# Properties

 Label 57.3.c.a.37.2 Level $57$ Weight $3$ Character 57.37 Analytic conductor $1.553$ 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] = [57,3,Mod(37,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("57.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 57.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.55313750685$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 37.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 57.37 Dual form 57.3.c.a.37.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205i q^{2} +1.73205i q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +8.66025i q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.73205i q^{2} +1.73205i q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +8.66025i q^{8} -3.00000 q^{9} +6.92820i q^{10} +10.0000 q^{11} +1.73205i q^{12} -24.2487i q^{13} -17.3205i q^{14} +6.92820i q^{15} -11.0000 q^{16} +10.0000 q^{17} -5.19615i q^{18} +19.0000 q^{19} +4.00000 q^{20} -17.3205i q^{21} +17.3205i q^{22} -20.0000 q^{23} -15.0000 q^{24} -9.00000 q^{25} +42.0000 q^{26} -5.19615i q^{27} -10.0000 q^{28} -34.6410i q^{29} -12.0000 q^{30} +17.3205i q^{31} +15.5885i q^{32} +17.3205i q^{33} +17.3205i q^{34} -40.0000 q^{35} -3.00000 q^{36} -10.3923i q^{37} +32.9090i q^{38} +42.0000 q^{39} +34.6410i q^{40} +34.6410i q^{41} +30.0000 q^{42} -10.0000 q^{43} +10.0000 q^{44} -12.0000 q^{45} -34.6410i q^{46} -80.0000 q^{47} -19.0526i q^{48} +51.0000 q^{49} -15.5885i q^{50} +17.3205i q^{51} -24.2487i q^{52} +41.5692i q^{53} +9.00000 q^{54} +40.0000 q^{55} -86.6025i q^{56} +32.9090i q^{57} +60.0000 q^{58} +34.6410i q^{59} +6.92820i q^{60} -10.0000 q^{61} -30.0000 q^{62} +30.0000 q^{63} -71.0000 q^{64} -96.9948i q^{65} -30.0000 q^{66} -76.2102i q^{67} +10.0000 q^{68} -34.6410i q^{69} -69.2820i q^{70} +103.923i q^{71} -25.9808i q^{72} -10.0000 q^{73} +18.0000 q^{74} -15.5885i q^{75} +19.0000 q^{76} -100.000 q^{77} +72.7461i q^{78} -17.3205i q^{79} -44.0000 q^{80} +9.00000 q^{81} -60.0000 q^{82} +70.0000 q^{83} -17.3205i q^{84} +40.0000 q^{85} -17.3205i q^{86} +60.0000 q^{87} +86.6025i q^{88} -103.923i q^{89} -20.7846i q^{90} +242.487i q^{91} -20.0000 q^{92} -30.0000 q^{93} -138.564i q^{94} +76.0000 q^{95} -27.0000 q^{96} +76.2102i q^{97} +88.3346i q^{98} -30.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 8 q^{5} - 6 q^{6} - 20 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 8 * q^5 - 6 * q^6 - 20 * q^7 - 6 * q^9 $$2 q + 2 q^{4} + 8 q^{5} - 6 q^{6} - 20 q^{7} - 6 q^{9} + 20 q^{11} - 22 q^{16} + 20 q^{17} + 38 q^{19} + 8 q^{20} - 40 q^{23} - 30 q^{24} - 18 q^{25} + 84 q^{26} - 20 q^{28} - 24 q^{30} - 80 q^{35} - 6 q^{36} + 84 q^{39} + 60 q^{42} - 20 q^{43} + 20 q^{44} - 24 q^{45} - 160 q^{47} + 102 q^{49} + 18 q^{54} + 80 q^{55} + 120 q^{58} - 20 q^{61} - 60 q^{62} + 60 q^{63} - 142 q^{64} - 60 q^{66} + 20 q^{68} - 20 q^{73} + 36 q^{74} + 38 q^{76} - 200 q^{77} - 88 q^{80} + 18 q^{81} - 120 q^{82} + 140 q^{83} + 80 q^{85} + 120 q^{87} - 40 q^{92} - 60 q^{93} + 152 q^{95} - 54 q^{96} - 60 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 8 * q^5 - 6 * q^6 - 20 * q^7 - 6 * q^9 + 20 * q^11 - 22 * q^16 + 20 * q^17 + 38 * q^19 + 8 * q^20 - 40 * q^23 - 30 * q^24 - 18 * q^25 + 84 * q^26 - 20 * q^28 - 24 * q^30 - 80 * q^35 - 6 * q^36 + 84 * q^39 + 60 * q^42 - 20 * q^43 + 20 * q^44 - 24 * q^45 - 160 * q^47 + 102 * q^49 + 18 * q^54 + 80 * q^55 + 120 * q^58 - 20 * q^61 - 60 * q^62 + 60 * q^63 - 142 * q^64 - 60 * q^66 + 20 * q^68 - 20 * q^73 + 36 * q^74 + 38 * q^76 - 200 * q^77 - 88 * q^80 + 18 * q^81 - 120 * q^82 + 140 * q^83 + 80 * q^85 + 120 * q^87 - 40 * q^92 - 60 * q^93 + 152 * q^95 - 54 * q^96 - 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.73205i 0.866025i 0.901388 + 0.433013i $$0.142549\pi$$
−0.901388 + 0.433013i $$0.857451\pi$$
$$3$$ 1.73205i 0.577350i
$$4$$ 1.00000 0.250000
$$5$$ 4.00000 0.800000 0.400000 0.916515i $$-0.369010\pi$$
0.400000 + 0.916515i $$0.369010\pi$$
$$6$$ −3.00000 −0.500000
$$7$$ −10.0000 −1.42857 −0.714286 0.699854i $$-0.753248\pi$$
−0.714286 + 0.699854i $$0.753248\pi$$
$$8$$ 8.66025i 1.08253i
$$9$$ −3.00000 −0.333333
$$10$$ 6.92820i 0.692820i
$$11$$ 10.0000 0.909091 0.454545 0.890724i $$-0.349802\pi$$
0.454545 + 0.890724i $$0.349802\pi$$
$$12$$ 1.73205i 0.144338i
$$13$$ − 24.2487i − 1.86529i −0.360801 0.932643i $$-0.617497\pi$$
0.360801 0.932643i $$-0.382503\pi$$
$$14$$ − 17.3205i − 1.23718i
$$15$$ 6.92820i 0.461880i
$$16$$ −11.0000 −0.687500
$$17$$ 10.0000 0.588235 0.294118 0.955769i $$-0.404974\pi$$
0.294118 + 0.955769i $$0.404974\pi$$
$$18$$ − 5.19615i − 0.288675i
$$19$$ 19.0000 1.00000
$$20$$ 4.00000 0.200000
$$21$$ − 17.3205i − 0.824786i
$$22$$ 17.3205i 0.787296i
$$23$$ −20.0000 −0.869565 −0.434783 0.900535i $$-0.643175\pi$$
−0.434783 + 0.900535i $$0.643175\pi$$
$$24$$ −15.0000 −0.625000
$$25$$ −9.00000 −0.360000
$$26$$ 42.0000 1.61538
$$27$$ − 5.19615i − 0.192450i
$$28$$ −10.0000 −0.357143
$$29$$ − 34.6410i − 1.19452i −0.802049 0.597259i $$-0.796256\pi$$
0.802049 0.597259i $$-0.203744\pi$$
$$30$$ −12.0000 −0.400000
$$31$$ 17.3205i 0.558726i 0.960186 + 0.279363i $$0.0901233\pi$$
−0.960186 + 0.279363i $$0.909877\pi$$
$$32$$ 15.5885i 0.487139i
$$33$$ 17.3205i 0.524864i
$$34$$ 17.3205i 0.509427i
$$35$$ −40.0000 −1.14286
$$36$$ −3.00000 −0.0833333
$$37$$ − 10.3923i − 0.280873i −0.990090 0.140437i $$-0.955149\pi$$
0.990090 0.140437i $$-0.0448506\pi$$
$$38$$ 32.9090i 0.866025i
$$39$$ 42.0000 1.07692
$$40$$ 34.6410i 0.866025i
