LMFDB - Embedded newform 912.3.o.a.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	912.721
Dual form			912.3.o.a.721.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (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 $

$2$	0	0
$2$	0	0
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	4.00000	0.800000	0.400000	−	0.916515i	$-0.369010\pi$
$5$	4.00000	0.800000	0.400000	+	0.916515i	$0.369010\pi$
$6$	0	0
$7$	10.0000	1.42857	0.714286	−	0.699854i	$-0.246752\pi$
$7$	10.0000	1.42857	0.714286	+	0.699854i	$0.246752\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−10.0000	−0.909091	−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	− 24.2487i	− 1.86529i	−0.360801	−	0.932643i	$-0.617497\pi$
$13$	− 24.2487i	− 1.86529i	0.360801	−	0.932643i	$-0.382503\pi$
$14$	0	0
$15$	− 6.92820i	− 0.461880i
$16$	0	0
$17$	10.0000	0.588235	0.294118	−	0.955769i	$-0.404974\pi$
$17$	10.0000	0.588235	0.294118	+	0.955769i	$0.404974\pi$
$18$	0	0
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	0	0
$21$	− 17.3205i	− 0.824786i
$22$	0	0
$23$	20.0000	0.869565	0.434783	−	0.900535i	$-0.356825\pi$
$23$	20.0000	0.869565	0.434783	+	0.900535i	$0.356825\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	− 34.6410i	− 1.19452i	−0.802049	−	0.597259i	$-0.796256\pi$
$29$	− 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	$-0.0901233\pi$
$32$	0	0
$33$	17.3205i	0.524864i
$34$	0	0
$35$	40.0000	1.14286
$36$	0	0
$37$	− 10.3923i	− 0.280873i	−0.990090	−	0.140437i	$-0.955149\pi$
$37$	− 10.3923i	− 0.280873i	0.990090	−	0.140437i	$-0.0448506\pi$
$38$	0	0
$39$	−42.0000	−1.07692
$40$	0	0
$41$	34.6410i	0.844903i	0.906386	+	0.422451i	$0.138830\pi$
$41$	34.6410i	0.844903i	−0.906386	+	0.422451i	$0.861170\pi$
$42$	0	0
$43$	10.0000	0.232558	0.116279	−	0.993217i	$-0.462903\pi$
$43$	10.0000	0.232558	0.116279	+	0.993217i	$0.462903\pi$
$44$	0	0
$45$	−12.0000	−0.266667
$46$	0	0
$47$	80.0000	1.70213	0.851064	−	0.525062i	$-0.175958\pi$
$47$	80.0000	1.70213	0.851064	+	0.525062i	$0.175958\pi$
$48$	0	0
$49$	51.0000	1.04082
$50$	0	0
$51$	− 17.3205i	− 0.339618i
$52$	0	0
$53$	41.5692i	0.784325i	0.919896	+	0.392162i	$0.128273\pi$
$53$	41.5692i	0.784325i	−0.919896	+	0.392162i	$0.871727\pi$
$54$	0	0
$55$	−40.0000	−0.727273
$56$	0	0
$57$	32.9090i	0.577350i
$58$	0	0
$59$	− 34.6410i	− 0.587136i	−0.955938	−	0.293568i	$-0.905157\pi$
$59$	− 34.6410i	− 0.587136i	0.955938	−	0.293568i	$-0.0948427\pi$
$60$	0	0
$61$	−10.0000	−0.163934	−0.0819672	−	0.996635i	$-0.526120\pi$
$61$	−10.0000	−0.163934	−0.0819672	+	0.996635i	$0.526120\pi$
$62$	0	0
$63$	−30.0000	−0.476190
$64$	0	0
$65$	− 96.9948i	− 1.49223i
$66$	0	0
$67$	76.2102i	1.13747i	0.822522	+	0.568733i	$0.192566\pi$
$67$	76.2102i	1.13747i	−0.822522	+	0.568733i	$0.807434\pi$
$68$	0	0
$69$	− 34.6410i	− 0.502044i
$70$	0	0
$71$	− 103.923i	− 1.46370i	−0.681463	−	0.731852i	$-0.738656\pi$
$71$	− 103.923i	− 1.46370i	0.681463	−	0.731852i	$-0.261344\pi$
$72$	0	0
$73$	−10.0000	−0.136986	−0.0684932	−	0.997652i	$-0.521819\pi$
$73$	−10.0000	−0.136986	−0.0684932	+	0.997652i	$0.521819\pi$
$74$	0	0
$75$	15.5885i	0.207846i
$76$	0	0
$77$	−100.000	−1.29870
$78$	0	0
$79$	17.3205i	0.219247i	0.993973	+	0.109623i	$0.0349645\pi$
$79$	17.3205i	0.219247i	−0.993973	+	0.109623i	$0.965035\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−70.0000	−0.843373	−0.421687	−	0.906742i	$-0.638562\pi$
$83$	−70.0000	−0.843373	−0.421687	+	0.906742i	$0.638562\pi$
$84$	0	0
$85$	40.0000	0.470588
$86$	0	0
$87$	−60.0000	−0.689655
$88$	0	0
$89$	− 103.923i	− 1.16767i	−0.811871	−	0.583837i	$-0.801551\pi$
$89$	− 103.923i	− 1.16767i	0.811871	−	0.583837i	$-0.198449\pi$
$90$	0	0
$91$	− 242.487i	− 2.66469i
$92$	0	0
$93$	−30.0000	−0.322581
$94$	0	0
$95$	−76.0000	−0.800000
$96$	0	0
$97$	76.2102i	0.785673i	0.919608	+	0.392836i	$0.128506\pi$
$97$	76.2102i	0.785673i	−0.919608	+	0.392836i	$0.871494\pi$
$98$	0	0
$99$	30.0000	0.303030
$100$	0	0
$101$	100.000	0.990099	0.495050	−	0.868865i	$-0.335150\pi$
$101$	100.000	0.990099	0.495050	+	0.868865i	$0.335150\pi$
$102$	0	0
$103$	− 183.597i	− 1.78250i	−0.453513	−	0.891249i	$-0.649830\pi$
$103$	− 183.597i	− 1.78250i	0.453513	−	0.891249i	$-0.350170\pi$
$104$	0	0
$105$	− 69.2820i	− 0.659829i
$106$	0	0
