LMFDB - Embedded newform 1380.2.n.a.689.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(689,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1380.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.n (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.27
Character	$\chi$	$=$	1380.689
Dual form			1380.2.n.a.689.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.403788 + 1.68433i) q^{3} +(-2.20116 + 0.393570i) q^{5} -3.85443 q^{7} +(-2.67391 + 1.36022i) q^{9} +O(q^{10})$	\(q+(0.403788 + 1.68433i) q^{3} +(-2.20116 + 0.393570i) q^{5} -3.85443 q^{7} +(-2.67391 + 1.36022i) q^{9} +3.62241 q^{11} -1.57565i q^{13} +(-1.55170 - 3.54855i) q^{15} -5.19481i q^{17} -6.52090i q^{19} +(-1.55637 - 6.49211i) q^{21} +(-4.03526 + 2.59166i) q^{23} +(4.69021 - 1.73262i) q^{25} +(-3.37075 - 3.95450i) q^{27} +9.95654i q^{29} +8.99350 q^{31} +(1.46269 + 6.10132i) q^{33} +(8.48420 - 1.51699i) q^{35} +5.60205 q^{37} +(2.65391 - 0.636229i) q^{39} -2.15604i q^{41} -8.84370 q^{43} +(5.35036 - 4.04643i) q^{45} +9.35089 q^{47} +7.85659 q^{49} +(8.74976 - 2.09760i) q^{51} +6.55833i q^{53} +(-7.97351 + 1.42567i) q^{55} +(10.9833 - 2.63306i) q^{57} -5.15719i q^{59} -10.7339i q^{61} +(10.3064 - 5.24287i) q^{63} +(0.620129 + 3.46826i) q^{65} -2.14099 q^{67} +(-5.99459 - 5.75021i) q^{69} -8.71690i q^{71} -10.6914i q^{73} +(4.81215 + 7.20023i) q^{75} -13.9623 q^{77} -9.40689i q^{79} +(5.29959 - 7.27422i) q^{81} +5.23889i q^{83} +(2.04452 + 11.4346i) q^{85} +(-16.7701 + 4.02033i) q^{87} +7.33287 q^{89} +6.07323i q^{91} +(3.63147 + 15.1480i) q^{93} +(2.56643 + 14.3536i) q^{95} +2.45455 q^{97} +(-9.68600 + 4.92728i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	0.403788	+	1.68433i	0.233127	+	0.972446i
$3$	0.403788	+	1.68433i	0.233127	+	0.972446i
$4$		0			0
$5$	−2.20116	+	0.393570i	−0.984388	+	0.176010i
$5$	−2.20116	+	0.393570i	−0.984388	+	0.176010i
$6$		0			0
$7$	−3.85443			−1.45684			−0.728418	−	0.685133i	$-0.759744\pi$
$7$	−3.85443			−1.45684			−0.728418	+	0.685133i	$0.759744\pi$
$8$		0			0
$9$	−2.67391	+	1.36022i	−0.891304	+	0.453407i
$10$		0			0
$11$	3.62241			1.09220			0.546099	−	0.837721i	$-0.316112\pi$
$11$	3.62241			1.09220			0.546099	+	0.837721i	$0.316112\pi$
$12$		0			0
$13$		−	1.57565i		−	0.437007i	−0.975836	−	0.218504i	$-0.929882\pi$
$13$		−	1.57565i		−	0.437007i	0.975836	−	0.218504i	$-0.0701175\pi$
$14$		0			0
$15$	−1.55170	−	3.54855i	−0.400648	−	0.916232i
$16$		0			0
$17$		−	5.19481i		−	1.25993i	−0.776625	−	0.629964i	$-0.783070\pi$
$17$		−	5.19481i		−	1.25993i	0.776625	−	0.629964i	$-0.216930\pi$
$18$		0			0
$19$		−	6.52090i		−	1.49600i	−0.663700	−	0.747999i	$-0.731015\pi$
$19$		−	6.52090i		−	1.49600i	0.663700	−	0.747999i	$-0.268985\pi$
$20$		0			0
$21$	−1.55637	−	6.49211i	−0.339628	−	1.41669i
$22$		0			0
$23$	−4.03526	+	2.59166i	−0.841409	+	0.540398i
$23$	−4.03526	+	2.59166i	−0.841409	+	0.540398i
$24$		0			0
$25$	4.69021	−	1.73262i	0.938041	−	0.346524i
$26$		0			0
$27$	−3.37075	−	3.95450i	−0.648701	−	0.761043i
$28$		0			0
$29$			9.95654i			1.84888i	0.381325	+	0.924441i	$0.375468\pi$
$29$			9.95654i			1.84888i	−0.381325	+	0.924441i	$0.624532\pi$
$30$		0			0
$31$	8.99350			1.61528			0.807641	−	0.589675i	$-0.200744\pi$
$31$	8.99350			1.61528			0.807641	+	0.589675i	$0.200744\pi$
$32$		0			0
$33$	1.46269	+	6.10132i	0.254621	+	1.06210i
$34$		0			0
$35$	8.48420	−	1.51699i	1.43409	−	0.256417i
$36$		0			0
$37$	5.60205			0.920972			0.460486	−	0.887667i	$-0.347675\pi$
$37$	5.60205			0.920972			0.460486	+	0.887667i	$0.347675\pi$
$38$		0			0
$39$	2.65391	−	0.636229i	0.424966	−	0.101878i
$40$		0			0
$41$		−	2.15604i		−	0.336716i	−0.985726	−	0.168358i	$-0.946153\pi$
$41$		−	2.15604i		−	0.336716i	0.985726	−	0.168358i	$-0.0538465\pi$
$42$		0			0
$43$	−8.84370			−1.34865			−0.674326	−	0.738434i	$-0.735566\pi$
$43$	−8.84370			−1.34865			−0.674326	+	0.738434i	$0.735566\pi$
$44$		0			0
$45$	5.35036	−	4.04643i	0.797585	−	0.603207i
$46$		0			0
$47$	9.35089			1.36397			0.681984	−	0.731367i	$-0.261117\pi$
$47$	9.35089			1.36397			0.681984	+	0.731367i	$0.261117\pi$
$48$		0			0
$49$	7.85659			1.12237
$50$		0			0
$51$	8.74976	−	2.09760i	1.22521	−	0.293723i
$52$		0			0
$53$			6.55833i			0.900856i	0.892813	+	0.450428i	$0.148729\pi$
$53$			6.55833i			0.900856i	−0.892813	+	0.450428i	$0.851271\pi$
$54$		0			0
$55$	−7.97351	+	1.42567i	−1.07515	+	0.192238i
$56$		0			0
$57$	10.9833	−	2.63306i	1.45478	−	0.348758i
$58$		0			0
$59$		−	5.15719i		−	0.671408i	−0.941967	−	0.335704i	$-0.891026\pi$
$59$		−	5.15719i		−	0.671408i	0.941967	−	0.335704i	$-0.108974\pi$
$60$		0			0
$61$		−	10.7339i		−	1.37434i	−0.726498	−	0.687168i	$-0.758854\pi$
$61$		−	10.7339i		−	1.37434i	0.726498	−	0.687168i	$-0.241146\pi$
$62$		0			0
$63$	10.3064	−	5.24287i	1.29848	−	0.660540i
$64$		0			0
$65$	0.620129	+	3.46826i	0.0769175	+	0.430185i
$66$		0			0
$67$	−2.14099			−0.261563			−0.130782	−	0.991411i	$-0.541749\pi$
$67$	−2.14099			−0.261563			−0.130782	+	0.991411i	$0.541749\pi$
$68$		0			0
$69$	−5.99459	−	5.75021i	−0.721664	−	0.692244i
$70$		0			0
$71$		−	8.71690i		−	1.03451i	−0.855833	−	0.517253i	$-0.826955\pi$
$71$		−	8.71690i		−	1.03451i	0.855833	−	0.517253i	$-0.173045\pi$
$72$		0			0
$73$		−	10.6914i		−	1.25134i	−0.780090	−	0.625668i	$-0.784827\pi$
$73$		−	10.6914i		−	1.25134i	0.780090	−	0.625668i	$-0.215173\pi$
$74$		0			0
$75$	4.81215	+	7.20023i	0.555659	+	0.831410i
$76$		0			0
$77$	−13.9623			−1.59115
$78$		0			0
$79$		−	9.40689i		−	1.05836i	−0.848510	−	0.529179i	$-0.822500\pi$
$79$		−	9.40689i		−	1.05836i	0.848510	−	0.529179i	$-0.177500\pi$
$80$		0			0
$81$	5.29959	−	7.27422i	0.588844	−	0.808247i
$82$		0			0
$83$			5.23889i			0.575043i	0.957774	+	0.287521i	$0.0928312\pi$
$83$			5.23889i			0.575043i	−0.957774	+	0.287521i	$0.907169\pi$
$84$		0			0
