# Properties

 Label 171.3.c.c.37.1 Level $171$ Weight $3$ Character 171.37 Analytic conductor $4.659$ 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] = [171,3,Mod(37,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, 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("171.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 171.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: $$4.65941252056$$ 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: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 37.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 171.37 Dual form 171.3.c.c.37.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{2} +1.00000 q^{4} -4.00000 q^{5} -10.0000 q^{7} -8.66025i q^{8} +O(q^{10})$$ $$q-1.73205i q^{2} +1.00000 q^{4} -4.00000 q^{5} -10.0000 q^{7} -8.66025i q^{8} +6.92820i q^{10} -10.0000 q^{11} -24.2487i q^{13} +17.3205i q^{14} -11.0000 q^{16} -10.0000 q^{17} +19.0000 q^{19} -4.00000 q^{20} +17.3205i q^{22} +20.0000 q^{23} -9.00000 q^{25} -42.0000 q^{26} -10.0000 q^{28} +34.6410i q^{29} +17.3205i q^{31} -15.5885i q^{32} +17.3205i q^{34} +40.0000 q^{35} -10.3923i q^{37} -32.9090i q^{38} +34.6410i q^{40} -34.6410i q^{41} -10.0000 q^{43} -10.0000 q^{44} -34.6410i q^{46} +80.0000 q^{47} +51.0000 q^{49} +15.5885i q^{50} -24.2487i q^{52} -41.5692i q^{53} +40.0000 q^{55} +86.6025i q^{56} +60.0000 q^{58} -34.6410i q^{59} -10.0000 q^{61} +30.0000 q^{62} -71.0000 q^{64} +96.9948i q^{65} -76.2102i q^{67} -10.0000 q^{68} -69.2820i q^{70} -103.923i q^{71} -10.0000 q^{73} -18.0000 q^{74} +19.0000 q^{76} +100.000 q^{77} -17.3205i q^{79} +44.0000 q^{80} -60.0000 q^{82} -70.0000 q^{83} +40.0000 q^{85} +17.3205i q^{86} +86.6025i q^{88} +103.923i q^{89} +242.487i q^{91} +20.0000 q^{92} -138.564i q^{94} -76.0000 q^{95} +76.2102i q^{97} -88.3346i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 8 q^{5} - 20 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 - 8 * q^5 - 20 * q^7 $$2 q + 2 q^{4} - 8 q^{5} - 20 q^{7} - 20 q^{11} - 22 q^{16} - 20 q^{17} + 38 q^{19} - 8 q^{20} + 40 q^{23} - 18 q^{25} - 84 q^{26} - 20 q^{28} + 80 q^{35} - 20 q^{43} - 20 q^{44} + 160 q^{47} + 102 q^{49} + 80 q^{55} + 120 q^{58} - 20 q^{61} + 60 q^{62} - 142 q^{64} - 20 q^{68} - 20 q^{73} - 36 q^{74} + 38 q^{76} + 200 q^{77} + 88 q^{80} - 120 q^{82} - 140 q^{83} + 80 q^{85} + 40 q^{92} - 152 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 - 8 * q^5 - 20 * q^7 - 20 * q^11 - 22 * q^16 - 20 * q^17 + 38 * q^19 - 8 * q^20 + 40 * q^23 - 18 * q^25 - 84 * q^26 - 20 * q^28 + 80 * q^35 - 20 * q^43 - 20 * q^44 + 160 * q^47 + 102 * q^49 + 80 * q^55 + 120 * q^58 - 20 * q^61 + 60 * q^62 - 142 * q^64 - 20 * q^68 - 20 * q^73 - 36 * q^74 + 38 * q^76 + 200 * q^77 + 88 * q^80 - 120 * q^82 - 140 * q^83 + 80 * q^85 + 40 * q^92 - 152 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$-0.857451\pi$$
0.901388 0.433013i $$-0.142549\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.250000
$$5$$ −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 $$-0.753248\pi$$
−0.714286 + 0.699854i $$0.753248\pi$$
$$8$$ − 8.66025i − 1.08253i
$$9$$ 0 0
$$10$$ 6.92820i 0.692820i
$$11$$ −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 $$-0.382503\pi$$
$$14$$ 17.3205i 1.23718i
$$15$$ 0 0
$$16$$ −11.0000 −0.687500
$$17$$ −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
$$20$$ −4.00000 −0.200000
$$21$$ 0 0
$$22$$ 17.3205i 0.787296i
$$23$$ 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
$$26$$ −42.0000 −1.61538
$$27$$ 0 0
$$28$$ −10.0000 −0.357143
$$29$$ 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 $$0.0901233\pi$$
−0.960186 + 0.279363i $$0.909877\pi$$
$$32$$ − 15.5885i − 0.487139i
$$33$$ 0 0
$$34$$ 17.3205i 0.509427i
