LMFDB - Embedded newform 1824.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1824,2,Mod(1,1824)] 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([0, 0, 0, 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("1824.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1824 = 2^{5} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1824.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,5,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.5647133287$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	1824.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.32340 q^{5} +1.39821 q^{7} +1.00000 q^{9} +1.39821 q^{11} -1.72161 q^{13} +4.32340 q^{15} +0.601793 q^{17} -1.00000 q^{19} +1.39821 q^{21} -8.36842 q^{23} +13.6918 q^{25} +1.00000 q^{27} +4.79641 q^{29} +9.57201 q^{31} +1.39821 q^{33} +6.04502 q^{35} -1.72161 q^{37} -1.72161 q^{39} +6.64681 q^{41} -10.0450 q^{43} +4.32340 q^{45} -9.76663 q^{47} -5.04502 q^{49} +0.601793 q^{51} -7.29362 q^{53} +6.04502 q^{55} -1.00000 q^{57} -12.0900 q^{59} +9.24860 q^{61} +1.39821 q^{63} -7.44322 q^{65} +0.646809 q^{67} -8.36842 q^{69} +12.0900 q^{71} -12.6918 q^{73} +13.6918 q^{75} +1.95498 q^{77} +2.12878 q^{79} +1.00000 q^{81} +2.60179 q^{85} +4.79641 q^{87} +3.59283 q^{89} -2.40717 q^{91} +9.57201 q^{93} -4.32340 q^{95} -5.05398 q^{97} +1.39821 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} + q^{11} + 6 q^{13} + 5 q^{15} + 5 q^{17} - 3 q^{19} + q^{21} + 2 q^{23} + 6 q^{25} + 3 q^{27} + 8 q^{29} + 8 q^{31} + q^{33} - q^{35} + 6 q^{37} + 6 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 + q^11 + 6 * q^13 + 5 * q^15 + 5 * q^17 - 3 * q^19 + q^21 + 2 * q^23 + 6 * q^25 + 3 * q^27 + 8 * q^29 + 8 * q^31 + q^33 - q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 11 * q^43 + 5 * q^45 + q^47 + 4 * q^49 + 5 * q^51 + 10 * q^53 - q^55 - 3 * q^57 + 2 * q^59 + 15 * q^61 + q^63 - 14 * q^67 + 2 * q^69 - 2 * q^71 - 3 * q^73 + 6 * q^75 + 25 * q^77 + 8 * q^79 + 3 * q^81 + 11 * q^85 + 8 * q^87 - 2 * q^89 - 20 * q^91 + 8 * q^93 - 5 * q^95 - 12 * q^97 + q^99

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.32340	1.93349	0.966743	−	0.255751i	$-0.0823229\pi$
$5$	4.32340	1.93349	0.966743	+	0.255751i	$0.0823229\pi$
$6$	0	0
$7$	1.39821	0.528473	0.264236	−	0.964458i	$-0.414880\pi$
$7$	1.39821	0.528473	0.264236	+	0.964458i	$0.414880\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.39821	0.421575	0.210788	−	0.977532i	$-0.432397\pi$
$11$	1.39821	0.421575	0.210788	+	0.977532i	$0.432397\pi$
$12$	0	0
$13$	−1.72161	−0.477489	−0.238745	−	0.971082i	$-0.576736\pi$
$13$	−1.72161	−0.477489	−0.238745	+	0.971082i	$0.576736\pi$
$14$	0	0
$15$	4.32340	1.11630
$16$	0	0
$17$	0.601793	0.145956	0.0729781	−	0.997334i	$-0.476750\pi$
$17$	0.601793	0.145956	0.0729781	+	0.997334i	$0.476750\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.39821	0.305114
$22$	0	0
$23$	−8.36842	−1.74494	−0.872468	−	0.488671i	$-0.837482\pi$
$23$	−8.36842	−1.74494	−0.872468	+	0.488671i	$0.837482\pi$
$24$	0	0
$25$	13.6918	2.73836
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.79641	0.890672	0.445336	−	0.895364i	$-0.353084\pi$
$29$	4.79641	0.890672	0.445336	+	0.895364i	$0.353084\pi$
$30$	0	0
$31$	9.57201	1.71918	0.859591	−	0.510982i	$-0.170718\pi$
$31$	9.57201	1.71918	0.859591	+	0.510982i	$0.170718\pi$
$32$	0	0
$33$	1.39821	0.243397
$34$	0	0
$35$	6.04502	1.02179
$36$	0	0
$37$	−1.72161	−0.283031	−0.141516	−	0.989936i	$-0.545198\pi$
$37$	−1.72161	−0.283031	−0.141516	+	0.989936i	$0.545198\pi$
$38$	0	0
$39$	−1.72161	−0.275679
$40$	0	0
$41$	6.64681	1.03806	0.519029	−	0.854757i	$-0.326294\pi$
$41$	6.64681	1.03806	0.519029	+	0.854757i	$0.326294\pi$
$42$	0	0
$43$	−10.0450	−1.53185	−0.765925	−	0.642930i	$-0.777719\pi$
$43$	−10.0450	−1.53185	−0.765925	+	0.642930i	$0.777719\pi$
$44$	0	0
$45$	4.32340	0.644495
$46$	0	0
$47$	−9.76663	−1.42461	−0.712305	−	0.701871i	$-0.752348\pi$
$47$	−9.76663	−1.42461	−0.712305	+	0.701871i	$0.752348\pi$
$48$	0	0
$49$	−5.04502	−0.720717
$50$	0	0
$51$	0.601793	0.0842678
$52$	0	0
$53$	−7.29362	−1.00186	−0.500928	−	0.865489i	$-0.667008\pi$
$53$	−7.29362	−1.00186	−0.500928	+	0.865489i	$0.667008\pi$
$54$	0	0
$55$	6.04502	0.815110
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−12.0900	−1.57399	−0.786994	−	0.616961i	$-0.788364\pi$
$59$	−12.0900	−1.57399	−0.786994	+	0.616961i	$0.788364\pi$
$60$	0	0
$61$	9.24860	1.18416	0.592081	−	0.805878i	$-0.298307\pi$
$61$	9.24860	1.18416	0.592081	+	0.805878i	$0.298307\pi$
$62$	0	0
$63$	1.39821	0.176158
$64$	0	0
$65$	−7.44322	−0.923218
$66$	0	0
$67$	0.646809	0.0790202	0.0395101	−	0.999219i	$-0.487420\pi$