$85$	2.04452	+	11.4346i	0.221760	+	1.24026i
$86$		0			0
$87$	−16.7701	+	4.02033i	−1.79794	+	0.431025i
$88$		0			0
$89$	7.33287			0.777282			0.388641	−	0.921389i	$-0.372945\pi$
$89$	7.33287			0.777282			0.388641	+	0.921389i	$0.372945\pi$
$90$		0			0
$91$			6.07323i			0.636648i
$92$		0			0
$93$	3.63147	+	15.1480i	0.376566	+	1.57077i
$94$		0			0
$95$	2.56643	+	14.3536i	0.263310	+	1.47264i
$96$		0			0
$97$	2.45455			0.249221			0.124611	−	0.992206i	$-0.460232\pi$
$97$	2.45455			0.249221			0.124611	+	0.992206i	$0.460232\pi$
$98$		0			0
$99$	−9.68600	+	4.92728i	−0.973480	+	0.495210i
$100$		0			0
$101$		−	0.883578i		−	0.0879193i	−0.999033	−	0.0439597i	$-0.986003\pi$
$101$		−	0.883578i		−	0.0879193i	0.999033	−	0.0439597i	$-0.0139973\pi$
$102$		0			0
$103$	−11.7691			−1.15964			−0.579821	−	0.814744i	$-0.696877\pi$
$103$	−11.7691			−1.15964			−0.579821	+	0.814744i	$0.696877\pi$
$104$		0			0
$105$	5.98092	+	13.6776i	0.583678	+	1.33480i
$106$		0			0
$107$		−	1.24813i		−	0.120662i	−0.998178	−	0.0603308i	$-0.980784\pi$
$107$		−	1.24813i		−	0.120662i	0.998178	−	0.0603308i	$-0.0192156\pi$
$108$		0			0
$109$		−	5.90303i		−	0.565407i	−0.959207	−	0.282704i	$-0.908769\pi$
$109$		−	5.90303i		−	0.565407i	0.959207	−	0.282704i	$-0.0912313\pi$
$110$		0			0
$111$	2.26204	+	9.43569i	0.214703	+	0.895596i
$112$		0			0
$113$		−	3.00542i		−	0.282726i	−0.989958	−	0.141363i	$-0.954851\pi$
$113$		−	3.00542i		−	0.282726i	0.989958	−	0.141363i	$-0.0451485\pi$
$114$		0			0
$115$	7.86225	−	7.29281i	0.733158	−	0.680058i
$116$		0			0
$117$	2.14324	+	4.21315i	0.198142	+	0.389506i
$118$		0			0
$119$			20.0230i			1.83551i
$120$		0			0
$121$	2.12187			0.192897
$122$		0			0
$123$	3.63147	−	0.870582i	0.327439	−	0.0784977i
$124$		0			0
$125$	−9.64198	+	5.65970i	−0.862405	+	0.506219i
$126$		0			0
$127$		−	16.1502i		−	1.43310i	−0.697534	−	0.716551i	$-0.745720\pi$
$127$		−	16.1502i		−	1.43310i	0.697534	−	0.716551i	$-0.254280\pi$
$128$		0			0
$129$	−3.57098	−	14.8957i	−0.314407	−	1.31149i
$130$		0			0
$131$		−	17.5863i		−	1.53652i	−0.640138	−	0.768260i	$-0.721123\pi$
$131$		−	17.5863i		−	1.53652i	0.640138	−	0.768260i	$-0.278877\pi$
$132$		0			0
$133$			25.1343i			2.17942i
$134$		0			0
$135$	8.97593	+	7.37785i	0.772525	+	0.634984i
$136$		0			0
$137$		−	10.9660i		−	0.936889i	−0.883493	−	0.468445i	$-0.844815\pi$
$137$		−	10.9660i		−	0.936889i	0.883493	−	0.468445i	$-0.155185\pi$
$138$		0			0
$139$	−8.07537			−0.684944			−0.342472	−	0.939528i	$-0.611264\pi$
$139$	−8.07537			−0.684944			−0.342472	+	0.939528i	$0.611264\pi$
$140$		0			0
$141$	3.77578	+	15.7500i	0.317978	+	1.32639i
$142$		0			0
$143$		−	5.70766i		−	0.477298i
$144$		0			0
$145$	−3.91859	−	21.9159i	−0.325421	−	1.82002i
$146$		0			0
$147$	3.17240	+	13.2331i	0.261655	+	1.09145i
$148$		0			0
$149$	12.5875			1.03121			0.515606	−	0.856826i	$-0.327567\pi$
$149$	12.5875			1.03121			0.515606	+	0.856826i	$0.327567\pi$
$150$		0			0
$151$	−17.8991			−1.45661			−0.728306	−	0.685252i	$-0.759692\pi$
$151$	−17.8991			−1.45661			−0.728306	+	0.685252i	$0.759692\pi$
$152$		0			0
$153$	7.06610	+	13.8905i	0.571260	+	1.12298i
$154$		0			0
$155$	−19.7961	+	3.53957i	−1.59006	+	0.284305i
$156$		0			0
$157$	7.44382			0.594082			0.297041	−	0.954865i	$-0.404000\pi$
$157$	7.44382			0.594082			0.297041	+	0.954865i	$0.404000\pi$
$158$		0			0
$159$	−11.0464	+	2.64818i	−0.876034	+	0.210014i
$160$		0			0
$161$	15.5536	−	9.98936i	1.22580	−	0.787272i
$162$		0			0
$163$		−	19.5357i		−	1.53015i	−0.643941	−	0.765075i	$-0.722702\pi$
$163$		−	19.5357i		−	1.53015i	0.643941	−	0.765075i	$-0.277298\pi$
$164$		0			0
$165$	−5.62090	−	12.8543i	−0.437587	−	1.00071i
$166$		0			0
$167$	−18.7702			−1.45248			−0.726242	−	0.687439i	$-0.758735\pi$
$167$	−18.7702			−1.45248			−0.726242	+	0.687439i	$0.758735\pi$
$168$		0			0
$169$	10.5173			0.809025
$170$		0			0
$171$	8.86987	+	17.4363i	0.678296	+	1.33339i
$172$		0			0
$173$	8.89789			0.676494			0.338247	−	0.941057i	$-0.390166\pi$
$173$	8.89789			0.676494			0.338247	+	0.941057i	$0.390166\pi$
$174$		0			0
$175$	−18.0780	+	6.67825i	−1.36657	+	0.504829i
$176$		0			0
$177$	8.68638	−	2.08241i	0.652909	−	0.156523i
$178$		0			0
$179$			4.51478i			0.337450i	0.985663	+	0.168725i	$0.0539650\pi$
$179$			4.51478i			0.337450i	−0.985663	+	0.168725i	$0.946035\pi$
$180$		0			0
$181$		−	1.64364i		−	0.122171i	−0.998133	−	0.0610853i	$-0.980544\pi$
$181$		−	1.64364i		−	0.122171i	0.998133	−	0.0610853i	$-0.0194562\pi$
$182$		0			0
$183$	18.0794	−	4.33422i	1.33647	−	0.320395i
$184$		0			0
$185$	−12.3310	+	2.20480i	−0.906594	+	0.162100i
$186$		0			0
$187$		−	18.8178i		−	1.37609i
$188$		0			0
$189$	12.9923	+	15.2423i	0.945051	+	1.10872i
$190$		0			0
$191$	−0.548393			−0.0396803			−0.0198402	−	0.999803i	$-0.506316\pi$
$191$	−0.548393			−0.0396803			−0.0198402	+	0.999803i	$0.506316\pi$
$192$		0			0
$193$		−	14.7429i		−	1.06122i	−0.847617	−	0.530608i	$-0.821964\pi$
$193$		−	14.7429i		−	1.06122i	0.847617	−	0.530608i	$-0.178036\pi$
$194$		0			0
$195$	−5.59128	+	2.44494i	−0.400400	+	0.175086i
$196$		0			0
$197$	−16.4470			−1.17180			−0.585899	−	0.810384i	$-0.699258\pi$
$197$	−16.4470			−1.17180			−0.585899	+	0.810384i	$0.699258\pi$
$198$		0			0
$199$			20.7568i			1.47141i	0.677304	+	0.735704i	$0.263148\pi$
$199$			20.7568i			1.47141i	−0.677304	+	0.735704i	$0.736852\pi$
$200$		0			0
$201$	−0.864506	−	3.60612i	−0.0609775	−	0.254356i
$202$		0			0
$203$		−	38.3767i		−	2.69352i
$204$		0			0
$205$	0.848551	+	4.74578i	0.0592654	+	0.331460i
$206$		0			0
$207$	7.26469	−	12.4187i	0.504931	−	0.863160i
$208$		0			0
$209$		−	23.6214i		−	1.63393i
$210$		0			0
$211$	4.80755			0.330965			0.165483	−	0.986213i	$-0.447082\pi$
$211$	4.80755			0.330965			0.165483	+	0.986213i	$0.447082\pi$
$212$		0			0
$213$	14.6821	−	3.51978i	1.00600	−	0.241171i
