LMFDB - Embedded newform 171.3.c.c.37.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \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	$0.142549\pi$
$2$	1.73205i	0.866025i	−0.901388	+	0.433013i	$0.857451\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.250000
$5$	−4.00000	−0.800000	−0.400000	−	0.916515i	$-0.630990\pi$
$5$	−4.00000	−0.800000	−0.400000	+	0.916515i	$0.630990\pi$
$6$	0	0
$7$	−10.0000	−1.42857	−0.714286	−	0.699854i	$-0.753248\pi$
$7$	−10.0000	−1.42857	−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	$-0.650198\pi$
$11$	−10.0000	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	24.2487i	1.86529i	0.360801	+	0.932643i	$0.382503\pi$
$13$	24.2487i	1.86529i	−0.360801	+	0.932643i	$0.617497\pi$
$14$	− 17.3205i	− 1.23718i
$15$	0	0
$16$	−11.0000	−0.687500
$17$	−10.0000	−0.588235	−0.294118	−	0.955769i	$-0.595026\pi$
$17$	−10.0000	−0.588235	−0.294118	+	0.955769i	$0.595026\pi$
$18$	0	0
$19$	19.0000	1.00000
$19$	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	$-0.356825\pi$
$23$	20.0000	0.869565	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	$-0.796256\pi$
$29$	− 34.6410i	− 1.19452i	0.802049	−	0.597259i	$-0.203744\pi$
$30$	0	0
$31$	− 17.3205i	− 0.558726i	−0.960186	−	0.279363i	$-0.909877\pi$
$31$	− 17.3205i	− 0.558726i	0.960186	−	0.279363i	$-0.0901233\pi$
$32$	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	$0.0448506\pi$
$37$	10.3923i	0.280873i	−0.990090	+	0.140437i	$0.955149\pi$
$38$	32.9090i	0.866025i
$39$	0	0
$40$	− 34.6410i	− 0.866025i
$41$	34.6410i	0.844903i	0.906386	+	0.422451i	$0.138830\pi$
$41$	34.6410i	0.844903i	−0.906386	+	0.422451i	$0.861170\pi$
$42$	0	0
$43$	−10.0000	−0.232558	−0.116279	−	0.993217i	$-0.537097\pi$
$43$	−10.0000	−0.232558	−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	$-0.175958\pi$
$47$	80.0000	1.70213	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	$0.128273\pi$
$53$	41.5692i	0.784325i	−0.919896	+	0.392162i	$0.871727\pi$
$54$	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	$0.0948427\pi$
$59$	34.6410i	0.587136i	−0.955938	+	0.293568i	$0.905157\pi$
$60$	0	0
$61$	−10.0000	−0.163934	−0.0819672	−	0.996635i	$-0.526120\pi$
$61$	−10.0000	−0.163934	−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	$0.192566\pi$
$67$	76.2102i	1.13747i	−0.822522	+	0.568733i	$0.807434\pi$
$68$	−10.0000	−0.147059
$69$	0	0
$70$	69.2820i	0.989743i
$71$	103.923i	1.46370i	0.681463	+	0.731852i	$0.261344\pi$
$71$	103.923i	1.46370i	−0.681463	+	0.731852i	$0.738656\pi$
$72$	0	0
$73$	−10.0000	−0.136986	−0.0684932	−	0.997652i	$-0.521819\pi$
$73$	−10.0000	−0.136986	−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	$0.0349645\pi$
$79$	17.3205i	0.219247i	−0.993973	+	0.109623i	$0.965035\pi$
$80$	44.0000	0.550000
$81$	0	0
$82$	−60.0000	−0.731707
$83$	−70.0000	−0.843373	−0.421687	−	0.906742i	$-0.638562\pi$
$83$	−70.0000	−0.843373	−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	$-0.801551\pi$
$89$	− 103.923i	− 1.16767i	0.811871	−	0.583837i	$-0.198449\pi$
$90$	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	$-0.871494\pi$
$97$	− 76.2102i	− 0.785673i	0.919608	−	0.392836i	$-0.128506\pi$
$98$	88.3346i	0.901373i
$99$	0	0
$100$	−9.00000	−0.0900000
$101$	−100.000	−0.990099	−0.495050	−	0.868865i	$-0.664850\pi$
$101$	−100.000	−0.990099	−0.495050	+	0.868865i	$0.664850\pi$
$102$	0	0
$103$	− 183.597i	− 1.78250i	−0.453513	−	0.891249i	$-0.649830\pi$
$103$	− 183.597i	− 1.78250i	0.453513	−	0.891249i	$-0.350170\pi$
$104$	−210.000	−2.01923
$105$	0	0
