LMFDB - Embedded newform 5184.2.c.l.5183.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(5183,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.5183");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.c (of order $2$, degree $1$, not 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:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 $ x^16 - 12x^14 + 49x^12 - 12x^10 - 600x^8 + 108x^6 + 4057x^4 + 18252*x^2 + 28561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 2592)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.16
Root			$2.39101 + 0.123030i$ of defining polynomial
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.l.5183.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.01832i q^{5} +3.08003i q^{7} -5.95017 q^{11} +1.34798 q^{13} +4.87346i q^{17} -7.49484i q^{19} -2.57858 q^{23} -11.1469 q^{25} -6.46781i q^{29} +1.48660i q^{31} -12.3765 q^{35} -1.91997 q^{37} +1.95737i q^{41} -7.84817i q^{43} -2.28527 q^{47} -2.48660 q^{49} +0.871059i q^{53} -23.9096i q^{55} +1.30103 q^{59} +6.59343 q^{61} +5.41662i q^{65} +9.74958i q^{67} -4.64914 q^{71} -2.14686 q^{73} -18.3267i q^{77} +9.33477i q^{79} -10.1956 q^{83} -19.5831 q^{85} +0.947609i q^{89} +4.15183i q^{91} +30.1166 q^{95} +10.3162 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{13} - 48 q^{25} - 72 q^{37} - 48 q^{49} + 56 q^{61} + 96 q^{73} + 24 q^{85} - 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^13 - 48 * q^25 - 72 * q^37 - 48 * q^49 + 56 * q^61 + 96 * q^73 + 24 * q^85 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	4.01832i	1.79705i	0.438927	+	0.898523i	$0.355359\pi$
$5$	4.01832i	1.79705i	−0.438927	+	0.898523i	$0.644641\pi$
$6$	0	0
$7$	3.08003i	1.16414i	0.813138	+	0.582072i	$0.197758\pi$
$7$	3.08003i	1.16414i	−0.813138	+	0.582072i	$0.802242\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.95017	−1.79404	−0.897021	−	0.441987i	$-0.854274\pi$
$11$	−5.95017	−1.79404	−0.897021	+	0.441987i	$0.854274\pi$
$12$	0	0
$13$	1.34798	0.373863	0.186932	−	0.982373i	$-0.440146\pi$
$13$	1.34798	0.373863	0.186932	+	0.982373i	$0.440146\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.87346i	1.18199i	0.806676	+	0.590994i	$0.201264\pi$
$17$	4.87346i	1.18199i	−0.806676	+	0.590994i	$0.798736\pi$
$18$	0	0
$19$	− 7.49484i	− 1.71943i	−0.510771	−	0.859717i	$-0.670640\pi$
$19$	− 7.49484i	− 1.71943i	0.510771	−	0.859717i	$-0.329360\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.57858	−0.537672	−0.268836	−	0.963186i	$-0.586639\pi$
$23$	−2.57858	−0.537672	−0.268836	+	0.963186i	$0.586639\pi$
$24$	0	0
$25$	−11.1469	−2.22937
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.46781i	− 1.20104i	−0.799609	−	0.600521i	$-0.794960\pi$
$29$	− 6.46781i	− 1.20104i	0.799609	−	0.600521i	$-0.205040\pi$
$30$	0	0
$31$	1.48660i	0.267002i	0.991049	+	0.133501i	$0.0426219\pi$
$31$	1.48660i	0.267002i	−0.991049	+	0.133501i	$0.957378\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.3765	−2.09202
$36$	0	0
$37$	−1.91997	−0.315641	−0.157820	−	0.987468i	$-0.550447\pi$
$37$	−1.91997	−0.315641	−0.157820	+	0.987468i	$0.550447\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.95737i	0.305690i	0.988250	+	0.152845i	$0.0488435\pi$
$41$	1.95737i	0.305690i	−0.988250	+	0.152845i	$0.951157\pi$
$42$	0	0
$43$	− 7.84817i	− 1.19683i	−0.801185	−	0.598417i	$-0.795796\pi$
$43$	− 7.84817i	− 1.19683i	0.801185	−	0.598417i	$-0.204204\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.28527	−0.333341	−0.166671	−	0.986013i	$-0.553302\pi$
$47$	−2.28527	−0.333341	−0.166671	+	0.986013i	$0.553302\pi$
$48$	0	0
$49$	−2.48660	−0.355229
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.871059i	0.119649i	0.998209	+	0.0598245i	$0.0190541\pi$
$53$	0.871059i	0.119649i	−0.998209	+	0.0598245i	$0.980946\pi$
$54$	0	0
$55$	− 23.9096i	− 3.22398i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.30103	0.169380	0.0846898	−	0.996407i	$-0.473010\pi$
$59$	1.30103	0.169380	0.0846898	+	0.996407i	$0.473010\pi$
$60$	0	0
$61$	6.59343	0.844202	0.422101	−	0.906549i	$-0.361293\pi$
$61$	6.59343	0.844202	0.422101	+	0.906549i	$0.361293\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.41662i	0.671849i
$66$	0	0
$67$	9.74958i	1.19110i	0.803318	+	0.595550i	$0.203066\pi$
$67$	9.74958i	1.19110i	−0.803318	+	0.595550i	$0.796934\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.64914	−0.551751	−0.275876	−	0.961193i	$-0.588968\pi$
$71$	−4.64914	−0.551751	−0.275876	+	0.961193i	$0.588968\pi$
$72$	0	0
$73$	−2.14686	−0.251271	−0.125635	−	0.992076i	$-0.540097\pi$
$73$	−2.14686	−0.251271	−0.125635	+	0.992076i	$0.540097\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 18.3267i	− 2.08852i
