LMFDB - Embedded newform 3300.2.c.k.1849.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1849,3300)] 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("3300.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,-4,0,0,0,0,0,0,0,-8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 660)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.3
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	3300.1849
Dual form			3300.2.c.k.1849.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.60555i q^{7} -1.00000 q^{9} -1.00000 q^{11} -0.605551i q^{13} +2.60555i q^{17} -2.00000 q^{19} +4.60555 q^{21} -1.00000i q^{27} +5.21110 q^{29} -9.21110 q^{31} -1.00000i q^{33} -7.21110i q^{37} +0.605551 q^{39} +5.21110 q^{41} +4.60555i q^{43} -5.21110i q^{47} -14.2111 q^{49} -2.60555 q^{51} -6.00000i q^{53} -2.00000i q^{57} -3.21110 q^{61} +4.60555i q^{63} +14.4222i q^{67} -12.0000 q^{71} +4.60555i q^{73} +4.60555i q^{77} -14.0000 q^{79} +1.00000 q^{81} -9.39445i q^{83} +5.21110i q^{87} -16.4222 q^{89} -2.78890 q^{91} -9.21110i q^{93} +10.0000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 4 q^{11} - 8 q^{19} + 4 q^{21} - 8 q^{29} - 8 q^{31} - 12 q^{39} - 8 q^{41} - 28 q^{49} + 4 q^{51} + 16 q^{61} - 48 q^{71} - 56 q^{79} + 4 q^{81} - 8 q^{89} - 40 q^{91} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^11 - 8 * q^19 + 4 * q^21 - 8 * q^29 - 8 * q^31 - 12 * q^39 - 8 * q^41 - 28 * q^49 + 4 * q^51 + 16 * q^61 - 48 * q^71 - 56 * q^79 + 4 * q^81 - 8 * q^89 - 40 * q^91 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.60555i	− 1.74073i	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	− 4.60555i	− 1.74073i	0.492403	−	0.870367i	$-0.336119\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 0.605551i	− 0.167950i	−0.996468	−	0.0839749i	$-0.973238\pi$
$13$	− 0.605551i	− 0.167950i	0.996468	−	0.0839749i	$-0.0267615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.60555i	0.631939i	0.948769	+	0.315970i	$0.102330\pi$
$17$	2.60555i	0.631939i	−0.948769	+	0.315970i	$0.897670\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	4.60555	1.00501
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.21110	0.967677	0.483839	−	0.875157i	$-0.339242\pi$
$29$	5.21110	0.967677	0.483839	+	0.875157i	$0.339242\pi$
$30$	0	0
$31$	−9.21110	−1.65436	−0.827181	−	0.561935i	$-0.810057\pi$
$31$	−9.21110	−1.65436	−0.827181	+	0.561935i	$0.810057\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.21110i	− 1.18550i	−0.805387	−	0.592749i	$-0.798043\pi$
$37$	− 7.21110i	− 1.18550i	0.805387	−	0.592749i	$-0.201957\pi$
$38$	0	0
$39$	0.605551	0.0969658
$40$	0	0
$41$	5.21110	0.813837	0.406919	−	0.913464i	$-0.366603\pi$
$41$	5.21110	0.813837	0.406919	+	0.913464i	$0.366603\pi$
$42$	0	0
$43$	4.60555i	0.702340i	0.936312	+	0.351170i	$0.114216\pi$
$43$	4.60555i	0.702340i	−0.936312	+	0.351170i	$0.885784\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.21110i	− 0.760117i	−0.924962	−	0.380059i	$-0.875904\pi$
$47$	− 5.21110i	− 0.760117i	0.924962	−	0.380059i	$-0.124096\pi$
$48$	0	0
$49$	−14.2111	−2.03016
$50$	0	0
$51$	−2.60555	−0.364850
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.21110	−0.411140	−0.205570	−	0.978642i	$-0.565905\pi$
$61$	−3.21110	−0.411140	−0.205570	+	0.978642i	$0.565905\pi$
$62$	0	0
$63$	4.60555i	0.580245i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.4222i	1.76195i	0.473160	+	0.880976i	$0.343113\pi$
$67$	14.4222i	1.76195i	−0.473160	+	0.880976i	$0.656887\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	4.60555i	0.539039i	0.962995	+	0.269520i	$0.0868649\pi$
$73$	4.60555i	0.539039i	−0.962995	+	0.269520i	$0.913135\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.60555i	0.524851i
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.39445i	− 1.03117i	−0.856837	−	0.515587i	$-0.827574\pi$
$83$	− 9.39445i	− 1.03117i	0.856837	−	0.515587i	$-0.172426\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.21110i	0.558689i
$88$	0	0
$89$	−16.4222	−1.74075	−0.870375	−	0.492389i	$-0.836124\pi$
$89$	−16.4222	−1.74075	−0.870375	+	0.492389i	$0.836124\pi$