$214$		0			0
$215$	19.4664	−	3.48061i	1.32760	−	0.237376i
$216$		0			0
$217$	−34.6648			−2.35320
$218$		0			0
$219$	18.0078	−	4.31706i	1.21686	−	0.291720i
$220$		0			0
$221$	−8.18522			−0.550597
$222$		0			0
$223$			1.47165i			0.0985488i	0.998785	+	0.0492744i	$0.0156909\pi$
$223$			1.47165i			0.0985488i	−0.998785	+	0.0492744i	$0.984309\pi$
$224$		0			0
$225$	−10.1844	+	11.0126i	−0.678963	+	0.734173i
$226$		0			0
$227$			26.0587i			1.72958i	0.502137	+	0.864788i	$0.332547\pi$
$227$			26.0587i			1.72958i	−0.502137	+	0.864788i	$0.667453\pi$
$228$		0			0
$229$			8.78901i			0.580794i	0.956906	+	0.290397i	$0.0937874\pi$
$229$			8.78901i			0.580794i	−0.956906	+	0.290397i	$0.906213\pi$
$230$		0			0
$231$	−5.63781	−	23.5171i	−0.370941	−	1.54731i
$232$		0			0
$233$	−8.58544			−0.562451			−0.281225	−	0.959642i	$-0.590741\pi$
$233$	−8.58544			−0.562451			−0.281225	+	0.959642i	$0.590741\pi$
$234$		0			0
$235$	−20.5828	+	3.68023i	−1.34267	+	0.240072i
$236$		0			0
$237$	15.8443	−	3.79839i	1.02920	−	0.246732i
$238$		0			0
$239$			20.5449i			1.32894i	0.747315	+	0.664470i	$0.231342\pi$
$239$			20.5449i			1.32894i	−0.747315	+	0.664470i	$0.768658\pi$
$240$		0			0
$241$			14.8622i			0.957361i	0.877989	+	0.478681i	$0.158885\pi$
$241$			14.8622i			0.957361i	−0.877989	+	0.478681i	$0.841115\pi$
$242$		0			0
$243$	14.3921	+	5.98900i	0.923252	+	0.384195i
$244$		0			0
$245$	−17.2936	+	3.09212i	−1.10485	+	0.197548i
$246$		0			0
$247$	−10.2747			−0.653762
$248$		0			0
$249$	−8.82400	+	2.11540i	−0.559198	+	0.134058i
$250$		0			0
$251$	−3.39169			−0.214081			−0.107041	−	0.994255i	$-0.534138\pi$
$251$	−3.39169			−0.214081			−0.107041	+	0.994255i	$0.534138\pi$
$252$		0			0
$253$	−14.6174	+	9.38806i	−0.918986	+	0.590222i
$254$		0			0
$255$	−18.4341	+	8.06080i	−1.15439	+	0.504787i
$256$		0			0
$257$	18.2553			1.13873			0.569367	−	0.822083i	$-0.307188\pi$
$257$	18.2553			1.13873			0.569367	+	0.822083i	$0.307188\pi$
$258$		0			0
$259$	−21.5927			−1.34170
$260$		0			0
$261$	−13.5431	−	26.6229i	−0.838297	−	1.64792i
$262$		0			0
$263$			7.42459i			0.457820i	0.973448	+	0.228910i	$0.0735162\pi$
$263$			7.42459i			0.457820i	−0.973448	+	0.228910i	$0.926484\pi$
$264$		0			0
$265$	−2.58116	−	14.4359i	−0.158560	−	0.886793i
$266$		0			0
$267$	2.96092	+	12.3509i	0.181206	+	0.755865i
$268$		0			0
$269$		−	5.96364i		−	0.363609i	−0.983335	−	0.181805i	$-0.941806\pi$
$269$		−	5.96364i		−	0.363609i	0.983335	−	0.181805i	$-0.0581938\pi$
$270$		0			0
$271$	1.79000			0.108735			0.0543674	−	0.998521i	$-0.482686\pi$
$271$	1.79000			0.108735			0.0543674	+	0.998521i	$0.482686\pi$
$272$		0			0
$273$	−10.2293	+	2.45230i	−0.619106	+	0.148420i
$274$		0			0
$275$	16.9899	−	6.27626i	1.02453	−	0.378473i
$276$		0			0
$277$			1.30335i			0.0783110i	0.999233	+	0.0391555i	$0.0124668\pi$
$277$			1.30335i			0.0783110i	−0.999233	+	0.0391555i	$0.987533\pi$
$278$		0			0
$279$	−24.0478	+	12.2332i	−1.43971	+	0.732380i
$280$		0			0
$281$	18.1310			1.08160			0.540801	−	0.841150i	$-0.318121\pi$
$281$	18.1310			1.08160			0.540801	+	0.841150i	$0.318121\pi$
$282$		0			0
$283$	3.07042			0.182518			0.0912588	−	0.995827i	$-0.470911\pi$
$283$	3.07042			0.182518			0.0912588	+	0.995827i	$0.470911\pi$
$284$		0			0
$285$	−23.1398	+	10.1185i	−1.37068	+	0.599368i
$286$		0			0
$287$			8.31028i			0.490540i
$288$		0			0
$289$	−9.98610			−0.587417
$290$		0			0
$291$	0.991117	+	4.13426i	0.0581003	+	0.242355i
$292$		0			0
$293$		−	32.8876i		−	1.92131i	−0.277737	−	0.960657i	$-0.589584\pi$
$293$		−	32.8876i		−	1.92131i	0.277737	−	0.960657i	$-0.410416\pi$
$294$		0			0
$295$	2.02971	+	11.3518i	0.118174	+	0.660927i
$296$		0			0
$297$	−12.2102	−	14.3248i	−0.708510	−	0.831210i
$298$		0			0
$299$	4.08355	+	6.35816i	0.236158	+	0.367702i
$300$		0			0
$301$	34.0874			1.96476
$302$		0			0
$303$	1.48823	−	0.356778i	0.0854968	−	0.0204964i
$304$		0			0
$305$	4.22454	+	23.6270i	0.241897	+	1.35288i
$306$		0			0
$307$			17.1903i			0.981100i	0.871413	+	0.490550i	$0.163204\pi$
$307$			17.1903i			0.981100i	−0.871413	+	0.490550i	$0.836796\pi$
$308$		0			0
$309$	−4.75221	−	19.8230i	−0.270344	−	1.12769i
$310$		0			0
$311$		−	33.1149i		−	1.87778i	−0.344224	−	0.938888i	$-0.611858\pi$
$311$		−	33.1149i		−	1.87778i	0.344224	−	0.938888i	$-0.388142\pi$
$312$		0			0
$313$	−4.09052			−0.231210			−0.115605	−	0.993295i	$-0.536881\pi$
$313$	−4.09052			−0.231210			−0.115605	+	0.993295i	$0.536881\pi$
$314$		0			0
$315$	−20.6226	+	15.5967i	−1.16195	+	0.878773i
$316$		0			0
$317$	30.0968			1.69041			0.845203	−	0.534445i	$-0.179479\pi$
$317$	30.0968			1.69041			0.845203	+	0.534445i	$0.179479\pi$
$318$		0			0
$319$			36.0667i			2.01935i
$320$		0			0
$321$	2.10226	−	0.503981i	0.117337	−	0.0281295i
$322$		0			0
$323$	−33.8749			−1.88485
$324$		0			0
$325$	−2.73001	−	7.39013i	−0.151433	−	0.409931i
$326$		0			0
$327$	9.94262	−	2.38357i	0.549828	−	0.131812i
$328$		0			0
$329$	−36.0423			−1.98708
$330$		0			0
$331$	2.88833			0.158757			0.0793784	−	0.996845i	$-0.474706\pi$
$331$	2.88833			0.158757			0.0793784	+	0.996845i	$0.474706\pi$
$332$		0			0
$333$	−14.9794	+	7.62003i	−0.820865	+	0.417575i
$334$		0			0
$335$	4.71266	−	0.842629i	0.257480	−	0.0460377i
$336$		0			0
$337$	3.48092			0.189618			0.0948088	−	0.995496i	$-0.469776\pi$
$337$	3.48092			0.189618			0.0948088	+	0.995496i	$0.469776\pi$
$338$		0			0
$339$	5.06211	−	1.21355i	0.274936	−	0.0659112i
$340$		0			0
$341$	32.5782			1.76421
$342$		0			0
$343$	−3.30168			−0.178274
$344$		0			0
$345$	15.4582	+	10.2978i	0.832239	+	0.554417i
$346$		0			0
$347$	−2.93765			−0.157701			−0.0788507	−	0.996886i	$-0.525125\pi$
$347$	−2.93765			−0.157701			−0.0788507	+	0.996886i	$0.525125\pi$
$348$		0			0
$349$	−0.531226			−0.0284359			−0.0142179	−	0.999899i	$-0.504526\pi$
$349$	−0.531226			−0.0284359			−0.0142179	+	0.999899i	$0.504526\pi$