$$35$$ 40.0000 1.14286
$$36$$ 0 0
$$37$$ − 10.3923i − 0.280873i −0.990090 0.140437i $$-0.955149\pi$$
0.990090 0.140437i $$-0.0448506\pi$$
$$38$$ − 32.9090i − 0.866025i
$$39$$ 0 0
$$40$$ 34.6410i 0.866025i
$$41$$ − 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 $$-0.537097\pi$$
−0.116279 + 0.993217i $$0.537097\pi$$
$$44$$ −10.0000 −0.227273
$$45$$ 0 0
$$46$$ − 34.6410i − 0.753066i
$$47$$ 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
$$50$$ 15.5885i 0.311769i
$$51$$ 0 0
$$52$$ − 24.2487i − 0.466321i
$$53$$ − 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
$$56$$ 86.6025i 1.54647i
$$57$$ 0 0
$$58$$ 60.0000 1.03448
$$59$$ − 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 $$0.526120\pi$$
$$62$$ 30.0000 0.483871
$$63$$ 0 0
$$64$$ −71.0000 −1.10938
$$65$$ 96.9948i 1.49223i
$$66$$ 0 0
$$67$$ − 76.2102i − 1.13747i −0.822522 0.568733i $$-0.807434\pi$$
0.822522 0.568733i $$-0.192566\pi$$
$$68$$ −10.0000 −0.147059
$$69$$ 0 0
$$70$$ − 69.2820i − 0.989743i
$$71$$ − 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 $$0.521819\pi$$
$$74$$ −18.0000 −0.243243
$$75$$ 0 0
$$76$$ 19.0000 0.250000
$$77$$ 100.000 1.29870
$$78$$ 0 0
$$79$$ − 17.3205i − 0.219247i −0.993973 0.109623i $$-0.965035\pi$$
0.993973 0.109623i $$-0.0349645\pi$$
$$80$$ 44.0000 0.550000
$$81$$ 0 0
$$82$$ −60.0000 −0.731707
$$83$$ −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
$$86$$ 17.3205i 0.201401i
$$87$$ 0 0
$$88$$ 86.6025i 0.984120i
$$89$$ 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
$$92$$ 20.0000 0.217391
$$93$$ 0 0
$$94$$ − 138.564i − 1.47409i
$$95$$ −76.0000 −0.800000
$$96$$ 0 0
$$97$$ 76.2102i 0.785673i 0.919608 + 0.392836i $$0.128506\pi$$
−0.919608 + 0.392836i $$0.871494\pi$$
$$98$$ − 88.3346i − 0.901373i
$$99$$ 0 0
$$100$$ −9.00000 −0.0900000
$$101$$ −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 $$0.350170\pi$$
−0.453513 + 0.891249i $$0.649830\pi$$
$$104$$ −210.000 −2.01923
$$105$$ 0 0
$$106$$ −72.0000 −0.679245
$$107$$ − 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 $$-0.253604\pi$$
$$110$$ − 69.2820i − 0.629837i
$$111$$ 0 0
$$112$$ 110.000 0.982143
$$113$$ 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
$$116$$ 34.6410i 0.298629i
$$117$$ 0 0
$$118$$ −60.0000 −0.508475
$$119$$ 100.000 0.840336
$$120$$ 0 0
$$121$$ −21.0000 −0.173554
$$122$$ 17.3205i 0.141971i
$$123$$ 0 0
$$124$$ 17.3205i 0.139682i
$$125$$ 136.000 1.08800
$$126$$ 0 0
$$127$$ − 114.315i − 0.900121i −0.892998 0.450060i $$-0.851402\pi$$
0.892998 0.450060i $$-0.148598\pi$$
$$128$$ 60.6218i 0.473608i
$$129$$ 0 0
$$130$$ 168.000 1.29231
$$131$$ 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
$$134$$ −132.000 −0.985075
$$135$$ 0 0
$$136$$ 86.6025i 0.636783i
$$137$$ −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 $$-0.442437\pi$$
0.179856 + 0.983693i $$0.442437\pi$$
$$140$$ 40.0000 0.285714
$$141$$ 0 0
$$142$$ −180.000 −1.26761
$$143$$ 242.487i 1.69571i
$$144$$ 0 0
$$145$$ − 138.564i − 0.955614i
$$146$$ 17.3205i 0.118634i
$$147$$ 0 0
$$148$$ − 10.3923i − 0.0702183i
$$149$$ 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 $$-0.732170\pi$$
0.666411 0.745585i $$-0.267830\pi$$
$$152$$ − 164.545i − 1.08253i
$$153$$ 0 0
$$154$$ − 173.205i − 1.12471i
$$155$$ − 69.2820i − 0.446981i
$$156$$ 0 0
$$157$$ 230.000 1.46497 0.732484 0.680784i $$-0.238361\pi$$
0.732484 + 0.680784i $$0.238361\pi$$
$$158$$ −30.0000 −0.189873
$$159$$ 0 0
$$160$$ 62.3538i 0.389711i
$$161$$ −200.000 −1.24224
$$162$$ 0 0
$$163$$ 170.000 1.04294 0.521472 0.853268i $$-0.325383\pi$$
0.521472 + 0.853268i $$0.325383\pi$$
$$164$$ − 34.6410i − 0.211226i
$$165$$ 0 0
$$166$$ 121.244i 0.730383i
$$167$$ 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
$$170$$ − 69.2820i − 0.407541i
