LMFDB - Embedded newform 1824.2.d.g.191.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1824,2,Mod(191,1824)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1824.191"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1824 = 2^{5} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1824.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$14.5647133287$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.23
Character	$\chi$	$=$	1824.191
Dual form			1824.2.d.g.191.24

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.911570 - 1.47277i) q^{3} -1.56046i q^{5} -5.19774i q^{7} +(-1.33808 - 2.68506i) q^{9} +3.94998 q^{11} -4.57497 q^{13} +(-2.29820 - 1.42247i) q^{15} +0.943764i q^{17} +1.00000i q^{19} +(-7.65505 - 4.73810i) q^{21} -0.478311 q^{23} +2.56496 q^{25} +(-5.17422 - 0.476937i) q^{27} -9.53573i q^{29} +2.67282i q^{31} +(3.60068 - 5.81740i) q^{33} -8.11087 q^{35} +6.67282 q^{37} +(-4.17040 + 6.73786i) q^{39} +0.718550i q^{41} +4.49735i q^{43} +(-4.18993 + 2.08802i) q^{45} +8.30239 q^{47} -20.0165 q^{49} +(1.38994 + 0.860307i) q^{51} +13.4492i q^{53} -6.16379i q^{55} +(1.47277 + 0.911570i) q^{57} +6.71376 q^{59} +9.09618 q^{61} +(-13.9562 + 6.95499i) q^{63} +7.13906i q^{65} +11.9465i q^{67} +(-0.436014 + 0.704441i) q^{69} -16.1835 q^{71} -2.58862 q^{73} +(2.33814 - 3.77759i) q^{75} -20.5309i q^{77} +9.22393i q^{79} +(-5.41908 + 7.18565i) q^{81} +4.07767 q^{83} +1.47271 q^{85} +(-14.0439 - 8.69248i) q^{87} -15.6078i q^{89} +23.7795i q^{91} +(3.93644 + 2.43646i) q^{93} +1.56046 q^{95} +1.01988 q^{97} +(-5.28539 - 10.6059i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{9} - 8 q^{13} + 12 q^{21} - 64 q^{25} + 32 q^{33} + 80 q^{37} - 88 q^{45} - 40 q^{49} + 4 q^{57} + 80 q^{61} - 60 q^{69} - 88 q^{73} + 84 q^{81} + 64 q^{85} - 48 q^{93} - 64 q^{97}+O(q^{100}) $ 32 * q - 4 * q^9 - 8 * q^13 + 12 * q^21 - 64 * q^25 + 32 * q^33 + 80 * q^37 - 88 * q^45 - 40 * q^49 + 4 * q^57 + 80 * q^61 - 60 * q^69 - 88 * q^73 + 84 * q^81 + 64 * q^85 - 48 * q^93 - 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1824\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$799$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0.911570	−	1.47277i	0.526295	−	0.850302i
$3$	0.911570	−	1.47277i	0.526295	−	0.850302i
$4$		0			0
$5$		−	1.56046i		−	0.697860i	−0.937149	−	0.348930i	$-0.886545\pi$
$5$		−	1.56046i		−	0.697860i	0.937149	−	0.348930i	$-0.113455\pi$
$6$		0			0
$7$		−	5.19774i		−	1.96456i	−0.187421	−	0.982280i	$-0.560013\pi$
$7$		−	5.19774i		−	1.96456i	0.187421	−	0.982280i	$-0.439987\pi$
$8$		0			0
$9$	−1.33808	−	2.68506i	−0.446027	−	0.895019i
$10$		0			0
$11$	3.94998			1.19096			0.595482	−	0.803369i	$-0.296961\pi$
$11$	3.94998			1.19096			0.595482	+	0.803369i	$0.296961\pi$
$12$		0			0
$13$	−4.57497			−1.26887			−0.634434	−	0.772977i	$-0.718767\pi$
$13$	−4.57497			−1.26887			−0.634434	+	0.772977i	$0.718767\pi$
$14$		0			0
$15$	−2.29820	−	1.42247i	−0.593391	−	0.367280i
$16$		0			0
$17$			0.943764i			0.228896i	0.993429	+	0.114448i	$0.0365100\pi$
$17$			0.943764i			0.228896i	−0.993429	+	0.114448i	$0.963490\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$	−7.65505	−	4.73810i	−1.67047	−	1.03394i
$22$		0			0
$23$	−0.478311			−0.0997348			−0.0498674	−	0.998756i	$-0.515880\pi$
$23$	−0.478311			−0.0997348			−0.0498674	+	0.998756i	$0.515880\pi$
$24$		0			0
$25$	2.56496			0.512992
$26$		0			0
$27$	−5.17422	−	0.476937i	−0.995779	−	0.0917866i
$28$		0			0
$29$		−	9.53573i		−	1.77074i	−0.464887	−	0.885370i	$-0.653905\pi$
$29$		−	9.53573i		−	1.77074i	0.464887	−	0.885370i	$-0.346095\pi$
$30$		0			0
$31$			2.67282i			0.480053i	0.970766	+	0.240026i	$0.0771561\pi$
$31$			2.67282i			0.480053i	−0.970766	+	0.240026i	$0.922844\pi$
$32$		0			0
$33$	3.60068	−	5.81740i	0.626798	−	1.01268i
$34$		0			0
$35$	−8.11087			−1.37099
$36$		0			0
$37$	6.67282			1.09701			0.548503	−	0.836149i	$-0.315198\pi$
$37$	6.67282			1.09701			0.548503	+	0.836149i	$0.315198\pi$
$38$		0			0
$39$	−4.17040	+	6.73786i	−0.667799	+	1.07892i
$40$		0			0
$41$			0.718550i			0.112219i	0.998425	+	0.0561093i	$0.0178695\pi$
$41$			0.718550i			0.112219i	−0.998425	+	0.0561093i	$0.982130\pi$
$42$		0			0
$43$			4.49735i			0.685839i	0.939365	+	0.342920i	$0.111416\pi$
$43$			4.49735i			0.685839i	−0.939365	+	0.342920i	$0.888584\pi$
$44$		0			0
$45$	−4.18993	+	2.08802i	−0.624598	+	0.311264i
$46$		0			0
$47$	8.30239			1.21103			0.605514	−	0.795835i	$-0.292968\pi$
$47$	8.30239			1.21103			0.605514	+	0.795835i	$0.292968\pi$
$48$		0			0
$49$	−20.0165			−2.85949
$50$		0			0
$51$	1.38994	+	0.860307i	0.194631	+	0.120467i
$52$		0			0
$53$			13.4492i			1.84739i	0.383124	+	0.923697i	$0.374848\pi$
$53$			13.4492i			1.84739i	−0.383124	+	0.923697i	$0.625152\pi$
$54$		0			0
$55$		−	6.16379i		−	0.831125i
$56$		0			0
$57$	1.47277	+	0.911570i	0.195073	+	0.120740i
$58$		0			0
$59$	6.71376			0.874057			0.437029	−	0.899448i	$-0.356031\pi$
$59$	6.71376			0.874057			0.437029	+	0.899448i	$0.356031\pi$
$60$		0			0
$61$	9.09618			1.16465			0.582323	−	0.812957i	$-0.302144\pi$
$61$	9.09618			1.16465			0.582323	+	0.812957i	$0.302144\pi$
$62$		0			0
$63$	−13.9562	+	6.95499i	−1.75832	+	0.876247i
$64$		0			0
$65$			7.13906i			0.885492i
$66$		0			0
$67$			11.9465i			1.45950i	0.683715	+	0.729750i	$0.260363\pi$
$67$			11.9465i			1.45950i	−0.683715	+	0.729750i	$0.739637\pi$
$68$		0			0
$69$	−0.436014	+	0.704441i	−0.0524899	+	0.0848047i
$70$		0			0
$71$	−16.1835			−1.92063			−0.960314	−	0.278921i	$-0.910023\pi$
$71$	−16.1835			−1.92063			−0.960314	+	0.278921i	$0.910023\pi$
$72$		0			0
$73$	−2.58862			−0.302975			−0.151487	−	0.988459i	$-0.548406\pi$