$78$	0	0
$79$	9.33477i	1.05024i	0.851027	+	0.525122i	$0.175980\pi$
$79$	9.33477i	1.05024i	−0.851027	+	0.525122i	$0.824020\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1956	−1.11911	−0.559555	−	0.828793i	$-0.689028\pi$
$83$	−10.1956	−1.11911	−0.559555	+	0.828793i	$0.689028\pi$
$84$	0	0
$85$	−19.5831	−2.12409
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.947609i	0.100446i	0.998738	+	0.0502232i	$0.0159933\pi$
$89$	0.947609i	0.100446i	−0.998738	+	0.0502232i	$0.984007\pi$
$90$	0	0
$91$	4.15183i	0.435230i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	30.1166	3.08990
$96$	0	0
$97$	10.3162	1.04745	0.523727	−	0.851886i	$-0.324541\pi$
$97$	10.3162	1.04745	0.523727	+	0.851886i	$0.324541\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.14027i	− 0.212964i	−0.994315	−	0.106482i	$-0.966041\pi$
$101$	− 2.14027i	− 0.212964i	0.994315	−	0.106482i	$-0.0339587\pi$
$102$	0	0
$103$	− 9.85641i	− 0.971181i	−0.874187	−	0.485590i	$-0.838605\pi$
$103$	− 9.85641i	− 0.971181i	0.874187	−	0.485590i	$-0.161395\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.38765	0.424170	0.212085	−	0.977251i	$-0.431975\pi$
$107$	4.38765	0.424170	0.212085	+	0.977251i	$0.431975\pi$
$108$	0	0
$109$	−16.4588	−1.57646	−0.788231	−	0.615379i	$-0.789003\pi$
$109$	−16.4588	−1.57646	−0.788231	+	0.615379i	$0.789003\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.4375i	1.35816i	0.734062	+	0.679082i	$0.237622\pi$
$113$	14.4375i	1.35816i	−0.734062	+	0.679082i	$0.762378\pi$
$114$	0	0
$115$	− 10.3616i	− 0.966221i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−15.0104	−1.37600
$120$	0	0
$121$	24.4045	2.21859
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 24.7000i	− 2.20924i
$126$	0	0
$127$	− 1.17862i	− 0.104586i	−0.998632	−	0.0522929i	$-0.983347\pi$
$127$	− 1.17862i	− 0.104586i	0.998632	−	0.0522929i	$-0.0166530\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.110415	−0.00964704	−0.00482352	−	0.999988i	$-0.501535\pi$
$131$	−0.110415	−0.00964704	−0.00482352	+	0.999988i	$0.501535\pi$
$132$	0	0
$133$	23.0844	2.00167
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.4606i	− 0.893710i	−0.894606	−	0.446855i	$-0.852544\pi$
$137$	− 10.4606i	− 0.893710i	0.894606	−	0.446855i	$-0.147456\pi$
$138$	0	0
$139$	− 14.6695i	− 1.24425i	−0.782916	−	0.622127i	$-0.786269\pi$
$139$	− 14.6695i	− 1.24425i	0.782916	−	0.622127i	$-0.213731\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.02072	−0.670726
$144$	0	0
$145$	25.9897	2.15833
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.41301i	− 0.279605i	−0.990179	−	0.139802i	$-0.955353\pi$
$149$	− 3.41301i	− 0.279605i	0.990179	−	0.139802i	$-0.0446467\pi$
$150$	0	0
$151$	− 8.82961i	− 0.718544i	−0.933233	−	0.359272i	$-0.883025\pi$
$151$	− 8.82961i	− 0.718544i	0.933233	−	0.359272i	$-0.116975\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.97364	−0.479814
$156$	0	0
$157$	20.3963	1.62780	0.813899	−	0.581006i	$-0.197341\pi$
$157$	20.3963	1.62780	0.813899	+	0.581006i	$0.197341\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.94213i	− 0.625927i
$162$	0	0
$163$	11.3880i	0.891978i	0.895038	+	0.445989i	$0.147148\pi$
$163$	11.3880i	0.891978i	−0.895038	+	0.445989i	$0.852852\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.2926	1.80244	0.901219	−	0.433364i	$-0.142673\pi$
$167$	23.2926	1.80244	0.901219	+	0.433364i	$0.142673\pi$
$168$	0	0
$169$	−11.1829	−0.860226
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.4257i	1.02074i	0.859956	+	0.510368i	$0.170491\pi$
$173$	13.4257i	1.02074i	−0.859956	+	0.510368i	$0.829509\pi$
$174$	0	0
$175$	− 34.3327i	− 2.59531i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.7141	−1.54824	−0.774120	−	0.633038i	$-0.781808\pi$
$179$	−20.7141	−1.54824	−0.774120	+	0.633038i	$0.781808\pi$
$180$	0	0
$181$	8.06148	0.599205	0.299602	−	0.954064i	$-0.403146\pi$
$181$	8.06148	0.599205	0.299602	+	0.954064i	$0.403146\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7.71503i	− 0.567220i
$186$	0	0
$187$	− 28.9979i	− 2.12054i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.7930	1.64924	0.824620	−	0.565687i	$-0.191389\pi$
$191$	22.7930	1.64924	0.824620	+	0.565687i	$0.191389\pi$
$192$	0	0
$193$	−16.4534	−1.18434	−0.592171	−	0.805812i	$-0.701729\pi$
$193$	−16.4534	−1.18434	−0.592171	+	0.805812i	$0.701729\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.9544i	1.56419i	0.623160	+	0.782095i	$0.285849\pi$
$197$	21.9544i	1.56419i	−0.623160	+	0.782095i	$0.714151\pi$
$198$	0	0
$199$	11.2933i	0.800564i	0.916392	+	0.400282i	$0.131088\pi$
$199$	11.2933i	0.800564i	−0.916392	+	0.400282i	$0.868912\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	19.9211	1.39818