$350$		0			0
$351$	−6.23091	+	5.31113i	−0.332581	+	0.283487i
$352$		0			0
$353$	6.40672			0.340996			0.170498	−	0.985358i	$-0.445462\pi$
$353$	6.40672			0.340996			0.170498	+	0.985358i	$0.445462\pi$
$354$		0			0
$355$	3.43071	+	19.1873i	0.182083	+	1.01836i
$356$		0			0
$357$	−33.7253	+	8.08506i	−1.78493	+	0.427907i
$358$		0			0
$359$	−33.4878			−1.76742			−0.883708	−	0.468038i	$-0.844961\pi$
$359$	−33.4878			−1.76742			−0.883708	+	0.468038i	$0.844961\pi$
$360$		0			0
$361$	−23.5222			−1.23801
$362$		0			0
$363$	0.856784	+	3.57391i	0.0449695	+	0.187582i
$364$		0			0
$365$	4.20782	+	23.5335i	0.220247	+	1.23180i
$366$		0			0
$367$	−10.7030			−0.558691			−0.279346	−	0.960191i	$-0.590118\pi$
$367$	−10.7030			−0.558691			−0.279346	+	0.960191i	$0.590118\pi$
$368$		0			0
$369$	2.93269	+	5.76505i	0.152670	+	0.300116i
$370$		0			0
$371$		−	25.2786i		−	1.31240i
$372$		0			0
$373$	23.1273			1.19748			0.598742	−	0.800942i	$-0.295667\pi$
$373$	23.1273			1.19748			0.598742	+	0.800942i	$0.295667\pi$
$374$		0			0
$375$	−13.4261	−	13.9549i	−0.693320	−	0.720629i
$376$		0			0
$377$	15.6880			0.807975
$378$		0			0
$379$		−	18.1959i		−	0.934661i	−0.884083	−	0.467330i	$-0.845216\pi$
$379$		−	18.1959i		−	0.934661i	0.884083	−	0.467330i	$-0.154784\pi$
$380$		0			0
$381$	27.2023	−	6.52127i	1.39362	−	0.334095i
$382$		0			0
$383$			24.6826i			1.26122i	0.776099	+	0.630611i	$0.217196\pi$
$383$			24.6826i			1.26122i	−0.776099	+	0.630611i	$0.782804\pi$
$384$		0			0
$385$	30.7333	−	5.49515i	1.56631	−	0.280059i
$386$		0			0
$387$	23.6473	−	12.0294i	1.20206	−	0.611488i
$388$		0			0
$389$	−0.549183			−0.0278447			−0.0139224	−	0.999903i	$-0.504432\pi$
$389$	−0.549183			−0.0278447			−0.0139224	+	0.999903i	$0.504432\pi$
$390$		0			0
$391$	13.4632	+	20.9624i	0.680863	+	1.06011i
$392$		0			0
$393$	29.6210	−	7.10113i	1.49418	−	0.358205i
$394$		0			0
$395$	3.70227	+	20.7061i	0.186281	+	1.04184i
$396$		0			0
$397$		−	23.8586i		−	1.19743i	−0.800962	−	0.598716i	$-0.795678\pi$
$397$		−	23.8586i		−	1.19743i	0.800962	−	0.598716i	$-0.204322\pi$
$398$		0			0
$399$	−42.3344	+	10.1489i	−2.11937	+	0.508083i
$400$		0			0
$401$	4.40232			0.219841			0.109921	−	0.993940i	$-0.464940\pi$
$401$	4.40232			0.219841			0.109921	+	0.993940i	$0.464940\pi$
$402$		0			0
$403$		−	14.1706i		−	0.705889i
$404$		0			0
$405$	−8.80234	+	18.0975i	−0.437392	+	0.899271i
$406$		0			0
$407$	20.2929			1.00588
$408$		0			0
$409$	−10.2444			−0.506554			−0.253277	−	0.967394i	$-0.581508\pi$
$409$	−10.2444			−0.506554			−0.253277	+	0.967394i	$0.581508\pi$
$410$		0			0
$411$	18.4703	−	4.42794i	0.911075	−	0.218414i
$412$		0			0
$413$			19.8780i			0.978132i
$414$		0			0
$415$	−2.06187	−	11.5316i	−0.101213	−	0.566065i
$416$		0			0
$417$	−3.26074	−	13.6016i	−0.159679	−	0.666071i
$418$		0			0
$419$	24.0221			1.17355			0.586777	−	0.809748i	$-0.300396\pi$
$419$	24.0221			1.17355			0.586777	+	0.809748i	$0.300396\pi$
$420$		0			0
$421$			3.37345i			0.164412i	0.996615	+	0.0822060i	$0.0261965\pi$
$421$			3.37345i			0.164412i	−0.996615	+	0.0822060i	$0.973803\pi$
$422$		0			0
$423$	−25.0034	+	12.7193i	−1.21571	+	0.618433i
$424$		0			0
$425$	−9.00064	−	24.3647i	−0.436595	−	1.18186i
$426$		0			0
$427$			41.3730i			2.00218i
$428$		0			0
$429$	9.61356	−	2.30468i	0.464147	−	0.111271i
$430$		0			0
$431$	−34.0488			−1.64008			−0.820038	−	0.572310i	$-0.806048\pi$
$431$	−34.0488			−1.64008			−0.820038	+	0.572310i	$0.806048\pi$
$432$		0			0
$433$	−15.0954			−0.725441			−0.362720	−	0.931898i	$-0.618152\pi$
$433$	−15.0954			−0.725441			−0.362720	+	0.931898i	$0.618152\pi$
$434$		0			0
$435$	35.3313	−	15.4496i	1.69401	−	0.740750i
$436$		0			0
$437$	16.9000	+	26.3135i	0.808435	+	1.25875i
$438$		0			0
$439$	−12.7435			−0.608214			−0.304107	−	0.952638i	$-0.598358\pi$
$439$	−12.7435			−0.608214			−0.304107	+	0.952638i	$0.598358\pi$
$440$		0			0
$441$	−21.0078	+	10.6867i	−1.00037	+	0.508891i
$442$		0			0
$443$	−25.6598			−1.21914			−0.609568	−	0.792734i	$-0.708657\pi$
$443$	−25.6598			−1.21914			−0.609568	+	0.792734i	$0.708657\pi$
$444$		0			0
$445$	−16.1408	+	2.88599i	−0.765148	+	0.136809i
$446$		0			0
$447$	5.08270	+	21.2015i	0.240403	+	1.00280i
$448$		0			0
$449$			20.1929i			0.952962i	0.879185	+	0.476481i	$0.158088\pi$
$449$			20.1929i			0.952962i	−0.879185	+	0.476481i	$0.841912\pi$
$450$		0			0
$451$		−	7.81005i		−	0.367761i
$452$		0			0
$453$	−7.22746	−	30.1480i	−0.339576	−	1.41648i
$454$		0			0
$455$	−2.39024	−	13.3681i	−0.112056	−	0.626709i
$456$		0			0
$457$	−10.0704			−0.471071			−0.235536	−	0.971866i	$-0.575684\pi$
$457$	−10.0704			−0.471071			−0.235536	+	0.971866i	$0.575684\pi$
$458$		0			0
$459$	−20.5429	+	17.5104i	−0.958859	+	0.817316i
$460$		0			0
$461$			9.98459i			0.465029i	0.972593	+	0.232514i	$0.0746952\pi$
$461$			9.98459i			0.465029i	−0.972593	+	0.232514i	$0.925305\pi$
$462$		0			0
$463$		−	11.3677i		−	0.528301i	−0.964482	−	0.264150i	$-0.914909\pi$
$463$		−	11.3677i		−	0.528301i	0.964482	−	0.264150i	$-0.0850915\pi$
$464$		0			0
$465$	−13.9552	−	31.9139i	−0.647159	−	1.47997i
$466$		0			0
$467$			8.31931i			0.384972i	0.981300	+	0.192486i	$0.0616550\pi$
$467$			8.31931i			0.384972i	−0.981300	+	0.192486i	$0.938345\pi$
$468$		0			0
$469$	8.25228			0.381055
$470$		0			0
$471$	3.00573	+	12.5378i	0.138497	+	0.577713i
$472$		0			0
$473$	−32.0355			−1.47300
$474$		0			0
$475$	−11.2982	−	30.5844i	−0.518399	−	1.40331i
$476$		0			0
$477$	−8.92079	−	17.5364i	−0.408455	−	0.802936i
$478$		0			0
$479$	−33.6309			−1.53664			−0.768318	−	0.640068i	$-0.778906\pi$
$479$	−33.6309			−1.53664			−0.768318	+	0.640068i	$0.778906\pi$
$480$		0			0
$481$		−	8.82688i		−	0.402471i
$482$		0			0
$483$	23.1057	+	22.1637i	1.05135	+	1.00849i
$484$		0			0
$485$	−5.40285	+	0.966036i	−0.245331	+	0.0438654i
