LMFDB - Embedded newform 198.3.d.a.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [198,3,Mod(109,198)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("198.109"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(198, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	198.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$5.39510923433$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	198.109
Dual form			198.3.d.a.109.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -4.00000 q^{5} +4.24264i q^{7} +2.82843i q^{8} +5.65685i q^{10} -11.0000 q^{11} +16.9706i q^{13} +6.00000 q^{14} +4.00000 q^{16} +4.24264i q^{17} +21.2132i q^{19} +8.00000 q^{20} +15.5563i q^{22} -10.0000 q^{23} -9.00000 q^{25} +24.0000 q^{26} -8.48528i q^{28} -21.2132i q^{29} +20.0000 q^{31} -5.65685i q^{32} +6.00000 q^{34} -16.9706i q^{35} -40.0000 q^{37} +30.0000 q^{38} -11.3137i q^{40} +63.6396i q^{41} -46.6690i q^{43} +22.0000 q^{44} +14.1421i q^{46} -40.0000 q^{47} +31.0000 q^{49} +12.7279i q^{50} -33.9411i q^{52} +20.0000 q^{53} +44.0000 q^{55} -12.0000 q^{56} -30.0000 q^{58} -70.0000 q^{59} -84.8528i q^{61} -28.2843i q^{62} -8.00000 q^{64} -67.8823i q^{65} -40.0000 q^{67} -8.48528i q^{68} -24.0000 q^{70} -40.0000 q^{71} -16.9706i q^{73} +56.5685i q^{74} -42.4264i q^{76} -46.6690i q^{77} -63.6396i q^{79} -16.0000 q^{80} +90.0000 q^{82} +101.823i q^{83} -16.9706i q^{85} -66.0000 q^{86} -31.1127i q^{88} +128.000 q^{89} -72.0000 q^{91} +20.0000 q^{92} +56.5685i q^{94} -84.8528i q^{95} +80.0000 q^{97} -43.8406i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 8 q^{5} - 22 q^{11} + 12 q^{14} + 8 q^{16} + 16 q^{20} - 20 q^{23} - 18 q^{25} + 48 q^{26} + 40 q^{31} + 12 q^{34} - 80 q^{37} + 60 q^{38} + 44 q^{44} - 80 q^{47} + 62 q^{49} + 40 q^{53}+ \cdots + 160 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 8 * q^5 - 22 * q^11 + 12 * q^14 + 8 * q^16 + 16 * q^20 - 20 * q^23 - 18 * q^25 + 48 * q^26 + 40 * q^31 + 12 * q^34 - 80 * q^37 + 60 * q^38 + 44 * q^44 - 80 * q^47 + 62 * q^49 + 40 * q^53 + 88 * q^55 - 24 * q^56 - 60 * q^58 - 140 * q^59 - 16 * q^64 - 80 * q^67 - 48 * q^70 - 80 * q^71 - 32 * q^80 + 180 * q^82 - 132 * q^86 + 256 * q^89 - 144 * q^91 + 40 * q^92 + 160 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/198\mathbb{Z}\right)^\times$.

$n$	$145$	$155$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	−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
$7$	4.24264i	0.606092i	0.952976	+	0.303046i	$0.0980035\pi$
$7$	4.24264i	0.606092i	−0.952976	+	0.303046i	$0.901996\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	5.65685i	0.565685i
$11$	−11.0000	−1.00000
$11$	−11.0000	−1.00000
$12$	0	0
$13$	16.9706i	1.30543i	0.757604	+	0.652714i	$0.226370\pi$
$13$	16.9706i	1.30543i	−0.757604	+	0.652714i	$0.773630\pi$
$14$	6.00000	0.428571
$15$	0	0
$16$	4.00000	0.250000
$17$	4.24264i	0.249567i	0.992184	+	0.124784i	$0.0398236\pi$
$17$	4.24264i	0.249567i	−0.992184	+	0.124784i	$0.960176\pi$
$18$	0	0
$19$	21.2132i	1.11648i	0.829678	+	0.558242i	$0.188524\pi$
$19$	21.2132i	1.11648i	−0.829678	+	0.558242i	$0.811476\pi$
$20$	8.00000	0.400000
$21$	0	0
$22$	15.5563i	0.707107i
$23$	−10.0000	−0.434783	−0.217391	−	0.976085i	$-0.569755\pi$
$23$	−10.0000	−0.434783	−0.217391	+	0.976085i	$0.569755\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	24.0000	0.923077
$27$	0	0
$28$	− 8.48528i	− 0.303046i
$29$	− 21.2132i	− 0.731490i	−0.930715	−	0.365745i	$-0.880814\pi$
$29$	− 21.2132i	− 0.731490i	0.930715	−	0.365745i	$-0.119186\pi$
$30$	0	0
$31$	20.0000	0.645161	0.322581	−	0.946542i	$-0.395450\pi$
$31$	20.0000	0.645161	0.322581	+	0.946542i	$0.395450\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	6.00000	0.176471
$35$	− 16.9706i	− 0.484873i
$36$	0	0
$37$	−40.0000	−1.08108	−0.540541	−	0.841318i	$-0.681780\pi$
$37$	−40.0000	−1.08108	−0.540541	+	0.841318i	$0.681780\pi$
$38$	30.0000	0.789474
$39$	0	0
$40$	− 11.3137i	− 0.282843i
$41$	63.6396i	1.55219i	0.630619	+	0.776093i	$0.282801\pi$
$41$	63.6396i	1.55219i	−0.630619	+	0.776093i	$0.717199\pi$
$42$	0	0
$43$	− 46.6690i	− 1.08533i	−0.839950	−	0.542663i	$-0.817416\pi$
$43$	− 46.6690i	− 1.08533i	0.839950	−	0.542663i	$-0.182584\pi$
$44$	22.0000	0.500000
$45$	0	0
$46$	14.1421i	0.307438i
$47$	−40.0000	−0.851064	−0.425532	−	0.904943i	$-0.639913\pi$
$47$	−40.0000	−0.851064	−0.425532	+	0.904943i	$0.639913\pi$
$48$	0	0
$49$	31.0000	0.632653
$50$	12.7279i	0.254558i
$51$	0	0
$52$	− 33.9411i	− 0.652714i
$53$	20.0000	0.377358	0.188679	−	0.982039i	$-0.439579\pi$
$53$	20.0000	0.377358	0.188679	+	0.982039i	$0.439579\pi$
$54$	0	0
$55$	44.0000	0.800000
$56$	−12.0000	−0.214286
$57$	0	0
$58$	−30.0000	−0.517241
$59$	−70.0000	−1.18644	−0.593220	−	0.805040i	$-0.702144\pi$
$59$	−70.0000	−1.18644	−0.593220	+	0.805040i	$0.702144\pi$
$60$	0	0
$61$	− 84.8528i	− 1.39103i	−0.718512	−	0.695515i	$-0.755176\pi$
$61$	− 84.8528i	− 1.39103i	0.718512	−	0.695515i	$-0.244824\pi$
$62$	− 28.2843i	− 0.456198i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 67.8823i	− 1.04434i
$66$	0	0
$67$	−40.0000	−0.597015	−0.298507	−	0.954407i	$-0.596489\pi$
$67$	−40.0000	−0.597015	−0.298507	+	0.954407i	$0.596489\pi$
$68$	− 8.48528i	− 0.124784i
$69$	0	0