$107$	− 62.3538i	− 0.582746i	−0.956610	−	0.291373i	$-0.905888\pi$
$107$	− 62.3538i	− 0.582746i	0.956610	−	0.291373i	$-0.0941121\pi$
$108$	0	0
$109$	− 155.885i	− 1.43013i	−0.699056	−	0.715067i	$-0.746396\pi$
$109$	− 155.885i	− 1.43013i	0.699056	−	0.715067i	$-0.253604\pi$
$110$	0	0
$111$	−18.0000	−0.162162
$112$	0	0
$113$	− 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
$116$	0	0
$117$	72.7461i	0.621762i
$118$	0	0
$119$	100.000	0.840336
$120$	0	0
$121$	−21.0000	−0.173554
$122$	0	0
$123$	60.0000	0.487805
$124$	0	0
$125$	−136.000	−1.08800
$126$	0	0
$127$	114.315i	0.900121i	0.892998	+	0.450060i	$0.148598\pi$
$127$	114.315i	0.900121i	−0.892998	+	0.450060i	$0.851402\pi$
$128$	0	0
$129$	− 17.3205i	− 0.134268i
$130$	0	0
$131$	38.0000	0.290076	0.145038	−	0.989426i	$-0.453670\pi$
$131$	38.0000	0.290076	0.145038	+	0.989426i	$0.453670\pi$
$132$	0	0
$133$	−190.000	−1.42857
$134$	0	0
$135$	20.7846i	0.153960i
$136$	0	0
$137$	190.000	1.38686	0.693431	−	0.720523i	$-0.256098\pi$
$137$	190.000	1.38686	0.693431	+	0.720523i	$0.256098\pi$
$138$	0	0
$139$	−50.0000	−0.359712	−0.179856	−	0.983693i	$-0.557563\pi$
$139$	−50.0000	−0.359712	−0.179856	+	0.983693i	$0.557563\pi$
$140$	0	0
$141$	− 138.564i	− 0.982724i
$142$	0	0
$143$	242.487i	1.69571i
$144$	0	0
$145$	− 138.564i	− 0.955614i
$146$	0	0
$147$	− 88.3346i	− 0.600916i
$148$	0	0
$149$	−20.0000	−0.134228	−0.0671141	−	0.997745i	$-0.521379\pi$
$149$	−20.0000	−0.134228	−0.0671141	+	0.997745i	$0.521379\pi$
$150$	0	0
$151$	225.167i	1.49117i	0.666411	+	0.745585i	$0.267830\pi$
$151$	225.167i	1.49117i	−0.666411	+	0.745585i	$0.732170\pi$
$152$	0	0
$153$	−30.0000	−0.196078
$154$	0	0
$155$	− 69.2820i	− 0.446981i
$156$	0	0
$157$	230.000	1.46497	0.732484	−	0.680784i	$-0.238361\pi$
$157$	230.000	1.46497	0.732484	+	0.680784i	$0.238361\pi$
$158$	0	0
$159$	72.0000	0.452830
$160$	0	0
$161$	200.000	1.24224
$162$	0	0
$163$	−170.000	−1.04294	−0.521472	−	0.853268i	$-0.674617\pi$
$163$	−170.000	−1.04294	−0.521472	+	0.853268i	$0.674617\pi$
$164$	0	0
$165$	69.2820i	0.419891i
$166$	0	0
$167$	131.636i	0.788239i	0.919059	+	0.394119i	$0.128950\pi$
$167$	131.636i	0.788239i	−0.919059	+	0.394119i	$0.871050\pi$
$168$	0	0
$169$	−419.000	−2.47929
$170$	0	0
$171$	57.0000	0.333333
$172$	0	0
$173$	235.559i	1.36161i	0.732464	+	0.680806i	$0.238370\pi$
$173$	235.559i	1.36161i	−0.732464	+	0.680806i	$0.761630\pi$
$174$	0	0
$175$	−90.0000	−0.514286
$176$	0	0
$177$	−60.0000	−0.338983
$178$	0	0
$179$	103.923i	0.580576i	0.956939	+	0.290288i	$0.0937511\pi$
$179$	103.923i	0.580576i	−0.956939	+	0.290288i	$0.906249\pi$
$180$	0	0
$181$	259.808i	1.43540i	0.696352	+	0.717701i	$0.254805\pi$
$181$	259.808i	1.43540i	−0.696352	+	0.717701i	$0.745195\pi$
$182$	0	0
$183$	17.3205i	0.0946476i
$184$	0	0
$185$	− 41.5692i	− 0.224698i
$186$	0	0
$187$	−100.000	−0.534759
$188$	0	0
$189$	51.9615i	0.274929i
$190$	0	0
$191$	332.000	1.73822	0.869110	−	0.494619i	$-0.164692\pi$
$191$	332.000	1.73822	0.869110	+	0.494619i	$0.164692\pi$
$192$	0	0
$193$	− 96.9948i	− 0.502564i	−0.967914	−	0.251282i	$-0.919148\pi$
$193$	− 96.9948i	− 0.502564i	0.967914	−	0.251282i	$-0.0808521\pi$
$194$	0	0
$195$	−168.000	−0.861538
$196$	0	0
$197$	160.000	0.812183	0.406091	−	0.913832i	$-0.366891\pi$
$197$	160.000	0.812183	0.406091	+	0.913832i	$0.366891\pi$
$198$	0	0
$199$	−98.0000	−0.492462	−0.246231	−	0.969211i	$-0.579192\pi$
$199$	−98.0000	−0.492462	−0.246231	+	0.969211i	$0.579192\pi$
$200$	0	0
$201$	132.000	0.656716
$202$	0	0
$203$	− 346.410i	− 1.70645i
$204$	0	0
$205$	138.564i	0.675922i
$206$	0	0
$207$	−60.0000	−0.289855
$208$	0	0
$209$	190.000	0.909091
$210$	0	0
$211$	− 173.205i	− 0.820877i	−0.911888	−	0.410439i	$-0.865376\pi$
$211$	− 173.205i	− 0.820877i	0.911888	−	0.410439i	$-0.134624\pi$
$212$	0	0
$213$	−180.000	−0.845070
$214$	0	0
$215$	40.0000	0.186047
$216$	0	0
$217$	− 173.205i	− 0.798180i
$218$	0	0
$219$	17.3205i	0.0790891i
$220$	0	0
$221$	− 242.487i	− 1.09723i
$222$	0	0
$223$	79.6743i	0.357284i	0.983914	+	0.178642i	$0.0571704\pi$
$223$	79.6743i	0.357284i	−0.983914	+	0.178642i	$0.942830\pi$
$224$	0	0
$225$	27.0000	0.120000
$226$	0	0
$227$	76.2102i	0.335728i	0.985810	+	0.167864i	$0.0536869\pi$
$227$	76.2102i	0.335728i	−0.985810	+	0.167864i	$0.946313\pi$