$67$	0.646809	0.0790202	0.0395101	+	0.999219i	$0.487420\pi$
$68$	0	0
$69$	−8.36842	−1.00744
$70$	0	0
$71$	12.0900	1.43482	0.717411	−	0.696650i	$-0.245327\pi$
$71$	12.0900	1.43482	0.717411	+	0.696650i	$0.245327\pi$
$72$	0	0
$73$	−12.6918	−1.48547	−0.742733	−	0.669588i	$-0.766471\pi$
$73$	−12.6918	−1.48547	−0.742733	+	0.669588i	$0.766471\pi$
$74$	0	0
$75$	13.6918	1.58100
$76$	0	0
$77$	1.95498	0.222791
$78$	0	0
$79$	2.12878	0.239507	0.119753	−	0.992804i	$-0.461790\pi$
$79$	2.12878	0.239507	0.119753	+	0.992804i	$0.461790\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	2.60179	0.282204
$86$	0	0
$87$	4.79641	0.514230
$88$	0	0
$89$	3.59283	0.380839	0.190420	−	0.981703i	$-0.439015\pi$
$89$	3.59283	0.380839	0.190420	+	0.981703i	$0.439015\pi$
$90$	0	0
$91$	−2.40717	−0.252340
$92$	0	0
$93$	9.57201	0.992571
$94$	0	0
$95$	−4.32340	−0.443572
$96$	0	0
$97$	−5.05398	−0.513154	−0.256577	−	0.966524i	$-0.582595\pi$
$97$	−5.05398	−0.513154	−0.256577	+	0.966524i	$0.582595\pi$
$98$	0	0
$99$	1.39821	0.140525
$100$	0	0
$101$	1.07480	0.106947	0.0534735	−	0.998569i	$-0.482971\pi$
$101$	1.07480	0.106947	0.0534735	+	0.998569i	$0.482971\pi$
$102$	0	0
$103$	−9.57201	−0.943158	−0.471579	−	0.881824i	$-0.656316\pi$
$103$	−9.57201	−0.943158	−0.471579	+	0.881824i	$0.656316\pi$
$104$	0	0
$105$	6.04502	0.589933
$106$	0	0
$107$	5.20359	0.503050	0.251525	−	0.967851i	$-0.419068\pi$
$107$	5.20359	0.503050	0.251525	+	0.967851i	$0.419068\pi$
$108$	0	0
$109$	16.5180	1.58214	0.791070	−	0.611726i	$-0.209524\pi$
$109$	16.5180	1.58214	0.791070	+	0.611726i	$0.209524\pi$
$110$	0	0
$111$	−1.72161	−0.163408
$112$	0	0
$113$	1.05398	0.0991500	0.0495750	−	0.998770i	$-0.484213\pi$
$113$	1.05398	0.0991500	0.0495750	+	0.998770i	$0.484213\pi$
$114$	0	0
$115$	−36.1801	−3.37381
$116$	0	0
$117$	−1.72161	−0.159163
$118$	0	0
$119$	0.841431	0.0771338
$120$	0	0
$121$	−9.04502	−0.822274
$122$	0	0
$123$	6.64681	0.599323
$124$	0	0
$125$	37.5783	3.36110
$126$	0	0
$127$	−14.2188	−1.26172	−0.630858	−	0.775898i	$-0.717297\pi$
$127$	−14.2188	−1.26172	−0.630858	+	0.775898i	$0.717297\pi$
$128$	0	0
$129$	−10.0450	−0.884414
$130$	0	0
$131$	−3.24860	−0.283832	−0.141916	−	0.989879i	$-0.545326\pi$
$131$	−3.24860	−0.283832	−0.141916	+	0.989879i	$0.545326\pi$
$132$	0	0
$133$	−1.39821	−0.121240
$134$	0	0
$135$	4.32340	0.372099
$136$	0	0
$137$	−7.39821	−0.632072	−0.316036	−	0.948747i	$-0.602352\pi$
$137$	−7.39821	−0.632072	−0.316036	+	0.948747i	$0.602352\pi$
$138$	0	0
$139$	−4.75140	−0.403008	−0.201504	−	0.979488i	$-0.564583\pi$
$139$	−4.75140	−0.403008	−0.201504	+	0.979488i	$0.564583\pi$
$140$	0	0
$141$	−9.76663	−0.822498
$142$	0	0
$143$	−2.40717	−0.201298
$144$	0	0
$145$	20.7368	1.72210
$146$	0	0
$147$	−5.04502	−0.416106
$148$	0	0
$149$	8.97021	0.734868	0.367434	−	0.930050i	$-0.380236\pi$
$149$	8.97021	0.734868	0.367434	+	0.930050i	$0.380236\pi$
$150$	0	0
$151$	−8.27839	−0.673686	−0.336843	−	0.941561i	$-0.609359\pi$
$151$	−8.27839	−0.673686	−0.336843	+	0.941561i	$0.609359\pi$
$152$	0	0
$153$	0.601793	0.0486520
$154$	0	0
$155$	41.3836	3.32401
$156$	0	0
$157$	6.55678	0.523288	0.261644	−	0.965164i	$-0.415735\pi$
$157$	6.55678	0.523288	0.261644	+	0.965164i	$0.415735\pi$
$158$	0	0
$159$	−7.29362	−0.578421
$160$	0	0
$161$	−11.7008	−0.922151
$162$	0	0
$163$	1.29362	0.101324	0.0506620	−	0.998716i	$-0.483867\pi$
$163$	1.29362	0.101324	0.0506620	+	0.998716i	$0.483867\pi$
$164$	0	0
$165$	6.04502	0.470604
$166$	0	0
$167$	9.85039	0.762246	0.381123	−	0.924524i	$-0.375537\pi$
$167$	9.85039	0.762246	0.381123	+	0.924524i	$0.375537\pi$
$168$	0	0
$169$	−10.0361	−0.772004
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	11.8504	0.900969	0.450484	−	0.892784i	$-0.351251\pi$
$173$	11.8504	0.900969	0.450484	+	0.892784i	$0.351251\pi$
$174$	0	0
$175$	19.1440	1.44715
$176$	0	0
$177$	−12.0900	−0.908742
$178$	0	0
$179$	7.70079	0.575584	0.287792	−	0.957693i	$-0.407079\pi$
$179$	7.70079	0.575584	0.287792	+	0.957693i	$0.407079\pi$
$180$	0	0
$181$	14.1109	1.04885	0.524426	−	0.851456i	$-0.324280\pi$
$181$	14.1109	1.04885	0.524426	+	0.851456i	$0.324280\pi$
$182$	0	0
$183$	9.24860	0.683676
$184$	0	0
$185$	−7.44322	−0.547237
$186$	0	0
$187$	0.841431	0.0615315
$188$	0	0
$189$	1.39821	0.101705
$190$	0	0
$191$	0.473011	0.0342258	0.0171129	−	0.999854i	$-0.494553\pi$
$191$	0.473011	0.0342258	0.0171129	+	0.999854i	$0.494553\pi$
$192$	0	0
$193$	1.44322	0.103885	0.0519427	−	0.998650i	$-0.483459\pi$
$193$	1.44322	0.103885	0.0519427	+	0.998650i	$0.483459\pi$
$194$	0	0
$195$	−7.44322	−0.533020
$196$	0	0
$197$	−2.66763	−0.190061	−0.0950305	−	0.995474i	$-0.530295\pi$