$73$	−2.58862			−0.302975			−0.151487	+	0.988459i	$0.548406\pi$
$74$		0			0
$75$	2.33814	−	3.77759i	0.269985	−	0.436198i
$76$		0			0
$77$		−	20.5309i		−	2.33972i
$78$		0			0
$79$			9.22393i			1.03777i	0.854843	+	0.518886i	$0.173653\pi$
$79$			9.22393i			1.03777i	−0.854843	+	0.518886i	$0.826347\pi$
$80$		0			0
$81$	−5.41908	+	7.18565i	−0.602120	+	0.798406i
$82$		0			0
$83$	4.07767			0.447583			0.223791	−	0.974637i	$-0.428157\pi$
$83$	4.07767			0.447583			0.223791	+	0.974637i	$0.428157\pi$
$84$		0			0
$85$	1.47271			0.159738
$86$		0			0
$87$	−14.0439	−	8.69248i	−1.50566	−	0.931932i
$88$		0			0
$89$		−	15.6078i		−	1.65443i	−0.561889	−	0.827213i	$-0.689925\pi$
$89$		−	15.6078i		−	1.65443i	0.561889	−	0.827213i	$-0.310075\pi$
$90$		0			0
$91$			23.7795i			2.49277i
$92$		0			0
$93$	3.93644	+	2.43646i	0.408190	+	0.252650i
$94$		0			0
$95$	1.56046			0.160100
$96$		0			0
$97$	1.01988			0.103554			0.0517768	−	0.998659i	$-0.483512\pi$
$97$	1.01988			0.103554			0.0517768	+	0.998659i	$0.483512\pi$
$98$		0			0
$99$	−5.28539	−	10.6059i	−0.531202	−	1.06594i
$100$		0			0
$101$		−	7.66119i		−	0.762317i	−0.924510	−	0.381158i	$-0.875525\pi$
$101$		−	7.66119i		−	0.762317i	0.924510	−	0.381158i	$-0.124475\pi$
$102$		0			0
$103$		−	15.7513i		−	1.55202i	−0.630719	−	0.776012i	$-0.717240\pi$
$103$		−	15.7513i		−	1.55202i	0.630719	−	0.776012i	$-0.282760\pi$
$104$		0			0
$105$	−7.39362	+	11.9454i	−0.721543	+	1.16575i
$106$		0			0
$107$	−1.10071			−0.106410			−0.0532051	−	0.998584i	$-0.516944\pi$
$107$	−1.10071			−0.106410			−0.0532051	+	0.998584i	$0.516944\pi$
$108$		0			0
$109$	−9.83599			−0.942117			−0.471059	−	0.882102i	$-0.656128\pi$
$109$	−9.83599			−0.942117			−0.471059	+	0.882102i	$0.656128\pi$
$110$		0			0
$111$	6.08274	−	9.82751i	0.577348	−	0.932786i
$112$		0			0
$113$		−	5.84994i		−	0.550316i	−0.961399	−	0.275158i	$-0.911270\pi$
$113$		−	5.84994i		−	0.550316i	0.961399	−	0.275158i	$-0.0887302\pi$
$114$		0			0
$115$			0.746386i			0.0696009i
$116$		0			0
$117$	6.12168	+	12.2841i	0.565949	+	1.13566i
$118$		0			0
$119$	4.90543			0.449680
$120$		0			0
$121$	4.60233			0.418394
$122$		0			0
$123$	1.05826	+	0.655009i	0.0954198	+	0.0590601i
$124$		0			0
$125$		−	11.8048i		−	1.05586i
$126$		0			0
$127$			4.38197i			0.388837i	0.980919	+	0.194418i	$0.0622819\pi$
$127$			4.38197i			0.388837i	−0.980919	+	0.194418i	$0.937718\pi$
$128$		0			0
$129$	6.62355	+	4.09965i	0.583171	+	0.360954i
$130$		0			0
$131$	7.84538			0.685454			0.342727	−	0.939435i	$-0.388649\pi$
$131$	7.84538			0.685454			0.342727	+	0.939435i	$0.388649\pi$
$132$		0			0
$133$	5.19774			0.450701
$134$		0			0
$135$	−0.744242	+	8.07417i	−0.0640541	+	0.694914i
$136$		0			0
$137$		−	4.62430i		−	0.395081i	−0.980295	−	0.197540i	$-0.936705\pi$
$137$		−	4.62430i		−	0.395081i	0.980295	−	0.197540i	$-0.0632953\pi$
$138$		0			0
$139$			12.1161i			1.02767i	0.857889	+	0.513836i	$0.171776\pi$
$139$			12.1161i			1.02767i	−0.857889	+	0.513836i	$0.828224\pi$
$140$		0			0
$141$	7.56821	−	12.2275i	0.637358	−	1.02974i
$142$		0			0
$143$	−18.0710			−1.51118
$144$		0			0
$145$	−14.8801			−1.23573
$146$		0			0
$147$	−18.2464	+	29.4796i	−1.50494	+	2.43143i
$148$		0			0
$149$		−	23.1466i		−	1.89624i	−0.317913	−	0.948120i	$-0.602982\pi$
$149$		−	23.1466i		−	1.89624i	0.317913	−	0.948120i	$-0.397018\pi$
$150$		0			0
$151$		−	13.8186i		−	1.12454i	−0.826952	−	0.562272i	$-0.809927\pi$
$151$		−	13.8186i		−	1.12454i	0.826952	−	0.562272i	$-0.190073\pi$
$152$		0			0
$153$	2.53406	−	1.26283i	0.204867	−	0.102094i
$154$		0			0
$155$	4.17084			0.335010
$156$		0			0
$157$	0.646129			0.0515667			0.0257834	−	0.999668i	$-0.491792\pi$
$157$	0.646129			0.0515667			0.0257834	+	0.999668i	$0.491792\pi$
$158$		0			0
$159$	19.8076	+	12.2599i	1.57084	+	0.972274i
$160$		0			0
$161$			2.48614i			0.195935i
$162$		0			0
$163$		−	7.98182i		−	0.625184i	−0.949887	−	0.312592i	$-0.898803\pi$
$163$		−	7.98182i		−	0.625184i	0.949887	−	0.312592i	$-0.101197\pi$
$164$		0			0
$165$	−9.07782	−	5.61872i	−0.706708	−	0.437417i
$166$		0			0
$167$	−9.91273			−0.767070			−0.383535	−	0.923526i	$-0.625294\pi$
$167$	−9.91273			−0.767070			−0.383535	+	0.923526i	$0.625294\pi$
$168$		0			0
$169$	7.93033			0.610026
$170$		0			0
$171$	2.68506	−	1.33808i	0.205332	−	0.102326i
$172$		0			0
$173$			8.86242i			0.673797i	0.941541	+	0.336899i	$0.109378\pi$
$173$			8.86242i			0.673797i	−0.941541	+	0.336899i	$0.890622\pi$
$174$		0			0
$175$		−	13.3320i		−	1.00780i
$176$		0			0
$177$	6.12006	−	9.88780i	0.460012	−	0.743213i
$178$		0			0
$179$	14.6128			1.09221			0.546107	−	0.837715i	$-0.316109\pi$
$179$	14.6128			1.09221			0.546107	+	0.837715i	$0.316109\pi$
$180$		0			0
$181$	4.52129			0.336065			0.168032	−	0.985781i	$-0.446259\pi$
$181$	4.52129			0.336065			0.168032	+	0.985781i	$0.446259\pi$
$182$		0			0
$183$	8.29180	−	13.3965i	0.612948	−	0.990301i
$184$		0			0
$185$		−	10.4127i		−	0.765556i
$186$		0			0
$187$			3.72785i			0.272607i
$188$		0			0
$189$	−2.47899	+	26.8942i	−0.180320	+	1.95627i
$190$		0			0
$191$	7.11497			0.514821			0.257410	−	0.966302i	$-0.417131\pi$
$191$	7.11497			0.514821			0.257410	+	0.966302i	$0.417131\pi$
$192$		0			0
$193$	−3.59540			−0.258802			−0.129401	−	0.991592i	$-0.541305\pi$
$193$	−3.59540			−0.258802			−0.129401	+	0.991592i	$0.541305\pi$
$194$		0			0
$195$	10.5142	+	6.50775i	0.752935	+	0.466030i
$196$		0			0
$197$		−	6.18698i		−	0.440804i	−0.975409	−	0.220402i	$-0.929263\pi$
$197$		−	6.18698i		−	0.440804i	0.975409	−	0.220402i	$-0.0707369\pi$
$198$		0			0
$199$			8.75679i			0.620752i	0.950614	+	0.310376i	$0.100455\pi$
$199$			8.75679i			0.620752i	−0.950614	+	0.310376i	$0.899545\pi$
$200$		0			0