$70$	−24.0000	−0.342857
$71$	−40.0000	−0.563380	−0.281690	−	0.959505i	$-0.590895\pi$
$71$	−40.0000	−0.563380	−0.281690	+	0.959505i	$0.590895\pi$
$72$	0	0
$73$	− 16.9706i	− 0.232473i	−0.993222	−	0.116237i	$-0.962917\pi$
$73$	− 16.9706i	− 0.232473i	0.993222	−	0.116237i	$-0.0370831\pi$
$74$	56.5685i	0.764440i
$75$	0	0
$76$	− 42.4264i	− 0.558242i
$77$	− 46.6690i	− 0.606092i
$78$	0	0
$79$	− 63.6396i	− 0.805565i	−0.915296	−	0.402782i	$-0.868043\pi$
$79$	− 63.6396i	− 0.805565i	0.915296	−	0.402782i	$-0.131957\pi$
$80$	−16.0000	−0.200000
$81$	0	0
$82$	90.0000	1.09756
$83$	101.823i	1.22679i	0.789777	+	0.613394i	$0.210196\pi$
$83$	101.823i	1.22679i	−0.789777	+	0.613394i	$0.789804\pi$
$84$	0	0
$85$	− 16.9706i	− 0.199654i
$86$	−66.0000	−0.767442
$87$	0	0
$88$	− 31.1127i	− 0.353553i
$89$	128.000	1.43820	0.719101	−	0.694905i	$-0.244554\pi$
$89$	128.000	1.43820	0.719101	+	0.694905i	$0.244554\pi$
$90$	0	0
$91$	−72.0000	−0.791209
$92$	20.0000	0.217391
$93$	0	0
$94$	56.5685i	0.601793i
$95$	− 84.8528i	− 0.893188i
$96$	0	0
$97$	80.0000	0.824742	0.412371	−	0.911016i	$-0.364701\pi$
$97$	80.0000	0.824742	0.412371	+	0.911016i	$0.364701\pi$
$98$	− 43.8406i	− 0.447353i
$99$	0	0
$100$	18.0000	0.180000
$101$	21.2132i	0.210032i	0.994471	+	0.105016i	$0.0334893\pi$
$101$	21.2132i	0.210032i	−0.994471	+	0.105016i	$0.966511\pi$
$102$	0	0
$103$	−100.000	−0.970874	−0.485437	−	0.874272i	$-0.661339\pi$
$103$	−100.000	−0.970874	−0.485437	+	0.874272i	$0.661339\pi$
$104$	−48.0000	−0.461538
$105$	0	0
$106$	− 28.2843i	− 0.266833i
$107$	186.676i	1.74464i	0.488938	+	0.872319i	$0.337384\pi$
$107$	186.676i	1.74464i	−0.488938	+	0.872319i	$0.662616\pi$
$108$	0	0
$109$	169.706i	1.55693i	0.627687	+	0.778466i	$0.284002\pi$
$109$	169.706i	1.55693i	−0.627687	+	0.778466i	$0.715998\pi$
$110$	− 62.2254i	− 0.565685i
$111$	0	0
$112$	16.9706i	0.151523i
$113$	200.000	1.76991	0.884956	−	0.465675i	$-0.154188\pi$
$113$	200.000	1.76991	0.884956	+	0.465675i	$0.154188\pi$
$114$	0	0
$115$	40.0000	0.347826
$116$	42.4264i	0.365745i
$117$	0	0
$118$	98.9949i	0.838940i
$119$	−18.0000	−0.151261
$120$	0	0
$121$	121.000	1.00000
$122$	−120.000	−0.983607
$123$	0	0
$124$	−40.0000	−0.322581
$125$	136.000	1.08800
$126$	0	0
$127$	207.889i	1.63692i	0.574560	+	0.818462i	$0.305173\pi$
$127$	207.889i	1.63692i	−0.574560	+	0.818462i	$0.694827\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−96.0000	−0.738462
$131$	− 254.558i	− 1.94319i	−0.236641	−	0.971597i	$-0.576047\pi$
$131$	− 254.558i	− 1.94319i	0.236641	−	0.971597i	$-0.423953\pi$
$132$	0	0
$133$	−90.0000	−0.676692
$134$	56.5685i	0.422153i
$135$	0	0
$136$	−12.0000	−0.0882353
$137$	80.0000	0.583942	0.291971	−	0.956427i	$-0.405689\pi$
$137$	80.0000	0.583942	0.291971	+	0.956427i	$0.405689\pi$
$138$	0	0
$139$	148.492i	1.06829i	0.845393	+	0.534145i	$0.179367\pi$
$139$	148.492i	1.06829i	−0.845393	+	0.534145i	$0.820633\pi$
$140$	33.9411i	0.242437i
$141$	0	0
$142$	56.5685i	0.398370i
$143$	− 186.676i	− 1.30543i
$144$	0	0
$145$	84.8528i	0.585192i
$146$	−24.0000	−0.164384
$147$	0	0
$148$	80.0000	0.540541
$149$	190.919i	1.28133i	0.767819	+	0.640667i	$0.221342\pi$
$149$	190.919i	1.28133i	−0.767819	+	0.640667i	$0.778658\pi$
$150$	0	0
$151$	− 106.066i	− 0.702424i	−0.936296	−	0.351212i	$-0.885770\pi$
$151$	− 106.066i	− 0.702424i	0.936296	−	0.351212i	$-0.114230\pi$
$152$	−60.0000	−0.394737
$153$	0	0
$154$	−66.0000	−0.428571
$155$	−80.0000	−0.516129
$156$	0	0
$157$	50.0000	0.318471	0.159236	−	0.987241i	$-0.449097\pi$
$157$	50.0000	0.318471	0.159236	+	0.987241i	$0.449097\pi$
$158$	−90.0000	−0.569620
$159$	0	0
$160$	22.6274i	0.141421i
$161$	− 42.4264i	− 0.263518i
$162$	0	0
$163$	200.000	1.22699	0.613497	−	0.789697i	$-0.289762\pi$
$163$	200.000	1.22699	0.613497	+	0.789697i	$0.289762\pi$
$164$	− 127.279i	− 0.776093i
$165$	0	0
$166$	144.000	0.867470
$167$	186.676i	1.11782i	0.829228	+	0.558911i	$0.188781\pi$
$167$	186.676i	1.11782i	−0.829228	+	0.558911i	$0.811219\pi$
$168$	0	0
$169$	−119.000	−0.704142
$170$	−24.0000	−0.141176
$171$	0	0
$172$	93.3381i	0.542663i
$173$	216.375i	1.25072i	0.780336	+	0.625360i	$0.215048\pi$
$173$	216.375i	1.25072i	−0.780336	+	0.625360i	$0.784952\pi$
$174$	0	0
$175$	− 38.1838i	− 0.218193i
$176$	−44.0000	−0.250000
$177$	0	0
$178$	− 181.019i	− 1.01696i
$179$	−208.000	−1.16201	−0.581006	−	0.813900i	$-0.697341\pi$
$179$	−208.000	−1.16201	−0.581006	+	0.813900i	$0.697341\pi$
$180$	0	0
$181$	−190.000	−1.04972	−0.524862	−	0.851187i	$-0.675883\pi$
$181$	−190.000	−1.04972	−0.524862	+	0.851187i	$0.675883\pi$
$182$	101.823i	0.559469i
$183$	0	0
$184$	− 28.2843i	− 0.153719i
$185$	160.000	0.864865
$186$	0	0
$187$	− 46.6690i	− 0.249567i
$188$	80.0000	0.425532
$189$	0	0
$190$	−120.000	−0.631579
$191$	−232.000	−1.21466	−0.607330	−	0.794450i	$-0.707759\pi$
$191$	−232.000	−1.21466	−0.607330	+	0.794450i	$0.707759\pi$
$192$	0	0
$193$	16.9706i	0.0879304i	0.999033	+	0.0439652i	$0.0139991\pi$
$193$	16.9706i	0.0879304i	−0.999033	+	0.0439652i	$0.986001\pi$
$194$	− 113.137i	− 0.583181i
$195$	0	0
$196$	−62.0000	−0.316327
$197$	− 292.742i	− 1.48600i	−0.669291	−	0.743001i	$-0.733402\pi$
$197$	− 292.742i	− 1.48600i	0.669291	−	0.743001i	$-0.266598\pi$
$198$	0	0