$197$	−2.66763	−0.190061	−0.0950305	+	0.995474i	$0.530295\pi$
$198$	0	0
$199$	−6.60179	−0.467989	−0.233994	−	0.972238i	$-0.575180\pi$
$199$	−6.60179	−0.467989	−0.233994	+	0.972238i	$0.575180\pi$
$200$	0	0
$201$	0.646809	0.0456224
$202$	0	0
$203$	6.70638	0.470696
$204$	0	0
$205$	28.7368	2.00707
$206$	0	0
$207$	−8.36842	−0.581645
$208$	0	0
$209$	−1.39821	−0.0967160
$210$	0	0
$211$	−9.20359	−0.633601	−0.316800	−	0.948492i	$-0.602609\pi$
$211$	−9.20359	−0.633601	−0.316800	+	0.948492i	$0.602609\pi$
$212$	0	0
$213$	12.0900	0.828395
$214$	0	0
$215$	−43.4287	−2.96181
$216$	0	0
$217$	13.3836	0.908541
$218$	0	0
$219$	−12.6918	−0.857634
$220$	0	0
$221$	−1.03605	−0.0696925
$222$	0	0
$223$	21.6620	1.45060	0.725299	−	0.688434i	$-0.241702\pi$
$223$	21.6620	1.45060	0.725299	+	0.688434i	$0.241702\pi$
$224$	0	0
$225$	13.6918	0.912788
$226$	0	0
$227$	2.70638	0.179629	0.0898145	−	0.995959i	$-0.471373\pi$
$227$	2.70638	0.179629	0.0898145	+	0.995959i	$0.471373\pi$
$228$	0	0
$229$	13.3386	0.881442	0.440721	−	0.897644i	$-0.354723\pi$
$229$	13.3386	0.881442	0.440721	+	0.897644i	$0.354723\pi$
$230$	0	0
$231$	1.95498	0.128629
$232$	0	0
$233$	−12.0450	−0.789095	−0.394548	−	0.918875i	$-0.629099\pi$
$233$	−12.0450	−0.789095	−0.394548	+	0.918875i	$0.629099\pi$
$234$	0	0
$235$	−42.2251	−2.75446
$236$	0	0
$237$	2.12878	0.138279
$238$	0	0
$239$	14.0242	0.907150	0.453575	−	0.891218i	$-0.350148\pi$
$239$	14.0242	0.907150	0.453575	+	0.891218i	$0.350148\pi$
$240$	0	0
$241$	−14.3476	−0.924210	−0.462105	−	0.886825i	$-0.652906\pi$
$241$	−14.3476	−0.924210	−0.462105	+	0.886825i	$0.652906\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−21.8116	−1.39349
$246$	0	0
$247$	1.72161	0.109544
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.6378	1.74449	0.872243	−	0.489073i	$-0.162665\pi$
$251$	27.6378	1.74449	0.872243	+	0.489073i	$0.162665\pi$
$252$	0	0
$253$	−11.7008	−0.735622
$254$	0	0
$255$	2.60179	0.162931
$256$	0	0
$257$	−24.4972	−1.52809	−0.764047	−	0.645161i	$-0.776790\pi$
$257$	−24.4972	−1.52809	−0.764047	+	0.645161i	$0.776790\pi$
$258$	0	0
$259$	−2.40717	−0.149574
$260$	0	0
$261$	4.79641	0.296891
$262$	0	0
$263$	26.1142	1.61027	0.805136	−	0.593090i	$-0.202092\pi$
$263$	26.1142	1.61027	0.805136	+	0.593090i	$0.202092\pi$
$264$	0	0
$265$	−31.5333	−1.93707
$266$	0	0
$267$	3.59283	0.219878
$268$	0	0
$269$	0.149606	0.00912166	0.00456083	−	0.999990i	$-0.498548\pi$
$269$	0.149606	0.00912166	0.00456083	+	0.999990i	$0.498548\pi$
$270$	0	0
$271$	−25.8504	−1.57030	−0.785150	−	0.619306i	$-0.787414\pi$
$271$	−25.8504	−1.57030	−0.785150	+	0.619306i	$0.787414\pi$
$272$	0	0
$273$	−2.40717	−0.145689
$274$	0	0
$275$	19.1440	1.15443
$276$	0	0
$277$	−29.4287	−1.76820	−0.884099	−	0.467300i	$-0.845227\pi$
$277$	−29.4287	−1.76820	−0.884099	+	0.467300i	$0.845227\pi$
$278$	0	0
$279$	9.57201	0.573061
$280$	0	0
$281$	20.0305	1.19492	0.597458	−	0.801900i	$-0.296177\pi$
$281$	20.0305	1.19492	0.597458	+	0.801900i	$0.296177\pi$
$282$	0	0
$283$	−6.04502	−0.359339	−0.179669	−	0.983727i	$-0.557503\pi$
$283$	−6.04502	−0.359339	−0.179669	+	0.983727i	$0.557503\pi$
$284$	0	0
$285$	−4.32340	−0.256096
$286$	0	0
$287$	9.29362	0.548585
$288$	0	0
$289$	−16.6378	−0.978697
$290$	0	0
$291$	−5.05398	−0.296269
$292$	0	0
$293$	−26.6468	−1.55672	−0.778362	−	0.627816i	$-0.783949\pi$
$293$	−26.6468	−1.55672	−0.778362	+	0.627816i	$0.783949\pi$
$294$	0	0
$295$	−52.2701	−3.04328
$296$	0	0
$297$	1.39821	0.0811322
$298$	0	0
$299$	14.4072	0.833188
$300$	0	0
$301$	−14.0450	−0.809541
$302$	0	0
$303$	1.07480	0.0617458
$304$	0	0
$305$	39.9854	2.28956
$306$	0	0
$307$	33.9404	1.93708	0.968541	−	0.248853i	$-0.0800537\pi$
$307$	33.9404	1.93708	0.968541	+	0.248853i	$0.0800537\pi$
$308$	0	0
$309$	−9.57201	−0.544532
$310$	0	0
$311$	−6.06584	−0.343962	−0.171981	−	0.985100i	$-0.555017\pi$
$311$	−6.06584	−0.343962	−0.171981	+	0.985100i	$0.555017\pi$
$312$	0	0
$313$	−1.26316	−0.0713980	−0.0356990	−	0.999363i	$-0.511366\pi$
$313$	−1.26316	−0.0713980	−0.0356990	+	0.999363i	$0.511366\pi$
$314$	0	0
$315$	6.04502	0.340598
$316$	0	0
$317$	9.44322	0.530384	0.265192	−	0.964196i	$-0.414565\pi$
$317$	9.44322	0.530384	0.265192	+	0.964196i	$0.414565\pi$
$318$	0	0
$319$	6.70638	0.375485
$320$	0	0
$321$	5.20359	0.290436
$322$	0	0
$323$	−0.601793	−0.0334846
$324$	0	0
$325$	−23.5720	−1.30754
$326$	0	0
$327$	16.5180	0.913449
$328$	0	0
$329$	−13.6558	−0.752867
$330$	0	0
$331$	−19.0540	−1.04730	−0.523651	−	0.851933i	$-0.675430\pi$
$331$	−19.0540	−1.04730	−0.523651	+	0.851933i	$0.675430\pi$
$332$	0	0
$333$	−1.72161	−0.0943437
$334$	0	0
$335$	2.79641	0.152784
$336$	0	0