$90$	0	0
$91$	−2.78890	−0.292356
$92$	0	0
$93$	− 9.21110i	− 0.955147i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 9.21110i	− 0.907597i	−0.891104	−	0.453798i	$-0.850069\pi$
$103$	− 9.21110i	− 0.907597i	0.891104	−	0.453798i	$-0.149931\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.8167i	1.91575i	0.287188	+	0.957874i	$0.407279\pi$
$107$	19.8167i	1.91575i	−0.287188	+	0.957874i	$0.592721\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	7.21110	0.684448
$112$	0	0
$113$	− 11.2111i	− 1.05465i	−0.849663	−	0.527326i	$-0.823195\pi$
$113$	− 11.2111i	− 1.05465i	0.849663	−	0.527326i	$-0.176805\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.605551i	0.0559832i
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.21110i	0.469869i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.81665i	− 0.871087i	−0.900168	−	0.435544i	$-0.856556\pi$
$127$	− 9.81665i	− 0.871087i	0.900168	−	0.435544i	$-0.143444\pi$
$128$	0	0
$129$	−4.60555	−0.405496
$130$	0	0
$131$	−17.2111	−1.50374	−0.751871	−	0.659311i	$-0.770848\pi$
$131$	−17.2111	−1.50374	−0.751871	+	0.659311i	$0.770848\pi$
$132$	0	0
$133$	9.21110i	0.798704i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−7.21110	−0.611638	−0.305819	−	0.952090i	$-0.598930\pi$
$139$	−7.21110	−0.611638	−0.305819	+	0.952090i	$0.598930\pi$
$140$	0	0
$141$	5.21110	0.438854
$142$	0	0
$143$	0.605551i	0.0506387i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.2111i	− 1.17211i
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	− 2.60555i	− 0.210646i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.78890i	0.382196i	0.981571	+	0.191098i	$0.0612048\pi$
$157$	4.78890i	0.382196i	−0.981571	+	0.191098i	$0.938795\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 10.7889i	− 0.845052i	−0.906351	−	0.422526i	$-0.861144\pi$
$163$	− 10.7889i	− 0.845052i	0.906351	−	0.422526i	$-0.138856\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.60555i	0.201624i	0.994906	+	0.100812i	$0.0321440\pi$
$167$	2.60555i	0.201624i	−0.994906	+	0.100812i	$0.967856\pi$
$168$	0	0
$169$	12.6333	0.971793
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	− 14.6056i	− 1.11044i	−0.831704	−	0.555220i	$-0.812634\pi$
$173$	− 14.6056i	− 1.11044i	0.831704	−	0.555220i	$-0.187366\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.4222	1.67591	0.837957	−	0.545736i	$-0.183750\pi$
$179$	22.4222	1.67591	0.837957	+	0.545736i	$0.183750\pi$
$180$	0	0
$181$	7.21110	0.535997	0.267999	−	0.963419i	$-0.413638\pi$
$181$	7.21110	0.535997	0.267999	+	0.963419i	$0.413638\pi$
$182$	0	0
$183$	− 3.21110i	− 0.237372i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.60555i	− 0.190537i
$188$	0	0
$189$	−4.60555	−0.335005
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 7.39445i	− 0.532264i	−0.963937	−	0.266132i	$-0.914254\pi$
$193$	− 7.39445i	− 0.532264i	0.963937	−	0.266132i	$-0.0857457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.8167i	− 1.41188i	−0.708273	−	0.705939i	$-0.750525\pi$
$197$	− 19.8167i	− 1.41188i	0.708273	−	0.705939i	$-0.249475\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−14.4222	−1.01726
$202$	0	0
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	42.4222i	2.87981i
$218$	0	0
$219$	−4.60555	−0.311214
$220$	0	0
$221$	1.57779	0.106134
$222$	0	0
$223$	− 19.6333i	− 1.31474i	−0.753566	−	0.657372i	$-0.771668\pi$
$223$	− 19.6333i	− 1.31474i	0.753566	−	0.657372i	$-0.228332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 19.8167i	− 1.31528i	−0.753333	−	0.657639i	$-0.771555\pi$
$227$	− 19.8167i	− 1.31528i	0.753333	−	0.657639i	$-0.228445\pi$
$228$	0	0
$229$	8.42221	0.556555	0.278277	−	0.960501i	$-0.410237\pi$
$229$	8.42221	0.556555	0.278277	+	0.960501i	$0.410237\pi$
$230$	0	0
$231$	−4.60555	−0.303023
$232$	0	0
$233$	14.6056i	0.956841i	0.878131	+	0.478421i	$0.158791\pi$
$233$	14.6056i	0.956841i	−0.878131	+	0.478421i	$0.841209\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 14.0000i	− 0.909398i
$238$	0	0
$239$	−27.6333	−1.78745	−0.893725	−	0.448615i	$-0.851917\pi$