$204$	0	0
$205$	−7.86532	−0.549338
$206$	0	0
$207$	0	0
$208$	0	0
$209$	44.5955i	3.08474i
$210$	0	0
$211$	− 14.1644i	− 0.975117i	−0.873090	−	0.487558i	$-0.837888\pi$
$211$	− 14.1644i	− 0.975117i	0.873090	−	0.487558i	$-0.162112\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	31.5364	2.15077
$216$	0	0
$217$	−4.57879	−0.310828
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.56934i	0.441902i
$222$	0	0
$223$	− 18.4230i	− 1.23370i	−0.787081	−	0.616849i	$-0.788409\pi$
$223$	− 18.4230i	− 1.23370i	0.787081	−	0.616849i	$-0.211591\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.06830	−0.469140	−0.234570	−	0.972099i	$-0.575368\pi$
$227$	−7.06830	−0.469140	−0.234570	+	0.972099i	$0.575368\pi$
$228$	0	0
$229$	−10.2723	−0.678811	−0.339405	−	0.940640i	$-0.610226\pi$
$229$	−10.2723	−0.678811	−0.339405	+	0.940640i	$0.610226\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.49420i	0.490961i	0.969401	+	0.245481i	$0.0789458\pi$
$233$	7.49420i	0.490961i	−0.969401	+	0.245481i	$0.921054\pi$
$234$	0	0
$235$	− 9.18294i	− 0.599029i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−0.618448	−0.0400041	−0.0200020	−	0.999800i	$-0.506367\pi$
$239$	−0.618448	−0.0400041	−0.0200020	+	0.999800i	$0.506367\pi$
$240$	0	0
$241$	−24.9043	−1.60423	−0.802113	−	0.597172i	$-0.796291\pi$
$241$	−24.9043	−1.60423	−0.802113	+	0.597172i	$0.796291\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 9.99196i	− 0.638363i
$246$	0	0
$247$	− 10.1029i	− 0.642833i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.54676	−0.160750	−0.0803751	−	0.996765i	$-0.525612\pi$
$251$	−2.54676	−0.160750	−0.0803751	+	0.996765i	$0.525612\pi$
$252$	0	0
$253$	15.3430	0.964607
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.94485i	− 0.557965i	−0.960296	−	0.278982i	$-0.910003\pi$
$257$	− 8.94485i	− 0.557965i	0.960296	−	0.278982i	$-0.0899971\pi$
$258$	0	0
$259$	− 5.91356i	− 0.367451i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.87190	0.177089	0.0885444	−	0.996072i	$-0.471778\pi$
$263$	2.87190	0.177089	0.0885444	+	0.996072i	$0.471778\pi$
$264$	0	0
$265$	−3.50019	−0.215015
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.51044i	− 0.275006i	−0.990501	−	0.137503i	$-0.956092\pi$
$269$	− 4.51044i	− 0.275006i	0.990501	−	0.137503i	$-0.0439077\pi$
$270$	0	0
$271$	2.72670i	0.165635i	0.996565	+	0.0828177i	$0.0263919\pi$
$271$	2.72670i	0.165635i	−0.996565	+	0.0828177i	$0.973608\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	66.3257	3.99959
$276$	0	0
$277$	−5.53627	−0.332642	−0.166321	−	0.986072i	$-0.553189\pi$
$277$	−5.53627	−0.332642	−0.166321	+	0.986072i	$0.553189\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 19.7417i	− 1.17769i	−0.808247	−	0.588844i	$-0.799583\pi$
$281$	− 19.7417i	− 1.17769i	0.808247	−	0.588844i	$-0.200417\pi$
$282$	0	0
$283$	− 13.0883i	− 0.778017i	−0.921234	−	0.389008i	$-0.872818\pi$
$283$	− 13.0883i	− 0.778017i	0.921234	−	0.389008i	$-0.127182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.02876	−0.355866
$288$	0	0
$289$	−6.75064	−0.397096
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.6224i	1.32162i	0.750555	+	0.660808i	$0.229786\pi$
$293$	22.6224i	1.32162i	−0.750555	+	0.660808i	$0.770214\pi$
$294$	0	0
$295$	5.22795i	0.304383i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.47589	−0.201016
$300$	0	0
$301$	24.1726	1.39329
$302$	0	0
$303$	0	0
$304$	0	0
$305$	26.4945i	1.51707i
$306$	0	0
$307$	7.64667i	0.436418i	0.975902	+	0.218209i	$0.0700215\pi$
$307$	7.64667i	0.436418i	−0.975902	+	0.218209i	$0.929978\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.3264	1.20931	0.604654	−	0.796488i	$-0.293311\pi$
$311$	21.3264	1.20931	0.604654	+	0.796488i	$0.293311\pi$
$312$	0	0
$313$	−17.7349	−1.00244	−0.501219	−	0.865320i	$-0.667115\pi$
$313$	−17.7349	−1.00244	−0.501219	+	0.865320i	$0.667115\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.24123i	− 0.182046i	−0.995849	−	0.0910228i	$-0.970986\pi$
$317$	− 3.24123i	− 0.182046i	0.995849	−	0.0910228i	$-0.0290136\pi$
$318$	0	0
$319$	38.4845i	2.15472i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	36.5258	2.03235
$324$	0	0
$325$	−15.0258	−0.833480
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.03871i	− 0.388057i
$330$	0	0
$331$	− 9.33477i	− 0.513086i	−0.966533	−	0.256543i	$-0.917417\pi$
$331$	− 9.33477i	− 0.513086i	0.966533	−	0.256543i	$-0.0825835\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−39.1769	−2.14046
$336$	0	0
$337$	14.0654	0.766191	0.383095	−	0.923709i	$-0.374858\pi$
$337$	14.0654	0.766191	0.383095	+	0.923709i	$0.374858\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.84554i	− 0.479013i
$342$	0	0