$201$	17.5944	+	10.8901i	1.24102	+	0.768127i
$202$		0			0
$203$	−49.5642			−3.47872
$204$		0			0
$205$	1.12127			0.0783129
$206$		0			0
$207$	0.640019	+	1.28429i	0.0444844	+	0.0892646i
$208$		0			0
$209$			3.94998i			0.273226i
$210$		0			0
$211$			8.94811i			0.616014i	0.951384	+	0.308007i	$0.0996620\pi$
$211$			8.94811i			0.616014i	−0.951384	+	0.308007i	$0.900338\pi$
$212$		0			0
$213$	−14.7524	+	23.8345i	−1.01082	+	1.63311i
$214$		0			0
$215$	7.01794			0.478620
$216$		0			0
$217$	13.8926			0.943093
$218$		0			0
$219$	−2.35970	+	3.81243i	−0.159454	+	0.257620i
$220$		0			0
$221$		−	4.31769i		−	0.290439i
$222$		0			0
$223$			6.09361i			0.408058i	0.978965	+	0.204029i	$0.0654037\pi$
$223$			6.09361i			0.408058i	−0.978965	+	0.204029i	$0.934596\pi$
$224$		0			0
$225$	−3.43212	−	6.88707i	−0.228808	−	0.459138i
$226$		0			0
$227$	17.5984			1.16805			0.584025	−	0.811736i	$-0.301477\pi$
$227$	17.5984			1.16805			0.584025	+	0.811736i	$0.301477\pi$
$228$		0			0
$229$	−3.41758			−0.225840			−0.112920	−	0.993604i	$-0.536020\pi$
$229$	−3.41758			−0.225840			−0.112920	+	0.993604i	$0.536020\pi$
$230$		0			0
$231$	−30.2373	−	18.7154i	−1.98947	−	1.23138i
$232$		0			0
$233$		−	2.84464i		−	0.186359i	−0.995649	−	0.0931794i	$-0.970297\pi$
$233$		−	2.84464i		−	0.186359i	0.995649	−	0.0931794i	$-0.0297030\pi$
$234$		0			0
$235$		−	12.9556i		−	0.845127i
$236$		0			0
$237$	13.5847	+	8.40825i	0.882420	+	0.546175i
$238$		0			0
$239$	1.75650			0.113618			0.0568091	−	0.998385i	$-0.481907\pi$
$239$	1.75650			0.113618			0.0568091	+	0.998385i	$0.481907\pi$
$240$		0			0
$241$	−22.8706			−1.47323			−0.736613	−	0.676314i	$-0.763576\pi$
$241$	−22.8706			−1.47323			−0.736613	+	0.676314i	$0.763576\pi$
$242$		0			0
$243$	5.64292	+	14.5313i	0.361994	+	0.932181i
$244$		0			0
$245$			31.2349i			1.99553i
$246$		0			0
$247$		−	4.57497i		−	0.291098i
$248$		0			0
$249$	3.71708	−	6.00546i	0.235560	−	0.380580i
$250$		0			0
$251$	22.1279			1.39670			0.698350	−	0.715756i	$-0.253918\pi$
$251$	22.1279			1.39670			0.698350	+	0.715756i	$0.253918\pi$
$252$		0			0
$253$	−1.88932			−0.118781
$254$		0			0
$255$	1.34248	−	2.16895i	0.0840691	−	0.135825i
$256$		0			0
$257$		−	13.0220i		−	0.812291i	−0.913809	−	0.406145i	$-0.866873\pi$
$257$		−	13.0220i		−	0.812291i	0.913809	−	0.406145i	$-0.133127\pi$
$258$		0			0
$259$		−	34.6836i		−	2.15513i
$260$		0			0
$261$	−25.6040	+	12.7596i	−1.58485	+	0.789798i
$262$		0			0
$263$	−6.61856			−0.408118			−0.204059	−	0.978959i	$-0.565413\pi$
$263$	−6.61856			−0.408118			−0.204059	+	0.978959i	$0.565413\pi$
$264$		0			0
$265$	20.9870			1.28922
$266$		0			0
$267$	−22.9867	−	14.2276i	−1.40676	−	0.870716i
$268$		0			0
$269$			5.92485i			0.361245i	0.983553	+	0.180622i	$0.0578112\pi$
$269$			5.92485i			0.361245i	−0.983553	+	0.180622i	$0.942189\pi$
$270$		0			0
$271$		−	9.04573i		−	0.549489i	−0.961517	−	0.274745i	$-0.911407\pi$
$271$		−	9.04573i		−	0.549489i	0.961517	−	0.274745i	$-0.0885933\pi$
$272$		0			0
$273$	35.0216	+	21.6766i	2.11960	+	1.31193i
$274$		0			0
$275$	10.1315			0.610955
$276$		0			0
$277$	−7.95300			−0.477849			−0.238925	−	0.971038i	$-0.576795\pi$
$277$	−7.95300			−0.477849			−0.238925	+	0.971038i	$0.576795\pi$
$278$		0			0
$279$	7.17668	−	3.57645i	0.429657	−	0.214117i
$280$		0			0
$281$			13.7321i			0.819187i	0.912268	+	0.409594i	$0.134330\pi$
$281$			13.7321i			0.819187i	−0.912268	+	0.409594i	$0.865670\pi$
$282$		0			0
$283$			15.8270i			0.940819i	0.882448	+	0.470410i	$0.155894\pi$
$283$			15.8270i			0.940819i	−0.882448	+	0.470410i	$0.844106\pi$
$284$		0			0
$285$	1.42247	−	2.29820i	0.0842598	−	0.136133i
$286$		0			0
$287$	3.73483			0.220460
$288$		0			0
$289$	16.1093			0.947606
$290$		0			0
$291$	0.929696	−	1.50205i	0.0544997	−	0.0880518i
$292$		0			0
$293$			19.2202i			1.12286i	0.827525	+	0.561428i	$0.189748\pi$
$293$			19.2202i			1.12286i	−0.827525	+	0.561428i	$0.810252\pi$
$294$		0			0
$295$		−	10.4766i		−	0.609969i
$296$		0			0
$297$	−20.4381	−	1.88389i	−1.18594	−	0.109314i
$298$		0			0
$299$	2.18826			0.126550
$300$		0			0
$301$	23.3760			1.34737
$302$		0			0
$303$	−11.2831	−	6.98371i	−0.648200	−	0.401204i
$304$		0			0
$305$		−	14.1942i		−	0.812760i
$306$		0			0
$307$		−	14.6217i		−	0.834506i	−0.908790	−	0.417253i	$-0.862993\pi$
$307$		−	14.6217i		−	0.834506i	0.908790	−	0.417253i	$-0.137007\pi$
$308$		0			0
$309$	−23.1980	−	14.3584i	−1.31969	−	0.816822i
$310$		0			0
$311$	14.8008			0.839274			0.419637	−	0.907692i	$-0.362157\pi$
$311$	14.8008			0.839274			0.419637	+	0.907692i	$0.362157\pi$
$312$		0			0
$313$	−14.2611			−0.806084			−0.403042	−	0.915181i	$-0.632047\pi$
$313$	−14.2611			−0.806084			−0.403042	+	0.915181i	$0.632047\pi$
$314$		0			0
$315$	10.8530	+	21.7782i	0.611497	+	1.22706i
$316$		0			0
$317$			17.6456i			0.991074i	0.868587	+	0.495537i	$0.165029\pi$
$317$			17.6456i			0.991074i	−0.868587	+	0.495537i	$0.834971\pi$
$318$		0			0
$319$		−	37.6659i		−	2.10889i
$320$		0			0
$321$	−1.00338	+	1.62110i	−0.0560031	+	0.0904808i
$322$		0			0
$323$	−0.943764			−0.0525124
$324$		0			0
$325$	−11.7346			−0.650919
$326$		0			0
$327$	−8.96619	+	14.4861i	−0.495832	+	0.801084i
$328$		0			0
$329$		−	43.1536i		−	2.37914i
$330$		0			0
$331$		−	30.5424i		−	1.67876i	−0.543545	−	0.839380i	$-0.682918\pi$
$331$		−	30.5424i		−	1.67876i	0.543545	−	0.839380i	$-0.317082\pi$
$332$		0			0
$333$	−8.92878	−	17.9169i	−0.489294	−	0.981841i
$334$		0			0
$335$	18.6421			1.01853
$336$		0			0
$337$	−1.27548			−0.0694797			−0.0347399	−	0.999396i	$-0.511060\pi$
$337$	−1.27548			−0.0694797			−0.0347399	+	0.999396i	$0.511060\pi$
$338$		0			0
$339$	−8.61560	−	5.33263i	−0.467935	−	0.289629i
$340$		0			0