$199$	−52.0000	−0.261307	−0.130653	−	0.991428i	$-0.541707\pi$
$199$	−52.0000	−0.261307	−0.130653	+	0.991428i	$0.541707\pi$
$200$	− 25.4558i	− 0.127279i
$201$	0	0
$202$	30.0000	0.148515
$203$	90.0000	0.443350
$204$	0	0
$205$	− 254.558i	− 1.24175i
$206$	141.421i	0.686511i
$207$	0	0
$208$	67.8823i	0.326357i
$209$	− 233.345i	− 1.11648i
$210$	0	0
$211$	− 190.919i	− 0.904829i	−0.891808	−	0.452414i	$-0.850563\pi$
$211$	− 190.919i	− 0.904829i	0.891808	−	0.452414i	$-0.149437\pi$
$212$	−40.0000	−0.188679
$213$	0	0
$214$	264.000	1.23364
$215$	186.676i	0.868261i
$216$	0	0
$217$	84.8528i	0.391027i
$218$	240.000	1.10092
$219$	0	0
$220$	−88.0000	−0.400000
$221$	−72.0000	−0.325792
$222$	0	0
$223$	140.000	0.627803	0.313901	−	0.949456i	$-0.398364\pi$
$223$	140.000	0.627803	0.313901	+	0.949456i	$0.398364\pi$
$224$	24.0000	0.107143
$225$	0	0
$226$	− 282.843i	− 1.25152i
$227$	− 101.823i	− 0.448561i	−0.974525	−	0.224281i	$-0.927997\pi$
$227$	− 101.823i	− 0.448561i	0.974525	−	0.224281i	$-0.0720032\pi$
$228$	0	0
$229$	8.00000	0.0349345	0.0174672	−	0.999847i	$-0.494440\pi$
$229$	8.00000	0.0349345	0.0174672	+	0.999847i	$0.494440\pi$
$230$	− 56.5685i	− 0.245950i
$231$	0	0
$232$	60.0000	0.258621
$233$	38.1838i	0.163879i	0.996637	+	0.0819394i	$0.0261114\pi$
$233$	38.1838i	0.163879i	−0.996637	+	0.0819394i	$0.973889\pi$
$234$	0	0
$235$	160.000	0.680851
$236$	140.000	0.593220
$237$	0	0
$238$	25.4558i	0.106957i
$239$	254.558i	1.06510i	0.846399	+	0.532549i	$0.178766\pi$
$239$	254.558i	1.06510i	−0.846399	+	0.532549i	$0.821234\pi$
$240$	0	0
$241$	− 424.264i	− 1.76043i	−0.474573	−	0.880216i	$-0.657398\pi$
$241$	− 424.264i	− 1.76043i	0.474573	−	0.880216i	$-0.342602\pi$
$242$	− 171.120i	− 0.707107i
$243$	0	0
$244$	169.706i	0.695515i
$245$	−124.000	−0.506122
$246$	0	0
$247$	−360.000	−1.45749
$248$	56.5685i	0.228099i
$249$	0	0
$250$	− 192.333i	− 0.769332i
$251$	−160.000	−0.637450	−0.318725	−	0.947847i	$-0.603255\pi$
$251$	−160.000	−0.637450	−0.318725	+	0.947847i	$0.603255\pi$
$252$	0	0
$253$	110.000	0.434783
$254$	294.000	1.15748
$255$	0	0
$256$	16.0000	0.0625000
$257$	440.000	1.71206	0.856031	−	0.516924i	$-0.172923\pi$
$257$	440.000	1.71206	0.856031	+	0.516924i	$0.172923\pi$
$258$	0	0
$259$	− 169.706i	− 0.655234i
$260$	135.765i	0.522171i
$261$	0	0
$262$	−360.000	−1.37405
$263$	− 186.676i	− 0.709795i	−0.934905	−	0.354898i	$-0.884516\pi$
$263$	− 186.676i	− 0.709795i	0.934905	−	0.354898i	$-0.115484\pi$
$264$	0	0
$265$	−80.0000	−0.301887
$266$	127.279i	0.478493i
$267$	0	0
$268$	80.0000	0.298507
$269$	−340.000	−1.26394	−0.631970	−	0.774993i	$-0.717754\pi$
$269$	−340.000	−1.26394	−0.631970	+	0.774993i	$0.717754\pi$
$270$	0	0
$271$	− 275.772i	− 1.01761i	−0.860883	−	0.508804i	$-0.830088\pi$
$271$	− 275.772i	− 1.01761i	0.860883	−	0.508804i	$-0.169912\pi$
$272$	16.9706i	0.0623918i
$273$	0	0
$274$	− 113.137i	− 0.412909i
$275$	99.0000	0.360000
$276$	0	0
$277$	237.588i	0.857718i	0.903371	+	0.428859i	$0.141084\pi$
$277$	237.588i	0.857718i	−0.903371	+	0.428859i	$0.858916\pi$
$278$	210.000	0.755396
$279$	0	0
$280$	48.0000	0.171429
$281$	275.772i	0.981394i	0.871330	+	0.490697i	$0.163258\pi$
$281$	275.772i	0.981394i	−0.871330	+	0.490697i	$0.836742\pi$
$282$	0	0
$283$	46.6690i	0.164908i	0.996595	+	0.0824541i	$0.0262758\pi$
$283$	46.6690i	0.164908i	−0.996595	+	0.0824541i	$0.973724\pi$
$284$	80.0000	0.281690
$285$	0	0
$286$	−264.000	−0.923077
$287$	−270.000	−0.940767
$288$	0	0
$289$	271.000	0.937716
$290$	120.000	0.413793
$291$	0	0
$292$	33.9411i	0.116237i
$293$	− 258.801i	− 0.883280i	−0.897192	−	0.441640i	$-0.854397\pi$
$293$	− 258.801i	− 0.883280i	0.897192	−	0.441640i	$-0.145603\pi$
$294$	0	0
$295$	280.000	0.949153
$296$	− 113.137i	− 0.382220i
$297$	0	0
$298$	270.000	0.906040
$299$	− 169.706i	− 0.567577i
$300$	0	0
$301$	198.000	0.657807
$302$	−150.000	−0.496689
$303$	0	0
$304$	84.8528i	0.279121i
$305$	339.411i	1.11282i
$306$	0	0
$307$	386.080i	1.25759i	0.777571	+	0.628795i	$0.216452\pi$
$307$	386.080i	1.25759i	−0.777571	+	0.628795i	$0.783548\pi$
$308$	93.3381i	0.303046i
$309$	0	0
$310$	113.137i	0.364958i
$311$	−10.0000	−0.0321543	−0.0160772	−	0.999871i	$-0.505118\pi$
$311$	−10.0000	−0.0321543	−0.0160772	+	0.999871i	$0.505118\pi$
$312$	0	0
$313$	−400.000	−1.27796	−0.638978	−	0.769225i	$-0.720642\pi$
$313$	−400.000	−1.27796	−0.638978	+	0.769225i	$0.720642\pi$
$314$	− 70.7107i	− 0.225193i
$315$	0	0
$316$	127.279i	0.402782i
$317$	−580.000	−1.82965	−0.914826	−	0.403847i	$-0.867673\pi$
$317$	−580.000	−1.82965	−0.914826	+	0.403847i	$0.867673\pi$
$318$	0	0
$319$	233.345i	0.731490i
$320$	32.0000	0.100000
$321$	0	0
$322$	−60.0000	−0.186335
$323$	−90.0000	−0.278638
$324$	0	0
$325$	− 152.735i	− 0.469954i
$326$	− 282.843i	− 0.867616i
$327$	0	0
$328$	−180.000	−0.548780
$329$	− 169.706i	− 0.515823i
$330$	0	0
$331$	−520.000	−1.57100	−0.785498	−	0.618864i	$-0.787593\pi$
$331$	−520.000	−1.57100	−0.785498	+	0.618864i	$0.787593\pi$
$332$	− 203.647i	− 0.613394i
$333$	0	0
$334$	264.000	0.790419
$335$	160.000	0.477612
$336$	0	0
$337$	526.087i	1.56109i	0.625099	+	0.780545i	$0.285058\pi$
$337$	526.087i	1.56109i	−0.625099	+	0.780545i	$0.714942\pi$
$338$	168.291i	0.497904i
$339$	0	0
$340$	33.9411i	0.0998268i
$341$	−220.000	−0.645161
$342$	0	0