$$171$$ 0 0
$$172$$ −10.0000 −0.0581395
$$173$$ − 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
$$176$$ 110.000 0.625000
$$177$$ 0 0
$$178$$ 180.000 1.01124
$$179$$ 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 $$0.745195\pi$$
$$182$$ 420.000 2.30769
$$183$$ 0 0
$$184$$ − 173.205i − 0.941332i
$$185$$ 41.5692i 0.224698i
$$186$$ 0 0
$$187$$ 100.000 0.534759
$$188$$ 80.0000 0.425532
$$189$$ 0 0
$$190$$ 131.636i 0.692820i
$$191$$ 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 $$-0.0808521\pi$$
$$194$$ 132.000 0.680412
$$195$$ 0 0
$$196$$ 51.0000 0.260204
$$197$$ −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 $$-0.420808\pi$$
0.246231 + 0.969211i $$0.420808\pi$$
$$200$$ 77.9423i 0.389711i
$$201$$ 0 0
$$202$$ 173.205i 0.857451i
$$203$$ − 346.410i − 1.70645i
$$204$$ 0 0
$$205$$ 138.564i 0.675922i
$$206$$ 318.000 1.54369
$$207$$ 0 0
$$208$$ 266.736i 1.28238i
$$209$$ −190.000 −0.909091
$$210$$ 0 0
$$211$$ 173.205i 0.820877i 0.911888 + 0.410439i $$0.134624\pi$$
−0.911888 + 0.410439i $$0.865376\pi$$
$$212$$ − 41.5692i − 0.196081i
$$213$$ 0 0
$$214$$ −108.000 −0.504673
$$215$$ 40.0000 0.186047
$$216$$ 0 0
$$217$$ − 173.205i − 0.798180i
$$218$$ −270.000 −1.23853
$$219$$ 0 0
$$220$$ 40.0000 0.181818
$$221$$ 242.487i 1.09723i
$$222$$ 0 0
$$223$$ − 79.6743i − 0.357284i −0.983914 0.178642i $$-0.942830\pi$$
0.983914 0.178642i $$-0.0571704\pi$$
$$224$$ 155.885i 0.695913i
$$225$$ 0 0
$$226$$ 12.0000 0.0530973
$$227$$ 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 $$0.422795\pi$$
$$230$$ 138.564i 0.602452i
$$231$$ 0 0
$$232$$ 300.000 1.29310
$$233$$ −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
$$236$$ − 34.6410i − 0.146784i
$$237$$ 0 0
$$238$$ − 173.205i − 0.727752i
$$239$$ 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 $$0.907183\pi$$
$$242$$ 36.3731i 0.150302i
$$243$$ 0 0
$$244$$ −10.0000 −0.0409836
$$245$$ −204.000 −0.832653
$$246$$ 0 0
$$247$$ − 460.726i − 1.86529i
$$248$$ 150.000 0.604839
$$249$$ 0 0
$$250$$ − 235.559i − 0.942236i
$$251$$ 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
$$254$$ −198.000 −0.779528
$$255$$ 0 0
$$256$$ −179.000 −0.699219
$$257$$ − 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
$$260$$ 96.9948i 0.373057i
$$261$$ 0 0
$$262$$ − 65.8179i − 0.251213i
$$263$$ 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
$$266$$ 329.090i 1.23718i
$$267$$ 0 0
$$268$$ − 76.2102i − 0.284367i
$$269$$ 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 $$-0.398447\pi$$
0.313653 + 0.949538i $$0.398447\pi$$
$$272$$ 110.000 0.404412
$$273$$ 0 0
$$274$$ 329.090i 1.20106i
$$275$$ 90.0000 0.327273
$$276$$ 0 0
$$277$$ −10.0000 −0.0361011 −0.0180505 0.999837i $$-0.505746\pi$$
−0.0180505 + 0.999837i $$0.505746\pi$$
$$278$$ − 86.6025i − 0.311520i
$$279$$ 0 0
$$280$$ − 346.410i − 1.23718i
$$281$$ − 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 $$-0.539468\pi$$
−0.123675 + 0.992323i $$0.539468\pi$$
$$284$$ − 103.923i − 0.365926i
$$285$$ 0 0
$$286$$ 420.000 1.46853
$$287$$ 346.410i 1.20700i
$$288$$ 0 0
$$289$$ −189.000 −0.653979
$$290$$ −240.000 −0.827586
$$291$$ 0 0
$$292$$ −10.0000 −0.0342466
$$293$$ − 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
$$296$$ −90.0000 −0.304054
$$297$$ 0 0
$$298$$ − 34.6410i − 0.116245i
$$299$$ − 484.974i − 1.62199i
$$300$$ 0 0
$$301$$ 100.000 0.332226
$$302$$ −390.000 −1.29139
$$303$$ 0 0
$$304$$ −209.000 −0.687500
$$305$$ 40.0000 0.131148
$$306$$ 0 0
$$307$$ − 145.492i − 0.473916i −0.971520 0.236958i $$-0.923850\pi$$
0.971520 0.236958i $$-0.0761504\pi$$
$$308$$ 100.000 0.324675
$$309$$ 0 0
$$310$$ −120.000 −0.387097
$$311$$ −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 $$0.701288\pi$$
$$314$$ − 398.372i − 1.26870i