$341$			10.5576i			0.571726i
$342$		0			0
$343$			67.6561i			3.65309i
$344$		0			0
$345$	1.09925	+	0.680383i	0.0591818	+	0.0366306i
$346$		0			0
$347$	27.5527			1.47910			0.739552	−	0.673099i	$-0.235037\pi$
$347$	27.5527			1.47910			0.739552	+	0.673099i	$0.235037\pi$
$348$		0			0
$349$	14.8750			0.796243			0.398121	−	0.917333i	$-0.369662\pi$
$349$	14.8750			0.796243			0.398121	+	0.917333i	$0.369662\pi$
$350$		0			0
$351$	23.6719	+	2.18197i	1.26351	+	0.116465i
$352$		0			0
$353$		−	28.0666i		−	1.49383i	−0.664918	−	0.746916i	$-0.731534\pi$
$353$		−	28.0666i		−	1.49383i	0.664918	−	0.746916i	$-0.268466\pi$
$354$		0			0
$355$			25.2537i			1.34033i
$356$		0			0
$357$	4.47165	−	7.22456i	0.236665	−	0.382364i
$358$		0			0
$359$	0.814123			0.0429678			0.0214839	−	0.999769i	$-0.493161\pi$
$359$	0.814123			0.0429678			0.0214839	+	0.999769i	$0.493161\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	4.19535	−	6.77816i	0.220199	−	0.355761i
$364$		0			0
$365$			4.03944i			0.211434i
$366$		0			0
$367$		−	7.92669i		−	0.413770i	−0.978365	−	0.206885i	$-0.933667\pi$
$367$		−	7.92669i		−	0.413770i	0.978365	−	0.206885i	$-0.0663326\pi$
$368$		0			0
$369$	1.92935	−	0.961479i	0.100438	−	0.0500526i
$370$		0			0
$371$	69.9055			3.62931
$372$		0			0
$373$	−24.6734			−1.27754			−0.638769	−	0.769399i	$-0.720556\pi$
$373$	−24.6734			−1.27754			−0.638769	+	0.769399i	$0.720556\pi$
$374$		0			0
$375$	−17.3858	−	10.7609i	−0.897797	−	0.555692i
$376$		0			0
$377$			43.6257i			2.24684i
$378$		0			0
$379$		−	4.17277i		−	0.214341i	−0.994241	−	0.107170i	$-0.965821\pi$
$379$		−	4.17277i		−	0.214341i	0.994241	−	0.107170i	$-0.0341790\pi$
$380$		0			0
$381$	6.45362	+	3.99447i	0.330629	+	0.204643i
$382$		0			0
$383$	23.2424			1.18763			0.593816	−	0.804601i	$-0.297621\pi$
$383$	23.2424			1.18763			0.593816	+	0.804601i	$0.297621\pi$
$384$		0			0
$385$	−32.0378			−1.63279
$386$		0			0
$387$	12.0756	−	6.01782i	0.613840	−	0.305903i
$388$		0			0
$389$			6.76247i			0.342871i	0.985195	+	0.171436i	$0.0548405\pi$
$389$			6.76247i			0.342871i	−0.985195	+	0.171436i	$0.945159\pi$
$390$		0			0
$391$		−	0.451413i		−	0.0228289i
$392$		0			0
$393$	7.15161	−	11.5544i	0.360751	−	0.582843i
$394$		0			0
$395$	14.3936			0.724220
$396$		0			0
$397$	18.5197			0.929476			0.464738	−	0.885448i	$-0.346149\pi$
$397$	18.5197			0.929476			0.464738	+	0.885448i	$0.346149\pi$
$398$		0			0
$399$	4.73810	−	7.65505i	0.237202	−	0.383232i
$400$		0			0
$401$			2.15044i			0.107388i	0.998557	+	0.0536940i	$0.0170996\pi$
$401$			2.15044i			0.107388i	−0.998557	+	0.0536940i	$0.982900\pi$
$402$		0			0
$403$		−	12.2281i		−	0.609124i
$404$		0			0
$405$	11.2129	+	8.45626i	0.557175	+	0.420195i
$406$		0			0
$407$	26.3575			1.30649
$408$		0			0
$409$	−5.05338			−0.249874			−0.124937	−	0.992165i	$-0.539873\pi$
$409$	−5.05338			−0.249874			−0.124937	+	0.992165i	$0.539873\pi$
$410$		0			0
$411$	−6.81051	−	4.21537i	−0.335938	−	0.207929i
$412$		0			0
$413$		−	34.8963i		−	1.71714i
$414$		0			0
$415$		−	6.36305i		−	0.312350i
$416$		0			0
$417$	17.8441	+	11.0446i	0.873831	+	0.540858i
$418$		0			0
$419$	16.7170			0.816677			0.408339	−	0.912831i	$-0.366108\pi$
$419$	16.7170			0.816677			0.408339	+	0.912831i	$0.366108\pi$
$420$		0			0
$421$	17.6475			0.860085			0.430042	−	0.902809i	$-0.358499\pi$
$421$	17.6475			0.860085			0.430042	+	0.902809i	$0.358499\pi$
$422$		0			0
$423$	−11.1093	−	22.2924i	−0.540151	−	1.08389i
$424$		0			0
$425$			2.42072i			0.117422i
$426$		0			0
$427$		−	47.2795i		−	2.28802i
$428$		0			0
$429$	−16.4730	+	26.6144i	−0.795324	+	1.28496i
$430$		0			0
$431$	−17.2969			−0.833161			−0.416581	−	0.909099i	$-0.636772\pi$
$431$	−17.2969			−0.833161			−0.416581	+	0.909099i	$0.636772\pi$
$432$		0			0
$433$	−5.57046			−0.267699			−0.133850	−	0.991002i	$-0.542734\pi$
$433$	−5.57046			−0.267699			−0.133850	+	0.991002i	$0.542734\pi$
$434$		0			0
$435$	−13.5643	+	21.9150i	−0.650358	+	1.05074i
$436$		0			0
$437$		−	0.478311i		−	0.0228807i
$438$		0			0
$439$		−	18.1982i		−	0.868554i	−0.900779	−	0.434277i	$-0.857004\pi$
$439$		−	18.1982i		−	0.868554i	0.900779	−	0.434277i	$-0.142996\pi$
$440$		0			0
$441$	26.7836	+	53.7453i	1.27541	+	2.55930i
$442$		0			0
$443$	−4.53497			−0.215463			−0.107731	−	0.994180i	$-0.534359\pi$
$443$	−4.53497			−0.215463			−0.107731	+	0.994180i	$0.534359\pi$
$444$		0			0
$445$	−24.3554			−1.15456
$446$		0			0
$447$	−34.0895	−	21.0997i	−1.61238	−	0.997981i
$448$		0			0
$449$			10.7240i			0.506095i	0.967454	+	0.253048i	$0.0814330\pi$
$449$			10.7240i			0.506095i	−0.967454	+	0.253048i	$0.918567\pi$
$450$		0			0
$451$			2.83826i			0.133648i
$452$		0			0
$453$	−20.3516	−	12.5966i	−0.956202	−	0.591842i
$454$		0			0
$455$	37.1070			1.73960
$456$		0			0
$457$	−11.5115			−0.538485			−0.269242	−	0.963072i	$-0.586773\pi$
$457$	−11.5115			−0.538485			−0.269242	+	0.963072i	$0.586773\pi$
$458$		0			0
$459$	0.450116	−	4.88324i	0.0210096	−	0.227930i
$460$		0			0
$461$		−	11.3375i		−	0.528040i	−0.964517	−	0.264020i	$-0.914952\pi$
$461$		−	11.3375i		−	0.528040i	0.964517	−	0.264020i	$-0.0850485\pi$
$462$		0			0
$463$			25.3906i			1.18000i	0.807402	+	0.590001i	$0.200873\pi$
$463$			25.3906i			1.18000i	−0.807402	+	0.590001i	$0.799127\pi$
$464$		0			0
$465$	3.80201	−	6.14267i	0.176314	−	0.284859i
$466$		0			0
$467$	13.5527			0.627146			0.313573	−	0.949564i	$-0.398474\pi$
$467$	13.5527			0.627146			0.313573	+	0.949564i	$0.398474\pi$
$468$		0			0
$469$	62.0948			2.86727
$470$		0			0
$471$	0.588992	−	0.951597i	0.0271393	−	0.0438473i
$472$		0			0
$473$			17.7644i			0.816810i
$474$		0			0
$475$			2.56496i			0.117688i
$476$		0			0
$477$	36.1120	−	17.9962i	1.65345	−	0.823988i