$343$	339.411i	0.989537i
$344$	132.000	0.383721
$345$	0	0
$346$	306.000	0.884393
$347$	− 101.823i	− 0.293439i	−0.989178	−	0.146720i	$-0.953129\pi$
$347$	− 101.823i	− 0.293439i	0.989178	−	0.146720i	$-0.0468715\pi$
$348$	0	0
$349$	339.411i	0.972525i	0.873813	+	0.486263i	$0.161640\pi$
$349$	339.411i	0.972525i	−0.873813	+	0.486263i	$0.838360\pi$
$350$	−54.0000	−0.154286
$351$	0	0
$352$	62.2254i	0.176777i
$353$	−40.0000	−0.113314	−0.0566572	−	0.998394i	$-0.518044\pi$
$353$	−40.0000	−0.113314	−0.0566572	+	0.998394i	$0.518044\pi$
$354$	0	0
$355$	160.000	0.450704
$356$	−256.000	−0.719101
$357$	0	0
$358$	294.156i	0.821666i
$359$	84.8528i	0.236359i	0.992992	+	0.118179i	$0.0377058\pi$
$359$	84.8528i	0.236359i	−0.992992	+	0.118179i	$0.962294\pi$
$360$	0	0
$361$	−89.0000	−0.246537
$362$	268.701i	0.742267i
$363$	0	0
$364$	144.000	0.395604
$365$	67.8823i	0.185979i
$366$	0	0
$367$	380.000	1.03542	0.517711	−	0.855555i	$-0.326784\pi$
$367$	380.000	1.03542	0.517711	+	0.855555i	$0.326784\pi$
$368$	−40.0000	−0.108696
$369$	0	0
$370$	− 226.274i	− 0.611552i
$371$	84.8528i	0.228714i
$372$	0	0
$373$	− 101.823i	− 0.272985i	−0.990641	−	0.136492i	$-0.956417\pi$
$373$	− 101.823i	− 0.272985i	0.990641	−	0.136492i	$-0.0435829\pi$
$374$	−66.0000	−0.176471
$375$	0	0
$376$	− 113.137i	− 0.300897i
$377$	360.000	0.954907
$378$	0	0
$379$	−592.000	−1.56201	−0.781003	−	0.624528i	$-0.785291\pi$
$379$	−592.000	−1.56201	−0.781003	+	0.624528i	$0.785291\pi$
$380$	169.706i	0.446594i
$381$	0	0
$382$	328.098i	0.858894i
$383$	410.000	1.07050	0.535248	−	0.844695i	$-0.320218\pi$
$383$	410.000	1.07050	0.535248	+	0.844695i	$0.320218\pi$
$384$	0	0
$385$	186.676i	0.484873i
$386$	24.0000	0.0621762
$387$	0	0
$388$	−160.000	−0.412371
$389$	−460.000	−1.18252	−0.591260	−	0.806481i	$-0.701369\pi$
$389$	−460.000	−1.18252	−0.591260	+	0.806481i	$0.701369\pi$
$390$	0	0
$391$	− 42.4264i	− 0.108507i
$392$	87.6812i	0.223677i
$393$	0	0
$394$	−414.000	−1.05076
$395$	254.558i	0.644452i
$396$	0	0
$397$	200.000	0.503778	0.251889	−	0.967756i	$-0.418948\pi$
$397$	200.000	0.503778	0.251889	+	0.967756i	$0.418948\pi$
$398$	73.5391i	0.184772i
$399$	0	0
$400$	−36.0000	−0.0900000
$401$	152.000	0.379052	0.189526	−	0.981876i	$-0.439305\pi$
$401$	152.000	0.379052	0.189526	+	0.981876i	$0.439305\pi$
$402$	0	0
$403$	339.411i	0.842212i
$404$	− 42.4264i	− 0.105016i
$405$	0	0
$406$	− 127.279i	− 0.313496i
$407$	440.000	1.08108
$408$	0	0
$409$	− 339.411i	− 0.829856i	−0.909854	−	0.414928i	$-0.863807\pi$
$409$	− 339.411i	− 0.829856i	0.909854	−	0.414928i	$-0.136193\pi$
$410$	−360.000	−0.878049
$411$	0	0
$412$	200.000	0.485437
$413$	− 296.985i	− 0.719092i
$414$	0	0
$415$	− 407.294i	− 0.981430i
$416$	96.0000	0.230769
$417$	0	0
$418$	−330.000	−0.789474
$419$	512.000	1.22196	0.610979	−	0.791647i	$-0.290776\pi$
$419$	512.000	1.22196	0.610979	+	0.791647i	$0.290776\pi$
$420$	0	0
$421$	770.000	1.82898	0.914489	−	0.404610i	$-0.132593\pi$
$421$	770.000	1.82898	0.914489	+	0.404610i	$0.132593\pi$
$422$	−270.000	−0.639810
$423$	0	0
$424$	56.5685i	0.133416i
$425$	− 38.1838i	− 0.0898442i
$426$	0	0
$427$	360.000	0.843091
$428$	− 373.352i	− 0.872319i
$429$	0	0
$430$	264.000	0.613953
$431$	− 339.411i	− 0.787497i	−0.919218	−	0.393749i	$-0.871178\pi$
$431$	− 339.411i	− 0.787497i	0.919218	−	0.393749i	$-0.128822\pi$
$432$	0	0
$433$	−160.000	−0.369515	−0.184758	−	0.982784i	$-0.559150\pi$
$433$	−160.000	−0.369515	−0.184758	+	0.982784i	$0.559150\pi$
$434$	120.000	0.276498
$435$	0	0
$436$	− 339.411i	− 0.778466i
$437$	− 212.132i	− 0.485428i
$438$	0	0
$439$	275.772i	0.628181i	0.949393	+	0.314091i	$0.101700\pi$
$439$	275.772i	0.628181i	−0.949393	+	0.314091i	$0.898300\pi$
$440$	124.451i	0.282843i
$441$	0	0
$442$	101.823i	0.230370i
$443$	170.000	0.383747	0.191874	−	0.981420i	$-0.438544\pi$
$443$	170.000	0.383747	0.191874	+	0.981420i	$0.438544\pi$
$444$	0	0
$445$	−512.000	−1.15056
$446$	− 197.990i	− 0.443924i
$447$	0	0
$448$	− 33.9411i	− 0.0757614i
$449$	248.000	0.552339	0.276169	−	0.961109i	$-0.410935\pi$
$449$	248.000	0.552339	0.276169	+	0.961109i	$0.410935\pi$
$450$	0	0
$451$	− 700.036i	− 1.55219i
$452$	−400.000	−0.884956
$453$	0	0
$454$	−144.000	−0.317181
$455$	288.000	0.632967
$456$	0	0
$457$	237.588i	0.519886i	0.965624	+	0.259943i	$0.0837038\pi$
$457$	237.588i	0.519886i	−0.965624	+	0.259943i	$0.916296\pi$
$458$	− 11.3137i	− 0.0247024i
$459$	0	0
$460$	−80.0000	−0.173913
$461$	530.330i	1.15039i	0.818016	+	0.575195i	$0.195074\pi$
$461$	530.330i	1.15039i	−0.818016	+	0.575195i	$0.804926\pi$
$462$	0	0
$463$	−100.000	−0.215983	−0.107991	−	0.994152i	$-0.534442\pi$
$463$	−100.000	−0.215983	−0.107991	+	0.994152i	$0.534442\pi$
$464$	− 84.8528i	− 0.182872i
$465$	0	0
$466$	54.0000	0.115880
$467$	710.000	1.52034	0.760171	−	0.649723i	$-0.225115\pi$
$467$	710.000	1.52034	0.760171	+	0.649723i	$0.225115\pi$
$468$	0	0
$469$	− 169.706i	− 0.361846i
$470$	− 226.274i	− 0.481434i
$471$	0	0
$472$	− 197.990i	− 0.419470i
$473$	513.360i	1.08533i
$474$	0	0
$475$	− 190.919i	− 0.401934i
$476$	36.0000	0.0756303
$477$	0	0
$478$	360.000	0.753138
$479$	169.706i	0.354291i	0.984185	+	0.177146i	$0.0566864\pi$
$479$	169.706i	0.354291i	−0.984185	+	0.177146i	$0.943314\pi$
$480$	0	0
