi$$
$$492$$ 0 0
$$493$$ − 346.410i − 0.702658i
$$494$$ 0 0
$$495$$ 120.000 0.242424
$$496$$ 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
$$501$$ 228.000 0.455090
$$502$$ 0 0
$$503$$ −100.000 −0.198807 −0.0994036 0.995047i $$-0.531693\pi$$
−0.0994036 + 0.995047i $$0.531693\pi$$
$$504$$ 0 0
$$505$$ 400.000 0.792079
$$506$$ 0 0
$$507$$ 725.729i 1.43142i
$$508$$ 0 0
$$509$$ 450.333i 0.884741i 0.896832 + 0.442371i $$0.145862\pi$$
−0.896832 + 0.442371i $$0.854138\pi$$
$$510$$ 0 0
$$511$$ −100.000 −0.195695
$$512$$ 0 0
$$513$$ − 98.7269i − 0.192450i
$$514$$ 0 0
$$515$$ − 734.390i − 1.42600i
$$516$$ 0 0
$$517$$ −800.000 −1.54739
$$518$$ 0 0
$$519$$ 408.000 0.786127
$$520$$ 0 0
$$521$$ 311.769i 0.598405i 0.954190 + 0.299203i $$0.0967207\pi$$
−0.954190 + 0.299203i $$0.903279\pi$$
$$522$$ 0 0
$$523$$ 789.815i 1.51016i 0.655631 + 0.755081i $$0.272403\pi$$
−0.655631 + 0.755081i $$0.727597\pi$$
$$524$$ 0 0
$$525$$ 155.885i 0.296923i
$$526$$ 0 0
$$527$$ − 173.205i − 0.328662i
$$528$$ 0 0
$$529$$ −129.000 −0.243856
$$530$$ 0 0
$$531$$ 103.923i 0.195712i
$$532$$ 0 0
$$533$$ 840.000 1.57598
$$534$$ 0 0
$$535$$ − 249.415i − 0.466197i
$$536$$ 0 0
$$537$$ 180.000 0.335196
$$538$$ 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
$$543$$ 450.000 0.828729
$$544$$ 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
$$549$$ 30.0000 0.0546448
$$550$$ 0 0
$$551$$ 658.179i 1.19452i
$$552$$ 0 0
$$553$$ 173.205i 0.313210i
$$554$$ 0 0
$$555$$ −72.0000 −0.129730
$$556$$ 0 0
$$557$$ −80.0000 −0.143627 −0.0718133 0.997418i $$-0.522879\pi$$
−0.0718133 + 0.997418i $$0.522879\pi$$
$$558$$ 0 0
$$559$$ − 242.487i − 0.433787i
$$560$$ 0 0
$$561$$ 173.205i 0.308743i
$$562$$ 0 0
$$563$$ 339.482i 0.602987i 0.953468 + 0.301494i $$0.0974852\pi$$
−0.953468 + 0.301494i $$0.902515\pi$$
$$564$$ 0 0
$$565$$ − 27.7128i − 0.0490492i
$$566$$ 0 0
$$567$$ 90.0000 0.158730
$$568$$ 0 0
$$569$$ − 658.179i − 1.15673i −0.815778 0.578365i $$-0.803691\pi$$
0.815778 0.578365i $$-0.196309\pi$$
$$570$$ 0 0
$$571$$ 610.000 1.06830 0.534151 0.845389i $$-0.320632\pi$$
0.534151 + 0.845389i $$0.320632\pi$$
$$572$$ 0 0
$$573$$ − 575.041i − 1.00356i
$$574$$ 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
$$579$$ −168.000 −0.290155
$$580$$ 0 0
$$581$$ −700.000 −1.20482
$$582$$ 0 0
$$583$$ − 415.692i − 0.713023i
$$584$$ 0 0
$$585$$ 290.985i 0.497409i
$$586$$ 0 0
$$587$$ 650.000 1.10733 0.553663 0.832741i $$-0.313230\pi$$
0.553663 + 0.832741i $$0.313230\pi$$
$$588$$ 0 0
$$589$$ 329.090i 0.558726i
$$590$$ 0 0
$$591$$ − 277.128i − 0.468914i
$$592$$ 0 0
$$593$$ 910.000 1.53457 0.767285 0.641306i $$-0.221607\pi$$
0.767285 + 0.641306i $$0.221607\pi$$
$$594$$ 0 0