$228$	0	0
$229$	110.000	0.480349	0.240175	−	0.970730i	$-0.422795\pi$
$229$	110.000	0.480349	0.240175	+	0.970730i	$0.422795\pi$
$230$	0	0
$231$	173.205i	0.749806i
$232$	0	0
$233$	190.000	0.815451	0.407725	−	0.913105i	$-0.366322\pi$
$233$	190.000	0.815451	0.407725	+	0.913105i	$0.366322\pi$
$234$	0	0
$235$	320.000	1.36170
$236$	0	0
$237$	30.0000	0.126582
$238$	0	0
$239$	128.000	0.535565	0.267782	−	0.963479i	$-0.413709\pi$
$239$	128.000	0.535565	0.267782	+	0.963479i	$0.413709\pi$
$240$	0	0
$241$	138.564i	0.574955i	0.957787	+	0.287477i	$0.0928166\pi$
$241$	138.564i	0.574955i	−0.957787	+	0.287477i	$0.907183\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	204.000	0.832653
$246$	0	0
$247$	460.726i	1.86529i
$248$	0	0
$249$	121.244i	0.486922i
$250$	0	0
$251$	2.00000	0.00796813	0.00398406	−	0.999992i	$-0.498732\pi$
$251$	2.00000	0.00796813	0.00398406	+	0.999992i	$0.498732\pi$
$252$	0	0
$253$	−200.000	−0.790514
$254$	0	0
$255$	− 69.2820i	− 0.271694i
$256$	0	0
$257$	491.902i	1.91402i	0.290059	+	0.957009i	$0.406325\pi$
$257$	491.902i	1.91402i	−0.290059	+	0.957009i	$0.593675\pi$
$258$	0	0
$259$	− 103.923i	− 0.401247i
$260$	0	0
$261$	103.923i	0.398173i
$262$	0	0
$263$	200.000	0.760456	0.380228	−	0.924893i	$-0.375845\pi$
$263$	200.000	0.760456	0.380228	+	0.924893i	$0.375845\pi$
$264$	0	0
$265$	166.277i	0.627460i
$266$	0	0
$267$	−180.000	−0.674157
$268$	0	0
$269$	− 415.692i	− 1.54532i	−0.634818	−	0.772662i	$-0.718925\pi$
$269$	− 415.692i	− 1.54532i	0.634818	−	0.772662i	$-0.281075\pi$
$270$	0	0
$271$	−170.000	−0.627306	−0.313653	−	0.949538i	$-0.601553\pi$
$271$	−170.000	−0.627306	−0.313653	+	0.949538i	$0.601553\pi$
$272$	0	0
$273$	−420.000	−1.53846
$274$	0	0
$275$	90.0000	0.327273
$276$	0	0
$277$	−10.0000	−0.0361011	−0.0180505	−	0.999837i	$-0.505746\pi$
$277$	−10.0000	−0.0361011	−0.0180505	+	0.999837i	$0.505746\pi$
$278$	0	0
$279$	51.9615i	0.186242i
$280$	0	0
$281$	381.051i	1.35605i	0.735037	+	0.678027i	$0.237165\pi$
$281$	381.051i	1.35605i	−0.735037	+	0.678027i	$0.762835\pi$
$282$	0	0
$283$	70.0000	0.247350	0.123675	−	0.992323i	$-0.460532\pi$
$283$	70.0000	0.247350	0.123675	+	0.992323i	$0.460532\pi$
$284$	0	0
$285$	131.636i	0.461880i
$286$	0	0
$287$	346.410i	1.20700i
$288$	0	0
$289$	−189.000	−0.653979
$290$	0	0
$291$	132.000	0.453608
$292$	0	0
$293$	180.133i	0.614789i	0.951582	+	0.307395i	$0.0994572\pi$
$293$	180.133i	0.614789i	−0.951582	+	0.307395i	$0.900543\pi$
$294$	0	0
$295$	− 138.564i	− 0.469709i
$296$	0	0
$297$	− 51.9615i	− 0.174955i
$298$	0	0
$299$	− 484.974i	− 1.62199i
$300$	0	0
$301$	100.000	0.332226
$302$	0	0
$303$	− 173.205i	− 0.571634i
$304$	0	0
$305$	−40.0000	−0.131148
$306$	0	0
$307$	145.492i	0.473916i	0.971520	+	0.236958i	$0.0761504\pi$
$307$	145.492i	0.473916i	−0.971520	+	0.236958i	$0.923850\pi$
$308$	0	0
$309$	−318.000	−1.02913
$310$	0	0
$311$	−580.000	−1.86495	−0.932476	−	0.361232i	$-0.882356\pi$
$311$	−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	$0.701288\pi$
$314$	0	0
$315$	−120.000	−0.380952
$316$	0	0
$317$	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
$320$	0	0
$321$	−108.000	−0.336449
$322$	0	0
$323$	−190.000	−0.588235
$324$	0	0
$325$	218.238i	0.671503i
$326$	0	0
$327$	−270.000	−0.825688
$328$	0	0
$329$	800.000	2.43161
$330$	0	0
$331$	173.205i	0.523278i	0.965166	+	0.261639i	$0.0842630\pi$
$331$	173.205i	0.523278i	−0.965166	+	0.261639i	$0.915737\pi$
$332$	0	0
$333$	31.1769i	0.0936244i
$334$	0	0
$335$	304.841i	0.909973i
$336$	0	0
$337$	339.482i	1.00736i	0.863889	+	0.503682i	$0.168022\pi$
$337$	339.482i	1.00736i	−0.863889	+	0.503682i	$0.831978\pi$
$338$	0	0
$339$	−12.0000	−0.0353982
$340$	0	0
$341$	173.205i	0.507933i
$342$	0	0
$343$	20.0000	0.0583090
$344$	0	0
$345$	− 138.564i	− 0.401635i
$346$	0	0
$347$	590.000	1.70029	0.850144	−	0.526550i	$-0.176515\pi$
$347$	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	$0.455161\pi$
$350$	0	0
$351$	126.000	0.358974
$352$	0	0
$353$	190.000	0.538244	0.269122	−	0.963106i	$-0.413267\pi$
$353$	190.000	0.538244	0.269122	+	0.963106i	$0.413267\pi$
$354$	0	0
$355$	− 415.692i	− 1.17096i
$356$	0	0
$357$	− 173.205i	− 0.485168i
$358$	0	0
$359$	200.000	0.557103	0.278552	−	0.960421i	$-0.410146\pi$
