LMFDB - Embedded newform 57.3.c.a.37.2

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

Display 100 coefficients 10 coefficients

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 (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	$0.857451\pi$
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	1.00000	0.250000
$5$	4.00000	0.800000	0.400000	−	0.916515i	$-0.369010\pi$
$5$	4.00000	0.800000	0.400000	+	0.916515i	$0.369010\pi$
$6$	−3.00000	−0.500000
$7$	−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
$9$	−3.00000	−0.333333
$10$	6.92820i	0.692820i
$11$	10.0000	0.909091	0.454545	−	0.890724i	$-0.349802\pi$
$11$	10.0000	0.909091	0.454545	+	0.890724i	$0.349802\pi$
$12$	1.73205i	0.144338i
$13$	− 24.2487i	− 1.86529i	−0.360801	−	0.932643i	$-0.617497\pi$
$13$	− 24.2487i	− 1.86529i	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	$-0.404974\pi$
$17$	10.0000	0.588235	0.294118	+	0.955769i	$0.404974\pi$
$18$	− 5.19615i	− 0.288675i
$19$	19.0000	1.00000
$19$	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	$-0.643175\pi$
$23$	−20.0000	−0.869565	−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	$-0.796256\pi$
$29$	− 34.6410i	− 1.19452i	0.802049	−	0.597259i	$-0.203744\pi$
$30$	−12.0000	−0.400000
$31$	17.3205i	0.558726i	0.960186	+	0.279363i	$0.0901233\pi$
$31$	17.3205i	0.558726i	−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	$-0.955149\pi$
$37$	− 10.3923i	− 0.280873i	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	$0.138830\pi$
$41$	34.6410i	0.844903i	−0.906386	+	0.422451i	$0.861170\pi$
$42$	30.0000	0.714286
$43$	−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
$45$	−12.0000	−0.266667
$46$	− 34.6410i	− 0.753066i
$47$	−80.0000	−1.70213	−0.851064	−	0.525062i	$-0.824042\pi$
$47$	−80.0000	−1.70213	−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	$0.128273\pi$
$53$	41.5692i	0.784325i	−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	$0.0948427\pi$
$59$	34.6410i	0.587136i	−0.955938	+	0.293568i	$0.905157\pi$
$60$	6.92820i	0.115470i
$61$	−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
$63$	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	$-0.807434\pi$
$67$	− 76.2102i	− 1.13747i	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	$0.261344\pi$
$71$	103.923i	1.46370i	−0.681463	+	0.731852i	$0.738656\pi$
$72$	− 25.9808i	− 0.360844i
$73$	−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
$75$	− 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	$-0.965035\pi$
$79$	− 17.3205i	− 0.219247i	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	$-0.361438\pi$
$83$	70.0000	0.843373	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	$-0.801551\pi$
$89$	− 103.923i	− 1.16767i	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	$0.128506\pi$
$97$	76.2102i	0.785673i	−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	$-0.335150\pi$
$101$	100.000	0.990099	0.495050	+	0.868865i	$0.335150\pi$
$102$	−30.0000	−0.294118
$103$	183.597i	1.78250i	0.453513	+	0.891249i	$0.350170\pi$
$103$	183.597i	1.78250i	−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	$0.0941121\pi$
$107$	62.3538i	0.582746i	−0.956610	+	0.291373i	$0.905888\pi$
$108$	− 5.19615i	− 0.0481125i
$109$	− 155.885i	− 1.43013i	−0.699056	−	0.715067i	$-0.746396\pi$
$109$	− 155.885i	− 1.43013i	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	$-0.990240\pi$
$113$	− 6.92820i	− 0.0613115i	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	$-0.851402\pi$
$127$	− 114.315i	− 0.900121i	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	$-0.546330\pi$
$131$	−38.0000	−0.290076	−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	$-0.256098\pi$
$137$	190.000	1.38686	0.693431	+	0.720523i	$0.256098\pi$
$138$	60.0000	0.434783
$139$	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
$141$	− 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	$-0.521379\pi$
$149$	−20.0000	−0.134228	−0.0671141	+	0.997745i	$0.521379\pi$
$150$	27.0000	0.180000
$151$	− 225.167i	− 1.49117i	−0.666411	−	0.745585i	$-0.732170\pi$
$151$	− 225.167i	− 1.49117i	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	$-0.238361\pi$
$157$	230.000	1.46497	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	$-0.325383\pi$
$163$	170.000	1.04294	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	$-0.871050\pi$
$167$	− 131.636i	− 0.788239i	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	$0.238370\pi$
$173$	235.559i	1.36161i	−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	$-0.906249\pi$
$179$	− 103.923i	− 0.580576i	0.956939	−	0.290288i	$-0.0937511\pi$