$$41$$ 34.6410i 0.844903i 0.906386 + 0.422451i $$0.138830\pi$$
−0.906386 + 0.422451i $$0.861170\pi$$
$$42$$ 30.0000 0.714286
$$43$$ −10.0000 −0.232558 −0.116279 0.993217i $$-0.537097\pi$$
−0.116279 + 0.993217i $$0.537097\pi$$
$$44$$ 10.0000 0.227273
$$45$$ −12.0000 −0.266667
$$46$$ − 34.6410i − 0.753066i
$$47$$ −80.0000 −1.70213 −0.851064 0.525062i $$-0.824042\pi$$
−0.851064 + 0.525062i $$0.824042\pi$$
$$48$$ − 19.0526i − 0.396928i
$$49$$ 51.0000 1.04082
$$50$$ − 15.5885i − 0.311769i
$$51$$ 17.3205i 0.339618i
$$52$$ − 24.2487i − 0.466321i
$$53$$ 41.5692i 0.784325i 0.919896 + 0.392162i $$0.128273\pi$$
−0.919896 + 0.392162i $$0.871727\pi$$
$$54$$ 9.00000 0.166667
$$55$$ 40.0000 0.727273
$$56$$ − 86.6025i − 1.54647i
$$57$$ 32.9090i 0.577350i
$$58$$ 60.0000 1.03448
$$59$$ 34.6410i 0.587136i 0.955938 + 0.293568i $$0.0948427\pi$$
−0.955938 + 0.293568i $$0.905157\pi$$
$$60$$ 6.92820i 0.115470i
$$61$$ −10.0000 −0.163934 −0.0819672 0.996635i $$-0.526120\pi$$
−0.0819672 + 0.996635i $$0.526120\pi$$
$$62$$ −30.0000 −0.483871
$$63$$ 30.0000 0.476190
$$64$$ −71.0000 −1.10938
$$65$$ − 96.9948i − 1.49223i
$$66$$ −30.0000 −0.454545
$$67$$ − 76.2102i − 1.13747i −0.822522 0.568733i $$-0.807434\pi$$
0.822522 0.568733i $$-0.192566\pi$$
$$68$$ 10.0000 0.147059
$$69$$ − 34.6410i − 0.502044i
$$70$$ − 69.2820i − 0.989743i
$$71$$ 103.923i 1.46370i 0.681463 + 0.731852i $$0.261344\pi$$
−0.681463 + 0.731852i $$0.738656\pi$$
$$72$$ − 25.9808i − 0.360844i
$$73$$ −10.0000 −0.136986 −0.0684932 0.997652i $$-0.521819\pi$$
−0.0684932 + 0.997652i $$0.521819\pi$$
$$74$$ 18.0000 0.243243
$$75$$ − 15.5885i − 0.207846i
$$76$$ 19.0000 0.250000
$$77$$ −100.000 −1.29870
$$78$$ 72.7461i 0.932643i
$$79$$ − 17.3205i − 0.219247i −0.993973 0.109623i $$-0.965035\pi$$
0.993973 0.109623i $$-0.0349645\pi$$
$$80$$ −44.0000 −0.550000
$$81$$ 9.00000 0.111111
$$82$$ −60.0000 −0.731707
$$83$$ 70.0000 0.843373 0.421687 0.906742i $$-0.361438\pi$$
0.421687 + 0.906742i $$0.361438\pi$$
$$84$$ − 17.3205i − 0.206197i
$$85$$ 40.0000 0.470588
$$86$$ − 17.3205i − 0.201401i
$$87$$ 60.0000 0.689655
$$88$$ 86.6025i 0.984120i
$$89$$ − 103.923i − 1.16767i −0.811871 0.583837i $$-0.801551\pi$$
0.811871 0.583837i $$-0.198449\pi$$
$$90$$ − 20.7846i − 0.230940i
$$91$$ 242.487i 2.66469i
$$92$$ −20.0000 −0.217391
$$93$$ −30.0000 −0.322581
$$94$$ − 138.564i − 1.47409i
$$95$$ 76.0000 0.800000
$$96$$ −27.0000 −0.281250
$$97$$ 76.2102i 0.785673i 0.919608 + 0.392836i $$0.128506\pi$$
−0.919608 + 0.392836i $$0.871494\pi$$
$$98$$ 88.3346i 0.901373i
$$99$$ −30.0000 −0.303030
$$100$$ −9.00000 −0.0900000
$$101$$ 100.000 0.990099 0.495050 0.868865i $$-0.335150\pi$$
0.495050 + 0.868865i $$0.335150\pi$$
$$102$$ −30.0000 −0.294118
$$103$$ 183.597i 1.78250i 0.453513 + 0.891249i $$0.350170\pi$$
−0.453513 + 0.891249i $$0.649830\pi$$
$$104$$ 210.000 2.01923
$$105$$ − 69.2820i − 0.659829i
$$106$$ −72.0000 −0.679245
$$107$$ 62.3538i 0.582746i 0.956610 + 0.291373i $$0.0941121\pi$$
−0.956610 + 0.291373i $$0.905888\pi$$
$$108$$ − 5.19615i − 0.0481125i
$$109$$ − 155.885i − 1.43013i −0.699056 0.715067i $$-0.746396\pi$$
0.699056 0.715067i $$-0.253604\pi$$
$$110$$ 69.2820i 0.629837i
$$111$$ 18.0000 0.162162
$$112$$ 110.000 0.982143
$$113$$ − 6.92820i − 0.0613115i −0.999530 0.0306558i $$-0.990240\pi$$
0.999530 0.0306558i $$-0.00975956\pi$$
$$114$$ −57.0000 −0.500000
$$115$$ −80.0000 −0.695652
$$116$$ − 34.6410i − 0.298629i
$$117$$ 72.7461i 0.621762i
$$118$$ −60.0000 −0.508475
$$119$$ −100.000 −0.840336
$$120$$ −60.0000 −0.500000
$$121$$ −21.0000 −0.173554
$$122$$ − 17.3205i − 0.141971i
$$123$$ −60.0000 −0.487805
$$124$$ 17.3205i 0.139682i
$$125$$ −136.000 −1.08800
$$126$$ 51.9615i 0.412393i
$$127$$ − 114.315i − 0.900121i −0.892998 0.450060i $$-0.851402\pi$$
0.892998 0.450060i $$-0.148598\pi$$
$$128$$ − 60.6218i − 0.473608i
$$129$$ − 17.3205i − 0.134268i
$$130$$ 168.000 1.29231
$$131$$ −38.0000 −0.290076 −0.145038 0.989426i $$-0.546330\pi$$
−0.145038 + 0.989426i $$0.546330\pi$$
$$132$$ 17.3205i 0.131216i
$$133$$ −190.000 −1.42857
$$134$$ 132.000 0.985075
$$135$$ − 20.7846i − 0.153960i
$$136$$ 86.6025i 0.636783i
$$137$$ 190.000 1.38686 0.693431 0.720523i $$-0.256098\pi$$
0.693431 + 0.720523i $$0.256098\pi$$
$$138$$ 60.0000 0.434783
$$139$$ 50.0000 0.359712 0.179856 0.983693i $$-0.442437\pi$$
0.179856 + 0.983693i $$0.442437\pi$$
$$140$$ −40.0000 −0.285714
$$141$$ − 138.564i − 0.982724i
$$142$$ −180.000 −1.26761
$$143$$ − 242.487i − 1.69571i
$$144$$ 33.0000 0.229167
$$145$$ − 138.564i − 0.955614i
$$146$$ − 17.3205i − 0.118634i
$$147$$ 88.3346i 0.600916i
$$148$$ − 10.3923i − 0.0702183i
$$149$$ −20.0000 −0.134228 −0.0671141 0.997745i $$-0.521379\pi$$
−0.0671141 + 0.997745i $$0.521379\pi$$
$$150$$ 27.0000 0.180000
$$151$$ − 225.167i − 1.49117i −0.666411 0.745585i $$-0.732170\pi$$
0.666411 0.745585i $$-0.267830\pi$$
$$152$$ 164.545i 1.08253i
$$153$$ −30.0000 −0.196078
$$154$$ − 173.205i − 1.12471i
$$155$$ 69.2820i 0.446981i
$$156$$ 42.0000 0.269231
$$157$$ 230.000 1.46497 0.732484 0.680784i $$-0.238361\pi$$
0.732484 + 0.680784i $$0.238361\pi$$
$$158$$ 30.0000 0.189873
$$159$$ −72.0000 −0.452830
$$160$$ 62.3538i 0.389711i
$$161$$ 200.000 1.24224
$$162$$ 15.5885i 0.0962250i
$$163$$ 170.000 1.04294 0.521472 0.853268i $$-0.325383\pi$$
0.521472 + 0.853268i $$0.325383\pi$$
$$164$$ 34.6410i 0.211226i
$$165$$ 69.2820i 0.419891i
$$166$$ 121.244i 0.730383i
$$167$$ − 131.636i − 0.788239i −0.919059 0.394119i $$-0.871050\pi$$
0.919059 0.394119i $$-0.128950\pi$$
$$168$$ 150.000 0.892857