$359$	200.000	0.557103	0.278552	+	0.960421i	$0.410146\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	36.3731i	0.100201i
$364$	0	0
$365$	−40.0000	−0.109589
$366$	0	0
$367$	−170.000	−0.463215	−0.231608	−	0.972809i	$-0.574399\pi$
$367$	−170.000	−0.463215	−0.231608	+	0.972809i	$0.574399\pi$
$368$	0	0
$369$	− 103.923i	− 0.281634i
$370$	0	0
$371$	415.692i	1.12046i
$372$	0	0
$373$	− 356.802i	− 0.956575i	−0.878203	−	0.478287i	$-0.841258\pi$
$373$	− 356.802i	− 0.956575i	0.878203	−	0.478287i	$-0.158742\pi$
$374$	0	0
$375$	235.559i	0.628157i
$376$	0	0
$377$	−840.000	−2.22812
$378$	0	0
$379$	− 207.846i	− 0.548407i	−0.961672	−	0.274203i	$-0.911586\pi$
$379$	− 207.846i	− 0.548407i	0.961672	−	0.274203i	$-0.0884141\pi$
$380$	0	0
$381$	198.000	0.519685
$382$	0	0
$383$	630.466i	1.64613i	0.567950	+	0.823063i	$0.307737\pi$
$383$	630.466i	1.64613i	−0.567950	+	0.823063i	$0.692263\pi$
$384$	0	0
$385$	−400.000	−1.03896
$386$	0	0
$387$	−30.0000	−0.0775194
$388$	0	0
$389$	−128.000	−0.329049	−0.164524	−	0.986373i	$-0.552609\pi$
$389$	−128.000	−0.329049	−0.164524	+	0.986373i	$0.552609\pi$
$390$	0	0
$391$	200.000	0.511509
$392$	0	0
$393$	− 65.8179i	− 0.167476i
$394$	0	0
$395$	69.2820i	0.175398i
$396$	0	0
$397$	650.000	1.63728	0.818640	−	0.574307i	$-0.194729\pi$
$397$	650.000	1.63728	0.818640	+	0.574307i	$0.194729\pi$
$398$	0	0
$399$	329.090i	0.824786i
$400$	0	0
$401$	− 173.205i	− 0.431933i	−0.976401	−	0.215966i	$-0.930710\pi$
$401$	− 173.205i	− 0.431933i	0.976401	−	0.215966i	$-0.0692902\pi$
$402$	0	0
$403$	−420.000	−1.04218
$404$	0	0
$405$	36.0000	0.0888889
$406$	0	0
$407$	103.923i	0.255339i
$408$	0	0
$409$	173.205i	0.423484i	0.977326	+	0.211742i	$0.0679137\pi$
$409$	173.205i	0.423484i	−0.977326	+	0.211742i	$0.932086\pi$
$410$	0	0
$411$	− 329.090i	− 0.800705i
$412$	0	0
$413$	− 346.410i	− 0.838766i
$414$	0	0
$415$	−280.000	−0.674699
$416$	0	0
$417$	86.6025i	0.207680i
$418$	0	0
$419$	38.0000	0.0906921	0.0453461	−	0.998971i	$-0.485561\pi$
$419$	38.0000	0.0906921	0.0453461	+	0.998971i	$0.485561\pi$
$420$	0	0
$421$	− 17.3205i	− 0.0411413i	−0.999788	−	0.0205707i	$-0.993452\pi$
$421$	− 17.3205i	− 0.0411413i	0.999788	−	0.0205707i	$-0.00654831\pi$
$422$	0	0
$423$	−240.000	−0.567376
$424$	0	0
$425$	−90.0000	−0.211765
$426$	0	0
$427$	−100.000	−0.234192
$428$	0	0
$429$	420.000	0.979021
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	− 353.338i	− 0.816024i	−0.912977	−	0.408012i	$-0.866222\pi$
$433$	− 353.338i	− 0.816024i	0.912977	−	0.408012i	$-0.133778\pi$
$434$	0	0
$435$	−240.000	−0.551724
$436$	0	0
$437$	−380.000	−0.869565
$438$	0	0
$439$	121.244i	0.276181i	0.990420	+	0.138091i	$0.0440965\pi$
$439$	121.244i	0.276181i	−0.990420	+	0.138091i	$0.955903\pi$
$440$	0	0
$441$	−153.000	−0.346939
$442$	0	0
$443$	110.000	0.248307	0.124153	−	0.992263i	$-0.460378\pi$
$443$	110.000	0.248307	0.124153	+	0.992263i	$0.460378\pi$
$444$	0	0
$445$	− 415.692i	− 0.934140i
$446$	0	0
$447$	34.6410i	0.0774967i
$448$	0	0
$449$	311.769i	0.694363i	0.937798	+	0.347182i	$0.112861\pi$
$449$	311.769i	0.694363i	−0.937798	+	0.347182i	$0.887139\pi$
$450$	0	0
$451$	− 346.410i	− 0.768093i
$452$	0	0
$453$	390.000	0.860927
$454$	0	0
$455$	− 969.948i	− 2.13175i
$456$	0	0
$457$	290.000	0.634573	0.317287	−	0.948330i	$-0.397228\pi$
$457$	290.000	0.634573	0.317287	+	0.948330i	$0.397228\pi$
$458$	0	0
$459$	51.9615i	0.113206i
$460$	0	0
$461$	−728.000	−1.57918	−0.789588	−	0.613638i	$-0.789706\pi$
$461$	−728.000	−1.57918	−0.789588	+	0.613638i	$0.789706\pi$
$462$	0	0
$463$	790.000	1.70626	0.853132	−	0.521696i	$-0.174700\pi$
$463$	790.000	1.70626	0.853132	+	0.521696i	$0.174700\pi$
$464$	0	0
$465$	−120.000	−0.258065
$466$	0	0
$467$	530.000	1.13490	0.567452	−	0.823407i	$-0.307929\pi$
$467$	530.000	1.13490	0.567452	+	0.823407i	$0.307929\pi$
$468$	0	0
$469$	762.102i	1.62495i
$470$	0	0
$471$	− 398.372i	− 0.845800i
$472$	0	0
$473$	−100.000	−0.211416
$474$	0	0
$475$	171.000	0.360000
$476$	0	0
$477$	− 124.708i	− 0.261442i
$478$	0	0
$479$	80.0000	0.167015	0.0835073	−	0.996507i	$-0.473388\pi$
$479$	80.0000	0.167015	0.0835073	+	0.996507i	$0.473388\pi$
$480$	0	0
$481$	−252.000	−0.523909
$482$	0	0
$483$	− 346.410i	− 0.717205i
$484$	0	0
$485$	304.841i	0.628538i
$486$	0	0
$487$	509.223i	1.04563i	0.852445	+	0.522816i	$0.175119\pi$