$180$	−12.0000	−0.0666667
$181$	259.808i	1.43540i	0.696352	+	0.717701i	$0.254805\pi$
$181$	259.808i	1.43540i	−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	$-0.835308\pi$
$191$	−332.000	−1.73822	−0.869110	+	0.494619i	$0.835308\pi$
$192$	− 122.976i	− 0.640498i
$193$	− 96.9948i	− 0.502564i	−0.967914	−	0.251282i	$-0.919148\pi$
$193$	− 96.9948i	− 0.502564i	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	$-0.366891\pi$
$197$	160.000	0.812183	0.406091	+	0.913832i	$0.366891\pi$
$198$	− 51.9615i	− 0.262432i
$199$	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
$201$	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	$0.134624\pi$
$211$	173.205i	0.820877i	−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	$-0.942830\pi$
$223$	− 79.6743i	− 0.357284i	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	$-0.946313\pi$
$227$	− 76.2102i	− 0.335728i	0.985810	−	0.167864i	$-0.0536869\pi$
$228$	32.9090i	0.144338i
$229$	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
$231$	− 173.205i	− 0.749806i
$232$	300.000	1.29310
$233$	190.000	0.815451	0.407725	−	0.913105i	$-0.366322\pi$
$233$	190.000	0.815451	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	$-0.586291\pi$
$239$	−128.000	−0.535565	−0.267782	+	0.963479i	$0.586291\pi$
$240$	− 76.2102i	− 0.317543i
$241$	138.564i	0.574955i	0.957787	+	0.287477i	$0.0928166\pi$
$241$	138.564i	0.574955i	−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	$-0.501268\pi$
$251$	−2.00000	−0.00796813	−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	$0.406325\pi$
$257$	491.902i	1.91402i	−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	$-0.624155\pi$
$263$	−200.000	−0.760456	−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	$-0.718925\pi$
$269$	− 415.692i	− 1.54532i	0.634818	−	0.772662i	$-0.281075\pi$
$270$	36.0000	0.133333
$271$	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
$273$	−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	$-0.505746\pi$
$277$	−10.0000	−0.0361011	−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	$0.237165\pi$
$281$	381.051i	1.35605i	−0.735037	+	0.678027i	$0.762835\pi$
$282$	240.000	0.851064
$283$	−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
$285$	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	$0.0994572\pi$
$293$	180.133i	0.614789i	−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	$-0.923850\pi$
$307$	− 145.492i	− 0.473916i	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	$-0.117644\pi$
$311$	580.000	1.86495	0.932476	+	0.361232i	$0.117644\pi$
$312$	363.731i	1.16580i
$313$	−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
$315$	120.000	0.380952
$316$	− 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	$0.986082\pi$
$318$	− 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	$-0.915737\pi$
$331$	− 173.205i	− 0.523278i	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	$0.168022\pi$
$337$	339.482i	1.00736i	−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	$-0.823485\pi$
$347$	−590.000	−1.70029	−0.850144	+	0.526550i	$0.823485\pi$
$348$	60.0000	0.172414
$349$	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
$351$	−126.000	−0.358974
$352$	155.885i	0.442854i
$353$	190.000	0.538244	0.269122	−	0.963106i	$-0.413267\pi$
$353$	190.000	0.538244	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	$-0.589854\pi$
$359$	−200.000	−0.557103	−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	$-0.425601\pi$
$367$	170.000	0.463215	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	$-0.841258\pi$
$373$	− 356.802i	− 0.956575i	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	$0.0884141\pi$
$379$	207.846i	0.548407i	−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	$-0.692263\pi$
$383$	− 630.466i	− 1.64613i	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	$-0.552609\pi$
$389$	−128.000	−0.329049	−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	$-0.194729\pi$
$397$	650.000	1.63728	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	$-0.930710\pi$
$401$	− 173.205i	− 0.431933i	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	$0.0679137\pi$
$409$	173.205i	0.423484i	−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	$-0.514439\pi$
$419$	−38.0000	−0.0906921	−0.0453461	+	0.998971i	$0.514439\pi$
$420$	− 69.2820i	− 0.164957i
$421$	− 17.3205i	− 0.0411413i	−0.999788	−	0.0205707i	$-0.993452\pi$
$421$	− 17.3205i	− 0.0411413i	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			$0$
$431$	0	0	−1.00000			$\pi$