$$315$$ 0 0
$$316$$ − 17.3205i − 0.0548117i
$$317$$ − 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
$$320$$ 284.000 0.887500
$$321$$ 0 0
$$322$$ 346.410i 1.07581i
$$323$$ −190.000 −0.588235
$$324$$ 0 0
$$325$$ 218.238i 0.671503i
$$326$$ − 294.449i − 0.903217i
$$327$$ 0 0
$$328$$ −300.000 −0.914634
$$329$$ −800.000 −2.43161
$$330$$ 0 0
$$331$$ − 173.205i − 0.523278i −0.965166 0.261639i $$-0.915737\pi$$
0.965166 0.261639i $$-0.0842630\pi$$
$$332$$ −70.0000 −0.210843
$$333$$ 0 0
$$334$$ 228.000 0.682635
$$335$$ 304.841i 0.909973i
$$336$$ 0 0
$$337$$ 339.482i 1.00736i 0.863889 + 0.503682i $$0.168022\pi$$
−0.863889 + 0.503682i $$0.831978\pi$$
$$338$$ 725.729i 2.14713i
$$339$$ 0 0
$$340$$ 40.0000 0.117647
$$341$$ − 173.205i − 0.507933i
$$342$$ 0 0
$$343$$ −20.0000 −0.0583090
$$344$$ 86.6025i 0.251752i
$$345$$ 0 0
$$346$$ −408.000 −1.17919
$$347$$ 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 $$0.455161\pi$$
$$350$$ − 155.885i − 0.445384i
$$351$$ 0 0
$$352$$ 155.885i 0.442854i
$$353$$ −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
$$356$$ 103.923i 0.291919i
$$357$$ 0 0
$$358$$ 180.000 0.502793
$$359$$ 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
$$362$$ 450.000 1.24309
$$363$$ 0 0
$$364$$ 242.487i 0.666173i
$$365$$ 40.0000 0.109589
$$366$$ 0 0
$$367$$ 170.000 0.463215 0.231608 0.972809i $$-0.425601\pi$$
0.231608 + 0.972809i $$0.425601\pi$$
$$368$$ −220.000 −0.597826
$$369$$ 0 0
$$370$$ 72.0000 0.194595
$$371$$ 415.692i 1.12046i
$$372$$ 0 0
$$373$$ − 356.802i − 0.956575i −0.878203 0.478287i $$-0.841258\pi$$
0.878203 0.478287i $$-0.158742\pi$$
$$374$$ − 173.205i − 0.463115i
$$375$$ 0 0
$$376$$ − 692.820i − 1.84261i
$$377$$ 840.000 2.22812
$$378$$ 0 0
$$379$$ 207.846i 0.548407i 0.961672 + 0.274203i $$0.0884141\pi$$
−0.961672 + 0.274203i $$0.911586\pi$$
$$380$$ −76.0000 −0.200000
$$381$$ 0 0
$$382$$ − 575.041i − 1.50534i
$$383$$ 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
$$386$$ −168.000 −0.435233
$$387$$ 0 0
$$388$$ 76.2102i 0.196418i
$$389$$ 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
$$392$$ − 441.673i − 1.12672i
$$393$$ 0 0
$$394$$ 277.128i 0.703371i
$$395$$ 69.2820i 0.175398i
$$396$$ 0 0
$$397$$ 650.000 1.63728 0.818640 0.574307i $$-0.194729\pi$$
0.818640 + 0.574307i $$0.194729\pi$$
$$398$$ − 169.741i − 0.426485i
$$399$$ 0 0
$$400$$ 99.0000 0.247500
$$401$$ 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
$$404$$ −100.000 −0.247525
$$405$$ 0 0
$$406$$ −600.000 −1.47783
$$407$$ 103.923i 0.255339i
$$408$$ 0 0
$$409$$ 173.205i 0.423484i 0.977326 + 0.211742i $$0.0679137\pi$$
−0.977326 + 0.211742i $$0.932086\pi$$
$$410$$ 240.000 0.585366
$$411$$ 0 0
$$412$$ 183.597i 0.445625i
$$413$$ 346.410i 0.838766i
$$414$$ 0 0
$$415$$ 280.000 0.674699
$$416$$ −378.000 −0.908654
$$417$$ 0 0
$$418$$ 329.090i 0.787296i
$$419$$ 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 $$-0.00654831\pi$$
$$422$$ 300.000 0.710900
$$423$$ 0 0
$$424$$ −360.000 −0.849057
$$425$$ 90.0000 0.211765
$$426$$ 0 0
$$427$$ 100.000 0.234192
$$428$$ − 62.3538i − 0.145687i
$$429$$ 0 0
$$430$$ − 69.2820i − 0.161121i
$$431$$ 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 $$-0.133778\pi$$
$$434$$ −300.000 −0.691244
$$435$$ 0 0
$$436$$ − 155.885i − 0.357533i
$$437$$ 380.000 0.869565
$$438$$ 0 0
$$439$$ − 121.244i − 0.276181i −0.990420 0.138091i $$-0.955903\pi$$
0.990420 0.138091i $$-0.0440965\pi$$
$$440$$ − 346.410i − 0.787296i
$$441$$ 0 0
$$442$$ 420.000 0.950226
$$443$$ 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
$$446$$ −138.000 −0.309417
$$447$$ 0 0
$$448$$ 710.000 1.58482
$$449$$ − 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
$$452$$ 6.92820i 0.0153279i
$$453$$ 0 0
$$454$$ 132.000 0.290749