$486$		0			0
$487$			28.7679i			1.30360i	0.758392	+	0.651798i	$0.225985\pi$
$487$			28.7679i			1.30360i	−0.758392	+	0.651798i	$0.774015\pi$
$488$		0			0
$489$	32.9044	−	7.88826i	1.48799	−	0.356720i
$490$		0			0
$491$			21.1190i			0.953088i	0.879151	+	0.476544i	$0.158111\pi$
$491$			21.1190i			0.953088i	−0.879151	+	0.476544i	$0.841889\pi$
$492$		0			0
$493$	51.7224			2.32946
$494$		0			0
$495$	19.3812	−	14.6579i	0.871121	−	0.658821i
$496$		0			0
$497$			33.5986i			1.50710i
$498$		0			0
$499$	16.4630			0.736983			0.368492	−	0.929631i	$-0.379874\pi$
$499$	16.4630			0.736983			0.368492	+	0.929631i	$0.379874\pi$
$500$		0			0
$501$	−7.57919	−	31.6152i	−0.338613	−	1.41246i
$502$		0			0
$503$		−	26.1029i		−	1.16387i	−0.813235	−	0.581936i	$-0.802295\pi$
$503$		−	26.1029i		−	1.16387i	0.813235	−	0.581936i	$-0.197705\pi$
$504$		0			0
$505$	0.347750	+	1.94490i	0.0154747	+	0.0865468i
$506$		0			0
$507$	4.24677	+	17.7146i	0.188606	+	0.786733i
$508$		0			0
$509$			13.3194i			0.590371i	0.955440	+	0.295186i	$0.0953815\pi$
$509$			13.3194i			0.590371i	−0.955440	+	0.295186i	$0.904618\pi$
$510$		0			0
$511$			41.2093i			1.82299i
$512$		0			0
$513$	−25.7869	+	21.9803i	−1.13852	+	0.970456i
$514$		0			0
$515$	25.9056	−	4.63195i	1.14154	−	0.204108i
$516$		0			0
$517$	33.8728			1.48972
$518$		0			0
$519$	3.59286	+	14.9870i	0.157709	+	0.657854i
$520$		0			0
$521$	−20.4699			−0.896802			−0.448401	−	0.893832i	$-0.648006\pi$
$521$	−20.4699			−0.896802			−0.448401	+	0.893832i	$0.648006\pi$
$522$		0			0
$523$	−1.93791			−0.0847388			−0.0423694	−	0.999102i	$-0.513491\pi$
$523$	−1.93791			−0.0847388			−0.0423694	+	0.999102i	$0.513491\pi$
$524$		0			0
$525$	−18.5481	−	27.7527i	−0.809504	−	1.21123i
$526$		0			0
$527$		−	46.7196i		−	2.03514i
$528$		0			0
$529$	9.56660	−	20.9160i	0.415939	−	0.909392i
$530$		0			0
$531$	7.01491	+	13.7899i	0.304421	+	0.598429i
$532$		0			0
$533$	−3.39716			−0.147147
$534$		0			0
$535$	0.491228	+	2.74734i	0.0212376	+	0.118778i
$536$		0			0
$537$	−7.60436	+	1.82301i	−0.328152	+	0.0786688i
$538$		0			0
$539$	28.4598			1.22585
$540$		0			0
$541$	−40.2890			−1.73216			−0.866080	−	0.499905i	$-0.833368\pi$
$541$	−40.2890			−1.73216			−0.866080	+	0.499905i	$0.833368\pi$
$542$		0			0
$543$	2.76842	−	0.663681i	0.118804	−	0.0284813i
$544$		0			0
$545$	2.32325	+	12.9935i	0.0995172	+	0.556580i
$546$		0			0
$547$		−	12.7493i		−	0.545120i	−0.962139	−	0.272560i	$-0.912130\pi$
$547$		−	12.7493i		−	0.545120i	0.962139	−	0.272560i	$-0.0878704\pi$
$548$		0			0
$549$	14.6005	+	28.7015i	0.623134	+	1.22495i
$550$		0			0
$551$	64.9256			2.76592
$552$		0			0
$553$			36.2582i			1.54185i
$554$		0			0
$555$	−8.69272	−	19.8792i	−0.368985	−	0.843824i
$556$		0			0
$557$		−	40.4426i		−	1.71361i	−0.515642	−	0.856804i	$-0.672447\pi$
$557$		−	40.4426i		−	1.71361i	0.515642	−	0.856804i	$-0.327553\pi$
$558$		0			0
$559$			13.9346i			0.589370i
$560$		0			0
$561$	31.6952	−	7.59838i	1.33817	−	0.320804i
$562$		0			0
$563$			7.66702i			0.323126i	0.986862	+	0.161563i	$0.0516535\pi$
$563$			7.66702i			0.323126i	−0.986862	+	0.161563i	$0.948346\pi$
$564$		0			0
$565$	1.18284	+	6.61542i	0.0497626	+	0.278313i
$566$		0			0
$567$	−20.4269	+	28.0379i	−0.857849	+	1.17748i
$568$		0			0
$569$	6.76808			0.283733			0.141866	−	0.989886i	$-0.454690\pi$
$569$	6.76808			0.283733			0.141866	+	0.989886i	$0.454690\pi$
$570$		0			0
$571$		−	31.2756i		−	1.30884i	−0.756130	−	0.654421i	$-0.772912\pi$
$571$		−	31.2756i		−	1.30884i	0.756130	−	0.654421i	$-0.227088\pi$
$572$		0			0
$573$	−0.221434	−	0.923672i	−0.00925056	−	0.0385870i
$574$		0			0
$575$	−14.4358	+	19.1470i	−0.602015	+	0.798484i
$576$		0			0
$577$			10.6638i			0.443938i	0.975054	+	0.221969i	$0.0712483\pi$
$577$			10.6638i			0.443938i	−0.975054	+	0.221969i	$0.928752\pi$
$578$		0			0
$579$	24.8318	−	5.95300i	1.03198	−	0.247398i
$580$		0			0
$581$		−	20.1929i		−	0.837743i
$582$		0			0
$583$			23.7570i			0.983914i
$584$		0			0
$585$	−6.37577	−	8.43031i	−0.263606	−	0.348550i
$586$		0			0
$587$	21.2614			0.877552			0.438776	−	0.898597i	$-0.355412\pi$
$587$	21.2614			0.877552			0.438776	+	0.898597i	$0.355412\pi$
$588$		0			0
$589$		−	58.6458i		−	2.41646i
$590$		0			0
$591$	−6.64109	−	27.7021i	−0.273178	−	1.13951i
$592$		0			0
$593$	−18.2613			−0.749901			−0.374950	−	0.927045i	$-0.622340\pi$
$593$	−18.2613			−0.749901			−0.374950	+	0.927045i	$0.622340\pi$
$594$		0			0
$595$	−7.88046	−	44.0739i	−0.323067	−	1.80685i
$596$		0			0
$597$	−34.9612	+	8.38133i	−1.43086	+	0.343025i
$598$		0			0
$599$		−	9.66719i		−	0.394991i	−0.980304	−	0.197495i	$-0.936719\pi$
$599$		−	9.66719i		−	0.394991i	0.980304	−	0.197495i	$-0.0632807\pi$
$600$		0			0
$601$	14.2939			0.583061			0.291531	−	0.956562i	$-0.405835\pi$
$601$	14.2939			0.583061			0.291531	+	0.956562i	$0.405835\pi$
$602$		0			0
$603$	5.72481	−	2.91222i	0.233132	−	0.118595i
$604$		0			0
$605$	−4.67056	+	0.835102i	−0.189885	+	0.0339517i
$606$		0			0
$607$		−	5.83706i		−	0.236919i	−0.992959	−	0.118459i	$-0.962204\pi$
$607$		−	5.83706i		−	0.236919i	0.992959	−	0.118459i	$-0.0377956\pi$
$608$		0			0
$609$	64.6389	−	15.4961i	2.61930	−	0.627932i
$610$		0			0
$611$		−	14.7337i		−	0.596064i
$612$		0			0
$613$	−22.0410			−0.890226			−0.445113	−	0.895474i	$-0.646836\pi$
$613$	−22.0410			−0.890226			−0.445113	+	0.895474i	$0.646836\pi$
$614$		0			0
$615$	−7.65081	+	3.34553i	−0.308510	+	0.134905i
$616$		0			0
$617$			11.8081i			0.475378i	0.971341	+	0.237689i	$0.0763898\pi$
$617$			11.8081i			0.475378i	−0.971341	+	0.237689i	$0.923610\pi$
$618$		0			0
$619$		−	27.3240i		−	1.09825i	−0.835742	−	0.549123i	$-0.814962\pi$
$619$		−	27.3240i		−	1.09825i	0.835742	−	0.549123i	$-0.185038\pi$
$620$		0			0
$621$	23.8506	+	7.22157i	0.957090	+	0.289792i
$622$		0			0
$623$	−28.2640			−1.13237