$432$	57.1577i	0.132309i
$433$	− 353.338i	− 0.816024i	−0.912977	−	0.408012i	$-0.866222\pi$
$433$	− 353.338i	− 0.816024i	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	$-0.955903\pi$
$439$	− 121.244i	− 0.276181i	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	$-0.539622\pi$
$443$	−110.000	−0.248307	−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	$0.112861\pi$
$449$	311.769i	0.694363i	−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	$-0.397228\pi$
$457$	290.000	0.634573	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	$-0.789706\pi$
$461$	−728.000	−1.57918	−0.789588	+	0.613638i	$0.789706\pi$
$462$	300.000	0.649351
$463$	−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
$465$	−120.000	−0.258065
$466$	329.090i	0.706201i
$467$	−530.000	−1.13490	−0.567452	−	0.823407i	$-0.692071\pi$
$467$	−530.000	−1.13490	−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	$-0.526612\pi$
$479$	−80.0000	−0.167015	−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	$-0.824881\pi$
$487$	− 509.223i	− 1.04563i	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	$-0.360041\pi$
$491$	418.000	0.851324	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	$-0.343914\pi$
$499$	470.000	0.941884	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	$-0.468307\pi$
$503$	100.000	0.198807	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	$0.145862\pi$
$509$	450.333i	0.884741i	−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	$0.0967207\pi$
$521$	311.769i	0.598405i	−0.954190	+	0.299203i	$0.903279\pi$
$522$	−180.000	−0.344828
$523$	− 789.815i	− 1.51016i	−0.655631	−	0.755081i	$-0.727597\pi$
$523$	− 789.815i	− 1.51016i	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	$-0.294873\pi$
$541$	650.000	1.20148	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	$0.183330\pi$
$547$	595.825i	1.08926i	−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	$-0.522879\pi$
$557$	−80.0000	−0.143627	−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	$-0.902515\pi$
$563$	− 339.482i	− 0.602987i	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	$-0.803691\pi$
$569$	− 658.179i	− 1.15673i	0.815778	−	0.578365i	$-0.196309\pi$
$570$	−228.000	−0.400000
$571$	−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
$573$	− 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	$-0.452937\pi$
$577$	170.000	0.294627	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	$-0.686770\pi$
$587$	−650.000	−1.10733	−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	$-0.221607\pi$
$593$	910.000	1.53457	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	$0.00920544\pi$
$599$	34.6410i	0.0578314i	−0.999582	+	0.0289157i	$0.990795\pi$
$600$	135.000	0.225000
$601$	− 173.205i	− 0.288195i	−0.989564	−	0.144097i	$-0.953972\pi$
$601$	− 173.205i	− 0.288195i	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	$0.196656\pi$
$607$	703.213i	1.15851i	−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	$-0.407847\pi$
$613$	350.000	0.570962	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	$-0.335414\pi$
$617$	610.000	0.988655	0.494327	+	0.869276i	$0.335414\pi$
$618$	− 550.792i	− 0.891249i
$619$	−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
$621$	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	$-0.410548\pi$
$631$	350.000	0.554675	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	$0.151921\pi$
$641$	588.897i	0.918716i	−0.888251	+	0.459358i	$0.848079\pi$
$642$	− 187.061i	− 0.291373i
$643$	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
$645$	− 69.2820i	− 0.107414i
$646$	329.090i	0.509427i
$647$	820.000	1.26739	0.633694	−	0.773584i	$-0.281538\pi$
$647$	820.000	1.26739	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	$-0.641060\pi$
$653$	−560.000	−0.857580	−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	$-0.889004\pi$
$659$	− 450.333i	− 0.683358i	0.939817	−	0.341679i	$-0.110996\pi$
$660$	69.2820i	0.104973i
$661$	− 398.372i	− 0.602680i	−0.953517	−	0.301340i	$-0.902566\pi$
$661$	− 398.372i	− 0.602680i	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	$-0.844831\pi$
$673$	− 630.466i	− 0.936800i	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	$0.127138\pi$
$677$	526.543i	0.777760i	−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	$-0.886195\pi$
$683$	− 478.046i	− 0.699921i	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	$-0.389544\pi$
$691$	470.000	0.680174	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	$-0.630792\pi$
$701$	−560.000	−0.798859	−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	$-0.743502\pi$