$106$	−72.0000	−0.679245
$107$	62.3538i	0.582746i	0.956610	+	0.291373i	$0.0941121\pi$
$107$	62.3538i	0.582746i	−0.956610	+	0.291373i	$0.905888\pi$
$108$	0	0
$109$	155.885i	1.43013i	0.699056	+	0.715067i	$0.253604\pi$
$109$	155.885i	1.43013i	−0.699056	+	0.715067i	$0.746396\pi$
$110$	69.2820i	0.629837i
$111$	0	0
$112$	110.000	0.982143
$113$	− 6.92820i	− 0.0613115i	−0.999530	−	0.0306558i	$-0.990240\pi$
$113$	− 6.92820i	− 0.0613115i	0.999530	−	0.0306558i	$-0.00975956\pi$
$114$	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	$0.148598\pi$
$127$	114.315i	0.900121i	−0.892998	+	0.450060i	$0.851402\pi$
$128$	− 60.6218i	− 0.473608i
$129$	0	0
$130$	168.000	1.29231
$131$	38.0000	0.290076	0.145038	−	0.989426i	$-0.453670\pi$
$131$	38.0000	0.290076	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	$-0.743902\pi$
$137$	−190.000	−1.38686	−0.693431	+	0.720523i	$0.743902\pi$
$138$	0	0
$139$	50.0000	0.359712	0.179856	−	0.983693i	$-0.442437\pi$
$139$	50.0000	0.359712	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	$-0.478621\pi$
$149$	20.0000	0.134228	0.0671141	+	0.997745i	$0.478621\pi$
$150$	0	0
$151$	225.167i	1.49117i	0.666411	+	0.745585i	$0.267830\pi$
$151$	225.167i	1.49117i	−0.666411	+	0.745585i	$0.732170\pi$
$152$	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	$-0.238361\pi$
$157$	230.000	1.46497	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	$-0.325383\pi$
$163$	170.000	1.04294	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	$-0.871050\pi$
$167$	− 131.636i	− 0.788239i	0.919059	−	0.394119i	$-0.128950\pi$
$168$	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	$0.238370\pi$
$173$	235.559i	1.36161i	−0.732464	+	0.680806i	$0.761630\pi$
$174$	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	$-0.906249\pi$
$179$	− 103.923i	− 0.580576i	0.956939	−	0.290288i	$-0.0937511\pi$
$180$	0	0
$181$	− 259.808i	− 1.43540i	−0.696352	−	0.717701i	$-0.745195\pi$
$181$	− 259.808i	− 1.43540i	0.696352	−	0.717701i	$-0.254805\pi$
$182$	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	$-0.164692\pi$
$191$	332.000	1.73822	0.869110	+	0.494619i	$0.164692\pi$
$192$	0	0
$193$	96.9948i	0.502564i	0.967914	+	0.251282i	$0.0808521\pi$
$193$	96.9948i	0.502564i	−0.967914	+	0.251282i	$0.919148\pi$
$194$	132.000	0.680412
$195$	0	0
$196$	51.0000	0.260204
$197$	−160.000	−0.812183	−0.406091	−	0.913832i	$-0.633109\pi$
$197$	−160.000	−0.812183	−0.406091	+	0.913832i	$0.633109\pi$
$198$	0	0
$199$	98.0000	0.492462	0.246231	−	0.969211i	$-0.420808\pi$
$199$	98.0000	0.492462	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	$-0.865376\pi$
$211$	− 173.205i	− 0.820877i	0.911888	−	0.410439i	$-0.134624\pi$
$212$	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	$0.0571704\pi$
$223$	79.6743i	0.357284i	−0.983914	+	0.178642i	$0.942830\pi$
$224$	− 155.885i	− 0.695913i
$225$	0	0
$226$	12.0000	0.0530973
$227$	− 76.2102i	− 0.335728i	−0.985810	−	0.167864i	$-0.946313\pi$
$227$	− 76.2102i	− 0.335728i	0.985810	−	0.167864i	$-0.0536869\pi$
$228$	0	0
$229$	110.000	0.480349	0.240175	−	0.970730i	$-0.422795\pi$
$229$	110.000	0.480349	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	$-0.633678\pi$
$233$	−190.000	−0.815451	−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	$-0.413709\pi$
$239$	128.000	0.535565	0.267782	+	0.963479i	$0.413709\pi$
$240$	0	0
$241$	− 138.564i	− 0.574955i	−0.957787	−	0.287477i	$-0.907183\pi$
$241$	− 138.564i	− 0.574955i	0.957787	−	0.287477i	$-0.0928166\pi$
$242$	− 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	$-0.498732\pi$