$343$	13.9014i	0.750606i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.56861	−0.352622	−0.176311	−	0.984335i	$-0.556416\pi$
$347$	−6.56861	−0.352622	−0.176311	+	0.984335i	$0.556416\pi$
$348$	0	0
$349$	−12.2547	−0.655981	−0.327991	−	0.944681i	$-0.606371\pi$
$349$	−12.2547	−0.655981	−0.327991	+	0.944681i	$0.606371\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 5.05563i	− 0.269084i	−0.990908	−	0.134542i	$-0.957044\pi$
$353$	− 5.05563i	− 0.269084i	0.990908	−	0.134542i	$-0.0429564\pi$
$354$	0	0
$355$	− 18.6817i	− 0.991522i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.8214	−1.67947	−0.839734	−	0.542998i	$-0.817289\pi$
$359$	−31.8214	−1.67947	−0.839734	+	0.542998i	$0.817289\pi$
$360$	0	0
$361$	−37.1726	−1.95645
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.62675i	− 0.451545i
$366$	0	0
$367$	− 2.51340i	− 0.131198i	−0.997846	−	0.0655991i	$-0.979104\pi$
$367$	− 2.51340i	− 0.131198i	0.997846	−	0.0655991i	$-0.0208959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.68289	−0.139289
$372$	0	0
$373$	−31.9179	−1.65265	−0.826323	−	0.563197i	$-0.809571\pi$
$373$	−31.9179	−1.65265	−0.826323	+	0.563197i	$0.809571\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.71849i	− 0.449025i
$378$	0	0
$379$	− 23.9053i	− 1.22793i	−0.789332	−	0.613967i	$-0.789573\pi$
$379$	− 23.9053i	− 1.22793i	0.789332	−	0.613967i	$-0.210427\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.26117	−0.166638	−0.0833189	−	0.996523i	$-0.526552\pi$
$383$	−3.26117	−0.166638	−0.0833189	+	0.996523i	$0.526552\pi$
$384$	0	0
$385$	73.6425	3.75317
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.1704i	− 0.769169i	−0.923090	−	0.384584i	$-0.874345\pi$
$389$	− 15.1704i	− 0.769169i	0.923090	−	0.384584i	$-0.125655\pi$
$390$	0	0
$391$	− 12.5666i	− 0.635522i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−37.5101	−1.88734
$396$	0	0
$397$	25.1683	1.26316	0.631580	−	0.775310i	$-0.282407\pi$
$397$	25.1683	1.26316	0.631580	+	0.775310i	$0.282407\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.44583i	0.321889i	0.986963	+	0.160945i	$0.0514540\pi$
$401$	6.44583i	0.321889i	−0.986963	+	0.160945i	$0.948546\pi$
$402$	0	0
$403$	2.00392i	0.0998221i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.4241	0.566273
$408$	0	0
$409$	−26.5167	−1.31116	−0.655582	−	0.755124i	$-0.727577\pi$
$409$	−26.5167	−1.31116	−0.655582	+	0.755124i	$0.727577\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00721i	0.197182i
$414$	0	0
$415$	− 40.9690i	− 2.01109i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.7270	−1.59882	−0.799410	−	0.600786i	$-0.794855\pi$
$419$	−32.7270	−1.59882	−0.799410	+	0.600786i	$0.794855\pi$
$420$	0	0
$421$	−23.6192	−1.15113	−0.575565	−	0.817756i	$-0.695218\pi$
$421$	−23.6192	−1.15113	−0.575565	+	0.817756i	$0.695218\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 54.3238i	− 2.63509i
$426$	0	0
$427$	20.3080i	0.982772i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.7052	−1.38268	−0.691340	−	0.722530i	$-0.742979\pi$
$431$	−28.7052	−1.38268	−0.691340	+	0.722530i	$0.742979\pi$
$432$	0	0
$433$	10.2444	0.492315	0.246158	−	0.969230i	$-0.420832\pi$
$433$	10.2444	0.492315	0.246158	+	0.969230i	$0.420832\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.3261i	0.924492i
$438$	0	0
$439$	− 39.3430i	− 1.87774i	−0.344273	−	0.938870i	$-0.611874\pi$
$439$	− 39.3430i	− 1.87774i	0.344273	−	0.938870i	$-0.388126\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.37221	−0.112707	−0.0563536	−	0.998411i	$-0.517947\pi$
$443$	−2.37221	−0.112707	−0.0563536	+	0.998411i	$0.517947\pi$
$444$	0	0
$445$	−3.80779	−0.180507
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33.6097i	− 1.58614i	−0.609129	−	0.793071i	$-0.708481\pi$
$449$	− 33.6097i	− 1.58614i	0.609129	−	0.793071i	$-0.291519\pi$
$450$	0	0
$451$	− 11.6467i	− 0.548420i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.6834	−0.782128
$456$	0	0
$457$	27.9536	1.30761	0.653807	−	0.756661i	$-0.273171\pi$
$457$	27.9536	1.30761	0.653807	+	0.756661i	$0.273171\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.04368i	0.141758i	0.997485	+	0.0708791i	$0.0225805\pi$
$461$	3.04368i	0.141758i	−0.997485	+	0.0708791i	$0.977420\pi$
$462$	0	0
$463$	− 19.9053i	− 0.925079i	−0.886599	−	0.462539i	$-0.846938\pi$
$463$	− 19.9053i	− 0.925079i	0.886599	−	0.462539i	$-0.153062\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.82289	−0.130628	−0.0653139	−	0.997865i	$-0.520805\pi$
$467$	−2.82289	−0.130628	−0.0653139	+	0.997865i	$0.520805\pi$
$468$	0	0
$469$	−30.0290	−1.38661
$470$	0	0
$471$	0	0
$472$	0	0
$473$	46.6979i	2.14717i
$474$	0	0
$475$	83.5439i	3.83326i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.3375	−1.38615	−0.693077	−	0.720864i	$-0.743745\pi$