$$169$$ −419.000 −2.47929
$$170$$ 69.2820i 0.407541i
$$171$$ −57.0000 −0.333333
$$172$$ −10.0000 −0.0581395
$$173$$ 235.559i 1.36161i 0.732464 + 0.680806i $$0.238370\pi$$
−0.732464 + 0.680806i $$0.761630\pi$$
$$174$$ 103.923i 0.597259i
$$175$$ 90.0000 0.514286
$$176$$ −110.000 −0.625000
$$177$$ −60.0000 −0.338983
$$178$$ 180.000 1.01124
$$179$$ − 103.923i − 0.580576i −0.956939 0.290288i $$-0.906249\pi$$
0.956939 0.290288i $$-0.0937511\pi$$
$$180$$ −12.0000 −0.0666667
$$181$$ 259.808i 1.43540i 0.696352 + 0.717701i $$0.254805\pi$$
−0.696352 + 0.717701i $$0.745195\pi$$
$$182$$ −420.000 −2.30769
$$183$$ − 17.3205i − 0.0946476i
$$184$$ − 173.205i − 0.941332i
$$185$$ − 41.5692i − 0.224698i
$$186$$ − 51.9615i − 0.279363i
$$187$$ 100.000 0.534759
$$188$$ −80.0000 −0.425532
$$189$$ 51.9615i 0.274929i
$$190$$ 131.636i 0.692820i
$$191$$ −332.000 −1.73822 −0.869110 0.494619i $$-0.835308\pi$$
−0.869110 + 0.494619i $$0.835308\pi$$
$$192$$ − 122.976i − 0.640498i
$$193$$ − 96.9948i − 0.502564i −0.967914 0.251282i $$-0.919148\pi$$
0.967914 0.251282i $$-0.0808521\pi$$
$$194$$ −132.000 −0.680412
$$195$$ 168.000 0.861538
$$196$$ 51.0000 0.260204
$$197$$ 160.000 0.812183 0.406091 0.913832i $$-0.366891\pi$$
0.406091 + 0.913832i $$0.366891\pi$$
$$198$$ − 51.9615i − 0.262432i
$$199$$ 98.0000 0.492462 0.246231 0.969211i $$-0.420808\pi$$
0.246231 + 0.969211i $$0.420808\pi$$
$$200$$ − 77.9423i − 0.389711i
$$201$$ 132.000 0.656716
$$202$$ 173.205i 0.857451i
$$203$$ 346.410i 1.70645i
$$204$$ 17.3205i 0.0849045i
$$205$$ 138.564i 0.675922i
$$206$$ −318.000 −1.54369
$$207$$ 60.0000 0.289855
$$208$$ 266.736i 1.28238i
$$209$$ 190.000 0.909091
$$210$$ 120.000 0.571429
$$211$$ 173.205i 0.820877i 0.911888 + 0.410439i $$0.134624\pi$$
−0.911888 + 0.410439i $$0.865376\pi$$
$$212$$ 41.5692i 0.196081i
$$213$$ −180.000 −0.845070
$$214$$ −108.000 −0.504673
$$215$$ −40.0000 −0.186047
$$216$$ 45.0000 0.208333
$$217$$ − 173.205i − 0.798180i
$$218$$ 270.000 1.23853
$$219$$ − 17.3205i − 0.0790891i
$$220$$ 40.0000 0.181818
$$221$$ − 242.487i − 1.09723i
$$222$$ 31.1769i 0.140437i
$$223$$ − 79.6743i − 0.357284i −0.983914 0.178642i $$-0.942830\pi$$
0.983914 0.178642i $$-0.0571704\pi$$
$$224$$ − 155.885i − 0.695913i
$$225$$ 27.0000 0.120000
$$226$$ 12.0000 0.0530973
$$227$$ − 76.2102i − 0.335728i −0.985810 0.167864i $$-0.946313\pi$$
0.985810 0.167864i $$-0.0536869\pi$$
$$228$$ 32.9090i 0.144338i
$$229$$ 110.000 0.480349 0.240175 0.970730i $$-0.422795\pi$$
0.240175 + 0.970730i $$0.422795\pi$$
$$230$$ − 138.564i − 0.602452i
$$231$$ − 173.205i − 0.749806i
$$232$$ 300.000 1.29310
$$233$$ 190.000 0.815451 0.407725 0.913105i $$-0.366322\pi$$
0.407725 + 0.913105i $$0.366322\pi$$
$$234$$ −126.000 −0.538462
$$235$$ −320.000 −1.36170
$$236$$ 34.6410i 0.146784i
$$237$$ 30.0000 0.126582
$$238$$ − 173.205i − 0.727752i
$$239$$ −128.000 −0.535565 −0.267782 0.963479i $$-0.586291\pi$$
−0.267782 + 0.963479i $$0.586291\pi$$
$$240$$ − 76.2102i − 0.317543i
$$241$$ 138.564i 0.574955i 0.957787 + 0.287477i $$0.0928166\pi$$
−0.957787 + 0.287477i $$0.907183\pi$$
$$242$$ − 36.3731i − 0.150302i
$$243$$ 15.5885i 0.0641500i
$$244$$ −10.0000 −0.0409836
$$245$$ 204.000 0.832653
$$246$$ − 103.923i − 0.422451i
$$247$$ − 460.726i − 1.86529i
$$248$$ −150.000 −0.604839
$$249$$ 121.244i 0.486922i
$$250$$ − 235.559i − 0.942236i
$$251$$ −2.00000 −0.00796813 −0.00398406 0.999992i $$-0.501268\pi$$
−0.00398406 + 0.999992i $$0.501268\pi$$
$$252$$ 30.0000 0.119048
$$253$$ −200.000 −0.790514
$$254$$ 198.000 0.779528
$$255$$ 69.2820i 0.271694i
$$256$$ −179.000 −0.699219
$$257$$ 491.902i 1.91402i 0.290059 + 0.957009i $$0.406325\pi$$
−0.290059 + 0.957009i $$0.593675\pi$$
$$258$$ 30.0000 0.116279
$$259$$ 103.923i 0.401247i
$$260$$ − 96.9948i − 0.373057i
$$261$$ 103.923i 0.398173i
$$262$$ − 65.8179i − 0.251213i
$$263$$ −200.000 −0.760456 −0.380228 0.924893i $$-0.624155\pi$$
−0.380228 + 0.924893i $$0.624155\pi$$
$$264$$ −150.000 −0.568182
$$265$$ 166.277i 0.627460i
$$266$$ − 329.090i − 1.23718i
$$267$$ 180.000 0.674157
$$268$$ − 76.2102i − 0.284367i
$$269$$ − 415.692i − 1.54532i −0.634818 0.772662i $$-0.718925\pi$$
0.634818 0.772662i $$-0.281075\pi$$
$$270$$ 36.0000 0.133333
$$271$$ 170.000 0.627306 0.313653 0.949538i $$-0.398447\pi$$
0.313653 + 0.949538i $$0.398447\pi$$
$$272$$ −110.000 −0.404412
$$273$$ −420.000 −1.53846
$$274$$ 329.090i 1.20106i
$$275$$ −90.0000 −0.327273
$$276$$ − 34.6410i − 0.125511i
$$277$$ −10.0000 −0.0361011 −0.0180505 0.999837i $$-0.505746\pi$$
−0.0180505 + 0.999837i $$0.505746\pi$$
$$278$$ 86.6025i 0.311520i
$$279$$ − 51.9615i − 0.186242i
$$280$$ − 346.410i − 1.23718i
$$281$$ 381.051i 1.35605i 0.735037 + 0.678027i $$0.237165\pi$$
−0.735037 + 0.678027i $$0.762835\pi$$
$$282$$ 240.000 0.851064
$$283$$ −70.0000 −0.247350 −0.123675 0.992323i $$-0.539468\pi$$
−0.123675 + 0.992323i $$0.539468\pi$$
$$284$$ 103.923i 0.365926i
$$285$$ 131.636i 0.461880i
$$286$$ 420.000 1.46853
$$287$$ − 346.410i − 1.20700i
$$288$$ − 46.7654i − 0.162380i
$$289$$ −189.000 −0.653979
$$290$$ 240.000 0.827586
$$291$$ −132.000 −0.453608
$$292$$ −10.0000 −0.0342466
$$293$$ 180.133i 0.614789i 0.951582 + 0.307395i $$0.0994572\pi$$
−0.951582 + 0.307395i $$0.900543\pi$$
$$294$$ −153.000 −0.520408
$$295$$ 138.564i 0.469709i
$$296$$ 90.0000 0.304054
$$297$$ − 51.9615i − 0.174955i
$$298$$ − 34.6410i − 0.116245i
$$299$$ 484.974i 1.62199i
$$300$$ − 15.5885i − 0.0519615i
$$301$$ 100.000 0.332226
$$302$$ 390.000 1.29139
$$303$$ 173.205i 0.571634i
$$304$$ −209.000 −0.687500