$478$		0			0
$479$	9.81816			0.448603			0.224302	−	0.974520i	$-0.427990\pi$
$479$	9.81816			0.448603			0.224302	+	0.974520i	$0.427990\pi$
$480$		0			0
$481$	−30.5279			−1.39195
$482$		0			0
$483$	3.66150	+	2.26629i	0.166604	+	0.103120i
$484$		0			0
$485$		−	1.59149i		−	0.0722659i
$486$		0			0
$487$			18.4184i			0.834620i	0.908764	+	0.417310i	$0.137027\pi$
$487$			18.4184i			0.834620i	−0.908764	+	0.417310i	$0.862973\pi$
$488$		0			0
$489$	−11.7554	−	7.27598i	−0.531595	−	0.329031i
$490$		0			0
$491$	−14.1060			−0.636594			−0.318297	−	0.947991i	$-0.603111\pi$
$491$	−14.1060			−0.636594			−0.318297	+	0.947991i	$0.603111\pi$
$492$		0			0
$493$	8.99948			0.405316
$494$		0			0
$495$	−16.5501	+	8.24765i	−0.743873	+	0.370704i
$496$		0			0
$497$			84.1175i			3.77319i
$498$		0			0
$499$			33.6421i			1.50603i	0.658006	+	0.753013i	$0.271400\pi$
$499$			33.6421i			1.50603i	−0.658006	+	0.753013i	$0.728600\pi$
$500$		0			0
$501$	−9.03615	+	14.5991i	−0.403705	+	0.652241i
$502$		0			0
$503$	−32.0656			−1.42973			−0.714867	−	0.699260i	$-0.753513\pi$
$503$	−32.0656			−1.42973			−0.714867	+	0.699260i	$0.753513\pi$
$504$		0			0
$505$	−11.9550			−0.531990
$506$		0			0
$507$	7.22905	−	11.6795i	0.321053	−	0.518706i
$508$		0			0
$509$			3.04414i			0.134929i	0.997722	+	0.0674645i	$0.0214909\pi$
$509$			3.04414i			0.134929i	−0.997722	+	0.0674645i	$0.978509\pi$
$510$		0			0
$511$			13.4549i			0.595212i
$512$		0			0
$513$	0.476937	−	5.17422i	0.0210573	−	0.228447i
$514$		0			0
$515$	−24.5793			−1.08309
$516$		0			0
$517$	32.7943			1.44229
$518$		0			0
$519$	13.0523	+	8.07871i	0.572931	+	0.354616i
$520$		0			0
$521$		−	15.9554i		−	0.699020i	−0.936933	−	0.349510i	$-0.886348\pi$
$521$		−	15.9554i		−	0.699020i	0.936933	−	0.349510i	$-0.113652\pi$
$522$		0			0
$523$			5.81054i			0.254077i	0.991898	+	0.127039i	$0.0405472\pi$
$523$			5.81054i			0.254077i	−0.991898	+	0.127039i	$0.959453\pi$
$524$		0			0
$525$	−19.6349	−	12.1530i	−0.856937	−	0.530402i
$526$		0			0
$527$	−2.52251			−0.109882
$528$		0			0
$529$	−22.7712			−0.990053
$530$		0			0
$531$	−8.98356	−	18.0268i	−0.389853	−	0.782298i
$532$		0			0
$533$		−	3.28734i		−	0.142391i
$534$		0			0
$535$			1.71762i			0.0742593i
$536$		0			0
$537$	13.3206	−	21.5213i	0.574827	−	0.928713i
$538$		0			0
$539$	−79.0646			−3.40555
$540$		0			0
$541$	33.5044			1.44047			0.720234	−	0.693731i	$-0.244034\pi$
$541$	33.5044			1.44047			0.720234	+	0.693731i	$0.244034\pi$
$542$		0			0
$543$	4.12147	−	6.65880i	0.176869	−	0.285756i
$544$		0			0
$545$			15.3487i			0.657466i
$546$		0			0
$547$			28.7552i			1.22948i	0.788728	+	0.614742i	$0.210740\pi$
$547$			28.7552i			1.22948i	−0.788728	+	0.614742i	$0.789260\pi$
$548$		0			0
$549$	−12.1714	−	24.4238i	−0.519464	−	1.04238i
$550$		0			0
$551$	9.53573			0.406236
$552$		0			0
$553$	47.9435			2.03877
$554$		0			0
$555$	−15.3354	−	9.49189i	−0.650954	−	0.402908i
$556$		0			0
$557$			43.3822i			1.83816i	0.394065	+	0.919082i	$0.371068\pi$
$557$			43.3822i			1.83816i	−0.394065	+	0.919082i	$0.628932\pi$
$558$		0			0
$559$		−	20.5752i		−	0.870240i
$560$		0			0
$561$	5.49025	+	3.39819i	0.231798	+	0.143472i
$562$		0			0
$563$	−22.6671			−0.955304			−0.477652	−	0.878549i	$-0.658512\pi$
$563$	−22.6671			−0.955304			−0.477652	+	0.878549i	$0.658512\pi$
$564$		0			0
$565$	−9.12861			−0.384044
$566$		0			0
$567$	37.3491	+	28.1669i	1.56852	+	1.18290i
$568$		0			0
$569$		−	14.0125i		−	0.587432i	−0.955893	−	0.293716i	$-0.905108\pi$
$569$		−	14.0125i		−	0.587432i	0.955893	−	0.293716i	$-0.0948921\pi$
$570$		0			0
$571$		−	8.24727i		−	0.345137i	−0.984997	−	0.172569i	$-0.944793\pi$
$571$		−	8.24727i		−	0.345137i	0.984997	−	0.172569i	$-0.0552067\pi$
$572$		0			0
$573$	6.48579	−	10.4787i	0.270948	−	0.437753i
$574$		0			0
$575$	−1.22685			−0.0511631
$576$		0			0
$577$	−9.95201			−0.414308			−0.207154	−	0.978308i	$-0.566420\pi$
$577$	−9.95201			−0.414308			−0.207154	+	0.978308i	$0.566420\pi$
$578$		0			0
$579$	−3.27745	+	5.29518i	−0.136206	+	0.220060i
$580$		0			0
$581$		−	21.1947i		−	0.879302i
$582$		0			0
$583$			53.1242i			2.20018i
$584$		0			0
$585$	19.1688	−	9.55264i	0.792532	−	0.394953i
$586$		0			0
$587$	4.87447			0.201191			0.100596	−	0.994927i	$-0.467925\pi$
$587$	4.87447			0.201191			0.100596	+	0.994927i	$0.467925\pi$
$588$		0			0
$589$	−2.67282			−0.110132
$590$		0			0
$591$	−9.11197	−	5.63986i	−0.374816	−	0.231993i
$592$		0			0
$593$		−	0.0890545i		−	0.00365703i	−0.999998	−	0.00182851i	$-0.999418\pi$
$593$		−	0.0890545i		−	0.00365703i	0.999998	−	0.00182851i	$-0.000582034\pi$
$594$		0			0
$595$		−	7.65474i		−	0.313814i
$596$		0			0
$597$	12.8967	+	7.98242i	0.527827	+	0.326699i
$598$		0			0
$599$	19.1498			0.782438			0.391219	−	0.920298i	$-0.372054\pi$
$599$	19.1498			0.782438			0.391219	+	0.920298i	$0.372054\pi$
$600$		0			0
$601$	32.1940			1.31322			0.656610	−	0.754231i	$-0.271990\pi$
$601$	32.1940			1.31322			0.656610	+	0.754231i	$0.271990\pi$
$602$		0			0
$603$	32.0771	−	15.9854i	1.30628	−	0.650976i
$604$		0			0
$605$		−	7.18176i		−	0.291980i
$606$		0			0
$607$			12.6053i			0.511635i	0.966725	+	0.255817i	$0.0823446\pi$
$607$			12.6053i			0.511635i	−0.966725	+	0.255817i	$0.917655\pi$
$608$		0			0
$609$	−45.1812	+	72.9965i	−1.83084	+	2.95797i
$610$		0			0
$611$	−37.9832			−1.53663
$612$		0			0
$613$	13.6810			0.552570			0.276285	−	0.961076i	$-0.410897\pi$
$613$	13.6810			0.552570			0.276285	+	0.961076i	$0.410897\pi$
$614$		0			0
$615$	1.02212	−	1.65137i	0.0412157	−	0.0665896i
$616$		0			0
$617$		−	24.1797i		−	0.973437i	−0.873559	−	0.486719i	$-0.838194\pi$
$617$		−	24.1797i		−	0.973437i	0.873559	−	0.486719i	$-0.161806\pi$