$239$	−27.6333	−1.78745	−0.893725	+	0.448615i	$0.851917\pi$
$240$	0	0
$241$	19.2111	1.23750	0.618748	−	0.785590i	$-0.287640\pi$
$241$	19.2111	1.23750	0.618748	+	0.785590i	$0.287640\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.21110i	0.0770606i
$248$	0	0
$249$	9.39445	0.595349
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.2111i	− 0.699329i	−0.936875	−	0.349665i	$-0.886296\pi$
$257$	− 11.2111i	− 0.699329i	0.936875	−	0.349665i	$-0.113704\pi$
$258$	0	0
$259$	−33.2111	−2.06364
$260$	0	0
$261$	−5.21110	−0.322559
$262$	0	0
$263$	2.60555i	0.160665i	0.996768	+	0.0803326i	$0.0255982\pi$
$263$	2.60555i	0.160665i	−0.996768	+	0.0803326i	$0.974402\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 16.4222i	− 1.00502i
$268$	0	0
$269$	4.42221	0.269627	0.134813	−	0.990871i	$-0.456957\pi$
$269$	4.42221	0.269627	0.134813	+	0.990871i	$0.456957\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	− 2.78890i	− 0.168792i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.605551i	0.0363840i	0.999835	+	0.0181920i	$0.00579102\pi$
$277$	0.605551i	0.0363840i	−0.999835	+	0.0181920i	$0.994209\pi$
$278$	0	0
$279$	9.21110	0.551454
$280$	0	0
$281$	29.2111	1.74259	0.871294	−	0.490761i	$-0.163281\pi$
$281$	29.2111	1.74259	0.871294	+	0.490761i	$0.163281\pi$
$282$	0	0
$283$	− 19.3944i	− 1.15288i	−0.817139	−	0.576440i	$-0.804441\pi$
$283$	− 19.3944i	− 1.15288i	0.817139	−	0.576440i	$-0.195559\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	10.2111	0.600653
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	2.60555i	0.152218i	0.997100	+	0.0761090i	$0.0242497\pi$
$293$	2.60555i	0.152218i	−0.997100	+	0.0761090i	$0.975750\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000i	0.0580259i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	21.2111	1.22259
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.8167i	1.01685i	0.861106	+	0.508425i	$0.169772\pi$
$307$	17.8167i	1.01685i	−0.861106	+	0.508425i	$0.830228\pi$
$308$	0	0
$309$	9.21110	0.524001
$310$	0	0
$311$	22.4222	1.27145	0.635723	−	0.771917i	$-0.280702\pi$
$311$	22.4222	1.27145	0.635723	+	0.771917i	$0.280702\pi$
$312$	0	0
$313$	19.2111i	1.08588i	0.839773	+	0.542938i	$0.182688\pi$
$313$	19.2111i	1.08588i	−0.839773	+	0.542938i	$0.817312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.6333i	− 1.21505i	−0.794301	−	0.607524i	$-0.792163\pi$
$317$	− 21.6333i	− 1.21505i	0.794301	−	0.607524i	$-0.207837\pi$
$318$	0	0
$319$	−5.21110	−0.291766
$320$	0	0
$321$	−19.8167	−1.10606
$322$	0	0
$323$	− 5.21110i	− 0.289954i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.0000i	− 0.774202i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	25.2111	1.38573	0.692864	−	0.721069i	$-0.256349\pi$
$331$	25.2111	1.38573	0.692864	+	0.721069i	$0.256349\pi$
$332$	0	0
$333$	7.21110i	0.395166i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.3944i	1.05648i	0.849094	+	0.528241i	$0.177148\pi$
$337$	19.3944i	1.05648i	−0.849094	+	0.528241i	$0.822852\pi$
$338$	0	0
$339$	11.2111	0.608904
$340$	0	0
$341$	9.21110	0.498809
$342$	0	0
$343$	33.2111i	1.79323i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.18335i	− 0.224574i	−0.993676	−	0.112287i	$-0.964182\pi$
$347$	− 4.18335i	− 0.224574i	0.993676	−	0.112287i	$-0.0358176\pi$
$348$	0	0
$349$	−17.6333	−0.943889	−0.471945	−	0.881628i	$-0.656448\pi$
$349$	−17.6333	−0.943889	−0.471945	+	0.881628i	$0.656448\pi$
$350$	0	0
$351$	−0.605551	−0.0323219
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	12.0000i	0.635107i
$358$	0	0
$359$	15.6333	0.825094	0.412547	−	0.910936i	$-0.364639\pi$
$359$	15.6333	0.825094	0.412547	+	0.910936i	$0.364639\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 14.7889i	− 0.771974i	−0.922504	−	0.385987i	$-0.873861\pi$
$367$	− 14.7889i	− 0.771974i	0.922504	−	0.385987i	$-0.126139\pi$
$368$	0	0
$369$	−5.21110	−0.271279
$370$	0	0
$371$	−27.6333	−1.43465
$372$	0	0
$373$	6.18335i	0.320162i	0.987104	+	0.160081i	$0.0511755\pi$
$373$	6.18335i	0.320162i	−0.987104	+	0.160081i	$0.948825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.15559i	− 0.162521i