$479$	−30.3375	−1.38615	−0.693077	+	0.720864i	$0.743745\pi$
$480$	0	0
$481$	−2.58808	−0.118006
$482$	0	0
$483$	0	0
$484$	0	0
$485$	41.4538i	1.88232i
$486$	0	0
$487$	− 11.9668i	− 0.542268i	−0.962542	−	0.271134i	$-0.912601\pi$
$487$	− 11.9668i	− 0.542268i	0.962542	−	0.271134i	$-0.0873986\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.2191	0.461179	0.230590	−	0.973051i	$-0.425935\pi$
$491$	10.2191	0.461179	0.230590	+	0.973051i	$0.425935\pi$
$492$	0	0
$493$	31.5206	1.41962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 14.3195i	− 0.642317i
$498$	0	0
$499$	− 11.2244i	− 0.502472i	−0.967926	−	0.251236i	$-0.919163\pi$
$499$	− 11.2244i	− 0.502472i	0.967926	−	0.251236i	$-0.0808371\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.4842	−0.913346	−0.456673	−	0.889635i	$-0.650959\pi$
$503$	−20.4842	−0.913346	−0.456673	+	0.889635i	$0.650959\pi$
$504$	0	0
$505$	8.60026	0.382707
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 32.2279i	− 1.42848i	−0.699903	−	0.714238i	$-0.746773\pi$
$509$	− 32.2279i	− 1.42848i	0.699903	−	0.714238i	$-0.253227\pi$
$510$	0	0
$511$	− 6.61239i	− 0.292515i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39.6061	1.74526
$516$	0	0
$517$	13.5978	0.598028
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 27.2971i	− 1.19591i	−0.801531	−	0.597953i	$-0.795981\pi$
$521$	− 27.2971i	− 1.19591i	0.801531	−	0.597953i	$-0.204019\pi$
$522$	0	0
$523$	− 26.4845i	− 1.15809i	−0.815297	−	0.579044i	$-0.803426\pi$
$523$	− 26.4845i	− 1.15809i	0.815297	−	0.579044i	$-0.196574\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.24491	−0.315593
$528$	0	0
$529$	−16.3509	−0.710909
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.63850i	0.114286i
$534$	0	0
$535$	17.6309i	0.762252i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.7957	0.637296
$540$	0	0
$541$	−7.15472	−0.307605	−0.153803	−	0.988102i	$-0.549152\pi$
$541$	−7.15472	−0.307605	−0.153803	+	0.988102i	$0.549152\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 66.1365i	− 2.83297i
$546$	0	0
$547$	10.2712i	0.439165i	0.975594	+	0.219583i	$0.0704696\pi$
$547$	10.2712i	0.439165i	−0.975594	+	0.219583i	$0.929530\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−48.4752	−2.06511
$552$	0	0
$553$	−28.7514	−1.22263
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.44585i	0.400234i	0.979772	+	0.200117i	$0.0641322\pi$
$557$	9.44585i	0.400234i	−0.979772	+	0.200117i	$0.935868\pi$
$558$	0	0
$559$	− 10.5792i	− 0.447452i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.1568	−1.27096	−0.635479	−	0.772118i	$-0.719197\pi$
$563$	−30.1568	−1.27096	−0.635479	+	0.772118i	$0.719197\pi$
$564$	0	0
$565$	−58.0144	−2.44068
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.2005i	1.05646i	0.849102	+	0.528230i	$0.177144\pi$
$569$	25.2005i	1.05646i	−0.849102	+	0.528230i	$0.822856\pi$
$570$	0	0
$571$	37.3059i	1.56120i	0.625029	+	0.780602i	$0.285087\pi$
$571$	37.3059i	1.56120i	−0.625029	+	0.780602i	$0.714913\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.7431	1.19867
$576$	0	0
$577$	−17.7799	−0.740189	−0.370094	−	0.928994i	$-0.620675\pi$
$577$	−17.7799	−0.740189	−0.370094	+	0.928994i	$0.620675\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 31.4027i	− 1.30280i
$582$	0	0
$583$	− 5.18294i	− 0.214656i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.5306	−1.17759	−0.588793	−	0.808284i	$-0.700397\pi$
$587$	−28.5306	−1.17759	−0.588793	+	0.808284i	$0.700397\pi$
$588$	0	0
$589$	11.1419	0.459092
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 31.3024i	− 1.28544i	−0.766102	−	0.642718i	$-0.777807\pi$
$593$	− 31.3024i	− 1.28544i	0.766102	−	0.642718i	$-0.222193\pi$
$594$	0	0
$595$	− 60.3166i	− 2.47274i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.45852	0.141311	0.0706557	−	0.997501i	$-0.477491\pi$
$599$	3.45852	0.141311	0.0706557	+	0.997501i	$0.477491\pi$
$600$	0	0
$601$	−13.5749	−0.553731	−0.276865	−	0.960909i	$-0.589296\pi$
$601$	−13.5749	−0.553731	−0.276865	+	0.960909i	$0.589296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	98.0649i	3.98691i
$606$	0	0
$607$	35.5728i	1.44385i	0.691969	+	0.721927i	$0.256744\pi$
$607$	35.5728i	1.44385i	−0.691969	+	0.721927i	$0.743256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.08051	−0.124624
$612$	0	0
$613$	23.5978	0.953104	0.476552	−	0.879146i	$-0.341886\pi$
$613$	23.5978	0.953104	0.476552	+	0.879146i	$0.341886\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13.3719i	− 0.538332i	−0.963094	−	0.269166i	$-0.913252\pi$
$617$	− 13.3719i	− 0.538332i	0.963094	−	0.269166i	$-0.0867480\pi$
$618$	0	0
$619$	37.1829i	1.49451i	0.664538	+	0.747254i	$0.268628\pi$