$337$	22.0900	1.20332	0.601660	−	0.798752i	$-0.294506\pi$
$337$	22.0900	1.20332	0.601660	+	0.798752i	$0.294506\pi$
$338$	0	0
$339$	1.05398	0.0572443
$340$	0	0
$341$	13.3836	0.724765
$342$	0	0
$343$	−16.8414	−0.909352
$344$	0	0
$345$	−36.1801	−1.94787
$346$	0	0
$347$	30.7819	1.65246	0.826228	−	0.563335i	$-0.190482\pi$
$347$	30.7819	1.65246	0.826228	+	0.563335i	$0.190482\pi$
$348$	0	0
$349$	−23.8775	−1.27813	−0.639066	−	0.769152i	$-0.720679\pi$
$349$	−23.8775	−1.27813	−0.639066	+	0.769152i	$0.720679\pi$
$350$	0	0
$351$	−1.72161	−0.0918928
$352$	0	0
$353$	−20.8864	−1.11167	−0.555837	−	0.831292i	$-0.687602\pi$
$353$	−20.8864	−1.11167	−0.555837	+	0.831292i	$0.687602\pi$
$354$	0	0
$355$	52.2701	2.77421
$356$	0	0
$357$	0.841431	0.0445332
$358$	0	0
$359$	−14.4134	−0.760712	−0.380356	−	0.924840i	$-0.624199\pi$
$359$	−14.4134	−0.760712	−0.380356	+	0.924840i	$0.624199\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.04502	−0.474740
$364$	0	0
$365$	−54.8719	−2.87213
$366$	0	0
$367$	−27.1440	−1.41691	−0.708453	−	0.705758i	$-0.750607\pi$
$367$	−27.1440	−1.41691	−0.708453	+	0.705758i	$0.750607\pi$
$368$	0	0
$369$	6.64681	0.346019
$370$	0	0
$371$	−10.1980	−0.529453
$372$	0	0
$373$	−8.42799	−0.436385	−0.218193	−	0.975906i	$-0.570016\pi$
$373$	−8.42799	−0.436385	−0.218193	+	0.975906i	$0.570016\pi$
$374$	0	0
$375$	37.5783	1.94053
$376$	0	0
$377$	−8.25756	−0.425286
$378$	0	0
$379$	−11.8325	−0.607793	−0.303897	−	0.952705i	$-0.598288\pi$
$379$	−11.8325	−0.607793	−0.303897	+	0.952705i	$0.598288\pi$
$380$	0	0
$381$	−14.2188	−0.728452
$382$	0	0
$383$	31.2757	1.59811	0.799057	−	0.601256i	$-0.205333\pi$
$383$	31.2757	1.59811	0.799057	+	0.601256i	$0.205333\pi$
$384$	0	0
$385$	8.45219	0.430763
$386$	0	0
$387$	−10.0450	−0.510617
$388$	0	0
$389$	−20.9702	−1.06323	−0.531616	−	0.846985i	$-0.678415\pi$
$389$	−20.9702	−1.06323	−0.531616	+	0.846985i	$0.678415\pi$
$390$	0	0
$391$	−5.03605	−0.254684
$392$	0	0
$393$	−3.24860	−0.163870
$394$	0	0
$395$	9.20359	0.463083
$396$	0	0
$397$	10.1946	0.511653	0.255827	−	0.966723i	$-0.417652\pi$
$397$	10.1946	0.511653	0.255827	+	0.966723i	$0.417652\pi$
$398$	0	0
$399$	−1.39821	−0.0699979
$400$	0	0
$401$	21.1440	1.05588	0.527941	−	0.849281i	$-0.322964\pi$
$401$	21.1440	1.05588	0.527941	+	0.849281i	$0.322964\pi$
$402$	0	0
$403$	−16.4793	−0.820891
$404$	0	0
$405$	4.32340	0.214832
$406$	0	0
$407$	−2.40717	−0.119319
$408$	0	0
$409$	−9.74244	−0.481732	−0.240866	−	0.970558i	$-0.577432\pi$
$409$	−9.74244	−0.481732	−0.240866	+	0.970558i	$0.577432\pi$
$410$	0	0
$411$	−7.39821	−0.364927
$412$	0	0
$413$	−16.9044	−0.831810
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.75140	−0.232677
$418$	0	0
$419$	−30.8448	−1.50687	−0.753433	−	0.657524i	$-0.771604\pi$
$419$	−30.8448	−1.50687	−0.753433	+	0.657524i	$0.771604\pi$
$420$	0	0
$421$	−28.5180	−1.38988	−0.694942	−	0.719066i	$-0.744570\pi$
$421$	−28.5180	−1.38988	−0.694942	+	0.719066i	$0.744570\pi$
$422$	0	0
$423$	−9.76663	−0.474870
$424$	0	0
$425$	8.23964	0.399681
$426$	0	0
$427$	12.9315	0.625797
$428$	0	0
$429$	−2.40717	−0.116219
$430$	0	0
$431$	−30.3297	−1.46093	−0.730464	−	0.682951i	$-0.760696\pi$
$431$	−30.3297	−1.46093	−0.730464	+	0.682951i	$0.760696\pi$
$432$	0	0
$433$	−32.4197	−1.55799	−0.778996	−	0.627029i	$-0.784271\pi$
$433$	−32.4197	−1.55799	−0.778996	+	0.627029i	$0.784271\pi$
$434$	0	0
$435$	20.7368	0.994255
$436$	0	0
$437$	8.36842	0.400316
$438$	0	0
$439$	9.96125	0.475425	0.237712	−	0.971336i	$-0.423602\pi$
$439$	9.96125	0.475425	0.237712	+	0.971336i	$0.423602\pi$
$440$	0	0
$441$	−5.04502	−0.240239
$442$	0	0
$443$	1.00896	0.0479373	0.0239686	−	0.999713i	$-0.492370\pi$
$443$	1.00896	0.0479373	0.0239686	+	0.999713i	$0.492370\pi$
$444$	0	0
$445$	15.5333	0.736347
$446$	0	0
$447$	8.97021	0.424277
$448$	0	0
$449$	−27.0844	−1.27819	−0.639097	−	0.769126i	$-0.720692\pi$
$449$	−27.0844	−1.27819	−0.639097	+	0.769126i	$0.720692\pi$
$450$	0	0
$451$	9.29362	0.437619
$452$	0	0
$453$	−8.27839	−0.388953
$454$	0	0
$455$	−10.4072	−0.487896
$456$	0	0
$457$	−18.1530	−0.849160	−0.424580	−	0.905390i	$-0.639578\pi$
$457$	−18.1530	−0.849160	−0.424580	+	0.905390i	$0.639578\pi$
$458$	0	0
$459$	0.601793	0.0280893
$460$	0	0
$461$	31.8567	1.48371	0.741856	−	0.670559i	$-0.233946\pi$
$461$	31.8567	1.48371	0.741856	+	0.670559i	$0.233946\pi$
$462$	0	0
$463$	32.6739	1.51848	0.759242	−	0.650808i	$-0.225570\pi$
$463$	32.6739	1.51848	0.759242	+	0.650808i	$0.225570\pi$
$464$	0	0
$465$	41.3836	1.91912
$466$	0	0
$467$	−24.8414	−1.14952	−0.574762	−	0.818321i	$-0.694905\pi$
$467$	−24.8414	−1.14952	−0.574762	+	0.818321i	$0.694905\pi$
$468$	0	0
$469$	0.904373	0.0417600
$470$	0	0