$$305$$ −40.0000 −0.131148
$$306$$ − 51.9615i − 0.169809i
$$307$$ − 145.492i − 0.473916i −0.971520 0.236958i $$-0.923850\pi$$
0.971520 0.236958i $$-0.0761504\pi$$
$$308$$ −100.000 −0.324675
$$309$$ −318.000 −1.02913
$$310$$ −120.000 −0.387097
$$311$$ 580.000 1.86495 0.932476 0.361232i $$-0.117644\pi$$
0.932476 + 0.361232i $$0.117644\pi$$
$$312$$ 363.731i 1.16580i
$$313$$ −370.000 −1.18211 −0.591054 0.806632i $$-0.701288\pi$$
−0.591054 + 0.806632i $$0.701288\pi$$
$$314$$ 398.372i 1.26870i
$$315$$ 120.000 0.380952
$$316$$ − 17.3205i − 0.0548117i
$$317$$ 27.7128i 0.0874221i 0.999044 + 0.0437111i $$0.0139181\pi$$
−0.999044 + 0.0437111i $$0.986082\pi$$
$$318$$ − 124.708i − 0.392162i
$$319$$ − 346.410i − 1.08593i
$$320$$ −284.000 −0.887500
$$321$$ −108.000 −0.336449
$$322$$ 346.410i 1.07581i
$$323$$ 190.000 0.588235
$$324$$ 9.00000 0.0277778
$$325$$ 218.238i 0.671503i
$$326$$ 294.449i 0.903217i
$$327$$ 270.000 0.825688
$$328$$ −300.000 −0.914634
$$329$$ 800.000 2.43161
$$330$$ −120.000 −0.363636
$$331$$ − 173.205i − 0.523278i −0.965166 0.261639i $$-0.915737\pi$$
0.965166 0.261639i $$-0.0842630\pi$$
$$332$$ 70.0000 0.210843
$$333$$ 31.1769i 0.0936244i
$$334$$ 228.000 0.682635
$$335$$ − 304.841i − 0.909973i
$$336$$ 190.526i 0.567040i
$$337$$ 339.482i 1.00736i 0.863889 + 0.503682i $$0.168022\pi$$
−0.863889 + 0.503682i $$0.831978\pi$$
$$338$$ − 725.729i − 2.14713i
$$339$$ 12.0000 0.0353982
$$340$$ 40.0000 0.117647
$$341$$ 173.205i 0.507933i
$$342$$ − 98.7269i − 0.288675i
$$343$$ −20.0000 −0.0583090
$$344$$ − 86.6025i − 0.251752i
$$345$$ − 138.564i − 0.401635i
$$346$$ −408.000 −1.17919
$$347$$ −590.000 −1.70029 −0.850144 0.526550i $$-0.823485\pi$$
−0.850144 + 0.526550i $$0.823485\pi$$
$$348$$ 60.0000 0.172414
$$349$$ 98.0000 0.280802 0.140401 0.990095i $$-0.455161\pi$$
0.140401 + 0.990095i $$0.455161\pi$$
$$350$$ 155.885i 0.445384i
$$351$$ −126.000 −0.358974
$$352$$ 155.885i 0.442854i
$$353$$ 190.000 0.538244 0.269122 0.963106i $$-0.413267\pi$$
0.269122 + 0.963106i $$0.413267\pi$$
$$354$$ − 103.923i − 0.293568i
$$355$$ 415.692i 1.17096i
$$356$$ − 103.923i − 0.291919i
$$357$$ − 173.205i − 0.485168i
$$358$$ 180.000 0.502793
$$359$$ −200.000 −0.557103 −0.278552 0.960421i $$-0.589854\pi$$
−0.278552 + 0.960421i $$0.589854\pi$$
$$360$$ − 103.923i − 0.288675i
$$361$$ 361.000 1.00000
$$362$$ −450.000 −1.24309
$$363$$ − 36.3731i − 0.100201i
$$364$$ 242.487i 0.666173i
$$365$$ −40.0000 −0.109589
$$366$$ 30.0000 0.0819672
$$367$$ 170.000 0.463215 0.231608 0.972809i $$-0.425601\pi$$
0.231608 + 0.972809i $$0.425601\pi$$
$$368$$ 220.000 0.597826
$$369$$ − 103.923i − 0.281634i
$$370$$ 72.0000 0.194595
$$371$$ − 415.692i − 1.12046i
$$372$$ −30.0000 −0.0806452
$$373$$ − 356.802i − 0.956575i −0.878203 0.478287i $$-0.841258\pi$$
0.878203 0.478287i $$-0.158742\pi$$
$$374$$ 173.205i 0.463115i
$$375$$ − 235.559i − 0.628157i
$$376$$ − 692.820i − 1.84261i
$$377$$ −840.000 −2.22812
$$378$$ −90.0000 −0.238095
$$379$$ 207.846i 0.548407i 0.961672 + 0.274203i $$0.0884141\pi$$
−0.961672 + 0.274203i $$0.911586\pi$$
$$380$$ 76.0000 0.200000
$$381$$ 198.000 0.519685
$$382$$ − 575.041i − 1.50534i
$$383$$ − 630.466i − 1.64613i −0.567950 0.823063i $$-0.692263\pi$$
0.567950 0.823063i $$-0.307737\pi$$
$$384$$ 105.000 0.273438
$$385$$ −400.000 −1.03896
$$386$$ 168.000 0.435233
$$387$$ 30.0000 0.0775194
$$388$$ 76.2102i 0.196418i
$$389$$ −128.000 −0.329049 −0.164524 0.986373i $$-0.552609\pi$$
−0.164524 + 0.986373i $$0.552609\pi$$
$$390$$ 290.985i 0.746114i
$$391$$ −200.000 −0.511509
$$392$$ 441.673i 1.12672i
$$393$$ − 65.8179i − 0.167476i
$$394$$ 277.128i 0.703371i
$$395$$ − 69.2820i − 0.175398i
$$396$$ −30.0000 −0.0757576
$$397$$ 650.000 1.63728 0.818640 0.574307i $$-0.194729\pi$$
0.818640 + 0.574307i $$0.194729\pi$$
$$398$$ 169.741i 0.426485i
$$399$$ − 329.090i − 0.824786i
$$400$$ 99.0000 0.247500
$$401$$ − 173.205i − 0.431933i −0.976401 0.215966i $$-0.930710\pi$$
0.976401 0.215966i $$-0.0692902\pi$$
$$402$$ 228.631i 0.568733i
$$403$$ 420.000 1.04218
$$404$$ 100.000 0.247525
$$405$$ 36.0000 0.0888889
$$406$$ −600.000 −1.47783
$$407$$ − 103.923i − 0.255339i
$$408$$ −150.000 −0.367647
$$409$$ 173.205i 0.423484i 0.977326 + 0.211742i $$0.0679137\pi$$
−0.977326 + 0.211742i $$0.932086\pi$$
$$410$$ −240.000 −0.585366
$$411$$ 329.090i 0.800705i
$$412$$ 183.597i 0.445625i
$$413$$ − 346.410i − 0.838766i
$$414$$ 103.923i 0.251022i
$$415$$ 280.000 0.674699
$$416$$ 378.000 0.908654
$$417$$ 86.6025i 0.207680i
$$418$$ 329.090i 0.787296i
$$419$$ −38.0000 −0.0906921 −0.0453461 0.998971i $$-0.514439\pi$$
−0.0453461 + 0.998971i $$0.514439\pi$$
$$420$$ − 69.2820i − 0.164957i
$$421$$ − 17.3205i − 0.0411413i −0.999788 0.0205707i $$-0.993452\pi$$
0.999788 0.0205707i $$-0.00654831\pi$$
$$422$$ −300.000 −0.710900
$$423$$ 240.000 0.567376
$$424$$ −360.000 −0.849057
$$425$$ −90.0000 −0.211765
$$426$$ − 311.769i − 0.731852i
$$427$$ 100.000 0.234192
$$428$$ 62.3538i 0.145687i
$$429$$ 420.000 0.979021
$$430$$ − 69.2820i − 0.161121i
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 57.1577i 0.132309i
$$433$$ − 353.338i − 0.816024i −0.912977 0.408012i $$-0.866222\pi$$
0.912977 0.408012i $$-0.133778\pi$$
$$434$$ 300.000 0.691244
$$435$$ 240.000 0.551724
$$436$$ − 155.885i − 0.357533i
$$437$$ −380.000 −0.869565
$$438$$ 30.0000 0.0684932
$$439$$ − 121.244i − 0.276181i −0.990420 0.138091i $$-0.955903\pi$$
0.990420 0.138091i $$-0.0440965\pi$$
$$440$$ 346.410i 0.787296i
$$441$$ −153.000 −0.346939
$$442$$ 420.000 0.950226