$624$		0			0
$625$	18.9961	−	16.2527i	0.759842	−	0.650107i
$626$		0			0
$627$	39.7861	−	9.53804i	1.58891	−	0.380913i
$628$		0			0
$629$		−	29.1016i		−	1.16036i
$630$		0			0
$631$		−	3.72151i		−	0.148151i	−0.997253	−	0.0740755i	$-0.976399\pi$
$631$		−	3.72151i		−	0.148151i	0.997253	−	0.0740755i	$-0.0236006\pi$
$632$		0			0
$633$	1.94123	+	8.09748i	0.0771570	+	0.321846i
$634$		0			0
$635$	6.35625	+	35.5493i	0.252240	+	1.41073i
$636$		0			0
$637$		−	12.3793i		−	0.490484i
$638$		0			0
$639$	11.8569	+	23.3082i	0.469052	+	0.922059i
$640$		0			0
$641$	22.4449			0.886521			0.443261	−	0.896393i	$-0.353822\pi$
$641$	22.4449			0.886521			0.443261	+	0.896393i	$0.353822\pi$
$642$		0			0
$643$	13.5307			0.533597			0.266799	−	0.963752i	$-0.414034\pi$
$643$	13.5307			0.533597			0.266799	+	0.963752i	$0.414034\pi$
$644$		0			0
$645$	13.7228	+	31.3823i	0.540334	+	1.23568i
$646$		0			0
$647$	22.5478			0.886444			0.443222	−	0.896412i	$-0.353835\pi$
$647$	22.5478			0.886444			0.443222	+	0.896412i	$0.353835\pi$
$648$		0			0
$649$		−	18.6814i		−	0.733311i
$650$		0			0
$651$	−13.9972	−	58.3868i	−0.548595	−	2.28836i
$652$		0			0
$653$	27.1467			1.06233			0.531166	−	0.847268i	$-0.321754\pi$
$653$	27.1467			1.06233			0.531166	+	0.847268i	$0.321754\pi$
$654$		0			0
$655$	6.92143	+	38.7102i	0.270443	+	1.51253i
$656$		0			0
$657$	14.5427	+	28.5879i	0.567364	+	1.11532i
$658$		0			0
$659$	−26.0104			−1.01322			−0.506611	−	0.862175i	$-0.669102\pi$
$659$	−26.0104			−1.01322			−0.506611	+	0.862175i	$0.669102\pi$
$660$		0			0
$661$			11.1835i			0.434988i	0.976062	+	0.217494i	$0.0697882\pi$
$661$			11.1835i			0.434988i	−0.976062	+	0.217494i	$0.930212\pi$
$662$		0			0
$663$	−3.30509	−	13.7866i	−0.128359	−	0.535426i
$664$		0			0
$665$	−9.89212	−	55.3247i	−0.383600	−	2.14540i
$666$		0			0
$667$	−25.8040	−	40.1772i	−0.999133	−	1.55567i
$668$		0			0
$669$	−2.47873	+	0.594234i	−0.0958334	+	0.0229744i
$670$		0			0
$671$		−	38.8826i		−	1.50105i
$672$		0			0
$673$			5.70314i			0.219840i	0.993940	+	0.109920i	$0.0350594\pi$
$673$			5.70314i			0.219840i	−0.993940	+	0.109920i	$0.964941\pi$
$674$		0			0
$675$	−22.6611	−	12.7072i	−0.872228	−	0.489099i
$676$		0			0
$677$		−	6.49317i		−	0.249553i	−0.992185	−	0.124776i	$-0.960179\pi$
$677$		−	6.49317i		−	0.249553i	0.992185	−	0.124776i	$-0.0398214\pi$
$678$		0			0
$679$	−9.46087			−0.363075
$680$		0			0
$681$	−43.8913	+	10.5222i	−1.68192	+	0.403211i
$682$		0			0
$683$	−0.0744250			−0.00284779			−0.00142390	−	0.999999i	$-0.500453\pi$
$683$	−0.0744250			−0.00284779			−0.00142390	+	0.999999i	$0.500453\pi$
$684$		0			0
$685$	4.31589	+	24.1379i	0.164902	+	0.922263i
$686$		0			0
$687$	−14.8036	+	3.54890i	−0.564791	+	0.135399i
$688$		0			0
$689$	10.3336			0.393681
$690$		0			0
$691$	9.10705			0.346449			0.173224	−	0.984882i	$-0.444581\pi$
$691$	9.10705			0.346449			0.173224	+	0.984882i	$0.444581\pi$
$692$		0			0
$693$	37.3340	−	18.9918i	1.41820	−	0.721440i
$694$		0			0
$695$	17.7752	−	3.17822i	0.674251	−	0.120557i
$696$		0			0
$697$	−11.2002			−0.424238
$698$		0			0
$699$	−3.46670	−	14.4607i	−0.131123	−	0.546953i
$700$		0			0
$701$	−14.3600			−0.542368			−0.271184	−	0.962527i	$-0.587415\pi$
$701$	−14.3600			−0.542368			−0.271184	+	0.962527i	$0.587415\pi$
$702$		0			0
$703$		−	36.5305i		−	1.37777i
$704$		0			0
$705$	−14.5098	−	33.1821i	−0.546470	−	1.24971i
$706$		0			0
$707$			3.40569i			0.128084i
$708$		0			0
$709$			47.8766i			1.79804i	0.437905	+	0.899021i	$0.355721\pi$
$709$			47.8766i			1.79804i	−0.437905	+	0.899021i	$0.644279\pi$
$710$		0			0
$711$	12.7955	+	25.1532i	0.479867	+	0.943318i
$712$		0			0
$713$	−36.2911	+	23.3081i	−1.35911	+	0.872895i
$714$		0			0
$715$	2.24636	+	12.5635i	0.0840092	+	0.469847i
$716$		0			0
$717$	−34.6043	+	8.29578i	−1.29232	+	0.309812i
$718$		0			0
$719$		−	34.0023i		−	1.26807i	−0.773304	−	0.634036i	$-0.781397\pi$
$719$		−	34.0023i		−	1.26807i	0.773304	−	0.634036i	$-0.218603\pi$
$720$		0			0
$721$	45.3630			1.68941
$722$		0			0
$723$	−25.0329	+	6.00120i	−0.930982	+	0.223187i
$724$		0			0
$725$	17.2509	+	46.6982i	0.640682	+	1.73433i
$726$		0			0
$727$	40.4234			1.49922			0.749610	−	0.661880i	$-0.230241\pi$
$727$	40.4234			1.49922			0.749610	+	0.661880i	$0.230241\pi$
$728$		0			0
$729$	−4.27609	+	26.6592i	−0.158374	+	0.987379i
$730$		0			0
$731$			45.9414i			1.69920i
$732$		0			0
$733$	27.0958			1.00081			0.500403	−	0.865793i	$-0.333185\pi$
$733$	27.0958			1.00081			0.500403	+	0.865793i	$0.333185\pi$
$734$		0			0
$735$	−12.1911	−	27.8795i	−0.449675	−	1.02835i
$736$		0			0
$737$	−7.75554			−0.285679
$738$		0			0
$739$	38.8420			1.42882			0.714412	−	0.699725i	$-0.246694\pi$
$739$	38.8420			1.42882			0.714412	+	0.699725i	$0.246694\pi$
$740$		0			0
$741$	−4.14879	−	17.3059i	−0.152410	−	0.635748i
$742$		0			0
$743$		−	8.63643i		−	0.316840i	−0.987372	−	0.158420i	$-0.949360\pi$
$743$		−	8.63643i		−	0.316840i	0.987372	−	0.158420i	$-0.0506400\pi$
$744$		0			0
$745$	−27.7072	+	4.95407i	−1.01511	+	0.181503i
$746$		0			0
$747$	−7.12605	−	14.0083i	−0.260729	−	0.512538i
$748$		0			0
$749$			4.81084i			0.175784i
$750$		0			0
$751$		−	30.1656i		−	1.10076i	−0.834915	−	0.550379i	$-0.814483\pi$
$751$		−	30.1656i		−	1.10076i	0.834915	−	0.550379i	$-0.185517\pi$
$752$		0			0
$753$	−1.36952	−	5.71271i	−0.0499082	−	0.208183i
$754$		0			0
$755$	39.3989	−	7.04456i	1.43387	−	0.256378i
$756$		0			0
$757$	47.5310			1.72754			0.863772	−	0.503883i	$-0.168096\pi$
$757$	47.5310			1.72754			0.863772	+	0.503883i	$0.168096\pi$
$758$		0			0
$759$	−21.7149	−	20.8296i	−0.788200	−	0.756067i
$760$		0			0
$761$			10.3590i			0.375514i	0.982215	+	0.187757i	$0.0601218\pi$
$761$			10.3590i			0.375514i	−0.982215	+	0.187757i	$0.939878\pi$
$762$		0			0
$763$			22.7528i			0.823706i
$764$		0			0
$765$	−21.0205	−	27.7941i	−0.759997	−	1.00490i