$481$	− 678.823i	− 1.41127i
$482$	−600.000	−1.24481
$483$	0	0
$484$	−242.000	−0.500000
$485$	−320.000	−0.659794
$486$	0	0
$487$	−580.000	−1.19097	−0.595483	−	0.803368i	$-0.703039\pi$
$487$	−580.000	−1.19097	−0.595483	+	0.803368i	$0.703039\pi$
$488$	240.000	0.491803
$489$	0	0
$490$	175.362i	0.357883i
$491$	− 84.8528i	− 0.172816i	−0.996260	−	0.0864082i	$-0.972461\pi$
$491$	− 84.8528i	− 0.172816i	0.996260	−	0.0864082i	$-0.0275389\pi$
$492$	0	0
$493$	90.0000	0.182556
$494$	509.117i	1.03060i
$495$	0	0
$496$	80.0000	0.161290
$497$	− 169.706i	− 0.341460i
$498$	0	0
$499$	152.000	0.304609	0.152305	−	0.988334i	$-0.451331\pi$
$499$	152.000	0.304609	0.152305	+	0.988334i	$0.451331\pi$
$500$	−272.000	−0.544000
$501$	0	0
$502$	226.274i	0.450745i
$503$	− 441.235i	− 0.877206i	−0.898681	−	0.438603i	$-0.855473\pi$
$503$	− 441.235i	− 0.877206i	0.898681	−	0.438603i	$-0.144527\pi$
$504$	0	0
$505$	− 84.8528i	− 0.168025i
$506$	− 155.563i	− 0.307438i
$507$	0	0
$508$	− 415.779i	− 0.818462i
$509$	−460.000	−0.903733	−0.451866	−	0.892086i	$-0.649242\pi$
$509$	−460.000	−0.903733	−0.451866	+	0.892086i	$0.649242\pi$
$510$	0	0
$511$	72.0000	0.140900
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	− 622.254i	− 1.21061i
$515$	400.000	0.776699
$516$	0	0
$517$	440.000	0.851064
$518$	−240.000	−0.463320
$519$	0	0
$520$	192.000	0.369231
$521$	392.000	0.752399	0.376200	−	0.926539i	$-0.377231\pi$
$521$	392.000	0.752399	0.376200	+	0.926539i	$0.377231\pi$
$522$	0	0
$523$	− 292.742i	− 0.559737i	−0.960038	−	0.279868i	$-0.909709\pi$
$523$	− 292.742i	− 0.559737i	0.960038	−	0.279868i	$-0.0902908\pi$
$524$	509.117i	0.971597i
$525$	0	0
$526$	−264.000	−0.501901
$527$	84.8528i	0.161011i
$528$	0	0
$529$	−429.000	−0.810964
$530$	113.137i	0.213466i
$531$	0	0
$532$	180.000	0.338346
$533$	−1080.00	−2.02627
$534$	0	0
$535$	− 746.705i	− 1.39571i
$536$	− 113.137i	− 0.211077i
$537$	0	0
$538$	480.833i	0.893741i
$539$	−341.000	−0.632653
$540$	0	0
$541$	− 848.528i	− 1.56844i	−0.620480	−	0.784222i	$-0.713062\pi$
$541$	− 848.528i	− 1.56844i	0.620480	−	0.784222i	$-0.286938\pi$
$542$	−390.000	−0.719557
$543$	0	0
$544$	24.0000	0.0441176
$545$	− 678.823i	− 1.24555i
$546$	0	0
$547$	767.918i	1.40387i	0.712240	+	0.701936i	$0.247681\pi$
$547$	767.918i	1.40387i	−0.712240	+	0.701936i	$0.752319\pi$
$548$	−160.000	−0.291971
$549$	0	0
$550$	− 140.007i	− 0.254558i
$551$	450.000	0.816697
$552$	0	0
$553$	270.000	0.488246
$554$	336.000	0.606498
$555$	0	0
$556$	− 296.985i	− 0.534145i
$557$	759.433i	1.36343i	0.731616	+	0.681717i	$0.238766\pi$
$557$	759.433i	1.36343i	−0.731616	+	0.681717i	$0.761234\pi$
$558$	0	0
$559$	792.000	1.41682
$560$	− 67.8823i	− 0.121218i
$561$	0	0
$562$	390.000	0.693950
$563$	780.646i	1.38658i	0.720658	+	0.693291i	$0.243840\pi$
$563$	780.646i	1.38658i	−0.720658	+	0.693291i	$0.756160\pi$
$564$	0	0
$565$	−800.000	−1.41593
$566$	66.0000	0.116608
$567$	0	0
$568$	− 113.137i	− 0.199185i
$569$	− 1124.30i	− 1.97592i	−0.154702	−	0.987961i	$-0.549442\pi$
$569$	− 1124.30i	− 1.97592i	0.154702	−	0.987961i	$-0.450558\pi$
$570$	0	0
$571$	− 190.919i	− 0.334359i	−0.985926	−	0.167179i	$-0.946534\pi$
$571$	− 190.919i	− 0.334359i	0.985926	−	0.167179i	$-0.0534659\pi$
$572$	373.352i	0.652714i
$573$	0	0
$574$	381.838i	0.665222i
$575$	90.0000	0.156522
$576$	0	0
$577$	−370.000	−0.641248	−0.320624	−	0.947207i	$-0.603893\pi$
$577$	−370.000	−0.641248	−0.320624	+	0.947207i	$0.603893\pi$
$578$	− 383.252i	− 0.663066i
$579$	0	0
$580$	− 169.706i	− 0.292596i
$581$	−432.000	−0.743546
$582$	0	0
$583$	−220.000	−0.377358
$584$	48.0000	0.0821918
$585$	0	0
$586$	−366.000	−0.624573
$587$	230.000	0.391823	0.195911	−	0.980622i	$-0.437233\pi$
$587$	230.000	0.391823	0.195911	+	0.980622i	$0.437233\pi$
$588$	0	0
$589$	424.264i	0.720313i
$590$	− 395.980i	− 0.671152i
$591$	0	0
$592$	−160.000	−0.270270
$593$	504.874i	0.851390i	0.904867	+	0.425695i	$0.139970\pi$
$593$	504.874i	0.851390i	−0.904867	+	0.425695i	$0.860030\pi$
$594$	0	0
$595$	72.0000	0.121008
$596$	− 381.838i	− 0.640667i
$597$	0	0
$598$	−240.000	−0.401338
$599$	−298.000	−0.497496	−0.248748	−	0.968568i	$-0.580019\pi$
$599$	−298.000	−0.497496	−0.248748	+	0.968568i	$0.580019\pi$
$600$	0	0
$601$	424.264i	0.705930i	0.935636	+	0.352965i	$0.114827\pi$
$601$	424.264i	0.705930i	−0.935636	+	0.352965i	$0.885173\pi$
$602$	− 280.014i	− 0.465140i
$603$	0	0
$604$	212.132i	0.351212i
$605$	−484.000	−0.800000
$606$	0	0
$607$	335.169i	0.552172i	0.961133	+	0.276086i	$0.0890375\pi$
$607$	335.169i	0.552172i	−0.961133	+	0.276086i	$0.910962\pi$
$608$	120.000	0.197368
$609$	0	0
$610$	480.000	0.786885
$611$	− 678.823i	− 1.11100i
$612$	0	0
$613$	356.382i	0.581373i	0.956818	+	0.290687i	$0.0938837\pi$
$613$	356.382i	0.581373i	−0.956818	+	0.290687i	$0.906116\pi$
$614$	546.000	0.889251
$615$	0	0
$616$	132.000	0.214286
$617$	440.000	0.713128	0.356564	−	0.934271i	$-0.383948\pi$
$617$	440.000	0.713128	0.356564	+	0.934271i	$0.383948\pi$
$618$	0	0
$619$	80.0000	0.129241	0.0646204	−	0.997910i	$-0.479416\pi$
$619$	80.0000	0.129241	0.0646204	+	0.997910i	$0.479416\pi$
$620$	160.000	0.258065
$621$	0	0
$622$	14.1421i	0.0227366i
$623$	543.058i	0.871682i
$624$	0	0
$625$	−319.000	−0.510400
$626$	565.685i	0.903651i
$627$	0	0