$$455$$ − 969.948i − 2.13175i
$$456$$ 0 0
$$457$$ 290.000 0.634573 0.317287 0.948330i $$-0.397228\pi$$
0.317287 + 0.948330i $$0.397228\pi$$
$$458$$ − 190.526i − 0.415995i
$$459$$ 0 0
$$460$$ −80.0000 −0.173913
$$461$$ 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 $$-0.825300\pi$$
−0.853132 + 0.521696i $$0.825300\pi$$
$$464$$ − 381.051i − 0.821231i
$$465$$ 0 0
$$466$$ 329.090i 0.706201i
$$467$$ 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
$$470$$ 554.256i 1.17927i
$$471$$ 0 0
$$472$$ −300.000 −0.635593
$$473$$ 100.000 0.211416
$$474$$ 0 0
$$475$$ −171.000 −0.360000
$$476$$ 100.000 0.210084
$$477$$ 0 0
$$478$$ − 221.703i − 0.463813i
$$479$$ 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
$$482$$ 240.000 0.497925
$$483$$ 0 0
$$484$$ −21.0000 −0.0433884
$$485$$ − 304.841i − 0.628538i
$$486$$ 0 0
$$487$$ − 509.223i − 1.04563i −0.852445 0.522816i $$-0.824881\pi$$
0.852445 0.522816i $$-0.175119\pi$$
$$488$$ 86.6025i 0.177464i
$$489$$ 0 0
$$490$$ 353.338i 0.721099i
$$491$$ −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
$$494$$ −798.000 −1.61538
$$495$$ 0 0
$$496$$ − 190.526i − 0.384124i
$$497$$ 1039.23i 2.09101i
$$498$$ 0 0
$$499$$ 470.000 0.941884 0.470942 0.882164i $$-0.343914\pi$$
0.470942 + 0.882164i $$0.343914\pi$$
$$500$$ 136.000 0.272000
$$501$$ 0 0
$$502$$ − 3.46410i − 0.00690060i
$$503$$ −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
$$506$$ 346.410i 0.684605i
$$507$$ 0 0
$$508$$ − 114.315i − 0.225030i
$$509$$ − 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
$$512$$ 552.524i 1.07915i
$$513$$ 0 0
$$514$$ −852.000 −1.65759
$$515$$ − 734.390i − 1.42600i
$$516$$ 0 0
$$517$$ −800.000 −1.54739
$$518$$ 180.000 0.347490
$$519$$ 0 0
$$520$$ 840.000 1.61538
$$521$$ − 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 $$-0.727597\pi$$
0.655631 0.755081i $$-0.272403\pi$$
$$524$$ 38.0000 0.0725191
$$525$$ 0 0
$$526$$ − 346.410i − 0.658574i
$$527$$ − 173.205i − 0.328662i
$$528$$ 0 0
$$529$$ −129.000 −0.243856
$$530$$ 288.000 0.543396
$$531$$ 0 0
$$532$$ −190.000 −0.357143
$$533$$ −840.000 −1.57598
$$534$$ 0 0
$$535$$ 249.415i 0.466197i
$$536$$ −660.000 −1.23134
$$537$$ 0 0
$$538$$ 720.000 1.33829
$$539$$ −510.000 −0.946197
$$540$$ 0 0
$$541$$ 650.000 1.20148 0.600739 0.799445i $$-0.294873\pi$$
0.600739 + 0.799445i $$0.294873\pi$$
$$542$$ − 294.449i − 0.543263i
$$543$$ 0 0
$$544$$ 155.885i 0.286553i
$$545$$ 623.538i 1.14411i
$$546$$ 0 0
$$547$$ 595.825i 1.08926i 0.838676 + 0.544630i $$0.183330\pi$$
−0.838676 + 0.544630i $$0.816670\pi$$
$$548$$ −190.000 −0.346715
$$549$$ 0 0
$$550$$ − 155.885i − 0.283426i
$$551$$ 658.179i 1.19452i
$$552$$ 0 0
$$553$$ 173.205i 0.313210i
$$554$$ 17.3205i 0.0312645i
$$555$$ 0 0
$$556$$ 50.0000 0.0899281
$$557$$ 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
$$560$$ −440.000 −0.785714
$$561$$ 0 0
$$562$$ −660.000 −1.17438
$$563$$ 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
$$566$$ 121.244i 0.214211i
$$567$$ 0 0
$$568$$ −900.000 −1.58451
$$569$$ 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 $$-0.679368\pi$$
−0.534151 + 0.845389i $$0.679368\pi$$
$$572$$ 242.487i 0.423929i
$$573$$ 0 0
$$574$$ 600.000 1.04530
$$575$$ −180.000 −0.313043
$$576$$ 0 0
$$577$$ 170.000 0.294627 0.147314 0.989090i $$-0.452937\pi$$
0.147314 + 0.989090i $$0.452937\pi$$
$$578$$ 327.358i 0.566363i
$$579$$ 0 0
$$580$$ − 138.564i − 0.238904i
$$581$$ 700.000 1.20482
$$582$$ 0 0
$$583$$ 415.692i 0.713023i
$$584$$ 86.6025i 0.148292i
$$585$$ 0 0
$$586$$ −312.000 −0.532423
$$587$$ 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
$$590$$ 240.000 0.406780
$$591$$ 0 0
$$592$$ 114.315i 0.193100i
$$593$$ −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