$$595$$ 400.000 0.672269
$$596$$ 0 0
$$597$$ 169.741i 0.284323i
$$598$$ 0 0
$$599$$ − 34.6410i − 0.0578314i −0.999582 0.0289157i $$-0.990795\pi$$
0.999582 0.0289157i $$-0.00920544\pi$$
$$600$$ 0 0
$$601$$ − 173.205i − 0.288195i −0.989564 0.144097i $$-0.953972\pi$$
0.989564 0.144097i $$-0.0460279\pi$$
$$602$$ 0 0
$$603$$ − 228.631i − 0.379155i
$$604$$ 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
$$609$$ −600.000 −0.985222
$$610$$ 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
$$615$$ 240.000 0.390244
$$616$$ 0 0
$$617$$ 610.000 0.988655 0.494327 0.869276i $$-0.335414\pi$$
0.494327 + 0.869276i $$0.335414\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.0161551 0.00807754 0.999967i $$-0.497429\pi$$
0.00807754 + 0.999967i $$0.497429\pi$$
$$620$$ 0 0
$$621$$ 103.923i 0.167348i
$$622$$ 0 0
$$623$$ − 1039.23i − 1.66811i
$$624$$ 0 0
$$625$$ −319.000 −0.510400
$$626$$ 0 0
$$627$$ − 329.090i − 0.524864i
$$628$$ 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
$$633$$ −300.000 −0.473934
$$634$$ 0 0
$$635$$ 457.261i 0.720097i
$$636$$ 0 0
$$637$$ − 1236.68i − 1.94142i
$$638$$ 0 0
$$639$$ 311.769i 0.487902i
$$640$$ 0 0
$$641$$ 588.897i 0.918716i 0.888251 + 0.459358i $$0.151921\pi$$
−0.888251 + 0.459358i $$0.848079\pi$$
$$642$$ 0 0
$$643$$ −650.000 −1.01089 −0.505443 0.862860i $$-0.668671\pi$$
−0.505443 + 0.862860i $$0.668671\pi$$
$$644$$ 0 0
$$645$$ − 69.2820i − 0.107414i
$$646$$ 0 0
$$647$$ −820.000 −1.26739 −0.633694 0.773584i $$-0.718462\pi$$
−0.633694 + 0.773584i $$0.718462\pi$$
$$648$$ 0 0
$$649$$ 346.410i 0.533760i
$$650$$ 0 0
$$651$$ −300.000 −0.460829
$$652$$ 0 0
$$653$$ −560.000 −0.857580 −0.428790 0.903404i $$-0.641060\pi$$
−0.428790 + 0.903404i $$0.641060\pi$$
$$654$$ 0 0
$$655$$ 152.000 0.232061
$$656$$ 0 0
$$657$$ 30.0000 0.0456621
$$658$$ 0 0
$$659$$ 450.333i 0.683358i 0.939817 + 0.341679i $$0.110996\pi$$
−0.939817 + 0.341679i $$0.889004\pi$$
$$660$$ 0 0
$$661$$ − 398.372i − 0.602680i −0.953517 0.301340i $$-0.902566\pi$$
0.953517 0.301340i $$-0.0974340\pi$$
$$662$$ 0 0
$$663$$ −420.000 −0.633484
$$664$$ 0 0
$$665$$ −760.000 −1.14286
$$666$$ 0 0
$$667$$ − 692.820i − 1.03871i
$$668$$ 0 0
$$669$$ 138.000 0.206278
$$670$$ 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
$$675$$ − 46.7654i − 0.0692820i
$$676$$ 0 0
$$677$$ 526.543i 0.777760i 0.921288 + 0.388880i $$0.127138\pi$$
−0.921288 + 0.388880i $$0.872862\pi$$
$$678$$ 0 0
$$679$$ 762.102i 1.12239i
$$680$$ 0 0
$$681$$ 132.000 0.193833
$$682$$ 0 0
$$683$$ 478.046i 0.699921i 0.936764 + 0.349960i $$0.113805\pi$$
−0.936764 + 0.349960i $$0.886195\pi$$
$$684$$ 0 0
$$685$$ 760.000 1.10949
$$686$$ 0 0
$$687$$ − 190.526i − 0.277330i