$471$	6.55678	0.302120
$472$	0	0
$473$	−14.0450	−0.645791
$474$	0	0
$475$	−13.6918	−0.628224
$476$	0	0
$477$	−7.29362	−0.333952
$478$	0	0
$479$	41.1053	1.87815	0.939074	−	0.343716i	$-0.111686\pi$
$479$	41.1053	1.87815	0.939074	+	0.343716i	$0.111686\pi$
$480$	0	0
$481$	2.96395	0.135144
$482$	0	0
$483$	−11.7008	−0.532404
$484$	0	0
$485$	−21.8504	−0.992175
$486$	0	0
$487$	18.3088	0.829653	0.414827	−	0.909901i	$-0.363842\pi$
$487$	18.3088	0.829653	0.414827	+	0.909901i	$0.363842\pi$
$488$	0	0
$489$	1.29362	0.0584994
$490$	0	0
$491$	−17.2936	−0.780450	−0.390225	−	0.920720i	$-0.627603\pi$
$491$	−17.2936	−0.780450	−0.390225	+	0.920720i	$0.627603\pi$
$492$	0	0
$493$	2.88645	0.129999
$494$	0	0
$495$	6.04502	0.271703
$496$	0	0
$497$	16.9044	0.758265
$498$	0	0
$499$	5.74580	0.257217	0.128609	−	0.991695i	$-0.458949\pi$
$499$	5.74580	0.257217	0.128609	+	0.991695i	$0.458949\pi$
$500$	0	0
$501$	9.85039	0.440083
$502$	0	0
$503$	−19.3324	−0.861988	−0.430994	−	0.902355i	$-0.641837\pi$
$503$	−19.3324	−0.861988	−0.430994	+	0.902355i	$0.641837\pi$
$504$	0	0
$505$	4.64681	0.206780
$506$	0	0
$507$	−10.0361	−0.445717
$508$	0	0
$509$	−0.239638	−0.0106218	−0.00531089	−	0.999986i	$-0.501691\pi$
$509$	−0.239638	−0.0106218	−0.00531089	+	0.999986i	$0.501691\pi$
$510$	0	0
$511$	−17.7458	−0.785028
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−41.3836	−1.82358
$516$	0	0
$517$	−13.6558	−0.600580
$518$	0	0
$519$	11.8504	0.520175
$520$	0	0
$521$	−16.8864	−0.739809	−0.369904	−	0.929070i	$-0.620610\pi$
$521$	−16.8864	−0.739809	−0.369904	+	0.929070i	$0.620610\pi$
$522$	0	0
$523$	−27.8325	−1.21703	−0.608514	−	0.793543i	$-0.708234\pi$
$523$	−27.8325	−1.21703	−0.608514	+	0.793543i	$0.708234\pi$
$524$	0	0
$525$	19.1440	0.835513
$526$	0	0
$527$	5.76036	0.250925
$528$	0	0
$529$	47.0305	2.04480
$530$	0	0
$531$	−12.0900	−0.524663
$532$	0	0
$533$	−11.4432	−0.495661
$534$	0	0
$535$	22.4972	0.972639
$536$	0	0
$537$	7.70079	0.332314
$538$	0	0
$539$	−7.05398	−0.303836
$540$	0	0
$541$	5.50617	0.236729	0.118364	−	0.992970i	$-0.462235\pi$
$541$	5.50617	0.236729	0.118364	+	0.992970i	$0.462235\pi$
$542$	0	0
$543$	14.1109	0.605555
$544$	0	0
$545$	71.4141	3.05904
$546$	0	0
$547$	3.26316	0.139523	0.0697613	−	0.997564i	$-0.477776\pi$
$547$	3.26316	0.139523	0.0697613	+	0.997564i	$0.477776\pi$
$548$	0	0
$549$	9.24860	0.394721
$550$	0	0
$551$	−4.79641	−0.204334
$552$	0	0
$553$	2.97648	0.126573
$554$	0	0
$555$	−7.44322	−0.315947
$556$	0	0
$557$	35.5574	1.50662	0.753309	−	0.657667i	$-0.228457\pi$
$557$	35.5574	1.50662	0.753309	+	0.657667i	$0.228457\pi$
$558$	0	0
$559$	17.2936	0.731442
$560$	0	0
$561$	0.841431	0.0355252
$562$	0	0
$563$	2.58723	0.109039	0.0545195	−	0.998513i	$-0.482637\pi$
$563$	2.58723	0.109039	0.0545195	+	0.998513i	$0.482637\pi$
$564$	0	0
$565$	4.55678	0.191705
$566$	0	0
$567$	1.39821	0.0587192
$568$	0	0
$569$	23.6829	0.992837	0.496419	−	0.868083i	$-0.334648\pi$
$569$	23.6829	0.992837	0.496419	+	0.868083i	$0.334648\pi$
$570$	0	0
$571$	23.8809	0.999383	0.499691	−	0.866204i	$-0.333447\pi$
$571$	23.8809	0.999383	0.499691	+	0.866204i	$0.333447\pi$
$572$	0	0
$573$	0.473011	0.0197603
$574$	0	0
$575$	−114.579	−4.77827
$576$	0	0
$577$	−6.45219	−0.268608	−0.134304	−	0.990940i	$-0.542880\pi$
$577$	−6.45219	−0.268608	−0.134304	+	0.990940i	$0.542880\pi$
$578$	0	0
$579$	1.44322	0.0599783
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.1980	−0.422358
$584$	0	0
$585$	−7.44322	−0.307739
$586$	0	0
$587$	34.3026	1.41582	0.707909	−	0.706303i	$-0.249638\pi$
$587$	34.3026	1.41582	0.707909	+	0.706303i	$0.249638\pi$
$588$	0	0
$589$	−9.57201	−0.394408
$590$	0	0
$591$	−2.66763	−0.109732
$592$	0	0
$593$	6.47928	0.266072	0.133036	−	0.991111i	$-0.457527\pi$
$593$	6.47928	0.266072	0.133036	+	0.991111i	$0.457527\pi$
$594$	0	0
$595$	3.63785	0.149137
$596$	0	0
$597$	−6.60179	−0.270193
$598$	0	0
$599$	−7.65240	−0.312669	−0.156334	−	0.987704i	$-0.549968\pi$
$599$	−7.65240	−0.312669	−0.156334	+	0.987704i	$0.549968\pi$
$600$	0	0
$601$	−16.5872	−0.676607	−0.338304	−	0.941037i	$-0.609853\pi$
$601$	−16.5872	−0.676607	−0.338304	+	0.941037i	$0.609853\pi$
$602$	0	0
$603$	0.646809	0.0263401
$604$	0	0
$605$	−39.1053	−1.58985
$606$	0	0
$607$	20.3268	0.825038	0.412519	−	0.910949i	$-0.364649\pi$
$607$	20.3268	0.825038	0.412519	+	0.910949i	$0.364649\pi$
$608$	0	0
$609$	6.70638	0.271756
$610$	0	0
$611$	16.8143	0.680235
$612$	0	0
$613$	13.8954	0.561230	0.280615	−	0.959820i	$-0.409462\pi$
$613$	13.8954	0.561230	0.280615	+	0.959820i	$0.409462\pi$
$614$	0	0
$615$	28.7368	1.15878
$616$	0	0
$617$	−47.0215	−1.89301	−0.946507	−	0.322683i	$-0.895415\pi$