$$443$$ −110.000 −0.248307 −0.124153 0.992263i $$-0.539622\pi$$
−0.124153 + 0.992263i $$0.539622\pi$$
$$444$$ 18.0000 0.0405405
$$445$$ − 415.692i − 0.934140i
$$446$$ 138.000 0.309417
$$447$$ − 34.6410i − 0.0774967i
$$448$$ 710.000 1.58482
$$449$$ 311.769i 0.694363i 0.937798 + 0.347182i $$0.112861\pi$$
−0.937798 + 0.347182i $$0.887139\pi$$
$$450$$ 46.7654i 0.103923i
$$451$$ 346.410i 0.768093i
$$452$$ − 6.92820i − 0.0153279i
$$453$$ 390.000 0.860927
$$454$$ 132.000 0.290749
$$455$$ 969.948i 2.13175i
$$456$$ −285.000 −0.625000
$$457$$ 290.000 0.634573 0.317287 0.948330i $$-0.397228\pi$$
0.317287 + 0.948330i $$0.397228\pi$$
$$458$$ 190.526i 0.415995i
$$459$$ − 51.9615i − 0.113206i
$$460$$ −80.0000 −0.173913
$$461$$ −728.000 −1.57918 −0.789588 0.613638i $$-0.789706\pi$$
−0.789588 + 0.613638i $$0.789706\pi$$
$$462$$ 300.000 0.649351
$$463$$ −790.000 −1.70626 −0.853132 0.521696i $$-0.825300\pi$$
−0.853132 + 0.521696i $$0.825300\pi$$
$$464$$ 381.051i 0.821231i
$$465$$ −120.000 −0.258065
$$466$$ 329.090i 0.706201i
$$467$$ −530.000 −1.13490 −0.567452 0.823407i $$-0.692071\pi$$
−0.567452 + 0.823407i $$0.692071\pi$$
$$468$$ 72.7461i 0.155440i
$$469$$ 762.102i 1.62495i
$$470$$ − 554.256i − 1.17927i
$$471$$ 398.372i 0.845800i
$$472$$ −300.000 −0.635593
$$473$$ −100.000 −0.211416
$$474$$ 51.9615i 0.109623i
$$475$$ −171.000 −0.360000
$$476$$ −100.000 −0.210084
$$477$$ − 124.708i − 0.261442i
$$478$$ − 221.703i − 0.463813i
$$479$$ −80.0000 −0.167015 −0.0835073 0.996507i $$-0.526612\pi$$
−0.0835073 + 0.996507i $$0.526612\pi$$
$$480$$ −108.000 −0.225000
$$481$$ −252.000 −0.523909
$$482$$ −240.000 −0.497925
$$483$$ 346.410i 0.717205i
$$484$$ −21.0000 −0.0433884
$$485$$ 304.841i 0.628538i
$$486$$ −27.0000 −0.0555556
$$487$$ − 509.223i − 1.04563i −0.852445 0.522816i $$-0.824881\pi$$
0.852445 0.522816i $$-0.175119\pi$$
$$488$$ − 86.6025i − 0.177464i
$$489$$ 294.449i 0.602144i
$$490$$ 353.338i 0.721099i
$$491$$ 418.000 0.851324 0.425662 0.904882i $$-0.360041\pi$$
0.425662 + 0.904882i $$0.360041\pi$$
$$492$$ −60.0000 −0.121951
$$493$$ − 346.410i − 0.702658i
$$494$$ 798.000 1.61538
$$495$$ −120.000 −0.242424
$$496$$ − 190.526i − 0.384124i
$$497$$ − 1039.23i − 2.09101i
$$498$$ −210.000 −0.421687
$$499$$ 470.000 0.941884 0.470942 0.882164i $$-0.343914\pi$$
0.470942 + 0.882164i $$0.343914\pi$$
$$500$$ −136.000 −0.272000
$$501$$ 228.000 0.455090
$$502$$ − 3.46410i − 0.00690060i
$$503$$ 100.000 0.198807 0.0994036 0.995047i $$-0.468307\pi$$
0.0994036 + 0.995047i $$0.468307\pi$$
$$504$$ 259.808i 0.515491i
$$505$$ 400.000 0.792079
$$506$$ − 346.410i − 0.684605i
$$507$$ − 725.729i − 1.43142i
$$508$$ − 114.315i − 0.225030i
$$509$$ 450.333i 0.884741i 0.896832 + 0.442371i $$0.145862\pi$$
−0.896832 + 0.442371i $$0.854138\pi$$
$$510$$ −120.000 −0.235294
$$511$$ 100.000 0.195695
$$512$$ − 552.524i − 1.07915i
$$513$$ − 98.7269i − 0.192450i
$$514$$ −852.000 −1.65759
$$515$$ 734.390i 1.42600i
$$516$$ − 17.3205i − 0.0335669i
$$517$$ −800.000 −1.54739
$$518$$ −180.000 −0.347490
$$519$$ −408.000 −0.786127
$$520$$ 840.000 1.61538
$$521$$ 311.769i 0.598405i 0.954190 + 0.299203i $$0.0967207\pi$$
−0.954190 + 0.299203i $$0.903279\pi$$
$$522$$ −180.000 −0.344828
$$523$$ − 789.815i − 1.51016i −0.655631 0.755081i $$-0.727597\pi$$
0.655631 0.755081i $$-0.272403\pi$$
$$524$$ −38.0000 −0.0725191
$$525$$ 155.885i 0.296923i
$$526$$ − 346.410i − 0.658574i
$$527$$ 173.205i 0.328662i
$$528$$ − 190.526i − 0.360844i
$$529$$ −129.000 −0.243856
$$530$$ −288.000 −0.543396
$$531$$ − 103.923i − 0.195712i
$$532$$ −190.000 −0.357143
$$533$$ 840.000 1.57598
$$534$$ 311.769i 0.583837i
$$535$$ 249.415i 0.466197i
$$536$$ 660.000 1.23134
$$537$$ 180.000 0.335196
$$538$$ 720.000 1.33829
$$539$$ 510.000 0.946197
$$540$$ − 20.7846i − 0.0384900i
$$541$$ 650.000 1.20148 0.600739 0.799445i $$-0.294873\pi$$
0.600739 + 0.799445i $$0.294873\pi$$
$$542$$ 294.449i 0.543263i
$$543$$ −450.000 −0.828729
$$544$$ 155.885i 0.286553i
$$545$$ − 623.538i − 1.14411i
$$546$$ − 727.461i − 1.33235i
$$547$$ 595.825i 1.08926i 0.838676 + 0.544630i $$0.183330\pi$$
−0.838676 + 0.544630i $$0.816670\pi$$
$$548$$ 190.000 0.346715
$$549$$ 30.0000 0.0546448
$$550$$ − 155.885i − 0.283426i
$$551$$ − 658.179i − 1.19452i
$$552$$ 300.000 0.543478
$$553$$ 173.205i 0.313210i
$$554$$ − 17.3205i − 0.0312645i
$$555$$ 72.0000 0.129730
$$556$$ 50.0000 0.0899281
$$557$$ −80.0000 −0.143627 −0.0718133 0.997418i $$-0.522879\pi$$
−0.0718133 + 0.997418i $$0.522879\pi$$
$$558$$ 90.0000 0.161290
$$559$$ 242.487i 0.433787i
$$560$$ 440.000 0.785714
$$561$$ 173.205i 0.308743i
$$562$$ −660.000 −1.17438
$$563$$ − 339.482i − 0.602987i −0.953468 0.301494i $$-0.902515\pi$$
0.953468 0.301494i $$-0.0974852\pi$$
$$564$$ − 138.564i − 0.245681i
$$565$$ − 27.7128i − 0.0490492i
$$566$$ − 121.244i − 0.214211i
$$567$$ −90.0000 −0.158730
$$568$$ −900.000 −1.58451
$$569$$ − 658.179i − 1.15673i −0.815778 0.578365i $$-0.803691\pi$$
0.815778 0.578365i $$-0.196309\pi$$
$$570$$ −228.000 −0.400000
$$571$$ −610.000 −1.06830 −0.534151 0.845389i $$-0.679368\pi$$
−0.534151 + 0.845389i $$0.679368\pi$$
$$572$$ − 242.487i − 0.423929i
$$573$$ − 575.041i − 1.00356i
$$574$$ 600.000 1.04530
$$575$$ 180.000 0.313043
$$576$$ 213.000 0.369792
$$577$$ 170.000 0.294627 0.147314 0.989090i $$-0.452937\pi$$
0.147314 + 0.989090i $$0.452937\pi$$
$$578$$ − 327.358i − 0.566363i
$$579$$ 168.000 0.290155
$$580$$ − 138.564i − 0.238904i
$$581$$ −700.000 −1.20482
$$582$$ − 228.631i − 0.392836i
$$583$$ 415.692i 0.713023i