$$688$$ 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
$$693$$ 300.000 0.432900
$$694$$ 0 0
$$695$$ −200.000 −0.287770
$$696$$ 0 0
$$697$$ 346.410i 0.497002i
$$698$$ 0 0
$$699$$ − 329.090i − 0.470801i
$$700$$ 0 0
$$701$$ −560.000 −0.798859 −0.399429 0.916764i $$-0.630792\pi$$
−0.399429 + 0.916764i $$0.630792\pi$$
$$702$$ 0 0
$$703$$ 197.454i 0.280873i
$$704$$ 0 0
$$705$$ − 554.256i − 0.786179i
$$706$$ 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
$$711$$ − 51.9615i − 0.0730823i
$$712$$ 0 0
$$713$$ − 346.410i − 0.485849i
$$714$$ 0 0
$$715$$ 969.948i 1.35657i
$$716$$ 0 0
$$717$$ − 221.703i − 0.309209i
$$718$$ 0 0
$$719$$ −520.000 −0.723227 −0.361613 0.932328i $$-0.617774\pi$$
−0.361613 + 0.932328i $$0.617774\pi$$
$$720$$ 0 0
$$721$$ − 1835.97i − 2.54643i
$$722$$ 0 0
$$723$$ 240.000 0.331950
$$724$$ 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
$$729$$ −27.0000 −0.0370370
$$730$$ 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
$$735$$ − 353.338i − 0.480732i
$$736$$ 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
$$741$$ 798.000 1.07692
$$742$$ 0 0
$$743$$ 235.559i 0.317038i 0.987356 + 0.158519i $$0.0506718\pi$$
−0.987356 + 0.158519i $$0.949328\pi$$
$$744$$ 0 0
$$745$$ −80.0000 −0.107383
$$746$$ 0 0
$$747$$ 210.000 0.281124
$$748$$ 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
$$753$$ − 3.46410i − 0.00460040i
$$754$$ 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
$$759$$ 346.410i 0.456403i
$$760$$ 0 0
$$761$$ −770.000 −1.01183 −0.505913 0.862584i $$-0.668844\pi$$
−0.505913 + 0.862584i $$0.668844\pi$$
$$762$$ 0 0
$$763$$ − 1558.85i − 2.04305i
$$764$$ 0 0
$$765$$ −120.000 −0.156863
$$766$$ 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
$$771$$ 852.000 1.10506
$$772$$ 0 0
$$773$$ − 145.492i − 0.188218i −0.995562 0.0941088i $$-0.970000\pi$$
0.995562 0.0941088i $$-0.0300002\pi$$
$$774$$ 0 0
$$775$$ 155.885i 0.201141i
$$776$$ 0 0
$$777$$ −180.000 −0.231660
$$778$$ 0 0
$$779$$ − 658.179i − 0.844903i
$$780$$ 0 0
$$781$$ 1039.23i 1.33064i
$$782$$ 0 0
$$783$$ 180.000 0.229885
$$784$$ 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
$$789$$ − 346.410i − 0.439050i
$$790$$ 0 0
$$791$$ − 69.2820i − 0.0875879i
$$792$$ 0 0
$$793$$ 242.487i 0.305785i
$$794$$ 0 0
$$795$$ 288.000 0.362264
$$796$$ 0 0
$$797$$ − 339.482i − 0.425950i −0.977058 0.212975i $$-0.931685\pi$$
0.977058 0.212975i $$-0.0683153\pi$$
$$798$$ 0 0
$$799$$ 800.000 1.00125
$$800$$ 0 0
$$801$$ 311.769i 0.389225i
$$802$$ 0 0
$$803$$ 100.000 0.124533
$$804$$ 0 0
$$805$$ 800.000 0.993789
$$806$$ 0 0
$$807$$ −720.000 −0.892193
$$808$$ 0 0
$$809$$ −182.000 −0.224969 −0.112485 0.993653i $$-0.535881\pi$$
−0.112485 + 0.993653i $$0.535881\pi$$