$618$		0			0
$619$		−	12.8882i		−	0.518020i	−0.965875	−	0.259010i	$-0.916604\pi$
$619$		−	12.8882i		−	0.518020i	0.965875	−	0.259010i	$-0.0833963\pi$
$620$		0			0
$621$	2.47489	+	0.228124i	0.0993138	+	0.00915431i
$622$		0			0
$623$	−81.1253			−3.25022
$624$		0			0
$625$	−5.59618			−0.223847
$626$		0			0
$627$	5.81740	+	3.60068i	0.232324	+	0.143797i
$628$		0			0
$629$			6.29757i			0.251100i
$630$		0			0
$631$		−	0.642645i		−	0.0255833i	−0.999918	−	0.0127916i	$-0.995928\pi$
$631$		−	0.642645i		−	0.0255833i	0.999918	−	0.0127916i	$-0.00407182\pi$
$632$		0			0
$633$	13.1785	+	8.15683i	0.523798	+	0.324205i
$634$		0			0
$635$	6.83789			0.271354
$636$		0			0
$637$	91.5746			3.62832
$638$		0			0
$639$	21.6548	+	43.4536i	0.856652	+	1.71900i
$640$		0			0
$641$			34.1623i			1.34933i	0.738123	+	0.674666i	$0.235712\pi$
$641$			34.1623i			1.34933i	−0.738123	+	0.674666i	$0.764288\pi$
$642$		0			0
$643$		−	30.0798i		−	1.18623i	−0.805117	−	0.593115i	$-0.797898\pi$
$643$		−	30.0798i		−	1.18623i	0.805117	−	0.593115i	$-0.202102\pi$
$644$		0			0
$645$	6.39734	−	10.3358i	0.251895	−	0.406971i
$646$		0			0
$647$	35.8344			1.40880			0.704398	−	0.709806i	$-0.251217\pi$
$647$	35.8344			1.40880			0.704398	+	0.709806i	$0.251217\pi$
$648$		0			0
$649$	26.5192			1.04097
$650$		0			0
$651$	12.6641	−	20.4606i	0.496345	−	0.801914i
$652$		0			0
$653$			16.8550i			0.659589i	0.944053	+	0.329794i	$0.106979\pi$
$653$			16.8550i			0.659589i	−0.944053	+	0.329794i	$0.893021\pi$
$654$		0			0
$655$		−	12.2424i		−	0.478351i
$656$		0			0
$657$	3.46378	+	6.95059i	0.135135	+	0.271168i
$658$		0			0
$659$	−26.8030			−1.04410			−0.522048	−	0.852916i	$-0.674832\pi$
$659$	−26.8030			−1.04410			−0.522048	+	0.852916i	$0.674832\pi$
$660$		0			0
$661$	1.47971			0.0575539			0.0287769	−	0.999586i	$-0.490839\pi$
$661$	1.47971			0.0575539			0.0287769	+	0.999586i	$0.490839\pi$
$662$		0			0
$663$	−6.35895	−	3.93587i	−0.246961	−	0.152857i
$664$		0			0
$665$		−	8.11087i		−	0.314526i
$666$		0			0
$667$			4.56105i			0.176604i
$668$		0			0
$669$	8.97446	+	5.55475i	0.346973	+	0.214759i
$670$		0			0
$671$	35.9297			1.38705
$672$		0			0
$673$	39.8636			1.53663			0.768315	−	0.640072i	$-0.221096\pi$
$673$	39.8636			1.53663			0.768315	+	0.640072i	$0.221096\pi$
$674$		0			0
$675$	−13.2717	−	1.22332i	−0.510826	−	0.0470858i
$676$		0			0
$677$		−	19.4838i		−	0.748823i	−0.927263	−	0.374412i	$-0.877845\pi$
$677$		−	19.4838i		−	0.748823i	0.927263	−	0.374412i	$-0.122155\pi$
$678$		0			0
$679$		−	5.30109i		−	0.203437i
$680$		0			0
$681$	16.0422	−	25.9184i	0.614739	−	0.993195i
$682$		0			0
$683$	29.8222			1.14111			0.570557	−	0.821258i	$-0.306727\pi$
$683$	29.8222			1.14111			0.570557	+	0.821258i	$0.306727\pi$
$684$		0			0
$685$	−7.21604			−0.275711
$686$		0			0
$687$	−3.11536	+	5.03330i	−0.118858	+	0.192032i
$688$		0			0
$689$		−	61.5298i		−	2.34410i
$690$		0			0
$691$		−	37.1878i		−	1.41469i	−0.706869	−	0.707344i	$-0.749893\pi$
$691$		−	37.1878i		−	1.41469i	0.706869	−	0.707344i	$-0.250107\pi$
$692$		0			0
$693$	−55.1268	+	27.4721i	−2.09409	+	1.04358i
$694$		0			0
$695$	18.9067			0.717170
$696$		0			0
$697$	−0.678142			−0.0256864
$698$		0			0
$699$	−4.18950	−	2.59309i	−0.158461	−	0.0980797i
$700$		0			0
$701$		−	23.7299i		−	0.896266i	−0.893967	−	0.448133i	$-0.852089\pi$
$701$		−	23.7299i		−	0.896266i	0.893967	−	0.448133i	$-0.147911\pi$
$702$		0			0
$703$			6.67282i			0.251670i
$704$		0			0
$705$	−19.0805	−	11.8099i	−0.718614	−	0.444786i
$706$		0			0
$707$	−39.8208			−1.49762
$708$		0			0
$709$	−42.6920			−1.60333			−0.801666	−	0.597772i	$-0.796053\pi$
$709$	−42.6920			−1.60333			−0.801666	+	0.597772i	$0.796053\pi$
$710$		0			0
$711$	24.7668	−	12.3424i	0.928827	−	0.462875i
$712$		0			0
$713$		−	1.27844i		−	0.0478780i
$714$		0			0
$715$			28.1991i			1.05459i
$716$		0			0
$717$	1.60117	−	2.58691i	0.0597967	−	0.0966099i
$718$		0			0
$719$	52.4059			1.95441			0.977205	−	0.212300i	$-0.0680953\pi$
$719$	52.4059			1.95441			0.977205	+	0.212300i	$0.0680953\pi$
$720$		0			0
$721$	−81.8712			−3.04904
$722$		0			0
$723$	−20.8482	+	33.6831i	−0.775352	+	1.25269i
$724$		0			0
$725$		−	24.4588i		−	0.908376i
$726$		0			0
$727$			22.3551i			0.829104i	0.910026	+	0.414552i	$0.136062\pi$
$727$			22.3551i			0.829104i	−0.910026	+	0.414552i	$0.863938\pi$
$728$		0			0
$729$	26.5451	+	4.93555i	0.983150	+	0.182798i
$730$		0			0
$731$	−4.24444			−0.156986
$732$		0			0
$733$	−35.1732			−1.29915			−0.649576	−	0.760297i	$-0.725053\pi$
$733$	−35.1732			−1.29915			−0.649576	+	0.760297i	$0.725053\pi$
$734$		0			0
$735$	46.0017	+	28.4728i	1.69680	+	1.05023i
$736$		0			0
$737$			47.1885i			1.73821i
$738$		0			0
$739$			3.00534i			0.110553i	0.998471	+	0.0552766i	$0.0176040\pi$
$739$			3.00534i			0.110553i	−0.998471	+	0.0552766i	$0.982396\pi$
$740$		0			0
$741$	−6.73786	−	4.17040i	−0.247521	−	0.153204i
$742$		0			0
$743$	−9.45589			−0.346903			−0.173451	−	0.984842i	$-0.555492\pi$
$743$	−9.45589			−0.346903			−0.173451	+	0.984842i	$0.555492\pi$
$744$		0			0
$745$	−36.1193			−1.32331
$746$		0			0
$747$	−5.45626	−	10.9488i	−0.199634	−	0.400595i
$748$		0			0
$749$			5.72123i			0.209049i
$750$		0			0
$751$		−	30.6226i		−	1.11744i	−0.829358	−	0.558718i	$-0.811293\pi$
$751$		−	30.6226i		−	1.11744i	0.829358	−	0.558718i	$-0.188707\pi$
$752$		0			0
$753$	20.1711	−	32.5892i	0.735076	−	1.18762i
$754$		0			0
$755$	−21.5634			−0.784774
$756$		0			0
$757$	−43.2437			−1.57172			−0.785859	−	0.618406i	$-0.787779\pi$
$757$	−43.2437			−1.57172			−0.785859	+	0.618406i	$0.787779\pi$
$758$		0			0
$759$	−1.72225	+	2.78253i	−0.0625136	+	0.100999i
$760$		0			0
$761$		−	40.2871i		−	1.46041i	−0.683230	−	0.730203i	$-0.739425\pi$