$251$	2.00000	0.00796813	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	$0.406325\pi$
$257$	491.902i	1.91402i	−0.290059	+	0.957009i	$0.593675\pi$
$258$	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	$-0.375845\pi$
$263$	200.000	0.760456	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	$-0.718925\pi$
$269$	− 415.692i	− 1.54532i	0.634818	−	0.772662i	$-0.281075\pi$
$270$	0	0
$271$	170.000	0.627306	0.313653	−	0.949538i	$-0.398447\pi$
$271$	170.000	0.627306	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	$-0.505746\pi$
$277$	−10.0000	−0.0361011	−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	$0.237165\pi$
$281$	381.051i	1.35605i	−0.735037	+	0.678027i	$0.762835\pi$
$282$	0	0
$283$	−70.0000	−0.247350	−0.123675	−	0.992323i	$-0.539468\pi$
$283$	−70.0000	−0.247350	−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	$0.0994572\pi$
$293$	180.133i	0.614789i	−0.951582	+	0.307395i	$0.900543\pi$
$294$	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	$0.0761504\pi$
$307$	145.492i	0.473916i	−0.971520	+	0.236958i	$0.923850\pi$
$308$	100.000	0.324675
$309$	0	0
$310$	−120.000	−0.387097
$311$	−580.000	−1.86495	−0.932476	−	0.361232i	$-0.882356\pi$
$311$	−580.000	−1.86495	−0.932476	+	0.361232i	$0.882356\pi$
$312$	0	0
$313$	−370.000	−1.18211	−0.591054	−	0.806632i	$-0.701288\pi$
$313$	−370.000	−1.18211	−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	$0.0139181\pi$
$317$	27.7128i	0.0874221i	−0.999044	+	0.0437111i	$0.986082\pi$
$318$	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	$0.0842630\pi$
$331$	173.205i	0.523278i	−0.965166	+	0.261639i	$0.915737\pi$
$332$	−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	$-0.831978\pi$
$337$	− 339.482i	− 1.00736i	0.863889	−	0.503682i	$-0.168022\pi$
$338$	− 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	$-0.176515\pi$
$347$	590.000	1.70029	0.850144	+	0.526550i	$0.176515\pi$
$348$	0	0
$349$	98.0000	0.280802	0.140401	−	0.990095i	$-0.455161\pi$
$349$	98.0000	0.280802	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	$-0.586733\pi$
$353$	−190.000	−0.538244	−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	$-0.410146\pi$
$359$	200.000	0.557103	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	$-0.425601\pi$
$367$	170.000	0.463215	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	$0.158742\pi$
$373$	356.802i	0.956575i	−0.878203	+	0.478287i	$0.841258\pi$
$374$	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	$-0.911586\pi$
$379$	− 207.846i	− 0.548407i	0.961672	−	0.274203i	$-0.0884141\pi$
$380$	−76.0000	−0.200000
$381$	0	0
$382$	575.041i	1.50534i
$383$	− 630.466i	− 1.64613i	−0.567950	−	0.823063i	$-0.692263\pi$
$383$	− 630.466i	− 1.64613i	0.567950	−	0.823063i	$-0.307737\pi$
$384$	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	$-0.447391\pi$
$389$	128.000	0.329049	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	$-0.194729\pi$
$397$	650.000	1.63728	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	$-0.930710\pi$
$401$	− 173.205i	− 0.431933i	0.976401	−	0.215966i	$-0.0692902\pi$
$402$	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	$-0.932086\pi$
$409$	− 173.205i	− 0.423484i	0.977326	−	0.211742i	$-0.0679137\pi$
$410$	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	$-0.485561\pi$
$419$	38.0000	0.0906921	0.0453461	+	0.998971i	$0.485561\pi$
$420$	0	0
$421$	17.3205i	0.0411413i	0.999788	+	0.0205707i	$0.00654831\pi$
$421$	17.3205i	0.0411413i	−0.999788	+	0.0205707i	$0.993452\pi$