$$584$$ − 86.6025i − 0.148292i
$$585$$ 290.985i 0.497409i
$$586$$ −312.000 −0.532423
$$587$$ −650.000 −1.10733 −0.553663 0.832741i $$-0.686770\pi$$
−0.553663 + 0.832741i $$0.686770\pi$$
$$588$$ 88.3346i 0.150229i
$$589$$ 329.090i 0.558726i
$$590$$ −240.000 −0.406780
$$591$$ 277.128i 0.468914i
$$592$$ 114.315i 0.193100i
$$593$$ 910.000 1.53457 0.767285 0.641306i $$-0.221607\pi$$
0.767285 + 0.641306i $$0.221607\pi$$
$$594$$ 90.0000 0.151515
$$595$$ −400.000 −0.672269
$$596$$ −20.0000 −0.0335570
$$597$$ 169.741i 0.284323i
$$598$$ −840.000 −1.40468
$$599$$ 34.6410i 0.0578314i 0.999582 + 0.0289157i $$0.00920544\pi$$
−0.999582 + 0.0289157i $$0.990795\pi$$
$$600$$ 135.000 0.225000
$$601$$ − 173.205i − 0.288195i −0.989564 0.144097i $$-0.953972\pi$$
0.989564 0.144097i $$-0.0460279\pi$$
$$602$$ 173.205i 0.287716i
$$603$$ 228.631i 0.379155i
$$604$$ − 225.167i − 0.372792i
$$605$$ −84.0000 −0.138843
$$606$$ −300.000 −0.495050
$$607$$ 703.213i 1.15851i 0.815148 + 0.579253i $$0.196656\pi$$
−0.815148 + 0.579253i $$0.803344\pi$$
$$608$$ 296.181i 0.487139i
$$609$$ −600.000 −0.985222
$$610$$ − 69.2820i − 0.113577i
$$611$$ 1939.90i 3.17495i
$$612$$ −30.0000 −0.0490196
$$613$$ 350.000 0.570962 0.285481 0.958384i $$-0.407847\pi$$
0.285481 + 0.958384i $$0.407847\pi$$
$$614$$ 252.000 0.410423
$$615$$ −240.000 −0.390244
$$616$$ − 866.025i − 1.40589i
$$617$$ 610.000 0.988655 0.494327 0.869276i $$-0.335414\pi$$
0.494327 + 0.869276i $$0.335414\pi$$
$$618$$ − 550.792i − 0.891249i
$$619$$ −10.0000 −0.0161551 −0.00807754 0.999967i $$-0.502571\pi$$
−0.00807754 + 0.999967i $$0.502571\pi$$
$$620$$ 69.2820i 0.111745i
$$621$$ 103.923i 0.167348i
$$622$$ 1004.59i 1.61510i
$$623$$ 1039.23i 1.66811i
$$624$$ −462.000 −0.740385
$$625$$ −319.000 −0.510400
$$626$$ − 640.859i − 1.02374i
$$627$$ 329.090i 0.524864i
$$628$$ 230.000 0.366242
$$629$$ − 103.923i − 0.165219i
$$630$$ 207.846i 0.329914i
$$631$$ 350.000 0.554675 0.277338 0.960773i $$-0.410548\pi$$
0.277338 + 0.960773i $$0.410548\pi$$
$$632$$ 150.000 0.237342
$$633$$ −300.000 −0.473934
$$634$$ −48.0000 −0.0757098
$$635$$ − 457.261i − 0.720097i
$$636$$ −72.0000 −0.113208
$$637$$ − 1236.68i − 1.94142i
$$638$$ 600.000 0.940439
$$639$$ − 311.769i − 0.487902i
$$640$$ − 242.487i − 0.378886i
$$641$$ 588.897i 0.918716i 0.888251 + 0.459358i $$0.151921\pi$$
−0.888251 + 0.459358i $$0.848079\pi$$
$$642$$ − 187.061i − 0.291373i
$$643$$ 650.000 1.01089 0.505443 0.862860i $$-0.331329\pi$$
0.505443 + 0.862860i $$0.331329\pi$$
$$644$$ 200.000 0.310559
$$645$$ − 69.2820i − 0.107414i
$$646$$ 329.090i 0.509427i
$$647$$ 820.000 1.26739 0.633694 0.773584i $$-0.281538\pi$$
0.633694 + 0.773584i $$0.281538\pi$$
$$648$$ 77.9423i 0.120281i
$$649$$ 346.410i 0.533760i
$$650$$ −378.000 −0.581538
$$651$$ 300.000 0.460829
$$652$$ 170.000 0.260736
$$653$$ −560.000 −0.857580 −0.428790 0.903404i $$-0.641060\pi$$
−0.428790 + 0.903404i $$0.641060\pi$$
$$654$$ 467.654i 0.715067i
$$655$$ −152.000 −0.232061
$$656$$ − 381.051i − 0.580871i
$$657$$ 30.0000 0.0456621
$$658$$ 1385.64i 2.10584i
$$659$$ − 450.333i − 0.683358i −0.939817 0.341679i $$-0.889004\pi$$
0.939817 0.341679i $$-0.110996\pi$$
$$660$$ 69.2820i 0.104973i
$$661$$ − 398.372i − 0.602680i −0.953517 0.301340i $$-0.902566\pi$$
0.953517 0.301340i $$-0.0974340\pi$$
$$662$$ 300.000 0.453172
$$663$$ 420.000 0.633484
$$664$$ 606.218i 0.912979i
$$665$$ −760.000 −1.14286
$$666$$ −54.0000 −0.0810811
$$667$$ 692.820i 1.03871i
$$668$$ − 131.636i − 0.197060i
$$669$$ 138.000 0.206278
$$670$$ 528.000 0.788060
$$671$$ −100.000 −0.149031
$$672$$ 270.000 0.401786
$$673$$ − 630.466i − 0.936800i −0.883516 0.468400i $$-0.844831\pi$$
0.883516 0.468400i $$-0.155169\pi$$
$$674$$ −588.000 −0.872404
$$675$$ 46.7654i 0.0692820i
$$676$$ −419.000 −0.619822
$$677$$ 526.543i 0.777760i 0.921288 + 0.388880i $$0.127138\pi$$
−0.921288 + 0.388880i $$0.872862\pi$$
$$678$$ 20.7846i 0.0306558i
$$679$$ − 762.102i − 1.12239i
$$680$$ 346.410i 0.509427i
$$681$$ 132.000 0.193833
$$682$$ −300.000 −0.439883
$$683$$ − 478.046i − 0.699921i −0.936764 0.349960i $$-0.886195\pi$$
0.936764 0.349960i $$-0.113805\pi$$
$$684$$ −57.0000 −0.0833333
$$685$$ 760.000 1.10949
$$686$$ − 34.6410i − 0.0504971i
$$687$$ 190.526i 0.277330i
$$688$$ 110.000 0.159884
$$689$$ 1008.00 1.46299
$$690$$ 240.000 0.347826
$$691$$ 470.000 0.680174 0.340087 0.940394i $$-0.389544\pi$$
0.340087 + 0.940394i $$0.389544\pi$$
$$692$$ 235.559i 0.340403i
$$693$$ 300.000 0.432900
$$694$$ − 1021.91i − 1.47249i
$$695$$ 200.000 0.287770
$$696$$ 519.615i 0.746574i
$$697$$ 346.410i 0.497002i
$$698$$ 169.741i 0.243182i
$$699$$ 329.090i 0.470801i
$$700$$ 90.0000 0.128571
$$701$$ −560.000 −0.798859 −0.399429 0.916764i $$-0.630792\pi$$
−0.399429 + 0.916764i $$0.630792\pi$$
$$702$$ − 218.238i − 0.310881i
$$703$$ − 197.454i − 0.280873i
$$704$$ −710.000 −1.00852
$$705$$ − 554.256i − 0.786179i
$$706$$ 329.090i 0.466133i
$$707$$ −1000.00 −1.41443
$$708$$ −60.0000 −0.0847458
$$709$$ −982.000 −1.38505 −0.692525 0.721394i $$-0.743502\pi$$
−0.692525 + 0.721394i $$0.743502\pi$$
$$710$$ −720.000 −1.01408
$$711$$ 51.9615i 0.0730823i
$$712$$ 900.000 1.26404
$$713$$ − 346.410i − 0.485849i
$$714$$ 300.000 0.420168
$$715$$ − 969.948i − 1.35657i
$$716$$ − 103.923i − 0.145144i
$$717$$ − 221.703i − 0.309209i
$$718$$ − 346.410i − 0.482465i
$$719$$ 520.000 0.723227 0.361613 0.932328i $$-0.382226\pi$$
0.361613 + 0.932328i $$0.382226\pi$$
$$720$$ 132.000 0.183333
$$721$$ − 1835.97i − 2.54643i
$$722$$ 625.270i 0.866025i