$761$		−	40.2871i		−	1.46041i	0.683230	−	0.730203i	$-0.260575\pi$
$762$		0			0
$763$			51.1249i			1.85085i
$764$		0			0
$765$	−1.97060	−	3.95430i	−0.0712473	−	0.142968i
$766$		0			0
$767$	−30.7152			−1.10906
$768$		0			0
$769$	51.4218			1.85432			0.927158	−	0.374671i	$-0.122244\pi$
$769$	51.4218			1.85432			0.927158	+	0.374671i	$0.122244\pi$
$770$		0			0
$771$	−19.1784	−	11.8705i	−0.690692	−	0.427504i
$772$		0			0
$773$			45.9251i			1.65181i	0.563808	+	0.825906i	$0.309336\pi$
$773$			45.9251i			1.65181i	−0.563808	+	0.825906i	$0.690664\pi$
$774$		0			0
$775$			6.85568i			0.246263i
$776$		0			0
$777$	−51.0808	−	31.6165i	−1.83251	−	1.13424i
$778$		0			0
$779$	−0.718550			−0.0257447
$780$		0			0
$781$	−63.9245			−2.28740
$782$		0			0
$783$	−4.54794	+	49.3399i	−0.162530	+	1.76327i
$784$		0			0
$785$		−	1.00826i		−	0.0359863i
$786$		0			0
$787$			9.83789i			0.350683i	0.984508	+	0.175341i	$0.0561030\pi$
$787$			9.83789i			0.350683i	−0.984508	+	0.175341i	$0.943897\pi$
$788$		0			0
$789$	−6.03328	+	9.74760i	−0.214791	+	0.347024i
$790$		0			0
$791$	−30.4065			−1.08113
$792$		0			0
$793$	−41.6147			−1.47778
$794$		0			0
$795$	19.1311	−	30.9090i	0.678511	−	1.09623i
$796$		0			0
$797$		−	15.2827i		−	0.541341i	−0.962672	−	0.270670i	$-0.912755\pi$
$797$		−	15.2827i		−	0.541341i	0.962672	−	0.270670i	$-0.0872453\pi$
$798$		0			0
$799$			7.83549i			0.277200i
$800$		0			0
$801$	−41.9079	+	20.8845i	−1.48074	+	0.737919i
$802$		0			0
$803$	−10.2250			−0.360832
$804$		0			0
$805$	3.87952			0.136735
$806$		0			0
$807$	8.72592	+	5.40092i	0.307167	+	0.190121i
$808$		0			0
$809$		−	40.5493i		−	1.42564i	−0.701349	−	0.712818i	$-0.747419\pi$
$809$		−	40.5493i		−	1.42564i	0.701349	−	0.712818i	$-0.252581\pi$
$810$		0			0
$811$		−	31.2472i		−	1.09724i	−0.836073	−	0.548618i	$-0.815154\pi$
$811$		−	31.2472i		−	1.09724i	0.836073	−	0.548618i	$-0.184846\pi$
$812$		0			0
$813$	−13.3223	−	8.24582i	−0.467232	−	0.289193i
$814$		0			0
$815$	−12.4553			−0.436291
$816$		0			0
$817$	−4.49735			−0.157342
$818$		0			0
$819$	63.8493	−	31.8189i	2.23107	−	1.11184i
$820$		0			0
$821$		−	44.2879i		−	1.54566i	−0.634613	−	0.772830i	$-0.718841\pi$
$821$		−	44.2879i		−	1.54566i	0.634613	−	0.772830i	$-0.281159\pi$
$822$		0			0
$823$		−	34.3913i		−	1.19881i	−0.800447	−	0.599403i	$-0.795405\pi$
$823$		−	34.3913i		−	1.19881i	0.800447	−	0.599403i	$-0.204595\pi$
$824$		0			0
$825$	9.23560	−	14.9214i	0.321542	−	0.519496i
$826$		0			0
$827$	9.91938			0.344931			0.172465	−	0.985016i	$-0.444827\pi$
$827$	9.91938			0.344931			0.172465	+	0.985016i	$0.444827\pi$
$828$		0			0
$829$	18.5294			0.643551			0.321776	−	0.946816i	$-0.395720\pi$
$829$	18.5294			0.643551			0.321776	+	0.946816i	$0.395720\pi$
$830$		0			0
$831$	−7.24971	+	11.7129i	−0.251490	+	0.406316i
$832$		0			0
$833$		−	18.8908i		−	0.654528i
$834$		0			0
$835$			15.4684i			0.535307i
$836$		0			0
$837$	1.27477	−	13.8298i	0.0440624	−	0.478027i
$838$		0			0
$839$	−47.2441			−1.63105			−0.815524	−	0.578723i	$-0.803551\pi$
$839$	−47.2441			−1.63105			−0.815524	+	0.578723i	$0.803551\pi$
$840$		0			0
$841$	−61.9301			−2.13552
$842$		0			0
$843$	20.2241	+	12.5178i	0.696557	+	0.431134i
$844$		0			0
$845$		−	12.3750i		−	0.425712i
$846$		0			0
$847$		−	23.9217i		−	0.821959i
$848$		0			0
$849$	23.3095	+	14.4274i	0.799981	+	0.495149i
$850$		0			0
$851$	−3.19169			−0.109410
$852$		0			0
$853$	−49.9700			−1.71094			−0.855469	−	0.517854i	$-0.826731\pi$
$853$	−49.9700			−1.71094			−0.855469	+	0.517854i	$0.826731\pi$
$854$		0			0
$855$	−2.08802	−	4.18993i	−0.0714089	−	0.143293i
$856$		0			0
$857$			25.6647i			0.876690i	0.898807	+	0.438345i	$0.144435\pi$
$857$			25.6647i			0.876690i	−0.898807	+	0.438345i	$0.855565\pi$
$858$		0			0
$859$			33.8458i			1.15481i	0.816460	+	0.577403i	$0.195934\pi$
$859$			33.8458i			1.15481i	−0.816460	+	0.577403i	$0.804066\pi$
$860$		0			0
$861$	3.40456	−	5.50054i	0.116027	−	0.187458i
$862$		0			0
$863$	−4.97167			−0.169238			−0.0846188	−	0.996413i	$-0.526967\pi$
$863$	−4.97167			−0.169238			−0.0846188	+	0.996413i	$0.526967\pi$
$864$		0			0
$865$	13.8295			0.470216
$866$		0			0
$867$	14.6848	−	23.7252i	0.498721	−	0.805752i
$868$		0			0
$869$			36.4343i			1.23595i
$870$		0			0
$871$		−	54.6549i		−	1.85191i
$872$		0			0
$873$	−1.36469	−	2.73845i	−0.0461877	−	0.0926825i
$874$		0			0
$875$	−61.3584			−2.07429
$876$		0			0
$877$	−4.41062			−0.148936			−0.0744680	−	0.997223i	$-0.523726\pi$
$877$	−4.41062			−0.148936			−0.0744680	+	0.997223i	$0.523726\pi$
$878$		0			0
$879$	28.3069	+	17.5206i	0.954767	+	0.590954i
$880$		0			0
$881$		−	19.9885i		−	0.673429i	−0.941607	−	0.336714i	$-0.890684\pi$
$881$		−	19.9885i		−	0.673429i	0.941607	−	0.336714i	$-0.109316\pi$
$882$		0			0
$883$		−	42.6504i		−	1.43530i	−0.696404	−	0.717650i	$-0.745218\pi$
$883$		−	42.6504i		−	1.43530i	0.696404	−	0.717650i	$-0.254782\pi$
$884$		0			0
$885$	−15.4295	−	9.55012i	−0.518658	−	0.321024i
$886$		0			0
$887$	−39.8221			−1.33710			−0.668548	−	0.743669i	$-0.733084\pi$
$887$	−39.8221			−1.33710			−0.668548	+	0.743669i	$0.733084\pi$
$888$		0			0
$889$	22.7763			0.763893
$890$		0			0
$891$	−21.4052	+	28.3832i	−0.717102	+	0.950872i
$892$		0			0
$893$			8.30239i			0.277829i
$894$		0			0
$895$		−	22.8028i		−	0.762213i
$896$		0			0
$897$	1.99475	−	3.22279i	0.0666028	−	0.107606i
$898$		0			0
$899$	25.4873			0.850049
$900$		0			0
$901$	−12.6929			−0.422862
$902$		0			0
$903$	21.3089	−	34.4274i	0.709115	−	1.14567i
$904$		0			0
$905$		−	7.05529i		−	0.234526i
$906$		0			0
$907$			55.6210i			1.84687i	0.383759	+	0.923433i	$0.374629\pi$
$907$			55.6210i			1.84687i	−0.383759	+	0.923433i	$0.625371\pi$