$619$	37.1829i	1.49451i	−0.664538	+	0.747254i	$0.731372\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.91867	−0.116934
$624$	0	0
$625$	43.5181	1.74073
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 9.35689i	− 0.373084i
$630$	0	0
$631$	37.8564i	1.50704i	0.657425	+	0.753520i	$0.271646\pi$
$631$	37.8564i	1.50704i	−0.657425	+	0.753520i	$0.728354\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.73608	0.187946
$636$	0	0
$637$	−3.35190	−0.132807
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.71440i	0.344198i	0.985080	+	0.172099i	$0.0550549\pi$
$641$	8.71440i	0.344198i	−0.985080	+	0.172099i	$0.944945\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.9855	−0.864339	−0.432169	−	0.901792i	$-0.642252\pi$
$647$	−21.9855	−0.864339	−0.432169	+	0.901792i	$0.642252\pi$
$648$	0	0
$649$	−7.74134	−0.303874
$650$	0	0
$651$	0	0
$652$	0	0
$653$	47.3916i	1.85458i	0.374347	+	0.927289i	$0.377867\pi$
$653$	47.3916i	1.85458i	−0.374347	+	0.927289i	$0.622133\pi$
$654$	0	0
$655$	− 0.443684i	− 0.0173362i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.9504	−0.971931	−0.485966	−	0.873978i	$-0.661532\pi$
$659$	−24.9504	−0.971931	−0.485966	+	0.873978i	$0.661532\pi$
$660$	0	0
$661$	42.0058	1.63384	0.816918	−	0.576753i	$-0.195681\pi$
$661$	42.0058	1.63384	0.816918	+	0.576753i	$0.195681\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	92.7602i	3.59709i
$666$	0	0
$667$	16.6778i	0.645766i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−39.2320	−1.51453
$672$	0	0
$673$	45.1852	1.74176	0.870880	−	0.491495i	$-0.163549\pi$
$673$	45.1852	1.74176	0.870880	+	0.491495i	$0.163549\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.5920i	− 1.09888i	−0.835533	−	0.549440i	$-0.814841\pi$
$677$	− 28.5920i	− 1.09888i	0.835533	−	0.549440i	$-0.185159\pi$
$678$	0	0
$679$	31.7743i	1.21939i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.4357	0.973269	0.486634	−	0.873606i	$-0.338224\pi$
$683$	25.4357	0.973269	0.486634	+	0.873606i	$0.338224\pi$
$684$	0	0
$685$	42.0340	1.60604
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.17417i	0.0447324i
$690$	0	0
$691$	− 33.6139i	− 1.27873i	−0.768902	−	0.639366i	$-0.779197\pi$
$691$	− 33.6139i	− 1.27873i	0.768902	−	0.639366i	$-0.220803\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	58.9469	2.23598
$696$	0	0
$697$	−9.53916	−0.361322
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.1433i	− 0.458648i	−0.973350	−	0.229324i	$-0.926349\pi$
$701$	− 12.1433i	− 0.458648i	0.973350	−	0.229324i	$-0.0736515\pi$
$702$	0	0
$703$	14.3898i	0.542723i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.59209	0.247921
$708$	0	0
$709$	−34.2723	−1.28712	−0.643561	−	0.765395i	$-0.722544\pi$
$709$	−34.2723	−1.28712	−0.643561	+	0.765395i	$0.722544\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.83333i	− 0.143559i
$714$	0	0
$715$	− 32.2298i	− 1.20533i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28.4119	1.05958	0.529792	−	0.848128i	$-0.322270\pi$
$719$	28.4119	1.05958	0.529792	+	0.848128i	$0.322270\pi$
$720$	0	0
$721$	30.3581	1.13059
$722$	0	0
$723$	0	0
$724$	0	0
$725$	72.0957i	2.67757i
$726$	0	0
$727$	26.5177i	0.983488i	0.870740	+	0.491744i	$0.163640\pi$
$727$	26.5177i	0.983488i	−0.870740	+	0.491744i	$0.836360\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.2478	1.41464
$732$	0	0
$733$	−24.0615	−0.888731	−0.444365	−	0.895846i	$-0.646571\pi$
$733$	−24.0615	−0.888731	−0.444365	+	0.895846i	$0.646571\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 58.0116i	− 2.13689i
$738$	0	0
$739$	20.9740i	0.771540i	0.922595	+	0.385770i	$0.126064\pi$
$739$	20.9740i	0.771540i	−0.922595	+	0.385770i	$0.873936\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.7240	−1.42064	−0.710322	−	0.703876i	$-0.751451\pi$
$743$	−38.7240	−1.42064	−0.710322	+	0.703876i	$0.751451\pi$
$744$	0	0
$745$	13.7146	0.502462
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.5141i	0.493794i
$750$	0	0
$751$	17.7574i	0.647977i	0.946061	+	0.323989i	$0.105024\pi$
$751$	17.7574i	0.647977i	−0.946061	+	0.323989i	$0.894976\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	35.4802	1.29126
$756$	0	0
$757$	26.7760	0.973191	0.486596	−	0.873627i	$-0.338239\pi$
$757$	26.7760	0.973191	0.486596	+	0.873627i	$0.338239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.55754i	− 0.201461i	−0.994914	−	0.100730i	$-0.967882\pi$
$761$	− 5.55754i	− 0.201461i	0.994914	−	0.100730i	$-0.0321179\pi$
$762$	0	0
$763$	− 50.6935i	− 1.83523i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.75376	0.0633248
$768$	0	0
$769$	−3.16007	−0.113955	−0.0569774	−	0.998375i	$-0.518146\pi$