$487$	509.223i	1.04563i	−0.852445	+	0.522816i	$0.824881\pi$
$488$	0	0
$489$	294.449i	0.602144i
$490$	0	0
$491$	−418.000	−0.851324	−0.425662	−	0.904882i	$-0.639959\pi$
$491$	−418.000	−0.851324	−0.425662	+	0.904882i	$0.639959\pi$
$492$	0	0
$493$	− 346.410i	− 0.702658i
$494$	0	0
$495$	120.000	0.242424
$496$	0	0
$497$	− 1039.23i	− 2.09101i
$498$	0	0
$499$	−470.000	−0.941884	−0.470942	−	0.882164i	$-0.656086\pi$
$499$	−470.000	−0.941884	−0.470942	+	0.882164i	$0.656086\pi$
$500$	0	0
$501$	228.000	0.455090
$502$	0	0
$503$	−100.000	−0.198807	−0.0994036	−	0.995047i	$-0.531693\pi$
$503$	−100.000	−0.198807	−0.0994036	+	0.995047i	$0.531693\pi$
$504$	0	0
$505$	400.000	0.792079
$506$	0	0
$507$	725.729i	1.43142i
$508$	0	0
$509$	450.333i	0.884741i	0.896832	+	0.442371i	$0.145862\pi$
$509$	450.333i	0.884741i	−0.896832	+	0.442371i	$0.854138\pi$
$510$	0	0
$511$	−100.000	−0.195695
$512$	0	0
$513$	− 98.7269i	− 0.192450i
$514$	0	0
$515$	− 734.390i	− 1.42600i
$516$	0	0
$517$	−800.000	−1.54739
$518$	0	0
$519$	408.000	0.786127
$520$	0	0
$521$	311.769i	0.598405i	0.954190	+	0.299203i	$0.0967207\pi$
$521$	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	$0.727597\pi$
$524$	0	0
$525$	155.885i	0.296923i
$526$	0	0
$527$	− 173.205i	− 0.328662i
$528$	0	0
$529$	−129.000	−0.243856
$530$	0	0
$531$	103.923i	0.195712i
$532$	0	0
$533$	840.000	1.57598
$534$	0	0
$535$	− 249.415i	− 0.466197i
$536$	0	0
$537$	180.000	0.335196
$538$	0	0
$539$	−510.000	−0.946197
$540$	0	0
$541$	650.000	1.20148	0.600739	−	0.799445i	$-0.294873\pi$
$541$	650.000	1.20148	0.600739	+	0.799445i	$0.294873\pi$
$542$	0	0
$543$	450.000	0.828729
$544$	0	0
$545$	− 623.538i	− 1.14411i
$546$	0	0
$547$	− 595.825i	− 1.08926i	−0.838676	−	0.544630i	$-0.816670\pi$
$547$	− 595.825i	− 1.08926i	0.838676	−	0.544630i	$-0.183330\pi$
$548$	0	0
$549$	30.0000	0.0546448
$550$	0	0
$551$	658.179i	1.19452i
$552$	0	0
$553$	173.205i	0.313210i
$554$	0	0
$555$	−72.0000	−0.129730
$556$	0	0
$557$	−80.0000	−0.143627	−0.0718133	−	0.997418i	$-0.522879\pi$
$557$	−80.0000	−0.143627	−0.0718133	+	0.997418i	$0.522879\pi$
$558$	0	0
$559$	− 242.487i	− 0.433787i
$560$	0	0
$561$	173.205i	0.308743i
$562$	0	0
$563$	339.482i	0.602987i	0.953468	+	0.301494i	$0.0974852\pi$
$563$	339.482i	0.602987i	−0.953468	+	0.301494i	$0.902515\pi$
$564$	0	0
$565$	− 27.7128i	− 0.0490492i
$566$	0	0
$567$	90.0000	0.158730
$568$	0	0
$569$	− 658.179i	− 1.15673i	−0.815778	−	0.578365i	$-0.803691\pi$
$569$	− 658.179i	− 1.15673i	0.815778	−	0.578365i	$-0.196309\pi$
$570$	0	0
$571$	610.000	1.06830	0.534151	−	0.845389i	$-0.320632\pi$
$571$	610.000	1.06830	0.534151	+	0.845389i	$0.320632\pi$
$572$	0	0
$573$	− 575.041i	− 1.00356i
$574$	0	0
$575$	−180.000	−0.313043
$576$	0	0
$577$	170.000	0.294627	0.147314	−	0.989090i	$-0.452937\pi$
$577$	170.000	0.294627	0.147314	+	0.989090i	$0.452937\pi$
$578$	0	0
$579$	−168.000	−0.290155
$580$	0	0
$581$	−700.000	−1.20482
$582$	0	0
$583$	− 415.692i	− 0.713023i
$584$	0	0
$585$	290.985i	0.497409i
$586$	0	0
$587$	650.000	1.10733	0.553663	−	0.832741i	$-0.313230\pi$
$587$	650.000	1.10733	0.553663	+	0.832741i	$0.313230\pi$
$588$	0	0
$589$	329.090i	0.558726i
$590$	0	0
$591$	− 277.128i	− 0.468914i
$592$	0	0
$593$	910.000	1.53457	0.767285	−	0.641306i	$-0.221607\pi$
$593$	910.000	1.53457	0.767285	+	0.641306i	$0.221607\pi$
$594$	0	0
$595$	400.000	0.672269
$596$	0	0
$597$	169.741i	0.284323i
$598$	0	0
$599$	− 34.6410i	− 0.0578314i	−0.999582	−	0.0289157i	$-0.990795\pi$
$599$	− 34.6410i	− 0.0578314i	0.999582	−	0.0289157i	$-0.00920544\pi$
$600$	0	0
$601$	− 173.205i	− 0.288195i	−0.989564	−	0.144097i	$-0.953972\pi$
$601$	− 173.205i	− 0.288195i	0.989564	−	0.144097i	$-0.0460279\pi$
$602$	0	0
$603$	− 228.631i	− 0.379155i
$604$	0	0
$605$	−84.0000	−0.138843
$606$	0	0
$607$	− 703.213i	− 1.15851i	−0.815148	−	0.579253i	$-0.803344\pi$
$607$	− 703.213i	− 1.15851i	0.815148	−	0.579253i	$-0.196656\pi$
$608$	0	0
$609$	−600.000	−0.985222
$610$	0	0
$611$	− 1939.90i	− 3.17495i
$612$	0	0
$613$	350.000	0.570962	0.285481	−	0.958384i	$-0.407847\pi$
$613$	350.000	0.570962	0.285481	+	0.958384i	$0.407847\pi$
$614$	0	0
$615$	240.000	0.390244
$616$	0	0
$617$	610.000	0.988655	0.494327	−	0.869276i	$-0.335414\pi$