$$810$$ 0 0
$$811$$ 831.384i 1.02513i 0.858647 + 0.512567i $$0.171306\pi$$
−0.858647 + 0.512567i $$0.828694\pi$$
$$812$$ 0 0
$$813$$ 294.449i 0.362175i
$$814$$ 0 0
$$815$$ −680.000 −0.834356
$$816$$ 0 0
$$817$$ −190.000 −0.232558
$$818$$ 0 0
$$819$$ 727.461i 0.888231i
$$820$$ 0 0
$$821$$ −8.00000 −0.00974421 −0.00487211 0.999988i $$-0.501551\pi$$
−0.00487211 + 0.999988i $$0.501551\pi$$
$$822$$ 0 0
$$823$$ −950.000 −1.15431 −0.577157 0.816633i $$-0.695838\pi$$
−0.577157 + 0.816633i $$0.695838\pi$$
$$824$$ 0 0
$$825$$ − 155.885i − 0.188951i
$$826$$ 0 0
$$827$$ 478.046i 0.578048i 0.957322 + 0.289024i $$0.0933308\pi$$
−0.957322 + 0.289024i $$0.906669\pi$$
$$828$$ 0 0
$$829$$ 1195.12i 1.44163i 0.693125 + 0.720817i $$0.256233\pi$$
−0.693125 + 0.720817i $$0.743767\pi$$
$$830$$ 0 0
$$831$$ 17.3205i 0.0208430i
$$832$$ 0 0
$$833$$ 510.000 0.612245
$$834$$ 0 0
$$835$$ 526.543i 0.630591i
$$836$$ 0 0
$$837$$ 90.0000 0.107527
$$838$$ 0 0
$$839$$ 1177.79i 1.40381i 0.712272 + 0.701904i $$0.247666\pi$$
−0.712272 + 0.701904i $$0.752334\pi$$
$$840$$ 0 0
$$841$$ −359.000 −0.426873
$$842$$ 0 0
$$843$$ 660.000 0.782918
$$844$$ 0 0
$$845$$ −1676.00 −1.98343
$$846$$ 0 0
$$847$$ −210.000 −0.247934
$$848$$ 0 0
$$849$$ − 121.244i − 0.142807i
$$850$$ 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
$$855$$ 228.000 0.266667
$$856$$ 0 0
$$857$$ 1254.00i 1.46325i 0.681708 + 0.731625i $$0.261238\pi$$
−0.681708 + 0.731625i $$0.738762\pi$$
$$858$$ 0 0
$$859$$ −182.000 −0.211874 −0.105937 0.994373i $$-0.533784\pi$$
−0.105937 + 0.994373i $$0.533784\pi$$
$$860$$ 0 0
$$861$$ 600.000 0.696864
$$862$$ 0 0
$$863$$ − 1080.80i − 1.25238i −0.779672 0.626188i $$-0.784614\pi$$
0.779672 0.626188i $$-0.215386\pi$$
$$864$$ 0 0
$$865$$ 942.236i 1.08929i
$$866$$ 0 0
$$867$$ 327.358i 0.377575i
$$868$$ 0 0
$$869$$ − 173.205i − 0.199315i
$$870$$ 0 0
$$871$$ 1848.00 2.12170
$$872$$ 0 0
$$873$$ − 228.631i − 0.261891i
$$874$$ 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
$$879$$ 312.000 0.354949
$$880$$ 0 0
$$881$$ 550.000 0.624291 0.312145 0.950034i $$-0.398952\pi$$
0.312145 + 0.950034i $$0.398952\pi$$
$$882$$ 0 0
$$883$$ 1450.00 1.64213 0.821065 0.570835i $$-0.193381\pi$$
0.821065 + 0.570835i $$0.193381\pi$$
$$884$$ 0 0
$$885$$ −240.000 −0.271186
$$886$$ 0 0
$$887$$ 1254.00i 1.41376i 0.707334 + 0.706880i $$0.249898\pi$$
−0.707334 + 0.706880i $$0.750102\pi$$
$$888$$ 0 0
$$889$$ 1143.15i 1.28589i
$$890$$ 0 0
$$891$$ −90.0000 −0.101010
$$892$$ 0 0
$$893$$ −1520.00 −1.70213
$$894$$ 0 0
$$895$$ 415.692i 0.464461i
$$896$$ 0 0
$$897$$ −840.000 −0.936455
$$898$$ 0 0
$$899$$ −600.000 −0.667408
$$900$$ 0 0
$$901$$ 415.692i 0.461368i
$$902$$ 0 0