$628$	−100.000	−0.159236
$629$	− 169.706i	− 0.269802i
$630$	0	0
$631$	188.000	0.297940	0.148970	−	0.988842i	$-0.452404\pi$
$631$	188.000	0.297940	0.148970	+	0.988842i	$0.452404\pi$
$632$	180.000	0.284810
$633$	0	0
$634$	820.244i	1.29376i
$635$	− 831.558i	− 1.30954i
$636$	0	0
$637$	526.087i	0.825883i
$638$	330.000	0.517241
$639$	0	0
$640$	− 45.2548i	− 0.0707107i
$641$	32.0000	0.0499220	0.0249610	−	0.999688i	$-0.492054\pi$
$641$	32.0000	0.0499220	0.0249610	+	0.999688i	$0.492054\pi$
$642$	0	0
$643$	440.000	0.684292	0.342146	−	0.939647i	$-0.388846\pi$
$643$	440.000	0.684292	0.342146	+	0.939647i	$0.388846\pi$
$644$	84.8528i	0.131759i
$645$	0	0
$646$	127.279i	0.197027i
$647$	650.000	1.00464	0.502318	−	0.864683i	$-0.332481\pi$
$647$	650.000	1.00464	0.502318	+	0.864683i	$0.332481\pi$
$648$	0	0
$649$	770.000	1.18644
$650$	−216.000	−0.332308
$651$	0	0
$652$	−400.000	−0.613497
$653$	−1060.00	−1.62328	−0.811639	−	0.584160i	$-0.801424\pi$
$653$	−1060.00	−1.62328	−0.811639	+	0.584160i	$0.801424\pi$
$654$	0	0
$655$	1018.23i	1.55456i
$656$	254.558i	0.388046i
$657$	0	0
$658$	−240.000	−0.364742
$659$	933.381i	1.41636i	0.706032	+	0.708180i	$0.250483\pi$
$659$	933.381i	1.41636i	−0.706032	+	0.708180i	$0.749517\pi$
$660$	0	0
$661$	−322.000	−0.487141	−0.243570	−	0.969883i	$-0.578319\pi$
$661$	−322.000	−0.487141	−0.243570	+	0.969883i	$0.578319\pi$
$662$	735.391i	1.11086i
$663$	0	0
$664$	−288.000	−0.433735
$665$	360.000	0.541353
$666$	0	0
$667$	212.132i	0.318039i
$668$	− 373.352i	− 0.558911i
$669$	0	0
$670$	− 226.274i	− 0.337723i
$671$	933.381i	1.39103i
$672$	0	0
$673$	916.410i	1.36168i	0.732432	+	0.680840i	$0.238385\pi$
$673$	916.410i	1.36168i	−0.732432	+	0.680840i	$0.761615\pi$
$674$	744.000	1.10386
$675$	0	0
$676$	238.000	0.352071
$677$	123.037i	0.181738i	0.995863	+	0.0908690i	$0.0289644\pi$
$677$	123.037i	0.181738i	−0.995863	+	0.0908690i	$0.971036\pi$
$678$	0	0
$679$	339.411i	0.499869i
$680$	48.0000	0.0705882
$681$	0	0
$682$	311.127i	0.456198i
$683$	−160.000	−0.234261	−0.117130	−	0.993117i	$-0.537370\pi$
$683$	−160.000	−0.234261	−0.117130	+	0.993117i	$0.537370\pi$
$684$	0	0
$685$	−320.000	−0.467153
$686$	480.000	0.699708
$687$	0	0
$688$	− 186.676i	− 0.271332i
$689$	339.411i	0.492614i
$690$	0	0
$691$	632.000	0.914616	0.457308	−	0.889308i	$-0.348814\pi$
$691$	632.000	0.914616	0.457308	+	0.889308i	$0.348814\pi$
$692$	− 432.749i	− 0.625360i
$693$	0	0
$694$	−144.000	−0.207493
$695$	− 593.970i	− 0.854633i
$696$	0	0
$697$	−270.000	−0.387374
$698$	480.000	0.687679
$699$	0	0
$700$	76.3675i	0.109096i
$701$	− 657.609i	− 0.938102i	−0.883171	−	0.469051i	$-0.844596\pi$
$701$	− 657.609i	− 0.938102i	0.883171	−	0.469051i	$-0.155404\pi$
$702$	0	0
$703$	− 848.528i	− 1.20701i
$704$	88.0000	0.125000
$705$	0	0
$706$	56.5685i	0.0801254i
$707$	−90.0000	−0.127298
$708$	0	0
$709$	920.000	1.29760	0.648801	−	0.760958i	$-0.275271\pi$
$709$	920.000	1.29760	0.648801	+	0.760958i	$0.275271\pi$
$710$	− 226.274i	− 0.318696i
$711$	0	0
$712$	362.039i	0.508481i
$713$	−200.000	−0.280505
$714$	0	0
$715$	746.705i	1.04434i
$716$	416.000	0.581006
$717$	0	0
$718$	120.000	0.167131
$719$	−712.000	−0.990264	−0.495132	−	0.868818i	$-0.664880\pi$
$719$	−712.000	−0.990264	−0.495132	+	0.868818i	$0.664880\pi$
$720$	0	0
$721$	− 424.264i	− 0.588438i
$722$	125.865i	0.174328i
$723$	0	0
$724$	380.000	0.524862
$725$	190.919i	0.263336i
$726$	0	0
$727$	−820.000	−1.12792	−0.563961	−	0.825801i	$-0.690723\pi$
$727$	−820.000	−1.12792	−0.563961	+	0.825801i	$0.690723\pi$
$728$	− 203.647i	− 0.279735i
$729$	0	0
$730$	96.0000	0.131507
$731$	198.000	0.270862
$732$	0	0
$733$	− 101.823i	− 0.138913i	−0.997585	−	0.0694566i	$-0.977873\pi$
$733$	− 101.823i	− 0.138913i	0.997585	−	0.0694566i	$-0.0221265\pi$
$734$	− 537.401i	− 0.732154i
$735$	0	0
$736$	56.5685i	0.0768594i
$737$	440.000	0.597015
$738$	0	0
$739$	530.330i	0.717632i	0.933408	+	0.358816i	$0.116819\pi$
$739$	530.330i	0.717632i	−0.933408	+	0.358816i	$0.883181\pi$
$740$	−320.000	−0.432432
$741$	0	0
$742$	120.000	0.161725
$743$	− 1289.76i	− 1.73589i	−0.496664	−	0.867943i	$-0.665442\pi$
$743$	− 1289.76i	− 1.73589i	0.496664	−	0.867943i	$-0.334558\pi$
$744$	0	0
$745$	− 763.675i	− 1.02507i
$746$	−144.000	−0.193029
$747$	0	0
$748$	93.3381i	0.124784i
$749$	−792.000	−1.05741
$750$	0	0
$751$	−1348.00	−1.79494	−0.897470	−	0.441075i	$-0.854597\pi$
$751$	−1348.00	−1.79494	−0.897470	+	0.441075i	$0.854597\pi$
$752$	−160.000	−0.212766
$753$	0	0
$754$	− 509.117i	− 0.675221i
$755$	424.264i	0.561939i
$756$	0	0
$757$	−610.000	−0.805812	−0.402906	−	0.915241i	$-0.632000\pi$
$757$	−610.000	−0.805812	−0.402906	+	0.915241i	$0.632000\pi$
$758$	837.214i	1.10450i
$759$	0	0
$760$	240.000	0.315789
$761$	445.477i	0.585384i	0.956207	+	0.292692i	$0.0945511\pi$
$761$	445.477i	0.585384i	−0.956207	+	0.292692i	$0.905449\pi$
$762$	0	0
$763$	−720.000	−0.943644
$764$	464.000	0.607330
$765$	0	0
$766$	− 579.828i	− 0.756955i
$767$	− 1187.94i	− 1.54881i
$768$	0	0
$769$	424.264i	0.551709i	0.961199	+	0.275854i	$0.0889608\pi$
$769$	424.264i	0.551709i	−0.961199	+	0.275854i	$0.911039\pi$
$770$	264.000	0.342857
$771$	0	0
$772$	− 33.9411i	− 0.0439652i
$773$	−700.000	−0.905563	−0.452781	−	0.891622i	$-0.649568\pi$