$617$	610.000	0.988655	0.494327	+	0.869276i	$0.335414\pi$
$618$	0	0
$619$	10.0000	0.0161551	0.00807754	−	0.999967i	$-0.497429\pi$
$619$	10.0000	0.0161551	0.00807754	+	0.999967i	$0.497429\pi$
$620$	0	0
$621$	103.923i	0.167348i
$622$	0	0
$623$	− 1039.23i	− 1.66811i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	− 329.090i	− 0.524864i
$628$	0	0
$629$	− 103.923i	− 0.165219i
$630$	0	0
$631$	−350.000	−0.554675	−0.277338	−	0.960773i	$-0.589452\pi$
$631$	−350.000	−0.554675	−0.277338	+	0.960773i	$0.589452\pi$
$632$	0	0
$633$	−300.000	−0.473934
$634$	0	0
$635$	457.261i	0.720097i
$636$	0	0
$637$	− 1236.68i	− 1.94142i
$638$	0	0
$639$	311.769i	0.487902i
$640$	0	0
$641$	588.897i	0.918716i	0.888251	+	0.459358i	$0.151921\pi$
$641$	588.897i	0.918716i	−0.888251	+	0.459358i	$0.848079\pi$
$642$	0	0
$643$	−650.000	−1.01089	−0.505443	−	0.862860i	$-0.668671\pi$
$643$	−650.000	−1.01089	−0.505443	+	0.862860i	$0.668671\pi$
$644$	0	0
$645$	− 69.2820i	− 0.107414i
$646$	0	0
$647$	−820.000	−1.26739	−0.633694	−	0.773584i	$-0.718462\pi$
$647$	−820.000	−1.26739	−0.633694	+	0.773584i	$0.718462\pi$
$648$	0	0
$649$	346.410i	0.533760i
$650$	0	0
$651$	−300.000	−0.460829
$652$	0	0
$653$	−560.000	−0.857580	−0.428790	−	0.903404i	$-0.641060\pi$
$653$	−560.000	−0.857580	−0.428790	+	0.903404i	$0.641060\pi$
$654$	0	0
$655$	152.000	0.232061
$656$	0	0
$657$	30.0000	0.0456621
$658$	0	0
$659$	450.333i	0.683358i	0.939817	+	0.341679i	$0.110996\pi$
$659$	450.333i	0.683358i	−0.939817	+	0.341679i	$0.889004\pi$
$660$	0	0
$661$	− 398.372i	− 0.602680i	−0.953517	−	0.301340i	$-0.902566\pi$
$661$	− 398.372i	− 0.602680i	0.953517	−	0.301340i	$-0.0974340\pi$
$662$	0	0
$663$	−420.000	−0.633484
$664$	0	0
$665$	−760.000	−1.14286
$666$	0	0
$667$	− 692.820i	− 1.03871i
$668$	0	0
$669$	138.000	0.206278
$670$	0	0
$671$	100.000	0.149031
$672$	0	0
$673$	− 630.466i	− 0.936800i	−0.883516	−	0.468400i	$-0.844831\pi$
$673$	− 630.466i	− 0.936800i	0.883516	−	0.468400i	$-0.155169\pi$
$674$	0	0
$675$	− 46.7654i	− 0.0692820i
$676$	0	0
$677$	526.543i	0.777760i	0.921288	+	0.388880i	$0.127138\pi$
$677$	526.543i	0.777760i	−0.921288	+	0.388880i	$0.872862\pi$
$678$	0	0
$679$	762.102i	1.12239i
$680$	0	0
$681$	132.000	0.193833
$682$	0	0
$683$	478.046i	0.699921i	0.936764	+	0.349960i	$0.113805\pi$
$683$	478.046i	0.699921i	−0.936764	+	0.349960i	$0.886195\pi$
$684$	0	0
$685$	760.000	1.10949
$686$	0	0
$687$	− 190.526i	− 0.277330i
$688$	0	0
$689$	1008.00	1.46299
$690$	0	0
$691$	−470.000	−0.680174	−0.340087	−	0.940394i	$-0.610456\pi$
$691$	−470.000	−0.680174	−0.340087	+	0.940394i	$0.610456\pi$
$692$	0	0
$693$	300.000	0.432900
$694$	0	0
$695$	−200.000	−0.287770
$696$	0	0
$697$	346.410i	0.497002i
$698$	0	0
$699$	− 329.090i	− 0.470801i
$700$	0	0
$701$	−560.000	−0.798859	−0.399429	−	0.916764i	$-0.630792\pi$
$701$	−560.000	−0.798859	−0.399429	+	0.916764i	$0.630792\pi$
$702$	0	0
$703$	197.454i	0.280873i
$704$	0	0
$705$	− 554.256i	− 0.786179i
$706$	0	0
$707$	1000.00	1.41443
$708$	0	0
$709$	−982.000	−1.38505	−0.692525	−	0.721394i	$-0.743502\pi$
$709$	−982.000	−1.38505	−0.692525	+	0.721394i	$0.743502\pi$
$710$	0	0
$711$	− 51.9615i	− 0.0730823i
$712$	0	0
$713$	− 346.410i	− 0.485849i
$714$	0	0
$715$	969.948i	1.35657i
$716$	0	0
$717$	− 221.703i	− 0.309209i
$718$	0	0
$719$	−520.000	−0.723227	−0.361613	−	0.932328i	$-0.617774\pi$
$719$	−520.000	−0.723227	−0.361613	+	0.932328i	$0.617774\pi$
$720$	0	0
$721$	− 1835.97i	− 2.54643i
$722$	0	0
$723$	240.000	0.331950
$724$	0	0
$725$	311.769i	0.430026i
$726$	0	0
$727$	790.000	1.08666	0.543329	−	0.839520i	$-0.317164\pi$
$727$	790.000	1.08666	0.543329	+	0.839520i	$0.317164\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	100.000	0.136799
$732$	0	0
$733$	−1150.00	−1.56889	−0.784447	−	0.620195i	$-0.787053\pi$
$733$	−1150.00	−1.56889	−0.784447	+	0.620195i	$0.787053\pi$
$734$	0	0
$735$	− 353.338i	− 0.480732i
$736$	0	0
$737$	− 762.102i	− 1.03406i
$738$	0	0
$739$	−578.000	−0.782138	−0.391069	−	0.920361i	$-0.627895\pi$
$739$	−578.000	−0.782138	−0.391069	+	0.920361i	$0.627895\pi$
$740$	0	0
$741$	798.000	1.07692
$742$	0	0
$743$	235.559i	0.317038i	0.987356	+	0.158519i	$0.0506718\pi$
$743$	235.559i	0.317038i	−0.987356	+	0.158519i	$0.949328\pi$
$744$	0	0
$745$	−80.0000	−0.107383