$$596$$ 20.0000 0.0335570
$$597$$ 0 0
$$598$$ −840.000 −1.40468
$$599$$ − 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 $$-0.0460279\pi$$
$$602$$ − 173.205i − 0.287716i
$$603$$ 0 0
$$604$$ − 225.167i − 0.372792i
$$605$$ 84.0000 0.138843
$$606$$ 0 0
$$607$$ 703.213i 1.15851i 0.815148 + 0.579253i $$0.196656\pi$$
−0.815148 + 0.579253i $$0.803344\pi$$
$$608$$ − 296.181i − 0.487139i
$$609$$ 0 0
$$610$$ − 69.2820i − 0.113577i
$$611$$ − 1939.90i − 3.17495i
$$612$$ 0 0
$$613$$ 350.000 0.570962 0.285481 0.958384i $$-0.407847\pi$$
0.285481 + 0.958384i $$0.407847\pi$$
$$614$$ −252.000 −0.410423
$$615$$ 0 0
$$616$$ − 866.025i − 1.40589i
$$617$$ −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 $$-0.502571\pi$$
−0.00807754 + 0.999967i $$0.502571\pi$$
$$620$$ − 69.2820i − 0.111745i
$$621$$ 0 0
$$622$$ 1004.59i 1.61510i
$$623$$ − 1039.23i − 1.66811i
$$624$$ 0 0
$$625$$ −319.000 −0.510400
$$626$$ 640.859i 1.02374i
$$627$$ 0 0
$$628$$ 230.000 0.366242
$$629$$ 103.923i 0.165219i
$$630$$ 0 0
$$631$$ 350.000 0.554675 0.277338 0.960773i $$-0.410548\pi$$
0.277338 + 0.960773i $$0.410548\pi$$
$$632$$ −150.000 −0.237342
$$633$$ 0 0
$$634$$ −48.0000 −0.0757098
$$635$$ 457.261i 0.720097i
$$636$$ 0 0
$$637$$ − 1236.68i − 1.94142i
$$638$$ −600.000 −0.940439
$$639$$ 0 0
$$640$$ − 242.487i − 0.378886i
$$641$$ − 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 $$-0.331329\pi$$
0.505443 + 0.862860i $$0.331329\pi$$
$$644$$ −200.000 −0.310559
$$645$$ 0 0
$$646$$ 329.090i 0.509427i
$$647$$ −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
$$650$$ 378.000 0.581538
$$651$$ 0 0
$$652$$ 170.000 0.260736
$$653$$ 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
$$656$$ 381.051i 0.580871i
$$657$$ 0 0
$$658$$ 1385.64i 2.10584i
$$659$$ 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 $$-0.0974340\pi$$
$$662$$ −300.000 −0.453172
$$663$$ 0 0
$$664$$ 606.218i 0.912979i
$$665$$ 760.000 1.14286
$$666$$ 0 0
$$667$$ 692.820i 1.03871i
$$668$$ 131.636i 0.197060i
$$669$$ 0 0
$$670$$ 528.000 0.788060
$$671$$ 100.000 0.149031
$$672$$ 0 0
$$673$$ − 630.466i − 0.936800i −0.883516 0.468400i $$-0.844831\pi$$
0.883516 0.468400i $$-0.155169\pi$$
$$674$$ 588.000 0.872404
$$675$$ 0 0
$$676$$ −419.000 −0.619822
$$677$$ − 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
$$680$$ − 346.410i − 0.509427i
$$681$$ 0 0
$$682$$ −300.000 −0.439883
$$683$$ 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
$$686$$ 34.6410i 0.0504971i
$$687$$ 0 0
$$688$$ 110.000 0.159884
$$689$$ −1008.00 −1.46299
$$690$$ 0 0
$$691$$ 470.000 0.680174 0.340087 0.940394i $$-0.389544\pi$$
0.340087 + 0.940394i $$0.389544\pi$$
$$692$$ − 235.559i − 0.340403i
$$693$$ 0 0
$$694$$ − 1021.91i − 1.47249i
$$695$$ −200.000 −0.287770
$$696$$ 0 0
$$697$$ 346.410i 0.497002i
$$698$$ − 169.741i − 0.243182i
$$699$$ 0 0
$$700$$ 90.0000 0.128571
$$701$$ 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
$$704$$ 710.000 1.00852
$$705$$ 0 0
$$706$$ 329.090i 0.466133i
$$707$$ 1000.00 1.41443
$$708$$ 0 0
$$709$$ −982.000 −1.38505 −0.692525 0.721394i $$-0.743502\pi$$
−0.692525 + 0.721394i $$0.743502\pi$$
$$710$$ 720.000 1.01408
$$711$$ 0 0
$$712$$ 900.000 1.26404
$$713$$ 346.410i 0.485849i
$$714$$ 0 0
$$715$$ − 969.948i − 1.35657i
$$716$$ 103.923i 0.145144i
$$717$$ 0 0
$$718$$ − 346.410i − 0.482465i
$$719$$ −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
$$722$$ − 625.270i − 0.866025i
$$723$$ 0 0
$$724$$ 259.808i 0.358850i
$$725$$ − 311.769i − 0.430026i
$$726$$ 0 0
$$727$$ −790.000 −1.08666 −0.543329 0.839520i $$-0.682836\pi$$
−0.543329 + 0.839520i $$0.682836\pi$$
$$728$$ 2100.00 2.88462
$$729$$ 0 0
$$730$$ − 69.2820i − 0.0949069i
$$731$$ 100.000 0.136799
$$732$$ 0 0
$$733$$ −1150.00 −1.56889 −0.784447 0.620195i $$-0.787053\pi$$