$$723$$ −240.000 −0.331950
$$724$$ 259.808i 0.358850i
$$725$$ 311.769i 0.430026i
$$726$$ 63.0000 0.0867769
$$727$$ −790.000 −1.08666 −0.543329 0.839520i $$-0.682836\pi$$
−0.543329 + 0.839520i $$0.682836\pi$$
$$728$$ −2100.00 −2.88462
$$729$$ −27.0000 −0.0370370
$$730$$ − 69.2820i − 0.0949069i
$$731$$ −100.000 −0.136799
$$732$$ − 17.3205i − 0.0236619i
$$733$$ −1150.00 −1.56889 −0.784447 0.620195i $$-0.787053\pi$$
−0.784447 + 0.620195i $$0.787053\pi$$
$$734$$ 294.449i 0.401156i
$$735$$ 353.338i 0.480732i
$$736$$ − 311.769i − 0.423599i
$$737$$ − 762.102i − 1.03406i
$$738$$ 180.000 0.243902
$$739$$ 578.000 0.782138 0.391069 0.920361i $$-0.372105\pi$$
0.391069 + 0.920361i $$0.372105\pi$$
$$740$$ − 41.5692i − 0.0561746i
$$741$$ 798.000 1.07692
$$742$$ 720.000 0.970350
$$743$$ − 235.559i − 0.317038i −0.987356 0.158519i $$-0.949328\pi$$
0.987356 0.158519i $$-0.0506718\pi$$
$$744$$ − 259.808i − 0.349204i
$$745$$ −80.0000 −0.107383
$$746$$ 618.000 0.828418
$$747$$ −210.000 −0.281124
$$748$$ 100.000 0.133690
$$749$$ − 623.538i − 0.832494i
$$750$$ 408.000 0.544000
$$751$$ 952.628i 1.26848i 0.773137 + 0.634240i $$0.218687\pi$$
−0.773137 + 0.634240i $$0.781313\pi$$
$$752$$ 880.000 1.17021
$$753$$ − 3.46410i − 0.00460040i
$$754$$ − 1454.92i − 1.92961i
$$755$$ − 900.666i − 1.19294i
$$756$$ 51.9615i 0.0687322i
$$757$$ −250.000 −0.330251 −0.165125 0.986273i $$-0.552803\pi$$
−0.165125 + 0.986273i $$0.552803\pi$$
$$758$$ −360.000 −0.474934
$$759$$ − 346.410i − 0.456403i
$$760$$ 658.179i 0.866025i
$$761$$ −770.000 −1.01183 −0.505913 0.862584i $$-0.668844\pi$$
−0.505913 + 0.862584i $$0.668844\pi$$
$$762$$ 342.946i 0.450060i
$$763$$ 1558.85i 2.04305i
$$764$$ −332.000 −0.434555
$$765$$ −120.000 −0.156863
$$766$$ 1092.00 1.42559
$$767$$ 840.000 1.09518
$$768$$ − 310.037i − 0.403694i
$$769$$ 110.000 0.143043 0.0715215 0.997439i $$-0.477215\pi$$
0.0715215 + 0.997439i $$0.477215\pi$$
$$770$$ − 692.820i − 0.899767i
$$771$$ −852.000 −1.10506
$$772$$ − 96.9948i − 0.125641i
$$773$$ − 145.492i − 0.188218i −0.995562 0.0941088i $$-0.970000\pi$$
0.995562 0.0941088i $$-0.0300002\pi$$
$$774$$ 51.9615i 0.0671338i
$$775$$ − 155.885i − 0.201141i
$$776$$ −660.000 −0.850515
$$777$$ −180.000 −0.231660
$$778$$ − 221.703i − 0.284965i
$$779$$ 658.179i 0.844903i
$$780$$ 168.000 0.215385
$$781$$ 1039.23i 1.33064i
$$782$$ − 346.410i − 0.442980i
$$783$$ −180.000 −0.229885
$$784$$ −561.000 −0.715561
$$785$$ 920.000 1.17197
$$786$$ 114.000 0.145038
$$787$$ − 96.9948i − 0.123246i −0.998099 0.0616232i $$-0.980372\pi$$
0.998099 0.0616232i $$-0.0196277\pi$$
$$788$$ 160.000 0.203046
$$789$$ − 346.410i − 0.439050i
$$790$$ 120.000 0.151899
$$791$$ 69.2820i 0.0875879i
$$792$$ − 259.808i − 0.328040i
$$793$$ 242.487i 0.305785i
$$794$$ 1125.83i 1.41793i
$$795$$ −288.000 −0.362264
$$796$$ 98.0000 0.123116
$$797$$ − 339.482i − 0.425950i −0.977058 0.212975i $$-0.931685\pi$$
0.977058 0.212975i $$-0.0683153\pi$$
$$798$$ 570.000 0.714286
$$799$$ −800.000 −1.00125
$$800$$ − 140.296i − 0.175370i
$$801$$ 311.769i 0.389225i
$$802$$ 300.000 0.374065
$$803$$ −100.000 −0.124533
$$804$$ 132.000 0.164179
$$805$$ 800.000 0.993789
$$806$$ 727.461i 0.902557i
$$807$$ 720.000 0.892193
$$808$$ 866.025i 1.07181i
$$809$$ −182.000 −0.224969 −0.112485 0.993653i $$-0.535881\pi$$
−0.112485 + 0.993653i $$0.535881\pi$$
$$810$$ 62.3538i 0.0769800i
$$811$$ − 831.384i − 1.02513i −0.858647 0.512567i $$-0.828694\pi$$
0.858647 0.512567i $$-0.171306\pi$$
$$812$$ 346.410i 0.426613i
$$813$$ 294.449i 0.362175i
$$814$$ 180.000 0.221130
$$815$$ 680.000 0.834356
$$816$$ − 190.526i − 0.233487i
$$817$$ −190.000 −0.232558
$$818$$ −300.000 −0.366748
$$819$$ − 727.461i − 0.888231i
$$820$$ 138.564i 0.168981i
$$821$$ −8.00000 −0.00974421 −0.00487211 0.999988i $$-0.501551\pi$$
−0.00487211 + 0.999988i $$0.501551\pi$$
$$822$$ −570.000 −0.693431
$$823$$ 950.000 1.15431 0.577157 0.816633i $$-0.304162\pi$$
0.577157 + 0.816633i $$0.304162\pi$$
$$824$$ −1590.00 −1.92961
$$825$$ − 155.885i − 0.188951i
$$826$$ 600.000 0.726392
$$827$$ − 478.046i − 0.578048i −0.957322 0.289024i $$-0.906669\pi$$
0.957322 0.289024i $$-0.0933308\pi$$
$$828$$ 60.0000 0.0724638
$$829$$ 1195.12i 1.44163i 0.693125 + 0.720817i $$0.256233\pi$$
−0.693125 + 0.720817i $$0.743767\pi$$
$$830$$ 484.974i 0.584306i
$$831$$ − 17.3205i − 0.0208430i
$$832$$ 1721.66i 2.06930i
$$833$$ 510.000 0.612245
$$834$$ −150.000 −0.179856
$$835$$ − 526.543i − 0.630591i
$$836$$ 190.000 0.227273
$$837$$ 90.0000 0.107527
$$838$$ − 65.8179i − 0.0785417i
$$839$$ − 1177.79i − 1.40381i −0.712272 0.701904i $$-0.752334\pi$$
0.712272 0.701904i $$-0.247666\pi$$
$$840$$ 600.000 0.714286
$$841$$ −359.000 −0.426873
$$842$$ 30.0000 0.0356295
$$843$$ −660.000 −0.782918
$$844$$ 173.205i 0.205219i
$$845$$ −1676.00 −1.98343
$$846$$ 415.692i 0.491362i
$$847$$ 210.000 0.247934
$$848$$ − 457.261i − 0.539223i
$$849$$ − 121.244i − 0.142807i
$$850$$ − 155.885i − 0.183394i
$$851$$ 207.846i 0.244237i
$$852$$ −180.000 −0.211268
$$853$$ 890.000 1.04338 0.521688 0.853136i $$-0.325302\pi$$
0.521688 + 0.853136i $$0.325302\pi$$
$$854$$ 173.205i 0.202816i
$$855$$ −228.000 −0.266667
$$856$$ −540.000 −0.630841
$$857$$ 1254.00i 1.46325i 0.681708 + 0.731625i $$0.261238\pi$$
−0.681708 + 0.731625i $$0.738762\pi$$
$$858$$ 727.461i 0.847857i
$$859$$ 182.000 0.211874 0.105937 0.994373i $$-0.466216\pi$$
0.105937 + 0.994373i $$0.466216\pi$$
$$860$$ −40.0000 −0.0465116
$$861$$ 600.000 0.696864
$$862$$ 0 0
$$863$$ 1080.80i 1.25238i 0.779672 + 0.626188i $$0.215386\pi$$