$422$	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			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	353.338i	0.816024i	0.912977	+	0.408012i	$0.133778\pi$
$433$	353.338i	0.816024i	−0.912977	+	0.408012i	$0.866222\pi$
$434$	−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	$0.0440965\pi$
$439$	121.244i	0.276181i	−0.990420	+	0.138091i	$0.955903\pi$
$440$	346.410i	0.787296i
$441$	0	0
$442$	420.000	0.950226
$443$	110.000	0.248307	0.124153	−	0.992263i	$-0.460378\pi$
$443$	110.000	0.248307	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	$0.112861\pi$
$449$	311.769i	0.694363i	−0.937798	+	0.347182i	$0.887139\pi$
$450$	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	$-0.397228\pi$
$457$	290.000	0.634573	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	$-0.210294\pi$
$461$	728.000	1.57918	0.789588	+	0.613638i	$0.210294\pi$
$462$	0	0
$463$	−790.000	−1.70626	−0.853132	−	0.521696i	$-0.825300\pi$
$463$	−790.000	−1.70626	−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	$-0.307929\pi$
$467$	530.000	1.13490	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	$-0.473388\pi$
$479$	80.0000	0.167015	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	$0.175119\pi$
$487$	509.223i	1.04563i	−0.852445	+	0.522816i	$0.824881\pi$
$488$	− 86.6025i	− 0.177464i
$489$	0	0
$490$	− 353.338i	− 0.721099i
$491$	−418.000	−0.851324	−0.425662	−	0.904882i	$-0.639959\pi$
$491$	−418.000	−0.851324	−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	$-0.343914\pi$
$499$	470.000	0.941884	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	$-0.531693\pi$
$503$	−100.000	−0.198807	−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	$0.145862\pi$
$509$	450.333i	0.884741i	−0.896832	+	0.442371i	$0.854138\pi$
$510$	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	$0.0967207\pi$
$521$	311.769i	0.598405i	−0.954190	+	0.299203i	$0.903279\pi$
$522$	0	0
$523$	789.815i	1.51016i	0.655631	+	0.755081i	$0.272403\pi$
$523$	789.815i	1.51016i	−0.655631	+	0.755081i	$0.727597\pi$
$524$	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	$-0.294873\pi$
$541$	650.000	1.20148	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	$-0.816670\pi$
$547$	− 595.825i	− 1.08926i	0.838676	−	0.544630i	$-0.183330\pi$
$548$	−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	$-0.477121\pi$
$557$	80.0000	0.143627	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	$-0.902515\pi$
$563$	− 339.482i	− 0.602987i	0.953468	−	0.301494i	$-0.0974852\pi$
$564$	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	$-0.803691\pi$
$569$	− 658.179i	− 1.15673i	0.815778	−	0.578365i	$-0.196309\pi$
$570$	0	0
$571$	−610.000	−1.06830	−0.534151	−	0.845389i	$-0.679368\pi$
$571$	−610.000	−1.06830	−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	$-0.452937\pi$
$577$	170.000	0.294627	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	$-0.313230\pi$
$587$	650.000	1.10733	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	$-0.778393\pi$
$593$	−910.000	−1.53457	−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	$0.00920544\pi$
$599$	34.6410i	0.0578314i	−0.999582	+	0.0289157i	$0.990795\pi$
$600$	0	0
$601$	173.205i	0.288195i	0.989564	+	0.144097i	$0.0460279\pi$
$601$	173.205i	0.288195i	−0.989564	+	0.144097i	$0.953972\pi$
$602$	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	$-0.803344\pi$
$607$	− 703.213i	− 1.15851i	0.815148	−	0.579253i	$-0.196656\pi$
$608$	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	$-0.407847\pi$