$908$		0			0
$909$	−20.5707	+	10.2513i	−0.682288	+	0.340014i
$910$		0			0
$911$	−34.4831			−1.14248			−0.571238	−	0.820784i	$-0.693537\pi$
$911$	−34.4831			−1.14248			−0.571238	+	0.820784i	$0.693537\pi$
$912$		0			0
$913$	16.1067			0.533054
$914$		0			0
$915$	−20.9048	−	12.9390i	−0.691091	−	0.427751i
$916$		0			0
$917$		−	40.7782i		−	1.34662i
$918$		0			0
$919$			53.7710i			1.77374i	0.462016	+	0.886872i	$0.347126\pi$
$919$			53.7710i			1.77374i	−0.462016	+	0.886872i	$0.652874\pi$
$920$		0			0
$921$	−21.5344	−	13.3287i	−0.709582	−	0.439196i
$922$		0			0
$923$	74.0390			2.43702
$924$		0			0
$925$	17.1155			0.562755
$926$		0			0
$927$	−42.2932	+	21.0765i	−1.38909	+	0.692244i
$928$		0			0
$929$		−	6.13198i		−	0.201184i	−0.994928	−	0.100592i	$-0.967926\pi$
$929$		−	6.13198i		−	0.201184i	0.994928	−	0.100592i	$-0.0320737\pi$
$930$		0			0
$931$		−	20.0165i		−	0.656013i
$932$		0			0
$933$	13.4919	−	21.7980i	0.441706	−	0.713636i
$934$		0			0
$935$	5.81716			0.190242
$936$		0			0
$937$	55.8334			1.82400			0.911999	−	0.410192i	$-0.134538\pi$
$937$	55.8334			1.82400			0.911999	+	0.410192i	$0.134538\pi$
$938$		0			0
$939$	−13.0000	+	21.0032i	−0.424238	+	0.685415i
$940$		0			0
$941$		−	19.6290i		−	0.639886i	−0.947437	−	0.319943i	$-0.896336\pi$
$941$		−	19.6290i		−	0.639886i	0.947437	−	0.319943i	$-0.103664\pi$
$942$		0			0
$943$		−	0.343691i		−	0.0111921i
$944$		0			0
$945$	41.9674	+	3.86837i	1.36520	+	0.125838i
$946$		0			0
$947$	4.58710			0.149061			0.0745303	−	0.997219i	$-0.476254\pi$
$947$	4.58710			0.149061			0.0745303	+	0.997219i	$0.476254\pi$
$948$		0			0
$949$	11.8428			0.384435
$950$		0			0
$951$	25.9878	+	16.0852i	0.842712	+	0.521597i
$952$		0			0
$953$		−	58.7629i		−	1.90352i	−0.306848	−	0.951758i	$-0.599274\pi$
$953$		−	58.7629i		−	1.90352i	0.306848	−	0.951758i	$-0.400726\pi$
$954$		0			0
$955$		−	11.1026i		−	0.359273i
$956$		0			0
$957$	−55.4731	−	34.3351i	−1.79319	−	1.10990i
$958$		0			0
$959$	−24.0359			−0.776159
$960$		0			0
$961$	23.8560			0.769549
$962$		0			0
$963$	1.47285	+	2.95548i	0.0474618	+	0.0952391i
$964$		0			0
$965$			5.61048i			0.180608i
$966$		0			0
$967$			30.6116i			0.984401i	0.870482	+	0.492201i	$0.163807\pi$
$967$			30.6116i			0.984401i	−0.870482	+	0.492201i	$0.836193\pi$
$968$		0			0
$969$	−0.860307	+	1.38994i	−0.0276370	+	0.0446514i
$970$		0			0
$971$	18.1059			0.581045			0.290523	−	0.956868i	$-0.406171\pi$
$971$	18.1059			0.581045			0.290523	+	0.956868i	$0.406171\pi$
$972$		0			0
$973$	62.9761			2.01892
$974$		0			0
$975$	−10.6969	+	17.2823i	−0.342575	+	0.553478i
$976$		0			0
$977$			19.6539i			0.628785i	0.949293	+	0.314393i	$0.101801\pi$
$977$			19.6539i			0.628785i	−0.949293	+	0.314393i	$0.898199\pi$
$978$		0			0
$979$		−	61.6506i		−	1.97036i
$980$		0			0
$981$	13.1614	+	26.4102i	0.420210	+	0.843213i
$982$		0			0
$983$	45.5366			1.45239			0.726196	−	0.687487i	$-0.241286\pi$
$983$	45.5366			1.45239			0.726196	+	0.687487i	$0.241286\pi$
$984$		0			0
$985$	−9.65454			−0.307619
$986$		0			0
$987$	−63.5552	−	39.3375i	−2.02298	−	1.25213i
$988$		0			0
$989$		−	2.15113i		−	0.0684021i
$990$		0			0
$991$			36.6902i			1.16550i	0.812650	+	0.582752i	$0.198024\pi$
$991$			36.6902i			1.16550i	−0.812650	+	0.582752i	$0.801976\pi$
$992$		0			0
$993$	−44.9817	−	27.8415i	−1.42745	−	0.883523i
$994$		0			0
$995$	13.6646			0.433198
$996$		0			0
$997$	−48.1862			−1.52607			−0.763036	−	0.646356i	$-0.776292\pi$
$997$	−48.1862			−1.52607			−0.763036	+	0.646356i	$0.776292\pi$
$998$		0			0
$999$	−34.5266	−	3.18252i	−1.09237	−	0.100690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.d.g.191.23	yes	32
3.2	odd	2	inner	1824.2.d.g.191.9	✓	32
4.3	odd	2	inner	1824.2.d.g.191.10	yes	32
12.11	even	2	inner	1824.2.d.g.191.24	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.d.g.191.9	✓	32	3.2	odd	2	inner
1824.2.d.g.191.10	yes	32	4.3	odd	2	inner
1824.2.d.g.191.23	yes	32	1.1	even	1	trivial
1824.2.d.g.191.24	yes	32	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1824.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.911570 - 1.47277i) q^{3} -1.56046i q^{5} -5.19774i q^{7} +(-1.33808 - 2.68506i) q^{9} +3.94998 q^{11} -4.57497 q^{13} +(-2.29820 - 1.42247i) q^{15} +0.943764i q^{17} +1.00000i q^{19} +(-7.65505 - 4.73810i) q^{21} -0.478311 q^{23} +2.56496 q^{25} +(-5.17422 - 0.476937i) q^{27} -9.53573i q^{29} +2.67282i q^{31} +(3.60068 - 5.81740i) q^{33} -8.11087 q^{35} +6.67282 q^{37} +(-4.17040 + 6.73786i) q^{39} +0.718550i q^{41} +4.49735i q^{43} +(-4.18993 + 2.08802i) q^{45} +8.30239 q^{47} -20.0165 q^{49} +(1.38994 + 0.860307i) q^{51} +13.4492i q^{53} -6.16379i q^{55} +(1.47277 + 0.911570i) q^{57} +6.71376 q^{59} +9.09618 q^{61} +(-13.9562 + 6.95499i) q^{63} +7.13906i q^{65} +11.9465i q^{67} +(-0.436014 + 0.704441i) q^{69} -16.1835 q^{71} -2.58862 q^{73} +(2.33814 - 3.77759i) q^{75} -20.5309i q^{77} +9.22393i q^{79} +(-5.41908 + 7.18565i) q^{81} +4.07767 q^{83} +1.47271 q^{85} +(-14.0439 - 8.69248i) q^{87} -15.6078i q^{89} +23.7795i q^{91} +(3.93644 + 2.43646i) q^{93} +1.56046 q^{95} +1.01988 q^{97} +(-5.28539 - 10.6059i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{9} - 8 q^{13} + 12 q^{21} - 64 q^{25} + 32 q^{33} + 80 q^{37} - 88 q^{45} - 40 q^{49} + 4 q^{57} + 80 q^{61} - 60 q^{69} - 88 q^{73} + 84 q^{81} + 64 q^{85} - 48 q^{93} - 64 q^{97}+O(q^{100}) \) 32 * q - 4 * q^9 - 8 * q^13 + 12 * q^21 - 64 * q^25 + 32 * q^33 + 80 * q^37 - 88 * q^45 - 40 * q^49 + 4 * q^57 + 80 * q^61 - 60 * q^69 - 88 * q^73 + 84 * q^81 + 64 * q^85 - 48 * q^93 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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