$746$	0	0
$747$	210.000	0.281124
$748$	0	0
$749$	− 623.538i	− 0.832494i
$750$	0	0
$751$	− 952.628i	− 1.26848i	−0.773137	−	0.634240i	$-0.781313\pi$
$751$	− 952.628i	− 1.26848i	0.773137	−	0.634240i	$-0.218687\pi$
$752$	0	0
$753$	− 3.46410i	− 0.00460040i
$754$	0	0
$755$	900.666i	1.19294i
$756$	0	0
$757$	−250.000	−0.330251	−0.165125	−	0.986273i	$-0.552803\pi$
$757$	−250.000	−0.330251	−0.165125	+	0.986273i	$0.552803\pi$
$758$	0	0
$759$	346.410i	0.456403i
$760$	0	0
$761$	−770.000	−1.01183	−0.505913	−	0.862584i	$-0.668844\pi$
$761$	−770.000	−1.01183	−0.505913	+	0.862584i	$0.668844\pi$
$762$	0	0
$763$	− 1558.85i	− 2.04305i
$764$	0	0
$765$	−120.000	−0.156863
$766$	0	0
$767$	−840.000	−1.09518
$768$	0	0
$769$	110.000	0.143043	0.0715215	−	0.997439i	$-0.477215\pi$
$769$	110.000	0.143043	0.0715215	+	0.997439i	$0.477215\pi$
$770$	0	0
$771$	852.000	1.10506
$772$	0	0
$773$	− 145.492i	− 0.188218i	−0.995562	−	0.0941088i	$-0.970000\pi$
$773$	− 145.492i	− 0.188218i	0.995562	−	0.0941088i	$-0.0300002\pi$
$774$	0	0
$775$	155.885i	0.201141i
$776$	0	0
$777$	−180.000	−0.231660
$778$	0	0
$779$	− 658.179i	− 0.844903i
$780$	0	0
$781$	1039.23i	1.33064i
$782$	0	0
$783$	180.000	0.229885
$784$	0	0
$785$	920.000	1.17197
$786$	0	0
$787$	96.9948i	0.123246i	0.998099	+	0.0616232i	$0.0196277\pi$
$787$	96.9948i	0.123246i	−0.998099	+	0.0616232i	$0.980372\pi$
$788$	0	0
$789$	− 346.410i	− 0.439050i
$790$	0	0
$791$	− 69.2820i	− 0.0875879i
$792$	0	0
$793$	242.487i	0.305785i
$794$	0	0
$795$	288.000	0.362264
$796$	0	0
$797$	− 339.482i	− 0.425950i	−0.977058	−	0.212975i	$-0.931685\pi$
$797$	− 339.482i	− 0.425950i	0.977058	−	0.212975i	$-0.0683153\pi$
$798$	0	0
$799$	800.000	1.00125
$800$	0	0
$801$	311.769i	0.389225i
$802$	0	0
$803$	100.000	0.124533
$804$	0	0
$805$	800.000	0.993789
$806$	0	0
$807$	−720.000	−0.892193
$808$	0	0
$809$	−182.000	−0.224969	−0.112485	−	0.993653i	$-0.535881\pi$
$809$	−182.000	−0.224969	−0.112485	+	0.993653i	$0.535881\pi$
$810$	0	0
$811$	831.384i	1.02513i	0.858647	+	0.512567i	$0.171306\pi$
$811$	831.384i	1.02513i	−0.858647	+	0.512567i	$0.828694\pi$
$812$	0	0
$813$	294.449i	0.362175i
$814$	0	0
$815$	−680.000	−0.834356
$816$	0	0
$817$	−190.000	−0.232558
$818$	0	0
$819$	727.461i	0.888231i
$820$	0	0
$821$	−8.00000	−0.00974421	−0.00487211	−	0.999988i	$-0.501551\pi$
$821$	−8.00000	−0.00974421	−0.00487211	+	0.999988i	$0.501551\pi$
$822$	0	0
$823$	−950.000	−1.15431	−0.577157	−	0.816633i	$-0.695838\pi$
$823$	−950.000	−1.15431	−0.577157	+	0.816633i	$0.695838\pi$
$824$	0	0
$825$	− 155.885i	− 0.188951i
$826$	0	0
$827$	478.046i	0.578048i	0.957322	+	0.289024i	$0.0933308\pi$
$827$	478.046i	0.578048i	−0.957322	+	0.289024i	$0.906669\pi$
$828$	0	0
$829$	1195.12i	1.44163i	0.693125	+	0.720817i	$0.256233\pi$
$829$	1195.12i	1.44163i	−0.693125	+	0.720817i	$0.743767\pi$
$830$	0	0
$831$	17.3205i	0.0208430i
$832$	0	0
$833$	510.000	0.612245
$834$	0	0
$835$	526.543i	0.630591i
$836$	0	0
$837$	90.0000	0.107527
$838$	0	0
$839$	1177.79i	1.40381i	0.712272	+	0.701904i	$0.247666\pi$
$839$	1177.79i	1.40381i	−0.712272	+	0.701904i	$0.752334\pi$
$840$	0	0
$841$	−359.000	−0.426873
$842$	0	0
$843$	660.000	0.782918
$844$	0	0
$845$	−1676.00	−1.98343
$846$	0	0
$847$	−210.000	−0.247934
$848$	0	0
$849$	− 121.244i	− 0.142807i
$850$	0	0
$851$	− 207.846i	− 0.244237i
$852$	0	0
$853$	890.000	1.04338	0.521688	−	0.853136i	$-0.325302\pi$
$853$	890.000	1.04338	0.521688	+	0.853136i	$0.325302\pi$
$854$	0	0
$855$	228.000	0.266667
$856$	0	0
$857$	1254.00i	1.46325i	0.681708	+	0.731625i	$0.261238\pi$
$857$	1254.00i	1.46325i	−0.681708	+	0.731625i	$0.738762\pi$
$858$	0	0
$859$	−182.000	−0.211874	−0.105937	−	0.994373i	$-0.533784\pi$
$859$	−182.000	−0.211874	−0.105937	+	0.994373i	$0.533784\pi$
$860$	0	0
$861$	600.000	0.696864
$862$	0	0
$863$	− 1080.80i	− 1.25238i	−0.779672	−	0.626188i	$-0.784614\pi$
$863$	− 1080.80i	− 1.25238i	0.779672	−	0.626188i	$-0.215386\pi$
$864$	0	0
$865$	942.236i	1.08929i
$866$	0	0
$867$	327.358i	0.377575i
$868$	0	0
$869$	− 173.205i	− 0.199315i
$870$	0	0
$871$	1848.00	2.12170
$872$	0	0
$873$	− 228.631i	− 0.261891i
$874$	0	0
$875$	−1360.00	−1.55429
$876$	0	0
$877$	− 1188.19i	− 1.35483i	−0.735601	−	0.677416i	$-0.763100\pi$