$773$	−700.000	−0.905563	−0.452781	+	0.891622i	$0.649568\pi$
$774$	0	0
$775$	−180.000	−0.232258
$776$	226.274i	0.291590i
$777$	0	0
$778$	650.538i	0.836167i
$779$	−1350.00	−1.73299
$780$	0	0
$781$	440.000	0.563380
$782$	−60.0000	−0.0767263
$783$	0	0
$784$	124.000	0.158163
$785$	−200.000	−0.254777
$786$	0	0
$787$	− 1234.61i	− 1.56875i	−0.620285	−	0.784376i	$-0.712983\pi$
$787$	− 1234.61i	− 1.56875i	0.620285	−	0.784376i	$-0.287017\pi$
$788$	585.484i	0.743001i
$789$	0	0
$790$	360.000	0.455696
$791$	848.528i	1.07273i
$792$	0	0
$793$	1440.00	1.81589
$794$	− 282.843i	− 0.356225i
$795$	0	0
$796$	104.000	0.130653
$797$	380.000	0.476788	0.238394	−	0.971169i	$-0.423379\pi$
$797$	380.000	0.476788	0.238394	+	0.971169i	$0.423379\pi$
$798$	0	0
$799$	− 169.706i	− 0.212398i
$800$	50.9117i	0.0636396i
$801$	0	0
$802$	− 214.960i	− 0.268031i
$803$	186.676i	0.232473i
$804$	0	0
$805$	169.706i	0.210814i
$806$	480.000	0.595533
$807$	0	0
$808$	−60.0000	−0.0742574
$809$	− 63.6396i	− 0.0786645i	−0.999226	−	0.0393323i	$-0.987477\pi$
$809$	− 63.6396i	− 0.0786645i	0.999226	−	0.0393323i	$-0.0125231\pi$
$810$	0	0
$811$	− 360.624i	− 0.444666i	−0.974971	−	0.222333i	$-0.928633\pi$
$811$	− 360.624i	− 0.444666i	0.974971	−	0.222333i	$-0.0713673\pi$
$812$	−180.000	−0.221675
$813$	0	0
$814$	− 622.254i	− 0.764440i
$815$	−800.000	−0.981595
$816$	0	0
$817$	990.000	1.21175
$818$	−480.000	−0.586797
$819$	0	0
$820$	509.117i	0.620874i
$821$	− 21.2132i	− 0.0258383i	−0.999917	−	0.0129191i	$-0.995888\pi$
$821$	− 21.2132i	− 0.0258383i	0.999917	−	0.0129191i	$-0.00411240\pi$
$822$	0	0
$823$	980.000	1.19077	0.595383	−	0.803442i	$-0.297000\pi$
$823$	980.000	1.19077	0.595383	+	0.803442i	$0.297000\pi$
$824$	− 282.843i	− 0.343256i
$825$	0	0
$826$	−420.000	−0.508475
$827$	− 16.9706i	− 0.0205206i	−0.999947	−	0.0102603i	$-0.996734\pi$
$827$	− 16.9706i	− 0.0205206i	0.999947	−	0.0102603i	$-0.00326602\pi$
$828$	0	0
$829$	−1000.00	−1.20627	−0.603136	−	0.797638i	$-0.706083\pi$
$829$	−1000.00	−1.20627	−0.603136	+	0.797638i	$0.706083\pi$
$830$	−576.000	−0.693976
$831$	0	0
$832$	− 135.765i	− 0.163178i
$833$	131.522i	0.157889i
$834$	0	0
$835$	− 746.705i	− 0.894257i
$836$	466.690i	0.558242i
$837$	0	0
$838$	− 724.077i	− 0.864054i
$839$	1130.00	1.34684	0.673421	−	0.739259i	$-0.264824\pi$
$839$	1130.00	1.34684	0.673421	+	0.739259i	$0.264824\pi$
$840$	0	0
$841$	391.000	0.464923
$842$	− 1088.94i	− 1.29328i
$843$	0	0
$844$	381.838i	0.452414i
$845$	476.000	0.563314
$846$	0	0
$847$	513.360i	0.606092i
$848$	80.0000	0.0943396
$849$	0	0
$850$	−54.0000	−0.0635294
$851$	400.000	0.470035
$852$	0	0
$853$	− 610.940i	− 0.716225i	−0.933678	−	0.358113i	$-0.883420\pi$
$853$	− 610.940i	− 0.716225i	0.933678	−	0.358113i	$-0.116580\pi$
$854$	− 509.117i	− 0.596156i
$855$	0	0
$856$	−528.000	−0.616822
$857$	1013.99i	1.18319i	0.806236	+	0.591593i	$0.201501\pi$
$857$	1013.99i	1.18319i	−0.806236	+	0.591593i	$0.798499\pi$
$858$	0	0
$859$	−760.000	−0.884750	−0.442375	−	0.896830i	$-0.645864\pi$
$859$	−760.000	−0.884750	−0.442375	+	0.896830i	$0.645864\pi$
$860$	− 373.352i	− 0.434131i
$861$	0	0
$862$	−480.000	−0.556845
$863$	−1510.00	−1.74971	−0.874855	−	0.484385i	$-0.839044\pi$
$863$	−1510.00	−1.74971	−0.874855	+	0.484385i	$0.839044\pi$
$864$	0	0
$865$	− 865.499i	− 1.00058i
$866$	226.274i	0.261287i
$867$	0	0
$868$	− 169.706i	− 0.195513i
$869$	700.036i	0.805565i
$870$	0	0
$871$	− 678.823i	− 0.779360i
$872$	−480.000	−0.550459
$873$	0	0
$874$	−300.000	−0.343249
$875$	576.999i	0.659428i
$876$	0	0
$877$	− 322.441i	− 0.367663i	−0.982958	−	0.183832i	$-0.941150\pi$
$877$	− 322.441i	− 0.367663i	0.982958	−	0.183832i	$-0.0588501\pi$
$878$	390.000	0.444191
$879$	0	0
$880$	176.000	0.200000
$881$	1160.00	1.31669	0.658343	−	0.752718i	$-0.271258\pi$
$881$	1160.00	1.31669	0.658343	+	0.752718i	$0.271258\pi$
$882$	0	0
$883$	−880.000	−0.996602	−0.498301	−	0.867004i	$-0.666043\pi$
$883$	−880.000	−0.996602	−0.498301	+	0.867004i	$0.666043\pi$
$884$	144.000	0.162896
$885$	0	0
$886$	− 240.416i	− 0.271350i
$887$	− 271.529i	− 0.306121i	−0.988217	−	0.153060i	$-0.951087\pi$
$887$	− 271.529i	− 0.306121i	0.988217	−	0.153060i	$-0.0489129\pi$
$888$	0	0
$889$	−882.000	−0.992126
$890$	724.077i	0.813570i
$891$	0	0
$892$	−280.000	−0.313901
$893$	− 848.528i	− 0.950199i
$894$	0	0
$895$	832.000	0.929609
$896$	−48.0000	−0.0535714
$897$	0	0
$898$	− 350.725i	− 0.390562i
$899$	− 424.264i	− 0.471929i
$900$	0	0
$901$	84.8528i	0.0941763i
$902$	−990.000	−1.09756
$903$	0	0
$904$	565.685i	0.625758i
$905$	760.000	0.839779
$906$	0	0
$907$	1040.00	1.14664	0.573319	−	0.819332i	$-0.305656\pi$
$907$	1040.00	1.14664	0.573319	+	0.819332i	$0.305656\pi$
$908$	203.647i	0.224281i
$909$	0	0
$910$	− 407.294i	− 0.447575i
$911$	−328.000	−0.360044	−0.180022	−	0.983663i	$-0.557617\pi$
$911$	−328.000	−0.360044	−0.180022	+	0.983663i	$0.557617\pi$
$912$	0	0
$913$	− 1120.06i	− 1.22679i
$914$	336.000	0.367615
$915$	0	0
$916$	−16.0000	−0.0174672
$917$	1080.00	1.17775
$918$	0	0
$919$	− 572.756i	− 0.623239i	−0.950207	−	0.311619i	$-0.899129\pi$
$919$	− 572.756i	− 0.623239i	0.950207	−	0.311619i	$-0.100871\pi$
$920$	113.137i	0.122975i
$921$	0	0
$922$	750.000	0.813449
$923$	− 678.823i	− 0.735452i
$924$	0	0
$925$	360.000	0.389189