−0.779672 + 0.626188i $$0.784614\pi$$
$$864$$ 81.0000 0.0937500
$$865$$ 942.236i 1.08929i
$$866$$ 612.000 0.706697
$$867$$ − 327.358i − 0.377575i
$$868$$ − 173.205i − 0.199545i
$$869$$ − 173.205i − 0.199315i
$$870$$ 415.692i 0.477807i
$$871$$ −1848.00 −2.12170
$$872$$ 1350.00 1.54817
$$873$$ − 228.631i − 0.261891i
$$874$$ − 658.179i − 0.753066i
$$875$$ 1360.00 1.55429
$$876$$ − 17.3205i − 0.0197723i
$$877$$ − 1188.19i − 1.35483i −0.735601 0.677416i $$-0.763100\pi$$
0.735601 0.677416i $$-0.236900\pi$$
$$878$$ 210.000 0.239180
$$879$$ −312.000 −0.354949
$$880$$ −440.000 −0.500000
$$881$$ 550.000 0.624291 0.312145 0.950034i $$-0.398952\pi$$
0.312145 + 0.950034i $$0.398952\pi$$
$$882$$ − 265.004i − 0.300458i
$$883$$ −1450.00 −1.64213 −0.821065 0.570835i $$-0.806619\pi$$
−0.821065 + 0.570835i $$0.806619\pi$$
$$884$$ − 242.487i − 0.274307i
$$885$$ −240.000 −0.271186
$$886$$ − 190.526i − 0.215040i
$$887$$ − 1254.00i − 1.41376i −0.707334 0.706880i $$-0.750102\pi$$
0.707334 0.706880i $$-0.249898\pi$$
$$888$$ 155.885i 0.175546i
$$889$$ 1143.15i 1.28589i
$$890$$ 720.000 0.808989
$$891$$ 90.0000 0.101010
$$892$$ − 79.6743i − 0.0893210i
$$893$$ −1520.00 −1.70213
$$894$$ 60.0000 0.0671141
$$895$$ − 415.692i − 0.464461i
$$896$$ 606.218i 0.676582i
$$897$$ −840.000 −0.936455
$$898$$ −540.000 −0.601336
$$899$$ 600.000 0.667408
$$900$$ 27.0000 0.0300000
$$901$$ 415.692i 0.461368i
$$902$$ −600.000 −0.665188
$$903$$ 173.205i 0.191811i
$$904$$ 60.0000 0.0663717
$$905$$ 1039.23i 1.14832i
$$906$$ 675.500i 0.745585i
$$907$$ − 110.851i − 0.122217i −0.998131 0.0611087i $$-0.980536\pi$$
0.998131 0.0611087i $$-0.0194636\pi$$
$$908$$ − 76.2102i − 0.0839320i
$$909$$ −300.000 −0.330033
$$910$$ −1680.00 −1.84615
$$911$$ 796.743i 0.874581i 0.899320 + 0.437291i $$0.144062\pi$$
−0.899320 + 0.437291i $$0.855938\pi$$
$$912$$ − 361.999i − 0.396928i
$$913$$ 700.000 0.766703
$$914$$ 502.295i 0.549557i
$$915$$ − 69.2820i − 0.0757181i
$$916$$ 110.000 0.120087
$$917$$ 380.000 0.414395
$$918$$ 90.0000 0.0980392
$$919$$ 62.0000 0.0674646 0.0337323 0.999431i $$-0.489261\pi$$
0.0337323 + 0.999431i $$0.489261\pi$$
$$920$$ − 692.820i − 0.753066i
$$921$$ 252.000 0.273616
$$922$$ − 1260.93i − 1.36761i
$$923$$ 2520.00 2.73023
$$924$$ − 173.205i − 0.187451i
$$925$$ 93.5307i 0.101114i
$$926$$ − 1368.32i − 1.47767i
$$927$$ − 550.792i − 0.594166i
$$928$$ 540.000 0.581897
$$929$$ −242.000 −0.260495 −0.130248 0.991482i $$-0.541577\pi$$
−0.130248 + 0.991482i $$0.541577\pi$$
$$930$$ − 207.846i − 0.223490i
$$931$$ 969.000 1.04082
$$932$$ 190.000 0.203863
$$933$$ 1004.59i 1.07673i
$$934$$ − 917.987i − 0.982855i
$$935$$ 400.000 0.427807
$$936$$ −630.000 −0.673077
$$937$$ 110.000 0.117396 0.0586980 0.998276i $$-0.481305\pi$$
0.0586980 + 0.998276i $$0.481305\pi$$
$$938$$ −1320.00 −1.40725
$$939$$ − 640.859i − 0.682491i
$$940$$ −320.000 −0.340426
$$941$$ − 796.743i − 0.846699i −0.905967 0.423349i $$-0.860854\pi$$
0.905967 0.423349i $$-0.139146\pi$$
$$942$$ −690.000 −0.732484
$$943$$ − 692.820i − 0.734698i
$$944$$ − 381.051i − 0.403656i
$$945$$ 207.846i 0.219943i
$$946$$ − 173.205i − 0.183092i
$$947$$ 1450.00 1.53115 0.765576 0.643346i $$-0.222454\pi$$
0.765576 + 0.643346i $$0.222454\pi$$
$$948$$ 30.0000 0.0316456
$$949$$ 242.487i 0.255519i
$$950$$ − 296.181i − 0.311769i
$$951$$ −48.0000 −0.0504732
$$952$$ − 866.025i − 0.909691i
$$953$$ 353.338i 0.370764i 0.982667 + 0.185382i $$0.0593523\pi$$
−0.982667 + 0.185382i $$0.940648\pi$$
$$954$$ 216.000 0.226415
$$955$$ −1328.00 −1.39058
$$956$$ −128.000 −0.133891
$$957$$ 600.000 0.626959
$$958$$ − 138.564i − 0.144639i
$$959$$ −1900.00 −1.98123
$$960$$ − 491.902i − 0.512398i
$$961$$ 661.000 0.687825
$$962$$ − 436.477i − 0.453718i
$$963$$ − 187.061i − 0.194249i
$$964$$ 138.564i 0.143739i
$$965$$ − 387.979i − 0.402051i
$$966$$ −600.000 −0.621118
$$967$$ 470.000 0.486039 0.243020 0.970021i $$-0.421862\pi$$
0.243020 + 0.970021i $$0.421862\pi$$
$$968$$ − 181.865i − 0.187877i
$$969$$ 329.090i 0.339618i
$$970$$ −528.000 −0.544330
$$971$$ − 519.615i − 0.535134i −0.963539 0.267567i $$-0.913780\pi$$
0.963539 0.267567i $$-0.0862197\pi$$
$$972$$ 15.5885i 0.0160375i
$$973$$ −500.000 −0.513875
$$974$$ 882.000 0.905544
$$975$$ −378.000 −0.387692
$$976$$ 110.000 0.112705
$$977$$ − 1669.70i − 1.70900i −0.519448 0.854502i $$-0.673862\pi$$
0.519448 0.854502i $$-0.326138\pi$$
$$978$$ −510.000 −0.521472
$$979$$ − 1039.23i − 1.06152i
$$980$$ 204.000 0.208163
$$981$$ 467.654i 0.476711i
$$982$$ 723.997i 0.737268i
$$983$$ 1690.48i 1.71972i 0.510533 + 0.859858i $$0.329448\pi$$
−0.510533 + 0.859858i $$0.670552\pi$$
$$984$$ − 519.615i − 0.528064i
$$985$$ 640.000 0.649746
$$986$$ 600.000 0.608519
$$987$$ 1385.64i 1.40389i
$$988$$ − 460.726i − 0.466321i
$$989$$ 200.000 0.202224
$$990$$ − 207.846i − 0.209946i
$$991$$ 571.577i 0.576768i 0.957515 + 0.288384i $$0.0931179\pi$$
−0.957515 + 0.288384i $$0.906882\pi$$
$$992$$ −270.000 −0.272177
$$993$$ 300.000 0.302115
$$994$$ 1800.00 1.81087
$$995$$ 392.000 0.393970
$$996$$ 121.244i 0.121730i
$$997$$ 1550.00 1.55466 0.777332 0.629091i $$-0.216573\pi$$
0.777332 + 0.629091i $$0.216573\pi$$
$$998$$ 814.064i 0.815695i
$$999$$ −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 57.3.c.a.37.2 yes 2
3.2 odd 2 171.3.c.c.37.1 2
4.3 odd 2 912.3.o.a.721.1 2
12.11 even 2 2736.3.o.e.721.1 2
19.18 odd 2 inner 57.3.c.a.37.1 2
57.56 even 2 171.3.c.c.37.2 2
76.75 even 2 912.3.o.a.721.2 2
228.227 odd 2 2736.3.o.e.721.2 2

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