$613$	350.000	0.570962	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	$-0.664586\pi$
$617$	−610.000	−0.988655	−0.494327	+	0.869276i	$0.664586\pi$
$618$	0	0
$619$	−10.0000	−0.0161551	−0.00807754	−	0.999967i	$-0.502571\pi$
$619$	−10.0000	−0.0161551	−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	$-0.410548\pi$
$631$	350.000	0.554675	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	$0.151921\pi$
$641$	588.897i	0.918716i	−0.888251	+	0.459358i	$0.848079\pi$
$642$	0	0
$643$	650.000	1.01089	0.505443	−	0.862860i	$-0.331329\pi$
$643$	650.000	1.01089	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	$-0.718462\pi$
$647$	−820.000	−1.26739	−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	$-0.358940\pi$
$653$	560.000	0.857580	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	$-0.889004\pi$
$659$	− 450.333i	− 0.683358i	0.939817	−	0.341679i	$-0.110996\pi$
$660$	0	0
$661$	398.372i	0.602680i	0.953517	+	0.301340i	$0.0974340\pi$
$661$	398.372i	0.602680i	−0.953517	+	0.301340i	$0.902566\pi$
$662$	−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	$0.155169\pi$
$673$	630.466i	0.936800i	−0.883516	+	0.468400i	$0.844831\pi$
$674$	588.000	0.872404
$675$	0	0
$676$	−419.000	−0.619822
$677$	526.543i	0.777760i	0.921288	+	0.388880i	$0.127138\pi$
$677$	526.543i	0.777760i	−0.921288	+	0.388880i	$0.872862\pi$
$678$	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	$-0.886195\pi$
$683$	− 478.046i	− 0.699921i	0.936764	−	0.349960i	$-0.113805\pi$
$684$	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	$-0.389544\pi$
$691$	470.000	0.680174	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	$-0.369208\pi$
$701$	560.000	0.798859	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	$-0.743502\pi$
$709$	−982.000	−1.38505	−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	$-0.617774\pi$
$719$	−520.000	−0.723227	−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	$-0.682836\pi$
$727$	−790.000	−1.08666	−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	$-0.787053\pi$
$733$	−1150.00	−1.56889	−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	$-0.372105\pi$
$739$	578.000	0.782138	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	$-0.949328\pi$
$743$	− 235.559i	− 0.317038i	0.987356	−	0.158519i	$-0.0506718\pi$
$744$	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	$-0.781313\pi$
$751$	− 952.628i	− 1.26848i	0.773137	−	0.634240i	$-0.218687\pi$
$752$	−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	$-0.552803\pi$
$757$	−250.000	−0.330251	−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	$-0.331156\pi$
$761$	770.000	1.01183	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	$-0.477215\pi$
$769$	110.000	0.143043	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	$-0.970000\pi$
$773$	− 145.492i	− 0.188218i	0.995562	−	0.0941088i	$-0.0300002\pi$
$774$	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	$0.0196277\pi$
$787$	96.9948i	0.123246i	−0.998099	+	0.0616232i	$0.980372\pi$
$788$	−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	$-0.931685\pi$
$797$	− 339.482i	− 0.425950i	0.977058	−	0.212975i	$-0.0683153\pi$
$798$	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	$-0.464119\pi$
$809$	182.000	0.224969	0.112485	+	0.993653i	$0.464119\pi$
$810$	0	0
$811$	831.384i	1.02513i	0.858647	+	0.512567i	$0.171306\pi$
$811$	831.384i	1.02513i	−0.858647	+	0.512567i	$0.828694\pi$
$812$	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	$-0.498449\pi$