$766$		0			0
$767$	−8.12593			−0.293410
$768$		0			0
$769$		−	4.78917i		−	0.172702i	−0.996265	−	0.0863509i	$-0.972479\pi$
$769$		−	4.78917i		−	0.172702i	0.996265	−	0.0863509i	$-0.0275206\pi$
$770$		0			0
$771$	7.37127	+	30.7479i	0.265470	+	1.10736i
$772$		0			0
$773$		−	0.772116i		−	0.0277711i	−0.999904	−	0.0138855i	$-0.995580\pi$
$773$		−	0.772116i		−	0.0277711i	0.999904	−	0.0138855i	$-0.00442005\pi$
$774$		0			0
$775$	42.1814	−	15.5823i	1.51520	−	0.559734i
$776$		0			0
$777$	−8.71887	−	36.3691i	−0.312788	−	1.30474i
$778$		0			0
$779$	−14.0593			−0.503727
$780$		0			0
$781$		−	31.5762i		−	1.12989i
$782$		0			0
$783$	39.3731	−	33.5610i	1.40708	−	1.19937i
$784$		0			0
$785$	−16.3850	+	2.92966i	−0.584807	+	0.104564i
$786$		0			0
$787$	17.0859			0.609045			0.304523	−	0.952505i	$-0.401503\pi$
$787$	17.0859			0.609045			0.304523	+	0.952505i	$0.401503\pi$
$788$		0			0
$789$	−12.5054	+	2.99796i	−0.445205	+	0.106730i
$790$		0			0
$791$			11.5842i			0.411886i
$792$		0			0
$793$	−16.9129			−0.600595
$794$		0			0
$795$	23.2726	−	10.1766i	0.825394	−	0.360926i
$796$		0			0
$797$			47.2162i			1.67248i	0.548361	+	0.836241i	$0.315252\pi$
$797$			47.2162i			1.67248i	−0.548361	+	0.836241i	$0.684748\pi$
$798$		0			0
$799$		−	48.5761i		−	1.71850i
$800$		0			0
$801$	−19.6074	+	9.97432i	−0.692794	+	0.352425i
$802$		0			0
$803$		−	38.7287i		−	1.36671i
$804$		0			0
$805$	−30.3044	+	28.1096i	−1.06809	+	0.990733i
$806$		0			0
$807$	10.0447	−	2.40804i	0.353590	−	0.0847672i
$808$		0			0
$809$		−	31.7441i		−	1.11606i	−0.829820	−	0.558031i	$-0.811557\pi$
$809$		−	31.7441i		−	1.11606i	0.829820	−	0.558031i	$-0.188443\pi$
$810$		0			0
$811$	1.17600			0.0412948			0.0206474	−	0.999787i	$-0.493427\pi$
$811$	1.17600			0.0412948			0.0206474	+	0.999787i	$0.493427\pi$
$812$		0			0
$813$	0.722781	+	3.01495i	0.0253491	+	0.105739i
$814$		0			0
$815$	7.68864	+	43.0011i	0.269321	+	1.50626i
$816$		0			0
$817$			57.6689i			2.01758i
$818$		0			0
$819$	−8.26094	−	16.2393i	−0.288661	−	0.567446i
$820$		0			0
$821$			30.4926i			1.06420i	0.846682	+	0.532100i	$0.178597\pi$
$821$			30.4926i			1.06420i	−0.846682	+	0.532100i	$0.821403\pi$
$822$		0			0
$823$		−	54.7241i		−	1.90756i	−0.300501	−	0.953781i	$-0.597154\pi$
$823$		−	54.7241i		−	1.90756i	0.300501	−	0.953781i	$-0.402846\pi$
$824$		0			0
$825$	17.4316	+	26.0822i	0.606889	+	0.908065i
$826$		0			0
$827$		−	30.5522i		−	1.06240i	−0.847245	−	0.531202i	$-0.821741\pi$
$827$		−	30.5522i		−	1.06240i	0.847245	−	0.531202i	$-0.178259\pi$
$828$		0			0
$829$	13.9785			0.485492			0.242746	−	0.970090i	$-0.421952\pi$
$829$	13.9785			0.485492			0.242746	+	0.970090i	$0.421952\pi$
$830$		0			0
$831$	−2.19527	+	0.526279i	−0.0761532	+	0.0182564i
$832$		0			0
$833$		−	40.8135i		−	1.41411i
$834$		0			0
$835$	41.3163	−	7.38739i	1.42981	−	0.255651i
$836$		0			0
$837$	−30.3149	−	35.5648i	−1.04783	−	1.22930i
$838$		0			0
$839$	−37.6423			−1.29956			−0.649779	−	0.760123i	$-0.725139\pi$
$839$	−37.6423			−1.29956			−0.649779	+	0.760123i	$0.725139\pi$
$840$		0			0
$841$	−70.1326			−2.41837
$842$		0			0
$843$	7.32106	+	30.5385i	0.252151	+	1.05180i
$844$		0			0
$845$	−23.1503	+	4.13930i	−0.796395	+	0.142396i
$846$		0			0
$847$	−8.17857			−0.281019
$848$		0			0
$849$	1.23980	+	5.17159i	0.0425498	+	0.177489i
$850$		0			0
$851$	−22.6057	+	14.5186i	−0.774914	+	0.497692i
$852$		0			0
$853$			21.1904i			0.725544i	0.931878	+	0.362772i	$0.118170\pi$
$853$			21.1904i			0.725544i	−0.931878	+	0.362772i	$0.881830\pi$
$854$		0			0
$855$	−26.3864	−	34.8892i	−0.902396	−	1.19319i
$856$		0			0
$857$	32.5880			1.11318			0.556592	−	0.830786i	$-0.312109\pi$
$857$	32.5880			1.11318			0.556592	+	0.830786i	$0.312109\pi$
$858$		0			0
$859$	45.8008			1.56270			0.781351	−	0.624092i	$-0.214531\pi$
$859$	45.8008			1.56270			0.781351	+	0.624092i	$0.214531\pi$
$860$		0			0
$861$	−13.9972	+	3.35559i	−0.477024	+	0.114358i
$862$		0			0
$863$	1.66726			0.0567541			0.0283771	−	0.999597i	$-0.490966\pi$
$863$	1.66726			0.0567541			0.0283771	+	0.999597i	$0.490966\pi$
$864$		0			0
$865$	−19.5857	+	3.50194i	−0.665933	+	0.119070i
$866$		0			0
$867$	−4.03227	−	16.8198i	−0.136943	−	0.571232i
$868$		0			0
$869$		−	34.0756i		−	1.15594i
$870$		0			0
$871$			3.37345i			0.114305i
$872$		0			0
$873$	−6.56324	+	3.33873i	−0.222132	+	0.112999i
$874$		0			0
$875$	37.1643	−	21.8149i	1.25638	−	0.737477i
$876$		0			0
$877$			45.6739i			1.54230i	0.636655	+	0.771149i	$0.280317\pi$
$877$			45.6739i			1.54230i	−0.636655	+	0.771149i	$0.719683\pi$
$878$		0			0
$879$	55.3935	−	13.2796i	1.86837	−	0.447910i
$880$		0			0
$881$	3.64912			0.122942			0.0614710	−	0.998109i	$-0.480421\pi$
$881$	3.64912			0.122942			0.0614710	+	0.998109i	$0.480421\pi$
$882$		0			0
$883$			26.5125i			0.892216i	0.894979	+	0.446108i	$0.147190\pi$
$883$			26.5125i			0.892216i	−0.894979	+	0.446108i	$0.852810\pi$
$884$		0			0
$885$	−18.3005	+	8.00241i	−0.615166	+	0.268998i
$886$		0			0
$887$	−35.8506			−1.20375			−0.601873	−	0.798592i	$-0.705579\pi$
$887$	−35.8506			−1.20375			−0.601873	+	0.798592i	$0.705579\pi$
$888$		0			0
$889$			62.2499i			2.08779i
$890$		0			0
$891$	19.1973	−	26.3502i	0.643134	−	0.882766i
$892$		0			0
$893$		−	60.9763i		−	2.04049i
$894$		0			0
$895$	−1.77688	−	9.93775i	−0.0593946	−	0.332182i
$896$		0			0
$897$	−9.06033	+	9.44538i	−0.302515	+	0.315372i
$898$		0			0
$899$			89.5442i			2.98646i
$900$		0			0
$901$	34.0693			1.13501
$902$		0			0
$903$	13.7641	+	57.4143i	0.458040	+	1.91063i
$904$		0			0
$905$	0.646886	+	3.61791i	0.0215032	+	0.120263i
$906$		0			0
$907$	−12.5499			−0.416714			−0.208357	−	0.978053i	$-0.566812\pi$
$907$	−12.5499			−0.416714			−0.208357	+	0.978053i	$0.566812\pi$
$908$		0			0
$909$	1.20186	+	2.36261i	0.0398632	+	0.0783628i
$910$		0			0