$769$	−3.16007	−0.113955	−0.0569774	+	0.998375i	$0.518146\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.55146i	0.307575i	0.988104	+	0.153787i	$0.0491471\pi$
$773$	8.55146i	0.307575i	−0.988104	+	0.153787i	$0.950853\pi$
$774$	0	0
$775$	− 16.5710i	− 0.595246i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.6702	0.525613
$780$	0	0
$781$	27.6631	0.989865
$782$	0	0
$783$	0	0
$784$	0	0
$785$	81.9586i	2.92523i
$786$	0	0
$787$	− 46.4348i	− 1.65522i	−0.561301	−	0.827612i	$-0.689699\pi$
$787$	− 46.4348i	− 1.65522i	0.561301	−	0.827612i	$-0.310301\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−44.4679	−1.58110
$792$	0	0
$793$	8.88783	0.315616
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.05474i	0.320735i	0.987057	+	0.160368i	$0.0512680\pi$
$797$	9.05474i	0.320735i	−0.987057	+	0.160368i	$0.948732\pi$
$798$	0	0
$799$	− 11.1372i	− 0.394005i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.7742	0.450790
$804$	0	0
$805$	31.9140	1.12482
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33.0623i	1.16241i	0.813758	+	0.581204i	$0.197418\pi$
$809$	33.0623i	1.16241i	−0.813758	+	0.581204i	$0.802582\pi$
$810$	0	0
$811$	20.5874i	0.722922i	0.932387	+	0.361461i	$0.117722\pi$
$811$	20.5874i	0.722922i	−0.932387	+	0.361461i	$0.882278\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−45.7606	−1.60292
$816$	0	0
$817$	−58.8208	−2.05788
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 49.2247i	− 1.71796i	−0.512013	−	0.858978i	$-0.671100\pi$
$821$	− 49.2247i	− 1.71796i	0.512013	−	0.858978i	$-0.328900\pi$
$822$	0	0
$823$	27.1126i	0.945087i	0.881307	+	0.472543i	$0.156664\pi$
$823$	27.1126i	0.945087i	−0.881307	+	0.472543i	$0.843336\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.5040	−1.30414	−0.652070	−	0.758159i	$-0.726099\pi$
$827$	−37.5040	−1.30414	−0.652070	+	0.758159i	$0.726099\pi$
$828$	0	0
$829$	6.78386	0.235613	0.117807	−	0.993037i	$-0.462414\pi$
$829$	6.78386	0.235613	0.117807	+	0.993037i	$0.462414\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 12.1184i	− 0.419877i
$834$	0	0
$835$	93.5972i	3.23906i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.0803	1.38373	0.691863	−	0.722029i	$-0.256790\pi$
$839$	40.0803	1.38373	0.691863	+	0.722029i	$0.256790\pi$
$840$	0	0
$841$	−12.8325	−0.442500
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 44.9366i	− 1.54587i
$846$	0	0
$847$	75.1666i	2.58276i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.95080	0.169711
$852$	0	0
$853$	15.4369	0.528551	0.264275	−	0.964447i	$-0.414867\pi$
$853$	15.4369	0.528551	0.264275	+	0.964447i	$0.414867\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.6093i	− 0.499043i	−0.968369	−	0.249522i	$-0.919727\pi$
$857$	− 14.6093i	− 0.499043i	0.968369	−	0.249522i	$-0.0802734\pi$
$858$	0	0
$859$	− 5.80674i	− 0.198123i	−0.995081	−	0.0990616i	$-0.968416\pi$
$859$	− 5.80674i	− 0.198123i	0.995081	−	0.0990616i	$-0.0315841\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.5724	0.666254	0.333127	−	0.942882i	$-0.391896\pi$
$863$	19.5724	0.666254	0.333127	+	0.942882i	$0.391896\pi$
$864$	0	0
$865$	−53.9487	−1.83431
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 55.5435i	− 1.88418i
$870$	0	0
$871$	13.1423i	0.445309i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	76.0769	2.57187
$876$	0	0
$877$	5.29618	0.178839	0.0894196	−	0.995994i	$-0.471499\pi$
$877$	5.29618	0.178839	0.0894196	+	0.995994i	$0.471499\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 16.5467i	− 0.557473i	−0.960368	−	0.278736i	$-0.910084\pi$
$881$	− 16.5467i	− 0.557473i	0.960368	−	0.278736i	$-0.0899155\pi$
$882$	0	0
$883$	24.9361i	0.839166i	0.907717	+	0.419583i	$0.137824\pi$
$883$	24.9361i	0.839166i	−0.907717	+	0.419583i	$0.862176\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.0101	−0.436837	−0.218419	−	0.975855i	$-0.570090\pi$
$887$	−13.0101	−0.436837	−0.218419	+	0.975855i	$0.570090\pi$
$888$	0	0
$889$	3.63020	0.121753
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.1277i	0.573158i
$894$	0	0
$895$	− 83.2356i	− 2.78226i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.61506	0.320680
$900$	0	0
$901$	−4.24507	−0.141424
$902$	0	0
$903$	0	0
$904$	0	0
$905$	32.3936i	1.07680i
$906$	0	0
$907$	− 35.5110i	− 1.17912i	−0.807724	−	0.589561i	$-0.799301\pi$
$907$	− 35.5110i	− 1.17912i	0.807724	−	0.589561i	$-0.200699\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.4146	1.40526	0.702629	−	0.711557i	$-0.252009\pi$
$911$	42.4146	1.40526	0.702629	+	0.711557i	$0.252009\pi$
$912$	0	0
$913$	60.6654	2.00773
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.340083i	− 0.0112305i
$918$	0	0
$919$	− 3.12106i	− 0.102954i	−0.998674	−	0.0514772i	$-0.983607\pi$