$617$	−47.0215	−1.89301	−0.946507	+	0.322683i	$0.895415\pi$
$618$	0	0
$619$	−1.41277	−0.0567839	−0.0283919	−	0.999597i	$-0.509039\pi$
$619$	−1.41277	−0.0567839	−0.0283919	+	0.999597i	$0.509039\pi$
$620$	0	0
$621$	−8.36842	−0.335813
$622$	0	0
$623$	5.02352	0.201263
$624$	0	0
$625$	94.0069	3.76028
$626$	0	0
$627$	−1.39821	−0.0558390
$628$	0	0
$629$	−1.03605	−0.0413101
$630$	0	0
$631$	−22.2251	−0.884766	−0.442383	−	0.896826i	$-0.645867\pi$
$631$	−22.2251	−0.884766	−0.442383	+	0.896826i	$0.645867\pi$
$632$	0	0
$633$	−9.20359	−0.365810
$634$	0	0
$635$	−61.4737	−2.43951
$636$	0	0
$637$	8.68556	0.344134
$638$	0	0
$639$	12.0900	0.478274
$640$	0	0
$641$	0.616351	0.0243444	0.0121722	−	0.999926i	$-0.496125\pi$
$641$	0.616351	0.0243444	0.0121722	+	0.999926i	$0.496125\pi$
$642$	0	0
$643$	3.24860	0.128112	0.0640562	−	0.997946i	$-0.479596\pi$
$643$	3.24860	0.128112	0.0640562	+	0.997946i	$0.479596\pi$
$644$	0	0
$645$	−43.4287	−1.71000
$646$	0	0
$647$	−15.5270	−0.610429	−0.305214	−	0.952284i	$-0.598728\pi$
$647$	−15.5270	−0.610429	−0.305214	+	0.952284i	$0.598728\pi$
$648$	0	0
$649$	−16.9044	−0.663555
$650$	0	0
$651$	13.3836	0.524547
$652$	0	0
$653$	30.7431	1.20307	0.601535	−	0.798846i	$-0.294556\pi$
$653$	30.7431	1.20307	0.601535	+	0.798846i	$0.294556\pi$
$654$	0	0
$655$	−14.0450	−0.548784
$656$	0	0
$657$	−12.6918	−0.495155
$658$	0	0
$659$	40.0900	1.56169	0.780843	−	0.624727i	$-0.214790\pi$
$659$	40.0900	1.56169	0.780843	+	0.624727i	$0.214790\pi$
$660$	0	0
$661$	19.8712	0.772901	0.386450	−	0.922310i	$-0.373701\pi$
$661$	19.8712	0.772901	0.386450	+	0.922310i	$0.373701\pi$
$662$	0	0
$663$	−1.03605	−0.0402370
$664$	0	0
$665$	−6.04502	−0.234416
$666$	0	0
$667$	−40.1384	−1.55417
$668$	0	0
$669$	21.6620	0.837503
$670$	0	0
$671$	12.9315	0.499214
$672$	0	0
$673$	−35.7312	−1.37734	−0.688669	−	0.725076i	$-0.741805\pi$
$673$	−35.7312	−1.37734	−0.688669	+	0.725076i	$0.741805\pi$
$674$	0	0
$675$	13.6918	0.526999
$676$	0	0
$677$	40.6773	1.56335	0.781677	−	0.623683i	$-0.214364\pi$
$677$	40.6773	1.56335	0.781677	+	0.623683i	$0.214364\pi$
$678$	0	0
$679$	−7.06651	−0.271188
$680$	0	0
$681$	2.70638	0.103709
$682$	0	0
$683$	43.0665	1.64789	0.823947	−	0.566667i	$-0.191767\pi$
$683$	43.0665	1.64789	0.823947	+	0.566667i	$0.191767\pi$
$684$	0	0
$685$	−31.9854	−1.22210
$686$	0	0
$687$	13.3386	0.508901
$688$	0	0
$689$	12.5568	0.478375
$690$	0	0
$691$	23.8179	0.906076	0.453038	−	0.891491i	$-0.350340\pi$
$691$	23.8179	0.906076	0.453038	+	0.891491i	$0.350340\pi$
$692$	0	0
$693$	1.95498	0.0742637
$694$	0	0
$695$	−20.5422	−0.779211
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−12.0450	−0.455584
$700$	0	0
$701$	−25.5124	−0.963591	−0.481796	−	0.876284i	$-0.660015\pi$
$701$	−25.5124	−0.963591	−0.481796	+	0.876284i	$0.660015\pi$
$702$	0	0
$703$	1.72161	0.0649318
$704$	0	0
$705$	−42.2251	−1.59029
$706$	0	0
$707$	1.50280	0.0565185
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	2.12878	0.0798356
$712$	0	0
$713$	−80.1026	−2.99986
$714$	0	0
$715$	−10.4072	−0.389206
$716$	0	0
$717$	14.0242	0.523743
$718$	0	0
$719$	−16.0838	−0.599823	−0.299912	−	0.953967i	$-0.596957\pi$
$719$	−16.0838	−0.599823	−0.299912	+	0.953967i	$0.596957\pi$
$720$	0	0
$721$	−13.3836	−0.498433
$722$	0	0
$723$	−14.3476	−0.533593
$724$	0	0
$725$	65.6717	2.43898
$726$	0	0
$727$	−42.1350	−1.56270	−0.781351	−	0.624092i	$-0.785469\pi$
$727$	−42.1350	−1.56270	−0.781351	+	0.624092i	$0.785469\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.04502	−0.223583
$732$	0	0
$733$	13.4432	0.496537	0.248268	−	0.968691i	$-0.420139\pi$
$733$	13.4432	0.496537	0.248268	+	0.968691i	$0.420139\pi$
$734$	0	0
$735$	−21.8116	−0.804535
$736$	0	0
$737$	0.904373	0.0333130
$738$	0	0
$739$	6.25420	0.230064	0.115032	−	0.993362i	$-0.463303\pi$
$739$	6.25420	0.230064	0.115032	+	0.993362i	$0.463303\pi$
$740$	0	0
$741$	1.72161	0.0632450
$742$	0	0
$743$	28.8269	1.05756	0.528778	−	0.848760i	$-0.322651\pi$
$743$	28.8269	1.05756	0.528778	+	0.848760i	$0.322651\pi$
$744$	0	0
$745$	38.7819	1.42086
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.27569	0.265848
$750$	0	0
$751$	39.6925	1.44840	0.724200	−	0.689590i	$-0.242209\pi$
$751$	39.6925	1.44840	0.724200	+	0.689590i	$0.242209\pi$
$752$	0	0
$753$	27.6378	1.00718
$754$	0	0
$755$	−35.7908	−1.30256
$756$	0	0
$757$	29.1711	1.06024	0.530121	−	0.847922i	$-0.322147\pi$
$757$	29.1711	1.06024	0.530121	+	0.847922i	$0.322147\pi$
$758$	0	0
$759$	−11.7008	−0.424712
$760$	0	0
$761$	−16.8719	−0.611605	−0.305803	−	0.952095i	$-0.598925\pi$
$761$	−16.8719	−0.611605	−0.305803	+	0.952095i	$0.598925\pi$
$762$	0	0
$763$	23.0956	0.836118
$764$	0	0