$821$	8.00000	0.00974421	0.00487211	+	0.999988i	$0.498449\pi$
$822$	0	0
$823$	950.000	1.15431	0.577157	−	0.816633i	$-0.304162\pi$
$823$	950.000	1.15431	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	$-0.906669\pi$
$827$	− 478.046i	− 0.578048i	0.957322	−	0.289024i	$-0.0933308\pi$
$828$	0	0
$829$	− 1195.12i	− 1.44163i	−0.693125	−	0.720817i	$-0.743767\pi$
$829$	− 1195.12i	− 1.44163i	0.693125	−	0.720817i	$-0.256233\pi$
$830$	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	$-0.752334\pi$
$839$	− 1177.79i	− 1.40381i	0.712272	−	0.701904i	$-0.247666\pi$
$840$	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	$-0.325302\pi$
$853$	890.000	1.04338	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	$0.261238\pi$
$857$	1254.00i	1.46325i	−0.681708	+	0.731625i	$0.738762\pi$
$858$	0	0
$859$	182.000	0.211874	0.105937	−	0.994373i	$-0.466216\pi$
$859$	182.000	0.211874	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	$0.215386\pi$
$863$	1080.80i	1.25238i	−0.779672	+	0.626188i	$0.784614\pi$
$864$	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	$0.236900\pi$
$877$	1188.19i	1.35483i	−0.735601	+	0.677416i	$0.763100\pi$
$878$	−210.000	−0.239180
$879$	0	0
$880$	−440.000	−0.500000
$881$	−550.000	−0.624291	−0.312145	−	0.950034i	$-0.601048\pi$
$881$	−550.000	−0.624291	−0.312145	+	0.950034i	$0.601048\pi$
$882$	0	0
$883$	−1450.00	−1.64213	−0.821065	−	0.570835i	$-0.806619\pi$
$883$	−1450.00	−1.64213	−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	$-0.750102\pi$
$887$	− 1254.00i	− 1.41376i	0.707334	−	0.706880i	$-0.249898\pi$
$888$	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	$0.0194636\pi$
$907$	110.851i	0.122217i	−0.998131	+	0.0611087i	$0.980536\pi$
$908$	− 76.2102i	− 0.0839320i
$909$	0	0
$910$	−1680.00	−1.84615
$911$	796.743i	0.874581i	0.899320	+	0.437291i	$0.144062\pi$
$911$	796.743i	0.874581i	−0.899320	+	0.437291i	$0.855938\pi$
$912$	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	$-0.489261\pi$
$919$	62.0000	0.0674646	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	$-0.458423\pi$
$929$	242.000	0.260495	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	$-0.481305\pi$
$937$	110.000	0.117396	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	$-0.860854\pi$
$941$	− 796.743i	− 0.846699i	0.905967	−	0.423349i	$-0.139146\pi$
$942$	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	$-0.777546\pi$
$947$	−1450.00	−1.53115	−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	$0.0593523\pi$
$953$	353.338i	0.370764i	−0.982667	+	0.185382i	$0.940648\pi$
$954$	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	$-0.421862\pi$
$967$	470.000	0.486039	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	$-0.913780\pi$
$971$	− 519.615i	− 0.535134i	0.963539	−	0.267567i	$-0.0862197\pi$
$972$	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	$-0.673862\pi$
$977$	− 1669.70i	− 1.70900i	0.519448	−	0.854502i	$-0.326138\pi$
$978$	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	$0.329448\pi$
$983$	1690.48i	1.71972i	−0.510533	+	0.859858i	$0.670552\pi$
$984$	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	$-0.906882\pi$
$991$	− 571.577i	− 0.576768i	0.957515	−	0.288384i	$-0.0931179\pi$
$992$	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	$-0.216573\pi$
$997$	1550.00	1.55466	0.777332	+	0.629091i	$0.216573\pi$
$998$	814.064i	0.815695i
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.c (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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