$378$	0	0
$379$	2.42221	0.124420	0.0622102	−	0.998063i	$-0.480185\pi$
$379$	2.42221	0.124420	0.0622102	+	0.998063i	$0.480185\pi$
$380$	0	0
$381$	9.81665	0.502922
$382$	0	0
$383$	− 15.6333i	− 0.798825i	−0.916771	−	0.399412i	$-0.869214\pi$
$383$	− 15.6333i	− 0.798825i	0.916771	−	0.399412i	$-0.130786\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.60555i	− 0.234113i
$388$	0	0
$389$	−4.42221	−0.224215	−0.112107	−	0.993696i	$-0.535760\pi$
$389$	−4.42221	−0.224215	−0.112107	+	0.993696i	$0.535760\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 17.2111i	− 0.868185i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	−9.21110	−0.461132
$400$	0	0
$401$	−16.4222	−0.820086	−0.410043	−	0.912066i	$-0.634486\pi$
$401$	−16.4222	−0.820086	−0.410043	+	0.912066i	$0.634486\pi$
$402$	0	0
$403$	5.57779i	0.277850i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.21110i	0.357441i
$408$	0	0
$409$	−24.4222	−1.20760	−0.603800	−	0.797136i	$-0.706348\pi$
$409$	−24.4222	−1.20760	−0.603800	+	0.797136i	$0.706348\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 7.21110i	− 0.353129i
$418$	0	0
$419$	1.57779	0.0770803	0.0385402	−	0.999257i	$-0.487729\pi$
$419$	1.57779	0.0770803	0.0385402	+	0.999257i	$0.487729\pi$
$420$	0	0
$421$	−3.21110	−0.156500	−0.0782498	−	0.996934i	$-0.524933\pi$
$421$	−3.21110	−0.156500	−0.0782498	+	0.996934i	$0.524933\pi$
$422$	0	0
$423$	5.21110i	0.253372i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.7889i	0.715685i
$428$	0	0
$429$	−0.605551	−0.0292363
$430$	0	0
$431$	32.8444	1.58206	0.791030	−	0.611778i	$-0.209545\pi$
$431$	32.8444	1.58206	0.791030	+	0.611778i	$0.209545\pi$
$432$	0	0
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−31.2111	−1.48962	−0.744812	−	0.667274i	$-0.767461\pi$
$439$	−31.2111	−1.48962	−0.744812	+	0.667274i	$0.767461\pi$
$440$	0	0
$441$	14.2111	0.676719
$442$	0	0
$443$	27.6333i	1.31290i	0.754370	+	0.656449i	$0.227942\pi$
$443$	27.6333i	1.31290i	−0.754370	+	0.656449i	$0.772058\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.0000i	− 0.567581i
$448$	0	0
$449$	7.57779	0.357618	0.178809	−	0.983884i	$-0.442776\pi$
$449$	7.57779	0.357618	0.178809	+	0.983884i	$0.442776\pi$
$450$	0	0
$451$	−5.21110	−0.245381
$452$	0	0
$453$	− 10.0000i	− 0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.6056i	1.15100i	0.817802	+	0.575500i	$0.195192\pi$
$457$	24.6056i	1.15100i	−0.817802	+	0.575500i	$0.804808\pi$
$458$	0	0
$459$	2.60555	0.121617
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	25.2111i	1.17166i	0.810434	+	0.585830i	$0.199231\pi$
$463$	25.2111i	1.17166i	−0.810434	+	0.585830i	$0.800769\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.8444i	− 1.51986i	−0.650006	−	0.759929i	$-0.725234\pi$
$467$	− 32.8444i	− 1.51986i	0.650006	−	0.759929i	$-0.274766\pi$
$468$	0	0
$469$	66.4222	3.06709
$470$	0	0
$471$	−4.78890	−0.220661
$472$	0	0
$473$	− 4.60555i	− 0.211763i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	3.63331	0.166010	0.0830050	−	0.996549i	$-0.473548\pi$
$479$	3.63331	0.166010	0.0830050	+	0.996549i	$0.473548\pi$
$480$	0	0
$481$	−4.36669	−0.199104
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	0	0
$489$	10.7889	0.487891
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	13.5778i	0.611513i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	55.2666i	2.47905i
$498$	0	0
$499$	−30.4222	−1.36188	−0.680942	−	0.732337i	$-0.738430\pi$
$499$	−30.4222	−1.36188	−0.680942	+	0.732337i	$0.738430\pi$
$500$	0	0
$501$	−2.60555	−0.116407
$502$	0	0
$503$	42.2389i	1.88334i	0.336541	+	0.941669i	$0.390743\pi$
$503$	42.2389i	1.88334i	−0.336541	+	0.941669i	$0.609257\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.6333i	0.561065i
$508$	0	0
$509$	7.57779	0.335880	0.167940	−	0.985797i	$-0.446289\pi$
$509$	7.57779	0.335880	0.167940	+	0.985797i	$0.446289\pi$
$510$	0	0
$511$	21.2111	0.938324
$512$	0	0
$513$	2.00000i	0.0883022i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.21110i	0.229184i
$518$	0	0
$519$	14.6056	0.641113
$520$	0	0
$521$	7.57779	0.331989	0.165995	−	0.986127i	$-0.446917\pi$