$765$	2.60179	0.0940680
$766$	0	0
$767$	20.8143	0.751562
$768$	0	0
$769$	−4.30258	−0.155155	−0.0775775	−	0.996986i	$-0.524719\pi$
$769$	−4.30258	−0.155155	−0.0775775	+	0.996986i	$0.524719\pi$
$770$	0	0
$771$	−24.4972	−0.882245
$772$	0	0
$773$	−45.6233	−1.64096	−0.820478	−	0.571678i	$-0.806293\pi$
$773$	−45.6233	−1.64096	−0.820478	+	0.571678i	$0.806293\pi$
$774$	0	0
$775$	131.058	4.70775
$776$	0	0
$777$	−2.40717	−0.0863568
$778$	0	0
$779$	−6.64681	−0.238147
$780$	0	0
$781$	16.9044	0.604886
$782$	0	0
$783$	4.79641	0.171410
$784$	0	0
$785$	28.3476	1.01177
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	26.1142	0.929691
$790$	0	0
$791$	1.47368	0.0523981
$792$	0	0
$793$	−15.9225	−0.565425
$794$	0	0
$795$	−31.5333	−1.11837
$796$	0	0
$797$	−42.6177	−1.50960	−0.754798	−	0.655957i	$-0.772265\pi$
$797$	−42.6177	−1.50960	−0.754798	+	0.655957i	$0.772265\pi$
$798$	0	0
$799$	−5.87748	−0.207930
$800$	0	0
$801$	3.59283	0.126946
$802$	0	0
$803$	−17.7458	−0.626236
$804$	0	0
$805$	−50.5872	−1.78297
$806$	0	0
$807$	0.149606	0.00526639
$808$	0	0
$809$	50.9798	1.79236	0.896178	−	0.443695i	$-0.146333\pi$
$809$	50.9798	1.79236	0.896178	+	0.443695i	$0.146333\pi$
$810$	0	0
$811$	20.5568	0.721846	0.360923	−	0.932596i	$-0.382462\pi$
$811$	20.5568	0.721846	0.360923	+	0.932596i	$0.382462\pi$
$812$	0	0
$813$	−25.8504	−0.906613
$814$	0	0
$815$	5.59283	0.195908
$816$	0	0
$817$	10.0450	0.351431
$818$	0	0
$819$	−2.40717	−0.0841133
$820$	0	0
$821$	−4.58097	−0.159877	−0.0799384	−	0.996800i	$-0.525472\pi$
$821$	−4.58097	−0.159877	−0.0799384	+	0.996800i	$0.525472\pi$
$822$	0	0
$823$	−32.8123	−1.14377	−0.571883	−	0.820335i	$-0.693787\pi$
$823$	−32.8123	−1.14377	−0.571883	+	0.820335i	$0.693787\pi$
$824$	0	0
$825$	19.1440	0.666509
$826$	0	0
$827$	20.3892	0.709004	0.354502	−	0.935055i	$-0.384650\pi$
$827$	20.3892	0.709004	0.354502	+	0.935055i	$0.384650\pi$
$828$	0	0
$829$	−15.2728	−0.530446	−0.265223	−	0.964187i	$-0.585446\pi$
$829$	−15.2728	−0.530446	−0.265223	+	0.964187i	$0.585446\pi$
$830$	0	0
$831$	−29.4287	−1.02087
$832$	0	0
$833$	−3.03605	−0.105193
$834$	0	0
$835$	42.5872	1.47379
$836$	0	0
$837$	9.57201	0.330857
$838$	0	0
$839$	−24.5277	−0.846789	−0.423394	−	0.905945i	$-0.639162\pi$
$839$	−24.5277	−0.846789	−0.423394	+	0.905945i	$0.639162\pi$
$840$	0	0
$841$	−5.99440	−0.206704
$842$	0	0
$843$	20.0305	0.689886
$844$	0	0
$845$	−43.3899	−1.49266
$846$	0	0
$847$	−12.6468	−0.434549
$848$	0	0
$849$	−6.04502	−0.207464
$850$	0	0
$851$	14.4072	0.493871
$852$	0	0
$853$	41.3657	1.41634	0.708168	−	0.706044i	$-0.249522\pi$
$853$	41.3657	1.41634	0.708168	+	0.706044i	$0.249522\pi$
$854$	0	0
$855$	−4.32340	−0.147857
$856$	0	0
$857$	−5.70079	−0.194735	−0.0973676	−	0.995248i	$-0.531042\pi$
$857$	−5.70079	−0.194735	−0.0973676	+	0.995248i	$0.531042\pi$
$858$	0	0
$859$	−8.93146	−0.304738	−0.152369	−	0.988324i	$-0.548690\pi$
$859$	−8.93146	−0.304738	−0.152369	+	0.988324i	$0.548690\pi$
$860$	0	0
$861$	9.29362	0.316726
$862$	0	0
$863$	−5.03605	−0.171429	−0.0857146	−	0.996320i	$-0.527317\pi$
$863$	−5.03605	−0.171429	−0.0857146	+	0.996320i	$0.527317\pi$
$864$	0	0
$865$	51.2340	1.74201
$866$	0	0
$867$	−16.6378	−0.565051
$868$	0	0
$869$	2.97648	0.100970
$870$	0	0
$871$	−1.11355	−0.0377313
$872$	0	0
$873$	−5.05398	−0.171051
$874$	0	0
$875$	52.5422	1.77625
$876$	0	0
$877$	−19.9196	−0.672637	−0.336319	−	0.941748i	$-0.609182\pi$
$877$	−19.9196	−0.672637	−0.336319	+	0.941748i	$0.609182\pi$
$878$	0	0
$879$	−26.6468	−0.898775
$880$	0	0
$881$	21.4578	0.722931	0.361466	−	0.932385i	$-0.382277\pi$
$881$	21.4578	0.722931	0.361466	+	0.932385i	$0.382277\pi$
$882$	0	0
$883$	−7.63785	−0.257034	−0.128517	−	0.991707i	$-0.541022\pi$
$883$	−7.63785	−0.257034	−0.128517	+	0.991707i	$0.541022\pi$
$884$	0	0
$885$	−52.2701	−1.75704
$886$	0	0
$887$	20.8269	0.699298	0.349649	−	0.936881i	$-0.386301\pi$
$887$	20.8269	0.699298	0.349649	+	0.936881i	$0.386301\pi$
$888$	0	0
$889$	−19.8809	−0.666782
$890$	0	0
$891$	1.39821	0.0468417
$892$	0	0
$893$	9.76663	0.326828
$894$	0	0
$895$	33.2936	1.11288
$896$	0	0
$897$	14.4072	0.481041
$898$	0	0
$899$	45.9113	1.53123
$900$	0	0
$901$	−4.38924	−0.146227
$902$	0	0
$903$	−14.0450	−0.467389
$904$	0	0
$905$	61.0069	2.02794
$906$	0	0
$907$	2.53885	0.0843012	0.0421506	−	0.999111i	$-0.486579\pi$
$907$	2.53885	0.0843012	0.0421506	+	0.999111i	$0.486579\pi$
$908$	0	0
$909$	1.07480	0.0356490
$910$	0	0
$911$	−4.64681	−0.153956	−0.0769778	−	0.997033i	$-0.524527\pi$
$911$	−4.64681	−0.153956	−0.0769778	+	0.997033i	$0.524527\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	39.9854	1.32188
$916$	0	0
$917$	−4.54222	−0.149997