$919$	− 3.12106i	− 0.102954i	0.998674	−	0.0514772i	$-0.0163929\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.26695	−0.206279
$924$	0	0
$925$	21.4016	0.703680
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 59.7151i	− 1.95919i	−0.200982	−	0.979595i	$-0.564413\pi$
$929$	− 59.7151i	− 1.95919i	0.200982	−	0.979595i	$-0.435587\pi$
$930$	0	0
$931$	18.6367i	0.610793i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	116.523	3.81070
$936$	0	0
$937$	−19.8793	−0.649428	−0.324714	−	0.945812i	$-0.605268\pi$
$937$	−19.8793	−0.649428	−0.324714	+	0.945812i	$0.605268\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 10.8236i	− 0.352840i	−0.984315	−	0.176420i	$-0.943548\pi$
$941$	− 10.8236i	− 0.352840i	0.984315	−	0.176420i	$-0.0564517\pi$
$942$	0	0
$943$	− 5.04724i	− 0.164361i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.1906	−0.753592	−0.376796	−	0.926296i	$-0.622974\pi$
$947$	−23.1906	−0.753592	−0.376796	+	0.926296i	$0.622974\pi$
$948$	0	0
$949$	−2.89393	−0.0939408
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 17.7414i	− 0.574700i	−0.957826	−	0.287350i	$-0.907226\pi$
$953$	− 17.7414i	− 0.574700i	0.957826	−	0.287350i	$-0.0927743\pi$
$954$	0	0
$955$	91.5893i	2.96376i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	32.2190	1.04041
$960$	0	0
$961$	28.7900	0.928710
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 66.1150i	− 2.12832i
$966$	0	0
$967$	− 13.4577i	− 0.432771i	−0.976308	−	0.216386i	$-0.930573\pi$
$967$	− 13.4577i	− 0.432771i	0.976308	−	0.216386i	$-0.0694269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.8995	0.574423	0.287211	−	0.957867i	$-0.407272\pi$
$971$	17.8995	0.574423	0.287211	+	0.957867i	$0.407272\pi$
$972$	0	0
$973$	45.1827	1.44849
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19.6540i	− 0.628787i	−0.949293	−	0.314393i	$-0.898199\pi$
$977$	− 19.6540i	− 0.628787i	0.949293	−	0.314393i	$-0.101801\pi$
$978$	0	0
$979$	− 5.63843i	− 0.180205i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−59.6527	−1.90263	−0.951314	−	0.308225i	$-0.900265\pi$
$983$	−59.6527	−1.90263	−0.951314	+	0.308225i	$0.900265\pi$
$984$	0	0
$985$	−88.2199	−2.81092
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.2372i	0.643505i
$990$	0	0
$991$	48.0373i	1.52595i	0.646426	+	0.762977i	$0.276263\pi$
$991$	48.0373i	1.52595i	−0.646426	+	0.762977i	$0.723737\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−45.3802	−1.43865
$996$	0	0
$997$	6.39227	0.202445	0.101223	−	0.994864i	$-0.467725\pi$
$997$	6.39227	0.202445	0.101223	+	0.994864i	$0.467725\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.l.5183.16		16
3.2	odd	2	inner	5184.2.c.l.5183.2		16
4.3	odd	2	inner	5184.2.c.l.5183.15		16
8.3	odd	2		2592.2.c.b.2591.1	✓	16
8.5	even	2		2592.2.c.b.2591.2	yes	16
12.11	even	2	inner	5184.2.c.l.5183.1		16
24.5	odd	2		2592.2.c.b.2591.16	yes	16
24.11	even	2		2592.2.c.b.2591.15	yes	16
72.5	odd	6		2592.2.s.i.1727.8		16
72.11	even	6		2592.2.s.i.863.1		16
72.13	even	6		2592.2.s.i.1727.1		16
72.29	odd	6		2592.2.s.j.863.1		16
72.43	odd	6		2592.2.s.i.863.8		16
72.59	even	6		2592.2.s.j.1727.8		16
72.61	even	6		2592.2.s.j.863.8		16
72.67	odd	6		2592.2.s.j.1727.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.c.b.2591.1	✓	16	8.3	odd	2
2592.2.c.b.2591.2	yes	16	8.5	even	2
2592.2.c.b.2591.15	yes	16	24.11	even	2
2592.2.c.b.2591.16	yes	16	24.5	odd	2
2592.2.s.i.863.1		16	72.11	even	6
2592.2.s.i.863.8		16	72.43	odd	6
2592.2.s.i.1727.1		16	72.13	even	6
2592.2.s.i.1727.8		16	72.5	odd	6
2592.2.s.j.863.1		16	72.29	odd	6
2592.2.s.j.863.8		16	72.61	even	6
2592.2.s.j.1727.1		16	72.67	odd	6
2592.2.s.j.1727.8		16	72.59	even	6
5184.2.c.l.5183.1		16	12.11	even	2	inner
5184.2.c.l.5183.2		16	3.2	odd	2	inner
5184.2.c.l.5183.15		16	4.3	odd	2	inner
5184.2.c.l.5183.16		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.01832i q^{5} +3.08003i q^{7} -5.95017 q^{11} +1.34798 q^{13} +4.87346i q^{17} -7.49484i q^{19} -2.57858 q^{23} -11.1469 q^{25} -6.46781i q^{29} +1.48660i q^{31} -12.3765 q^{35} -1.91997 q^{37} +1.95737i q^{41} -7.84817i q^{43} -2.28527 q^{47} -2.48660 q^{49} +0.871059i q^{53} -23.9096i q^{55} +1.30103 q^{59} +6.59343 q^{61} +5.41662i q^{65} +9.74958i q^{67} -4.64914 q^{71} -2.14686 q^{73} -18.3267i q^{77} +9.33477i q^{79} -10.1956 q^{83} -19.5831 q^{85} +0.947609i q^{89} +4.15183i q^{91} +30.1166 q^{95} +10.3162 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{13} - 48 q^{25} - 72 q^{37} - 48 q^{49} + 56 q^{61} + 96 q^{73} + 24 q^{85} - 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^13 - 48 * q^25 - 72 * q^37 - 48 * q^49 + 56 * q^61 + 96 * q^73 + 24 * q^85 - 32 * q^97

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