$521$	7.57779	0.331989	0.165995	+	0.986127i	$0.446917\pi$
$522$	0	0
$523$	− 35.0278i	− 1.53166i	−0.643045	−	0.765828i	$-0.722329\pi$
$523$	− 35.0278i	− 1.53166i	0.643045	−	0.765828i	$-0.277671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.15559i	− 0.136684i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.4222i	0.967590i
$538$	0	0
$539$	14.2111	0.612116
$540$	0	0
$541$	−20.4222	−0.878019	−0.439010	−	0.898482i	$-0.644671\pi$
$541$	−20.4222	−0.878019	−0.439010	+	0.898482i	$0.644671\pi$
$542$	0	0
$543$	7.21110i	0.309458i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.6056i	1.05206i	0.850467	+	0.526029i	$0.176320\pi$
$547$	24.6056i	1.05206i	−0.850467	+	0.526029i	$0.823680\pi$
$548$	0	0
$549$	3.21110	0.137047
$550$	0	0
$551$	−10.4222	−0.444001
$552$	0	0
$553$	64.4777i	2.74187i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.1833i	0.685710i	0.939388	+	0.342855i	$0.111394\pi$
$557$	16.1833i	0.685710i	−0.939388	+	0.342855i	$0.888606\pi$
$558$	0	0
$559$	2.78890	0.117958
$560$	0	0
$561$	2.60555	0.110006
$562$	0	0
$563$	− 23.4500i	− 0.988298i	−0.869377	−	0.494149i	$-0.835480\pi$
$563$	− 23.4500i	− 0.988298i	0.869377	−	0.494149i	$-0.164520\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.60555i	− 0.193415i
$568$	0	0
$569$	32.8444	1.37691	0.688455	−	0.725279i	$-0.258289\pi$
$569$	32.8444	1.37691	0.688455	+	0.725279i	$0.258289\pi$
$570$	0	0
$571$	−15.2111	−0.636565	−0.318282	−	0.947996i	$-0.603106\pi$
$571$	−15.2111	−0.636565	−0.318282	+	0.947996i	$0.603106\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.2111i	1.13281i	0.824126	+	0.566407i	$0.191667\pi$
$577$	27.2111i	1.13281i	−0.824126	+	0.566407i	$0.808333\pi$
$578$	0	0
$579$	7.39445	0.307303
$580$	0	0
$581$	−43.2666	−1.79500
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.4222i	− 0.925463i	−0.886498	−	0.462732i	$-0.846869\pi$
$587$	− 22.4222i	− 0.925463i	0.886498	−	0.462732i	$-0.153131\pi$
$588$	0	0
$589$	18.4222	0.759074
$590$	0	0
$591$	19.8167	0.815148
$592$	0	0
$593$	9.39445i	0.385784i	0.981220	+	0.192892i	$0.0617867\pi$
$593$	9.39445i	0.385784i	−0.981220	+	0.192892i	$0.938213\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	−20.8444	−0.851680	−0.425840	−	0.904799i	$-0.640021\pi$
$599$	−20.8444	−0.851680	−0.425840	+	0.904799i	$0.640021\pi$
$600$	0	0
$601$	−37.6333	−1.53509	−0.767547	−	0.640992i	$-0.778523\pi$
$601$	−37.6333	−1.53509	−0.767547	+	0.640992i	$0.778523\pi$
$602$	0	0
$603$	− 14.4222i	− 0.587318i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 44.2389i	− 1.79560i	−0.440404	−	0.897800i	$-0.645165\pi$
$607$	− 44.2389i	− 1.79560i	0.440404	−	0.897800i	$-0.354835\pi$
$608$	0	0
$609$	24.0000	0.972529
$610$	0	0
$611$	−3.15559	−0.127661
$612$	0	0
$613$	− 31.3944i	− 1.26801i	−0.773329	−	0.634005i	$-0.781410\pi$
$613$	− 31.3944i	− 1.26801i	0.773329	−	0.634005i	$-0.218590\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 33.6333i	− 1.35403i	−0.735971	−	0.677013i	$-0.763274\pi$
$617$	− 33.6333i	− 1.35403i	0.735971	−	0.677013i	$-0.236726\pi$
$618$	0	0
$619$	−1.21110	−0.0486783	−0.0243392	−	0.999704i	$-0.507748\pi$
$619$	−1.21110	−0.0486783	−0.0243392	+	0.999704i	$0.507748\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	75.6333i	3.03018i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.00000i	0.0798723i
$628$	0	0
$629$	18.7889	0.749162
$630$	0	0
$631$	−2.42221	−0.0964265	−0.0482132	−	0.998837i	$-0.515353\pi$
$631$	−2.42221	−0.0964265	−0.0482132	+	0.998837i	$0.515353\pi$
$632$	0	0
$633$	2.00000i	0.0794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.60555i	0.340964i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	40.4222	1.59658	0.798291	−	0.602273i	$-0.205738\pi$
$641$	40.4222	1.59658	0.798291	+	0.602273i	$0.205738\pi$
$642$	0	0
$643$	− 42.0555i	− 1.65851i	−0.558872	−	0.829254i	$-0.688766\pi$
$643$	− 42.0555i	− 1.65851i	0.558872	−	0.829254i	$-0.311234\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.21110i	0.204870i	0.994740	+	0.102435i	$0.0326633\pi$