$911$	36.2713			1.20172			0.600861	−	0.799354i	$-0.294825\pi$
$911$	36.2713			1.20172			0.600861	+	0.799354i	$0.294825\pi$
$912$		0			0
$913$			18.9774i			0.628061i
$914$		0			0
$915$	−38.0898	+	16.6558i	−1.25921	+	0.550625i
$916$		0			0
$917$			67.7850i			2.23846i
$918$		0			0
$919$			29.8092i			0.983313i	0.870789	+	0.491657i	$0.163608\pi$
$919$			29.8092i			0.983313i	−0.870789	+	0.491657i	$0.836392\pi$
$920$		0			0
$921$	−28.9540	+	6.94122i	−0.954067	+	0.228721i
$922$		0			0
$923$	−13.7348			−0.452086
$924$		0			0
$925$	26.2748	−	9.70623i	0.863909	−	0.319139i
$926$		0			0
$927$	31.4695	−	16.0085i	1.03359	−	0.525790i
$928$		0			0
$929$		−	44.6592i		−	1.46522i	−0.680648	−	0.732611i	$-0.738302\pi$
$929$		−	44.6592i		−	1.46522i	0.680648	−	0.732611i	$-0.261698\pi$
$930$		0			0
$931$		−	51.2321i		−	1.67906i
$932$		0			0
$933$	55.7763	−	13.3714i	1.82604	−	0.437760i
$934$		0			0
$935$	7.40610	+	41.4209i	0.242205	+	1.35461i
$936$		0			0
$937$	9.56496			0.312474			0.156237	−	0.987720i	$-0.450064\pi$
$937$	9.56496			0.312474			0.156237	+	0.987720i	$0.450064\pi$
$938$		0			0
$939$	−1.65170	−	6.88977i	−0.0539013	−	0.224839i
$940$		0			0
$941$	0.391211			0.0127531			0.00637656	−	0.999980i	$-0.497970\pi$
$941$	0.391211			0.0127531			0.00637656	+	0.999980i	$0.497970\pi$
$942$		0			0
$943$	5.58771	+	8.70016i	0.181961	+	0.283316i
$944$		0			0
$945$	−34.5970	−	28.4374i	−1.12544	−	0.925068i
$946$		0			0
$947$	−44.3716			−1.44188			−0.720942	−	0.692996i	$-0.756290\pi$
$947$	−44.3716			−1.44188			−0.720942	+	0.692996i	$0.756290\pi$
$948$		0			0
$949$	−16.8459			−0.546842
$950$		0			0
$951$	12.1527	+	50.6929i	0.394080	+	1.64383i
$952$		0			0
$953$		−	16.3734i		−	0.530386i	−0.964195	−	0.265193i	$-0.914564\pi$
$953$		−	16.3734i		−	0.530386i	0.964195	−	0.265193i	$-0.0854357\pi$
$954$		0			0
$955$	1.20710	−	0.215831i	0.0390608	−	0.00698412i
$956$		0			0
$957$	−60.7480	+	14.5633i	−1.96371	+	0.470764i
$958$		0			0
$959$			42.2677i			1.36489i
$960$		0			0
$961$	49.8831			1.60913
$962$		0			0
$963$	1.69774	+	3.33740i	0.0547088	+	0.107546i
$964$		0			0
$965$	5.80235	+	32.4514i	0.186784	+	1.04465i
$966$		0			0
$967$			4.53370i			0.145794i	0.997339	+	0.0728970i	$0.0232244\pi$
$967$			4.53370i			0.145794i	−0.997339	+	0.0728970i	$0.976776\pi$
$968$		0			0
$969$	−13.6783	−	57.0564i	−0.439409	−	1.83291i
$970$		0			0
$971$	3.46255			0.111119			0.0555593	−	0.998455i	$-0.482306\pi$
$971$	3.46255			0.111119			0.0555593	+	0.998455i	$0.482306\pi$
$972$		0			0
$973$	31.1259			0.997851
$974$		0			0
$975$	11.3450	−	7.58227i	0.363332	−	0.242827i
$976$		0			0
$977$		−	6.08953i		−	0.194821i	−0.995244	−	0.0974107i	$-0.968944\pi$
$977$		−	6.08953i		−	0.194821i	0.995244	−	0.0974107i	$-0.0310560\pi$
$978$		0			0
$979$	26.5627			0.848946
$980$		0			0
$981$	8.02942	+	15.7842i	0.256360	+	0.503950i
$982$		0			0
$983$		−	34.3403i		−	1.09529i	−0.836712	−	0.547643i	$-0.815525\pi$
$983$		−	34.3403i		−	1.09529i	0.836712	−	0.547643i	$-0.184475\pi$
$984$		0			0
$985$	36.2024	−	6.47303i	1.15350	−	0.206248i
$986$		0			0
$987$	−14.5535	−	60.7070i	−0.463241	−	1.93233i
$988$		0			0
$989$	35.6866	−	22.9199i	1.13477	−	0.728809i
$990$		0			0
$991$	15.6161			0.496062			0.248031	−	0.968752i	$-0.420217\pi$
$991$	15.6161			0.496062			0.248031	+	0.968752i	$0.420217\pi$
$992$		0			0
$993$	1.16627	+	4.86488i	0.0370105	+	0.154382i
$994$		0			0
$995$	−8.16923	−	45.6889i	−0.258982	−	1.44844i
$996$		0			0
$997$			13.4949i			0.427386i	0.976901	+	0.213693i	$0.0685493\pi$
$997$			13.4949i			0.427386i	−0.976901	+	0.213693i	$0.931451\pi$
$998$		0			0
$999$	−18.8831	−	22.1533i	−0.597435	−	0.700899i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.n.a.689.27	yes	48
3.2	odd	2	inner	1380.2.n.a.689.24	yes	48
5.4	even	2	inner	1380.2.n.a.689.21	✓	48
15.14	odd	2	inner	1380.2.n.a.689.26	yes	48
23.22	odd	2	inner	1380.2.n.a.689.28	yes	48
69.68	even	2	inner	1380.2.n.a.689.23	yes	48
115.114	odd	2	inner	1380.2.n.a.689.22	yes	48
345.344	even	2	inner	1380.2.n.a.689.25	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.n.a.689.21	✓	48	5.4	even	2	inner
1380.2.n.a.689.22	yes	48	115.114	odd	2	inner
1380.2.n.a.689.23	yes	48	69.68	even	2	inner
1380.2.n.a.689.24	yes	48	3.2	odd	2	inner
1380.2.n.a.689.25	yes	48	345.344	even	2	inner
1380.2.n.a.689.26	yes	48	15.14	odd	2	inner
1380.2.n.a.689.27	yes	48	1.1	even	1	trivial
1380.2.n.a.689.28	yes	48	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.403788 + 1.68433i) q^{3} +(-2.20116 + 0.393570i) q^{5} -3.85443 q^{7} +(-2.67391 + 1.36022i) q^{9} +O(q^{10})\)	\(q+(0.403788 + 1.68433i) q^{3} +(-2.20116 + 0.393570i) q^{5} -3.85443 q^{7} +(-2.67391 + 1.36022i) q^{9} +3.62241 q^{11} -1.57565i q^{13} +(-1.55170 - 3.54855i) q^{15} -5.19481i q^{17} -6.52090i q^{19} +(-1.55637 - 6.49211i) q^{21} +(-4.03526 + 2.59166i) q^{23} +(4.69021 - 1.73262i) q^{25} +(-3.37075 - 3.95450i) q^{27} +9.95654i q^{29} +8.99350 q^{31} +(1.46269 + 6.10132i) q^{33} +(8.48420 - 1.51699i) q^{35} +5.60205 q^{37} +(2.65391 - 0.636229i) q^{39} -2.15604i q^{41} -8.84370 q^{43} +(5.35036 - 4.04643i) q^{45} +9.35089 q^{47} +7.85659 q^{49} +(8.74976 - 2.09760i) q^{51} +6.55833i q^{53} +(-7.97351 + 1.42567i) q^{55} +(10.9833 - 2.63306i) q^{57} -5.15719i q^{59} -10.7339i q^{61} +(10.3064 - 5.24287i) q^{63} +(0.620129 + 3.46826i) q^{65} -2.14099 q^{67} +(-5.99459 - 5.75021i) q^{69} -8.71690i q^{71} -10.6914i q^{73} +(4.81215 + 7.20023i) q^{75} -13.9623 q^{77} -9.40689i q^{79} +(5.29959 - 7.27422i) q^{81} +5.23889i q^{83} +(2.04452 + 11.4346i) q^{85} +(-16.7701 + 4.02033i) q^{87} +7.33287 q^{89} +6.07323i q^{91} +(3.63147 + 15.1480i) q^{93} +(2.56643 + 14.3536i) q^{95} +2.45455 q^{97} +(-9.68600 + 4.92728i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

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