$709$	−982.000	−1.38505	−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	$-0.382226\pi$
$719$	520.000	0.723227	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	$-0.682836\pi$
$727$	−790.000	−1.08666	−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	$-0.787053\pi$
$733$	−1150.00	−1.56889	−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	$-0.372105\pi$
$739$	578.000	0.782138	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	$-0.949328\pi$
$743$	− 235.559i	− 0.317038i	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	$0.218687\pi$
$751$	952.628i	1.26848i	−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	$-0.552803\pi$
$757$	−250.000	−0.330251	−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	$-0.668844\pi$
$761$	−770.000	−1.01183	−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	$-0.477215\pi$
$769$	110.000	0.143043	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	$-0.970000\pi$
$773$	− 145.492i	− 0.188218i	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	$-0.980372\pi$
$787$	− 96.9948i	− 0.123246i	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	$-0.931685\pi$
$797$	− 339.482i	− 0.425950i	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	$-0.535881\pi$
$809$	−182.000	−0.224969	−0.112485	+	0.993653i	$0.535881\pi$
$810$	62.3538i	0.0769800i
$811$	− 831.384i	− 1.02513i	−0.858647	−	0.512567i	$-0.828694\pi$
$811$	− 831.384i	− 1.02513i	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	$-0.501551\pi$
$821$	−8.00000	−0.00974421	−0.00487211	+	0.999988i	$0.501551\pi$
$822$	−570.000	−0.693431
$823$	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
$825$	− 155.885i	− 0.188951i
$826$	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	$-0.0933308\pi$
$828$	60.0000	0.0724638
$829$	1195.12i	1.44163i	0.693125	+	0.720817i	$0.256233\pi$
$829$	1195.12i	1.44163i	−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	$-0.752334\pi$
$839$	− 1177.79i	− 1.40381i	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	$-0.325302\pi$
$853$	890.000	1.04338	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	$0.261238\pi$
$857$	1254.00i	1.46325i	−0.681708	+	0.731625i	$0.738762\pi$
$858$	727.461i	0.847857i
$859$	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
$861$	600.000	0.696864
$862$	0	0
$863$	1080.80i	1.25238i	0.779672	+	0.626188i	$0.215386\pi$
$863$	1080.80i	1.25238i	−0.779672	+	0.626188i	$0.784614\pi$
$864$	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	$-0.763100\pi$
$877$	− 1188.19i	− 1.35483i	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	$-0.398952\pi$
$881$	550.000	0.624291	0.312145	+	0.950034i	$0.398952\pi$
$882$	− 265.004i	− 0.300458i
$883$	−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
$885$	−240.000	−0.271186
$886$	− 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	$-0.249898\pi$
$888$	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	$-0.980536\pi$
$907$	− 110.851i	− 0.122217i	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	$0.144062\pi$
$911$	796.743i	0.874581i	−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	$-0.489261\pi$
$919$	62.0000	0.0674646	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	$-0.541577\pi$
$929$	−242.000	−0.260495	−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	$-0.481305\pi$
$937$	110.000	0.117396	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	$-0.860854\pi$
$941$	− 796.743i	− 0.846699i	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	$-0.222454\pi$
$947$	1450.00	1.53115	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	$0.0593523\pi$
$953$	353.338i	0.370764i	−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	$-0.421862\pi$
$967$	470.000	0.486039	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	$-0.913780\pi$
$971$	− 519.615i	− 0.535134i	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	$-0.673862\pi$
$977$	− 1669.70i	− 1.70900i	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	$0.329448\pi$
$983$	1690.48i	1.71972i	−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	$0.0931179\pi$
$991$	571.577i	0.576768i	−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	$-0.216573\pi$
$997$	1550.00	1.55466	0.777332	+	0.629091i	$0.216573\pi$
$998$	814.064i	0.815695i
$999$	−54.0000	−0.0540541

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

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

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

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