$647$	5.21110i	0.204870i	−0.994740	+	0.102435i	$0.967337\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−42.4222	−1.66266
$652$	0	0
$653$	− 45.6333i	− 1.78577i	−0.450285	−	0.892885i	$-0.648678\pi$
$653$	− 45.6333i	− 1.78577i	0.450285	−	0.892885i	$-0.351322\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.60555i	− 0.179680i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	24.4222	0.949914	0.474957	−	0.880009i	$-0.342464\pi$
$661$	24.4222	0.949914	0.474957	+	0.880009i	$0.342464\pi$
$662$	0	0
$663$	1.57779i	0.0612765i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	19.6333	0.759068
$670$	0	0
$671$	3.21110	0.123963
$672$	0	0
$673$	51.0278i	1.96698i	0.180975	+	0.983488i	$0.442075\pi$
$673$	51.0278i	1.96698i	−0.180975	+	0.983488i	$0.557925\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.6056i	− 0.561337i	−0.959805	−	0.280668i	$-0.909444\pi$
$677$	− 14.6056i	− 0.561337i	0.959805	−	0.280668i	$-0.0905561\pi$
$678$	0	0
$679$	46.0555	1.76745
$680$	0	0
$681$	19.8167	0.759376
$682$	0	0
$683$	6.78890i	0.259770i	0.991529	+	0.129885i	$0.0414608\pi$
$683$	6.78890i	0.259770i	−0.991529	+	0.129885i	$0.958539\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.42221i	0.321327i
$688$	0	0
$689$	−3.63331	−0.138418
$690$	0	0
$691$	39.2666	1.49377	0.746886	−	0.664952i	$-0.231548\pi$
$691$	39.2666	1.49377	0.746886	+	0.664952i	$0.231548\pi$
$692$	0	0
$693$	− 4.60555i	− 0.174950i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.5778i	0.514296i
$698$	0	0
$699$	−14.6056	−0.552433
$700$	0	0
$701$	27.6333	1.04370	0.521848	−	0.853039i	$-0.325243\pi$
$701$	27.6333	1.04370	0.521848	+	0.853039i	$0.325243\pi$
$702$	0	0
$703$	14.4222i	0.543944i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−32.7889	−1.23141	−0.615706	−	0.787976i	$-0.711129\pi$
$709$	−32.7889	−1.23141	−0.615706	+	0.787976i	$0.711129\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 27.6333i	− 1.03198i
$718$	0	0
$719$	44.8444	1.67241	0.836207	−	0.548414i	$-0.184768\pi$
$719$	44.8444	1.67241	0.836207	+	0.548414i	$0.184768\pi$
$720$	0	0
$721$	−42.4222	−1.57989
$722$	0	0
$723$	19.2111i	0.714469i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 52.8444i	− 1.95989i	−0.199266	−	0.979945i	$-0.563856\pi$
$727$	− 52.8444i	− 1.95989i	0.199266	−	0.979945i	$-0.436144\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	− 16.2389i	− 0.599796i	−0.953971	−	0.299898i	$-0.903047\pi$
$733$	− 16.2389i	− 0.599796i	0.953971	−	0.299898i	$-0.0969526\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.4222i	− 0.531249i
$738$	0	0
$739$	−8.78890	−0.323305	−0.161652	−	0.986848i	$-0.551682\pi$
$739$	−8.78890	−0.323305	−0.161652	+	0.986848i	$0.551682\pi$
$740$	0	0
$741$	−1.21110	−0.0444910
$742$	0	0
$743$	− 30.2389i	− 1.10936i	−0.832065	−	0.554678i	$-0.812841\pi$
$743$	− 30.2389i	− 1.10936i	0.832065	−	0.554678i	$-0.187159\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.39445i	0.343725i
$748$	0	0
$749$	91.2666	3.33481
$750$	0	0
$751$	−47.2666	−1.72478	−0.862392	−	0.506242i	$-0.831034\pi$
$751$	−47.2666	−1.72478	−0.862392	+	0.506242i	$0.831034\pi$
$752$	0	0
$753$	− 24.0000i	− 0.874609i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.78890i	0.174055i	0.996206	+	0.0870277i	$0.0277369\pi$
$757$	4.78890i	0.174055i	−0.996206	+	0.0870277i	$0.972263\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	43.2666	1.56841	0.784207	−	0.620500i	$-0.213070\pi$
$761$	43.2666	1.56841	0.784207	+	0.620500i	$0.213070\pi$
$762$	0	0
$763$	64.4777i	2.33425i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	3.21110	0.115795	0.0578977	−	0.998323i	$-0.481560\pi$
$769$	3.21110	0.115795	0.0578977	+	0.998323i	$0.481560\pi$
$770$	0	0
$771$	11.2111	0.403758
$772$	0	0
$773$	2.36669i	0.0851240i	0.999094	+	0.0425620i	$0.0135520\pi$
$773$	2.36669i	0.0851240i	−0.999094	+	0.0425620i	$0.986448\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 33.2111i	− 1.19144i
$778$	0	0
$779$	−10.4222	−0.373414
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	− 5.21110i	− 0.186230i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 15.0278i	− 0.535682i	−0.963463	−	0.267841i	$-0.913690\pi$