$877$	− 1188.19i	− 1.35483i	0.735601	−	0.677416i	$-0.236900\pi$
$878$	0	0
$879$	312.000	0.354949
$880$	0	0
$881$	550.000	0.624291	0.312145	−	0.950034i	$-0.398952\pi$
$881$	550.000	0.624291	0.312145	+	0.950034i	$0.398952\pi$
$882$	0	0
$883$	1450.00	1.64213	0.821065	−	0.570835i	$-0.193381\pi$
$883$	1450.00	1.64213	0.821065	+	0.570835i	$0.193381\pi$
$884$	0	0
$885$	−240.000	−0.271186
$886$	0	0
$887$	1254.00i	1.41376i	0.707334	+	0.706880i	$0.249898\pi$
$887$	1254.00i	1.41376i	−0.707334	+	0.706880i	$0.750102\pi$
$888$	0	0
$889$	1143.15i	1.28589i
$890$	0	0
$891$	−90.0000	−0.101010
$892$	0	0
$893$	−1520.00	−1.70213
$894$	0	0
$895$	415.692i	0.464461i
$896$	0	0
$897$	−840.000	−0.936455
$898$	0	0
$899$	−600.000	−0.667408
$900$	0	0
$901$	415.692i	0.461368i
$902$	0	0
$903$	− 173.205i	− 0.191811i
$904$	0	0
$905$	1039.23i	1.14832i
$906$	0	0
$907$	110.851i	0.122217i	0.998131	+	0.0611087i	$0.0194636\pi$
$907$	110.851i	0.122217i	−0.998131	+	0.0611087i	$0.980536\pi$
$908$	0	0
$909$	−300.000	−0.330033
$910$	0	0
$911$	− 796.743i	− 0.874581i	−0.899320	−	0.437291i	$-0.855938\pi$
$911$	− 796.743i	− 0.874581i	0.899320	−	0.437291i	$-0.144062\pi$
$912$	0	0
$913$	700.000	0.766703
$914$	0	0
$915$	69.2820i	0.0757181i
$916$	0	0
$917$	380.000	0.414395
$918$	0	0
$919$	−62.0000	−0.0674646	−0.0337323	−	0.999431i	$-0.510739\pi$
$919$	−62.0000	−0.0674646	−0.0337323	+	0.999431i	$0.510739\pi$
$920$	0	0
$921$	252.000	0.273616
$922$	0	0
$923$	−2520.00	−2.73023
$924$	0	0
$925$	93.5307i	0.101114i
$926$	0	0
$927$	550.792i	0.594166i
$928$	0	0
$929$	−242.000	−0.260495	−0.130248	−	0.991482i	$-0.541577\pi$
$929$	−242.000	−0.260495	−0.130248	+	0.991482i	$0.541577\pi$
$930$	0	0
$931$	−969.000	−1.04082
$932$	0	0
$933$	1004.59i	1.07673i
$934$	0	0
$935$	−400.000	−0.427807
$936$	0	0
$937$	110.000	0.117396	0.0586980	−	0.998276i	$-0.481305\pi$
$937$	110.000	0.117396	0.0586980	+	0.998276i	$0.481305\pi$
$938$	0	0
$939$	640.859i	0.682491i
$940$	0	0
$941$	− 796.743i	− 0.846699i	−0.905967	−	0.423349i	$-0.860854\pi$
$941$	− 796.743i	− 0.846699i	0.905967	−	0.423349i	$-0.139146\pi$
$942$	0	0
$943$	692.820i	0.734698i
$944$	0	0
$945$	207.846i	0.219943i
$946$	0	0
$947$	−1450.00	−1.53115	−0.765576	−	0.643346i	$-0.777546\pi$
$947$	−1450.00	−1.53115	−0.765576	+	0.643346i	$0.777546\pi$
$948$	0	0
$949$	242.487i	0.255519i
$950$	0	0
$951$	48.0000	0.0504732
$952$	0	0
$953$	353.338i	0.370764i	0.982667	+	0.185382i	$0.0593523\pi$
$953$	353.338i	0.370764i	−0.982667	+	0.185382i	$0.940648\pi$
$954$	0	0
$955$	1328.00	1.39058
$956$	0	0
$957$	600.000	0.626959
$958$	0	0
$959$	1900.00	1.98123
$960$	0	0
$961$	661.000	0.687825
$962$	0	0
$963$	187.061i	0.194249i
$964$	0	0
$965$	− 387.979i	− 0.402051i
$966$	0	0
$967$	−470.000	−0.486039	−0.243020	−	0.970021i	$-0.578138\pi$
$967$	−470.000	−0.486039	−0.243020	+	0.970021i	$0.578138\pi$
$968$	0	0
$969$	329.090i	0.339618i
$970$	0	0
$971$	519.615i	0.535134i	0.963539	+	0.267567i	$0.0862197\pi$
$971$	519.615i	0.535134i	−0.963539	+	0.267567i	$0.913780\pi$
$972$	0	0
$973$	−500.000	−0.513875
$974$	0	0
$975$	378.000	0.387692
$976$	0	0
$977$	− 1669.70i	− 1.70900i	−0.519448	−	0.854502i	$-0.673862\pi$
$977$	− 1669.70i	− 1.70900i	0.519448	−	0.854502i	$-0.326138\pi$
$978$	0	0
$979$	1039.23i	1.06152i
$980$	0	0
$981$	467.654i	0.476711i
$982$	0	0
$983$	− 1690.48i	− 1.71972i	−0.510533	−	0.859858i	$-0.670552\pi$
$983$	− 1690.48i	− 1.71972i	0.510533	−	0.859858i	$-0.329448\pi$
$984$	0	0
$985$	640.000	0.649746
$986$	0	0
$987$	− 1385.64i	− 1.40389i
$988$	0	0
$989$	200.000	0.202224
$990$	0	0
$991$	− 571.577i	− 0.576768i	−0.957515	−	0.288384i	$-0.906882\pi$
$991$	− 571.577i	− 0.576768i	0.957515	−	0.288384i	$-0.0931179\pi$
$992$	0	0
$993$	300.000	0.302115
$994$	0	0
$995$	−392.000	−0.393970
$996$	0	0
$997$	1550.00	1.55466	0.777332	−	0.629091i	$-0.216573\pi$
$997$	1550.00	1.55466	0.777332	+	0.629091i	$0.216573\pi$
$998$	0	0
$999$	54.0000	0.0540541

Twists

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

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

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.o (of order \(2\), degree \(1\), not minimal)

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

Introduction

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