$$903$$ − 173.205i − 0.191811i
$$904$$ 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
$$909$$ −300.000 −0.330033
$$910$$ 0 0
$$911$$ − 796.743i − 0.874581i −0.899320 0.437291i $$-0.855938\pi$$
0.899320 0.437291i $$-0.144062\pi$$
$$912$$ 0 0
$$913$$ 700.000 0.766703
$$914$$ 0 0
$$915$$ 69.2820i 0.0757181i
$$916$$ 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
$$921$$ 252.000 0.273616
$$922$$ 0 0
$$923$$ −2520.00 −2.73023
$$924$$ 0 0
$$925$$ 93.5307i 0.101114i
$$926$$ 0 0
$$927$$ 550.792i 0.594166i
$$928$$ 0 0
$$929$$ −242.000 −0.260495 −0.130248 0.991482i $$-0.541577\pi$$
−0.130248 + 0.991482i $$0.541577\pi$$
$$930$$ 0 0
$$931$$ −969.000 −1.04082
$$932$$ 0 0
$$933$$ 1004.59i 1.07673i
$$934$$ 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
$$939$$ 640.859i 0.682491i
$$940$$ 0 0
$$941$$ − 796.743i − 0.846699i −0.905967 0.423349i $$-0.860854\pi$$
0.905967 0.423349i $$-0.139146\pi$$
$$942$$ 0 0
$$943$$ 692.820i 0.734698i
$$944$$ 0 0
$$945$$ 207.846i 0.219943i
$$946$$ 0 0
$$947$$ −1450.00 −1.53115 −0.765576 0.643346i $$-0.777546\pi$$
−0.765576 + 0.643346i $$0.777546\pi$$
$$948$$ 0 0
$$949$$ 242.487i 0.255519i
$$950$$ 0 0
$$951$$ 48.0000 0.0504732
$$952$$ 0 0
$$953$$ 353.338i 0.370764i 0.982667 + 0.185382i $$0.0593523\pi$$
−0.982667 + 0.185382i $$0.940648\pi$$
$$954$$ 0 0
$$955$$ 1328.00 1.39058
$$956$$ 0 0
$$957$$ 600.000 0.626959
$$958$$ 0 0
$$959$$ 1900.00 1.98123
$$960$$ 0 0
$$961$$ 661.000 0.687825
$$962$$ 0 0
$$963$$ 187.061i 0.194249i
$$964$$ 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
$$969$$ 329.090i 0.339618i
$$970$$ 0 0
$$971$$ 519.615i 0.535134i 0.963539 + 0.267567i $$0.0862197\pi$$
−0.963539 + 0.267567i $$0.913780\pi$$
$$972$$ 0 0
$$973$$ −500.000 −0.513875
$$974$$ 0 0
$$975$$ 378.000 0.387692
$$976$$ 0 0
$$977$$ − 1669.70i − 1.70900i −0.519448 0.854502i $$-0.673862\pi$$
0.519448 0.854502i $$-0.326138\pi$$
$$978$$ 0 0
$$979$$ 1039.23i 1.06152i
$$980$$ 0 0
$$981$$ 467.654i 0.476711i
$$982$$ 0 0
$$983$$ − 1690.48i − 1.71972i −0.510533 0.859858i $$-0.670552\pi$$
0.510533 0.859858i $$-0.329448\pi$$
$$984$$ 0 0
$$985$$ 640.000 0.649746
$$986$$ 0 0
$$987$$ − 1385.64i − 1.40389i
$$988$$ 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
$$993$$ 300.000 0.302115
$$994$$ 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
$$999$$ 54.0000 0.0540541
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 912.3.o.a.721.1 2
3.2 odd 2 2736.3.o.e.721.1 2
4.3 odd 2 57.3.c.a.37.2 yes 2
12.11 even 2 171.3.c.c.37.1 2
19.18 odd 2 inner 912.3.o.a.721.2 2
57.56 even 2 2736.3.o.e.721.2 2
76.75 even 2 57.3.c.a.37.1 2
228.227 odd 2 171.3.c.c.37.2 2

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