−0.784447 + 0.620195i $$0.787053\pi$$
$$734$$ − 294.449i − 0.401156i
$$735$$ 0 0
$$736$$ − 311.769i − 0.423599i
$$737$$ 762.102i 1.03406i
$$738$$ 0 0
$$739$$ 578.000 0.782138 0.391069 0.920361i $$-0.372105\pi$$
0.391069 + 0.920361i $$0.372105\pi$$
$$740$$ 41.5692i 0.0561746i
$$741$$ 0 0
$$742$$ 720.000 0.970350
$$743$$ 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
$$746$$ −618.000 −0.828418
$$747$$ 0 0
$$748$$ 100.000 0.133690
$$749$$ 623.538i 0.832494i
$$750$$ 0 0
$$751$$ 952.628i 1.26848i 0.773137 + 0.634240i $$0.218687\pi$$
−0.773137 + 0.634240i $$0.781313\pi$$
$$752$$ −880.000 −1.17021
$$753$$ 0 0
$$754$$ − 1454.92i − 1.92961i
$$755$$ 900.666i 1.19294i
$$756$$ 0 0
$$757$$ −250.000 −0.330251 −0.165125 0.986273i $$-0.552803\pi$$
−0.165125 + 0.986273i $$0.552803\pi$$
$$758$$ 360.000 0.474934
$$759$$ 0 0
$$760$$ 658.179i 0.866025i
$$761$$ 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
$$764$$ 332.000 0.434555
$$765$$ 0 0
$$766$$ 1092.00 1.42559
$$767$$ −840.000 −1.09518
$$768$$ 0 0
$$769$$ 110.000 0.143043 0.0715215 0.997439i $$-0.477215\pi$$
0.0715215 + 0.997439i $$0.477215\pi$$
$$770$$ 692.820i 0.899767i
$$771$$ 0 0
$$772$$ − 96.9948i − 0.125641i
$$773$$ 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
$$776$$ 660.000 0.850515
$$777$$ 0 0
$$778$$ − 221.703i − 0.284965i
$$779$$ − 658.179i − 0.844903i
$$780$$ 0 0
$$781$$ 1039.23i 1.33064i
$$782$$ 346.410i 0.442980i
$$783$$ 0 0
$$784$$ −561.000 −0.715561
$$785$$ −920.000 −1.17197
$$786$$ 0 0
$$787$$ − 96.9948i − 0.123246i −0.998099 0.0616232i $$-0.980372\pi$$
0.998099 0.0616232i $$-0.0196277\pi$$
$$788$$ −160.000 −0.203046
$$789$$ 0 0
$$790$$ 120.000 0.151899
$$791$$ − 69.2820i − 0.0875879i
$$792$$ 0 0
$$793$$ 242.487i 0.305785i
$$794$$ − 1125.83i − 1.41793i
$$795$$ 0 0
$$796$$ 98.0000 0.123116
$$797$$ 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
$$800$$ 140.296i 0.175370i
$$801$$ 0 0
$$802$$ 300.000 0.374065
$$803$$ 100.000 0.124533
$$804$$ 0 0
$$805$$ 800.000 0.993789
$$806$$ − 727.461i − 0.902557i
$$807$$ 0 0
$$808$$ 866.025i 1.07181i
$$809$$ 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 $$-0.828694\pi$$
0.858647 0.512567i $$-0.171306\pi$$
$$812$$ − 346.410i − 0.426613i
$$813$$ 0 0
$$814$$ 180.000 0.221130
$$815$$ −680.000 −0.834356
$$816$$ 0 0
$$817$$ −190.000 −0.232558
$$818$$ 300.000 0.366748
$$819$$ 0 0
$$820$$ 138.564i 0.168981i
$$821$$ 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 $$-0.304162\pi$$
0.577157 + 0.816633i $$0.304162\pi$$
$$824$$ 1590.00 1.92961
$$825$$ 0 0
$$826$$ 600.000 0.726392
$$827$$ 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 $$0.743767\pi$$
$$830$$ − 484.974i − 0.584306i
$$831$$ 0 0
$$832$$ 1721.66i 2.06930i
$$833$$ −510.000 −0.612245
$$834$$ 0 0
$$835$$ − 526.543i − 0.630591i
$$836$$ −190.000 −0.227273
$$837$$ 0 0
$$838$$ − 65.8179i − 0.0785417i
$$839$$ 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
$$842$$ −30.0000 −0.0356295
$$843$$ 0 0
$$844$$ 173.205i 0.205219i
$$845$$ 1676.00 1.98343
$$846$$ 0 0
$$847$$ 210.000 0.247934
$$848$$ 457.261i 0.539223i
$$849$$ 0 0
$$850$$ − 155.885i − 0.183394i
$$851$$ − 207.846i − 0.244237i
$$852$$ 0 0
$$853$$ 890.000 1.04338 0.521688 0.853136i $$-0.325302\pi$$
0.521688 + 0.853136i $$0.325302\pi$$
$$854$$ − 173.205i − 0.202816i
$$855$$ 0 0
$$856$$ −540.000 −0.630841
$$857$$ − 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 $$-0.466216\pi$$
0.105937 + 0.994373i $$0.466216\pi$$
$$860$$ 40.0000 0.0465116
$$861$$ 0 0
$$862$$ 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
$$866$$ −612.000 −0.706697
$$867$$ 0 0
$$868$$ − 173.205i − 0.199545i
$$869$$ 173.205i 0.199315i
$$870$$ 0 0
$$871$$ −1848.00 −2.12170
$$872$$ −1350.00 −1.54817