$787$	− 15.0278i	− 0.535682i	0.963463	−	0.267841i	$-0.0863101\pi$
$788$	0	0
$789$	−2.60555	−0.0927601
$790$	0	0
$791$	−51.6333	−1.83587
$792$	0	0
$793$	1.94449i	0.0690508i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.2111i	0.822179i	0.911595	+	0.411090i	$0.134852\pi$
$797$	23.2111i	0.822179i	−0.911595	+	0.411090i	$0.865148\pi$
$798$	0	0
$799$	13.5778	0.480348
$800$	0	0
$801$	16.4222	0.580250
$802$	0	0
$803$	− 4.60555i	− 0.162526i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.42221i	0.155669i
$808$	0	0
$809$	−27.6333	−0.971535	−0.485768	−	0.874088i	$-0.661460\pi$
$809$	−27.6333	−0.971535	−0.485768	+	0.874088i	$0.661460\pi$
$810$	0	0
$811$	−37.6333	−1.32148	−0.660742	−	0.750613i	$-0.729758\pi$
$811$	−37.6333	−1.32148	−0.660742	+	0.750613i	$0.729758\pi$
$812$	0	0
$813$	2.00000i	0.0701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.21110i	− 0.322256i
$818$	0	0
$819$	2.78890	0.0974520
$820$	0	0
$821$	15.6333	0.545606	0.272803	−	0.962070i	$-0.412049\pi$
$821$	15.6333	0.545606	0.272803	+	0.962070i	$0.412049\pi$
$822$	0	0
$823$	35.6333i	1.24210i	0.783771	+	0.621050i	$0.213293\pi$
$823$	35.6333i	1.24210i	−0.783771	+	0.621050i	$0.786707\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.2389i	1.88607i	0.332694	+	0.943035i	$0.392042\pi$
$827$	54.2389i	1.88607i	−0.332694	+	0.943035i	$0.607958\pi$
$828$	0	0
$829$	−16.0555	−0.557631	−0.278816	−	0.960345i	$-0.589942\pi$
$829$	−16.0555	−0.557631	−0.278816	+	0.960345i	$0.589942\pi$
$830$	0	0
$831$	−0.605551	−0.0210063
$832$	0	0
$833$	− 37.0278i	− 1.28294i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.21110i	0.318382i
$838$	0	0
$839$	46.4222	1.60267	0.801336	−	0.598214i	$-0.204123\pi$
$839$	46.4222	1.60267	0.801336	+	0.598214i	$0.204123\pi$
$840$	0	0
$841$	−1.84441	−0.0636004
$842$	0	0
$843$	29.2111i	1.00608i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.60555i	− 0.158249i
$848$	0	0
$849$	19.3944	0.665616
$850$	0	0
$851$	0	0
$852$	0	0
$853$	3.02776i	0.103668i	0.998656	+	0.0518342i	$0.0165067\pi$
$853$	3.02776i	0.103668i	−0.998656	+	0.0518342i	$0.983493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31.8167i	− 1.08684i	−0.839462	−	0.543418i	$-0.817130\pi$
$857$	− 31.8167i	− 1.08684i	0.839462	−	0.543418i	$-0.182870\pi$
$858$	0	0
$859$	43.6333	1.48875	0.744375	−	0.667762i	$-0.232748\pi$
$859$	43.6333	1.48875	0.744375	+	0.667762i	$0.232748\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	− 15.6333i	− 0.532164i	−0.963950	−	0.266082i	$-0.914271\pi$
$863$	− 15.6333i	− 0.532164i	0.963950	−	0.266082i	$-0.0857292\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.2111i	0.346787i
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	8.73338	0.295919
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 51.0278i	− 1.72308i	−0.507686	−	0.861542i	$-0.669499\pi$
$877$	− 51.0278i	− 1.72308i	0.507686	−	0.861542i	$-0.330501\pi$
$878$	0	0
$879$	−2.60555	−0.0878831
$880$	0	0
$881$	−4.42221	−0.148988	−0.0744939	−	0.997221i	$-0.523734\pi$
$881$	−4.42221	−0.148988	−0.0744939	+	0.997221i	$0.523734\pi$
$882$	0	0
$883$	− 38.4222i	− 1.29301i	−0.762910	−	0.646505i	$-0.776230\pi$
$883$	− 38.4222i	− 1.29301i	0.762910	−	0.646505i	$-0.223770\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.8722i	1.54024i	0.637901	+	0.770118i	$0.279803\pi$
$887$	45.8722i	1.54024i	−0.637901	+	0.770118i	$0.720197\pi$
$888$	0	0
$889$	−45.2111	−1.51633
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	10.4222i	0.348766i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	15.6333	0.520821
$902$	0	0
$903$	21.2111i	0.705861i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 26.7889i	− 0.889511i	−0.895652	−	0.444755i	$-0.853291\pi$
$907$	− 26.7889i	− 0.889511i	0.895652	−	0.444755i	$-0.146709\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	9.39445i	0.310911i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	79.2666i	2.61761i
$918$	0	0
$919$	−52.0555	−1.71715	−0.858576	−	0.512686i	$-0.828651\pi$