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	33.9404	1.11838
$922$	0	0
$923$	−20.8143	−0.685112
$924$	0	0
$925$	−23.5720	−0.775043
$926$	0	0
$927$	−9.57201	−0.314386
$928$	0	0
$929$	−47.4320	−1.55619	−0.778097	−	0.628144i	$-0.783815\pi$
$929$	−47.4320	−1.55619	−0.778097	+	0.628144i	$0.783815\pi$
$930$	0	0
$931$	5.04502	0.165344
$932$	0	0
$933$	−6.06584	−0.198587
$934$	0	0
$935$	3.63785	0.118970
$936$	0	0
$937$	26.1946	0.855741	0.427871	−	0.903840i	$-0.359264\pi$
$937$	26.1946	0.855741	0.427871	+	0.903840i	$0.359264\pi$
$938$	0	0
$939$	−1.26316	−0.0412217
$940$	0	0
$941$	−11.0361	−0.359765	−0.179883	−	0.983688i	$-0.557572\pi$
$941$	−11.0361	−0.359765	−0.179883	+	0.983688i	$0.557572\pi$
$942$	0	0
$943$	−55.6233	−1.81134
$944$	0	0
$945$	6.04502	0.196644
$946$	0	0
$947$	2.40717	0.0782225	0.0391113	−	0.999235i	$-0.487547\pi$
$947$	2.40717	0.0782225	0.0391113	+	0.999235i	$0.487547\pi$
$948$	0	0
$949$	21.8504	0.709294
$950$	0	0
$951$	9.44322	0.306217
$952$	0	0
$953$	42.9169	1.39021	0.695107	−	0.718906i	$-0.255357\pi$
$953$	42.9169	1.39021	0.695107	+	0.718906i	$0.255357\pi$
$954$	0	0
$955$	2.04502	0.0661752
$956$	0	0
$957$	6.70638	0.216787
$958$	0	0
$959$	−10.3442	−0.334033
$960$	0	0
$961$	60.6233	1.95559
$962$	0	0
$963$	5.20359	0.167683
$964$	0	0
$965$	6.23964	0.200861
$966$	0	0
$967$	56.7368	1.82453	0.912267	−	0.409596i	$-0.134330\pi$
$967$	56.7368	1.82453	0.912267	+	0.409596i	$0.134330\pi$
$968$	0	0
$969$	−0.601793	−0.0193324
$970$	0	0
$971$	−11.4849	−0.368567	−0.184284	−	0.982873i	$-0.558996\pi$
$971$	−11.4849	−0.368567	−0.184284	+	0.982873i	$0.558996\pi$
$972$	0	0
$973$	−6.64344	−0.212979
$974$	0	0
$975$	−23.5720	−0.754908
$976$	0	0
$977$	24.1496	0.772614	0.386307	−	0.922370i	$-0.373750\pi$
$977$	24.1496	0.772614	0.386307	+	0.922370i	$0.373750\pi$
$978$	0	0
$979$	5.02352	0.160552
$980$	0	0
$981$	16.5180	0.527380
$982$	0	0
$983$	43.8809	1.39958	0.699791	−	0.714348i	$-0.253276\pi$
$983$	43.8809	1.39958	0.699791	+	0.714348i	$0.253276\pi$
$984$	0	0
$985$	−11.5333	−0.367480
$986$	0	0
$987$	−13.6558	−0.434668
$988$	0	0
$989$	84.0609	2.67298
$990$	0	0
$991$	−40.2909	−1.27988	−0.639942	−	0.768424i	$-0.721041\pi$
$991$	−40.2909	−1.27988	−0.639942	+	0.768424i	$0.721041\pi$
$992$	0	0
$993$	−19.0540	−0.604660
$994$	0	0
$995$	−28.5422	−0.904849
$996$	0	0
$997$	44.9619	1.42396	0.711979	−	0.702201i	$-0.247799\pi$
$997$	44.9619	1.42396	0.711979	+	0.702201i	$0.247799\pi$
$998$	0	0
$999$	−1.72161	−0.0544694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.a.v.1.3	yes	3
3.2	odd	2		5472.2.a.bn.1.1		3
4.3	odd	2		1824.2.a.t.1.3	✓	3
8.3	odd	2		3648.2.a.by.1.1		3
8.5	even	2		3648.2.a.bw.1.1		3
12.11	even	2		5472.2.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.a.t.1.3	✓	3	4.3	odd	2
1824.2.a.v.1.3	yes	3	1.1	even	1	trivial
3648.2.a.bw.1.1		3	8.5	even	2
3648.2.a.by.1.1		3	8.3	odd	2
5472.2.a.bm.1.1		3	12.11	even	2
5472.2.a.bn.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1824.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.32340 q^{5} +1.39821 q^{7} +1.00000 q^{9} +1.39821 q^{11} -1.72161 q^{13} +4.32340 q^{15} +0.601793 q^{17} -1.00000 q^{19} +1.39821 q^{21} -8.36842 q^{23} +13.6918 q^{25} +1.00000 q^{27} +4.79641 q^{29} +9.57201 q^{31} +1.39821 q^{33} +6.04502 q^{35} -1.72161 q^{37} -1.72161 q^{39} +6.64681 q^{41} -10.0450 q^{43} +4.32340 q^{45} -9.76663 q^{47} -5.04502 q^{49} +0.601793 q^{51} -7.29362 q^{53} +6.04502 q^{55} -1.00000 q^{57} -12.0900 q^{59} +9.24860 q^{61} +1.39821 q^{63} -7.44322 q^{65} +0.646809 q^{67} -8.36842 q^{69} +12.0900 q^{71} -12.6918 q^{73} +13.6918 q^{75} +1.95498 q^{77} +2.12878 q^{79} +1.00000 q^{81} +2.60179 q^{85} +4.79641 q^{87} +3.59283 q^{89} -2.40717 q^{91} +9.57201 q^{93} -4.32340 q^{95} -5.05398 q^{97} +1.39821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} + q^{11} + 6 q^{13} + 5 q^{15} + 5 q^{17} - 3 q^{19} + q^{21} + 2 q^{23} + 6 q^{25} + 3 q^{27} + 8 q^{29} + 8 q^{31} + q^{33} - q^{35} + 6 q^{37} + 6 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 5 * q^5 + q^7 + 3 * q^9 + q^11 + 6 * q^13 + 5 * q^15 + 5 * q^17 - 3 * q^19 + q^21 + 2 * q^23 + 6 * q^25 + 3 * q^27 + 8 * q^29 + 8 * q^31 + q^33 - q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 11 * q^43 + 5 * q^45 + q^47 + 4 * q^49 + 5 * q^51 + 10 * q^53 - q^55 - 3 * q^57 + 2 * q^59 + 15 * q^61 + q^63 - 14 * q^67 + 2 * q^69 - 2 * q^71 - 3 * q^73 + 6 * q^75 + 25 * q^77 + 8 * q^79 + 3 * q^81 + 11 * q^85 + 8 * q^87 - 2 * q^89 - 20 * q^91 + 8 * q^93 - 5 * q^95 - 12 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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