$926$	141.421i	0.152723i
$927$	0	0
$928$	−120.000	−0.129310
$929$	680.000	0.731970	0.365985	−	0.930621i	$-0.380732\pi$
$929$	680.000	0.731970	0.365985	+	0.930621i	$0.380732\pi$
$930$	0	0
$931$	657.609i	0.706347i
$932$	− 76.3675i	− 0.0819394i
$933$	0	0
$934$	− 1004.09i	− 1.07504i
$935$	186.676i	0.199654i
$936$	0	0
$937$	− 67.8823i	− 0.0724464i	−0.999344	−	0.0362232i	$-0.988467\pi$
$937$	− 67.8823i	− 0.0724464i	0.999344	−	0.0362232i	$-0.0115327\pi$
$938$	−240.000	−0.255864
$939$	0	0
$940$	−320.000	−0.340426
$941$	− 1718.27i	− 1.82600i	−0.407955	−	0.913002i	$-0.633758\pi$
$941$	− 1718.27i	− 1.82600i	0.407955	−	0.913002i	$-0.366242\pi$
$942$	0	0
$943$	− 636.396i	− 0.674863i
$944$	−280.000	−0.296610
$945$	0	0
$946$	726.000	0.767442
$947$	1280.00	1.35164	0.675818	−	0.737068i	$-0.263790\pi$
$947$	1280.00	1.35164	0.675818	+	0.737068i	$0.263790\pi$
$948$	0	0
$949$	288.000	0.303477
$950$	−270.000	−0.284211
$951$	0	0
$952$	− 50.9117i	− 0.0534787i
$953$	− 207.889i	− 0.218142i	−0.994034	−	0.109071i	$-0.965212\pi$
$953$	− 207.889i	− 0.218142i	0.994034	−	0.109071i	$-0.0347876\pi$
$954$	0	0
$955$	928.000	0.971728
$956$	− 509.117i	− 0.532549i
$957$	0	0
$958$	240.000	0.250522
$959$	339.411i	0.353922i
$960$	0	0
$961$	−561.000	−0.583767
$962$	−960.000	−0.997921
$963$	0	0
$964$	848.528i	0.880216i
$965$	− 67.8823i	− 0.0703443i
$966$	0	0
$967$	377.595i	0.390481i	0.980755	+	0.195240i	$0.0625487\pi$
$967$	377.595i	0.390481i	−0.980755	+	0.195240i	$0.937451\pi$
$968$	342.240i	0.353553i
$969$	0	0
$970$	452.548i	0.466545i
$971$	−1258.00	−1.29557	−0.647786	−	0.761823i	$-0.724305\pi$
$971$	−1258.00	−1.29557	−0.647786	+	0.761823i	$0.724305\pi$
$972$	0	0
$973$	−630.000	−0.647482
$974$	820.244i	0.842139i
$975$	0	0
$976$	− 339.411i	− 0.347757i
$977$	1520.00	1.55578	0.777892	−	0.628399i	$-0.216289\pi$
$977$	1520.00	1.55578	0.777892	+	0.628399i	$0.216289\pi$
$978$	0	0
$979$	−1408.00	−1.43820
$980$	248.000	0.253061
$981$	0	0
$982$	−120.000	−0.122200
$983$	1430.00	1.45473	0.727365	−	0.686251i	$-0.240745\pi$
$983$	1430.00	1.45473	0.727365	+	0.686251i	$0.240745\pi$
$984$	0	0
$985$	1170.97i	1.18880i
$986$	− 127.279i	− 0.129086i
$987$	0	0
$988$	720.000	0.728745
$989$	466.690i	0.471881i
$990$	0	0
$991$	740.000	0.746720	0.373360	−	0.927686i	$-0.378206\pi$
$991$	740.000	0.746720	0.373360	+	0.927686i	$0.378206\pi$
$992$	− 113.137i	− 0.114049i
$993$	0	0
$994$	−240.000	−0.241449
$995$	208.000	0.209045
$996$	0	0
$997$	− 576.999i	− 0.578735i	−0.957218	−	0.289368i	$-0.906555\pi$
$997$	− 576.999i	− 0.578735i	0.957218	−	0.289368i	$-0.0934450\pi$
$998$	− 214.960i	− 0.215391i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.3.d.a.109.1	✓	2
3.2	odd	2		198.3.d.c.109.2	yes	2
4.3	odd	2		1584.3.j.b.1297.1		2
11.10	odd	2	inner	198.3.d.a.109.2	yes	2
12.11	even	2		1584.3.j.e.1297.1		2
33.32	even	2		198.3.d.c.109.1	yes	2
44.43	even	2		1584.3.j.b.1297.2		2
132.131	odd	2		1584.3.j.e.1297.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.3.d.a.109.1	✓	2	1.1	even	1	trivial
198.3.d.a.109.2	yes	2	11.10	odd	2	inner
198.3.d.c.109.1	yes	2	33.32	even	2
198.3.d.c.109.2	yes	2	3.2	odd	2
1584.3.j.b.1297.1		2	4.3	odd	2
1584.3.j.b.1297.2		2	44.43	even	2
1584.3.j.e.1297.1		2	12.11	even	2
1584.3.j.e.1297.2		2	132.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	198.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -4.00000 q^{5} +4.24264i q^{7} +2.82843i q^{8} +5.65685i q^{10} -11.0000 q^{11} +16.9706i q^{13} +6.00000 q^{14} +4.00000 q^{16} +4.24264i q^{17} +21.2132i q^{19} +8.00000 q^{20} +15.5563i q^{22} -10.0000 q^{23} -9.00000 q^{25} +24.0000 q^{26} -8.48528i q^{28} -21.2132i q^{29} +20.0000 q^{31} -5.65685i q^{32} +6.00000 q^{34} -16.9706i q^{35} -40.0000 q^{37} +30.0000 q^{38} -11.3137i q^{40} +63.6396i q^{41} -46.6690i q^{43} +22.0000 q^{44} +14.1421i q^{46} -40.0000 q^{47} +31.0000 q^{49} +12.7279i q^{50} -33.9411i q^{52} +20.0000 q^{53} +44.0000 q^{55} -12.0000 q^{56} -30.0000 q^{58} -70.0000 q^{59} -84.8528i q^{61} -28.2843i q^{62} -8.00000 q^{64} -67.8823i q^{65} -40.0000 q^{67} -8.48528i q^{68} -24.0000 q^{70} -40.0000 q^{71} -16.9706i q^{73} +56.5685i q^{74} -42.4264i q^{76} -46.6690i q^{77} -63.6396i q^{79} -16.0000 q^{80} +90.0000 q^{82} +101.823i q^{83} -16.9706i q^{85} -66.0000 q^{86} -31.1127i q^{88} +128.000 q^{89} -72.0000 q^{91} +20.0000 q^{92} +56.5685i q^{94} -84.8528i q^{95} +80.0000 q^{97} -43.8406i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 8 q^{5} - 22 q^{11} + 12 q^{14} + 8 q^{16} + 16 q^{20} - 20 q^{23} - 18 q^{25} + 48 q^{26} + 40 q^{31} + 12 q^{34} - 80 q^{37} + 60 q^{38} + 44 q^{44} - 80 q^{47} + 62 q^{49} + 40 q^{53}+ \cdots + 160 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 8 * q^5 - 22 * q^11 + 12 * q^14 + 8 * q^16 + 16 * q^20 - 20 * q^23 - 18 * q^25 + 48 * q^26 + 40 * q^31 + 12 * q^34 - 80 * q^37 + 60 * q^38 + 44 * q^44 - 80 * q^47 + 62 * q^49 + 40 * q^53 + 88 * q^55 - 24 * q^56 - 60 * q^58 - 140 * q^59 - 16 * q^64 - 80 * q^67 - 48 * q^70 - 80 * q^71 - 32 * q^80 + 180 * q^82 - 132 * q^86 + 256 * q^89 - 144 * q^91 + 40 * q^92 + 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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