$919$	−52.0555	−1.71715	−0.858576	+	0.512686i	$0.828651\pi$
$920$	0	0
$921$	−17.8167	−0.587079
$922$	0	0
$923$	7.26662i	0.239184i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.21110i	0.302532i
$928$	0	0
$929$	4.42221	0.145088	0.0725439	−	0.997365i	$-0.476888\pi$
$929$	4.42221	0.145088	0.0725439	+	0.997365i	$0.476888\pi$
$930$	0	0
$931$	28.4222	0.931500
$932$	0	0
$933$	22.4222i	0.734070i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.2389i	− 1.05320i	−0.850114	−	0.526599i	$-0.823467\pi$
$937$	− 32.2389i	− 1.05320i	0.850114	−	0.526599i	$-0.176533\pi$
$938$	0	0
$939$	−19.2111	−0.626931
$940$	0	0
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.6333i	− 0.897962i	−0.893541	−	0.448981i	$-0.851787\pi$
$947$	− 27.6333i	− 0.897962i	0.893541	−	0.448981i	$-0.148213\pi$
$948$	0	0
$949$	2.78890	0.0905314
$950$	0	0
$951$	21.6333	0.701508
$952$	0	0
$953$	16.1833i	0.524230i	0.965037	+	0.262115i	$0.0844200\pi$
$953$	16.1833i	0.524230i	−0.965037	+	0.262115i	$0.915580\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.21110i	− 0.168451i
$958$	0	0
$959$	27.6333	0.892326
$960$	0	0
$961$	53.8444	1.73692
$962$	0	0
$963$	− 19.8167i	− 0.638583i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.6056i	0.791261i	0.918410	+	0.395631i	$0.129474\pi$
$967$	24.6056i	0.791261i	−0.918410	+	0.395631i	$0.870526\pi$
$968$	0	0
$969$	5.21110	0.167405
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	33.2111i	1.06470i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	16.4222	0.524856
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	50.0555i	1.59652i	0.602311	+	0.798261i	$0.294247\pi$
$983$	50.0555i	1.59652i	−0.602311	+	0.798261i	$0.705753\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35.6333	1.13193	0.565965	−	0.824430i	$-0.308504\pi$
$991$	35.6333	1.13193	0.565965	+	0.824430i	$0.308504\pi$
$992$	0	0
$993$	25.2111i	0.800050i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.23886i	0.134246i	0.997745	+	0.0671230i	$0.0213820\pi$
$997$	4.23886i	0.134246i	−0.997745	+	0.0671230i	$0.978618\pi$
$998$	0	0
$999$	−7.21110	−0.228149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.c.k.1849.3		4
3.2	odd	2		9900.2.c.v.5149.1		4
5.2	odd	4		660.2.a.f.1.2	✓	2
5.3	odd	4		3300.2.a.t.1.1		2
5.4	even	2	inner	3300.2.c.k.1849.2		4
15.2	even	4		1980.2.a.j.1.2		2
15.8	even	4		9900.2.a.bn.1.1		2
15.14	odd	2		9900.2.c.v.5149.4		4
20.7	even	4		2640.2.a.y.1.1		2
55.32	even	4		7260.2.a.z.1.1		2
60.47	odd	4		7920.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
660.2.a.f.1.2	✓	2	5.2	odd	4
1980.2.a.j.1.2		2	15.2	even	4
2640.2.a.y.1.1		2	20.7	even	4
3300.2.a.t.1.1		2	5.3	odd	4
3300.2.c.k.1849.2		4	5.4	even	2	inner
3300.2.c.k.1849.3		4	1.1	even	1	trivial
7260.2.a.z.1.1		2	55.32	even	4
7920.2.a.bn.1.1		2	60.47	odd	4
9900.2.a.bn.1.1		2	15.8	even	4
9900.2.c.v.5149.1		4	3.2	odd	2
9900.2.c.v.5149.4		4	15.14	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:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.60555i q^{7} -1.00000 q^{9} -1.00000 q^{11} -0.605551i q^{13} +2.60555i q^{17} -2.00000 q^{19} +4.60555 q^{21} -1.00000i q^{27} +5.21110 q^{29} -9.21110 q^{31} -1.00000i q^{33} -7.21110i q^{37} +0.605551 q^{39} +5.21110 q^{41} +4.60555i q^{43} -5.21110i q^{47} -14.2111 q^{49} -2.60555 q^{51} -6.00000i q^{53} -2.00000i q^{57} -3.21110 q^{61} +4.60555i q^{63} +14.4222i q^{67} -12.0000 q^{71} +4.60555i q^{73} +4.60555i q^{77} -14.0000 q^{79} +1.00000 q^{81} -9.39445i q^{83} +5.21110i q^{87} -16.4222 q^{89} -2.78890 q^{91} -9.21110i q^{93} +10.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 4 q^{11} - 8 q^{19} + 4 q^{21} - 8 q^{29} - 8 q^{31} - 12 q^{39} - 8 q^{41} - 28 q^{49} + 4 q^{51} + 16 q^{61} - 48 q^{71} - 56 q^{79} + 4 q^{81} - 8 q^{89} - 40 q^{91} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^11 - 8 * q^19 + 4 * q^21 - 8 * q^29 - 8 * q^31 - 12 * q^39 - 8 * q^41 - 28 * q^49 + 4 * q^51 + 16 * q^61 - 48 * q^71 - 56 * q^79 + 4 * q^81 - 8 * q^89 - 40 * q^91 + 4 * q^99

Introduction

L-functions

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