$$873$$ 0 0
$$874$$ − 658.179i − 0.753066i
$$875$$ −1360.00 −1.55429
$$876$$ 0 0
$$877$$ − 1188.19i − 1.35483i −0.735601 0.677416i $$-0.763100\pi$$
0.735601 0.677416i $$-0.236900\pi$$
$$878$$ −210.000 −0.239180
$$879$$ 0 0
$$880$$ −440.000 −0.500000
$$881$$ −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 $$-0.806619\pi$$
−0.821065 + 0.570835i $$0.806619\pi$$
$$884$$ 242.487i 0.274307i
$$885$$ 0 0
$$886$$ − 190.526i − 0.215040i
$$887$$ 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
$$890$$ −720.000 −0.808989
$$891$$ 0 0
$$892$$ − 79.6743i − 0.0893210i
$$893$$ 1520.00 1.70213
$$894$$ 0 0
$$895$$ − 415.692i − 0.464461i
$$896$$ − 606.218i − 0.676582i
$$897$$ 0 0
$$898$$ −540.000 −0.601336
$$899$$ −600.000 −0.667408
$$900$$ 0 0
$$901$$ 415.692i 0.461368i
$$902$$ 600.000 0.665188
$$903$$ 0 0
$$904$$ 60.0000 0.0663717
$$905$$ − 1039.23i − 1.14832i
$$906$$ 0 0
$$907$$ − 110.851i − 0.122217i −0.998131 0.0611087i $$-0.980536\pi$$
0.998131 0.0611087i $$-0.0194636\pi$$
$$908$$ 76.2102i 0.0839320i
$$909$$ 0 0
$$910$$ −1680.00 −1.84615
$$911$$ − 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
$$914$$ − 502.295i − 0.549557i
$$915$$ 0 0
$$916$$ 110.000 0.120087
$$917$$ −380.000 −0.414395
$$918$$ 0 0
$$919$$ 62.0000 0.0674646 0.0337323 0.999431i $$-0.489261\pi$$
0.0337323 + 0.999431i $$0.489261\pi$$
$$920$$ 692.820i 0.753066i
$$921$$ 0 0
$$922$$ − 1260.93i − 1.36761i
$$923$$ −2520.00 −2.73023
$$924$$ 0 0
$$925$$ 93.5307i 0.101114i
$$926$$ 1368.32i 1.47767i
$$927$$ 0 0
$$928$$ 540.000 0.581897
$$929$$ 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
$$932$$ −190.000 −0.203863
$$933$$ 0 0
$$934$$ − 917.987i − 0.982855i
$$935$$ −400.000 −0.427807
$$936$$ 0 0
$$937$$ 110.000 0.117396 0.0586980 0.998276i $$-0.481305\pi$$
0.0586980 + 0.998276i $$0.481305\pi$$
$$938$$ 1320.00 1.40725
$$939$$ 0 0
$$940$$ −320.000 −0.340426
$$941$$ 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
$$944$$ 381.051i 0.403656i
$$945$$ 0 0
$$946$$ − 173.205i − 0.183092i
$$947$$ −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
$$950$$ 296.181i 0.311769i
$$951$$ 0 0
$$952$$ − 866.025i − 0.909691i
$$953$$ − 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
$$956$$ 128.000 0.133891
$$957$$ 0 0
$$958$$ − 138.564i − 0.144639i
$$959$$ 1900.00 1.98123
$$960$$ 0 0
$$961$$ 661.000 0.687825
$$962$$ 436.477i 0.453718i
$$963$$ 0 0
$$964$$ 138.564i 0.143739i
$$965$$ 387.979i 0.402051i
$$966$$ 0 0
$$967$$ 470.000 0.486039 0.243020 0.970021i $$-0.421862\pi$$
0.243020 + 0.970021i $$0.421862\pi$$
$$968$$ 181.865i 0.187877i
$$969$$ 0 0
$$970$$ −528.000 −0.544330
$$971$$ 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
$$974$$ −882.000 −0.905544
$$975$$ 0 0
$$976$$ 110.000 0.112705
$$977$$ 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
$$980$$ −204.000 −0.208163
$$981$$ 0 0
$$982$$ 723.997i 0.737268i
$$983$$ − 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
$$986$$ −600.000 −0.608519
$$987$$ 0 0
$$988$$ − 460.726i − 0.466321i
$$989$$ −200.000 −0.202224
$$990$$ 0 0
$$991$$ 571.577i 0.576768i 0.957515 + 0.288384i $$0.0931179\pi$$
−0.957515 + 0.288384i $$0.906882\pi$$
$$992$$ 270.000 0.272177
$$993$$ 0 0
$$994$$ 1800.00 1.81087
$$995$$ −392.000 −0.393970
$$996$$ 0 0
$$997$$ 1550.00 1.55466 0.777332 0.629091i $$-0.216573\pi$$
0.777332 + 0.629091i $$0.216573\pi$$
$$998$$ − 814.064i − 0.815695i
$$999$$ 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 171.3.c.c.37.1 2
3.2 odd 2 57.3.c.a.37.2 yes 2
4.3 odd 2 2736.3.o.e.721.1 2
12.11 even 2 912.3.o.a.721.1 2
19.18 odd 2 inner 171.3.c.c.37.2 2
57.56 even 2 57.3.c.a.37.1 2
76.75 even 2 2736.3.o.e.721.2 2
228.227 odd 2 912.3.o.a.721.2 2

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