LMFDB - Embedded newform 6912.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6912,2,Mod(1,6912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6912.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6912 = 2^{8} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6912.a (trivial)

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:	yes
Analytic conductor:	$55.1925978771$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1728)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{5} -1.73205 q^{7} +O(q^{10})$	$q+1.73205 q^{5} -1.73205 q^{7} +3.00000 q^{11} -4.00000 q^{19} +6.92820 q^{23} -2.00000 q^{25} -6.92820 q^{29} -1.73205 q^{31} -3.00000 q^{35} -6.92820 q^{37} -12.0000 q^{41} -4.00000 q^{43} -6.92820 q^{47} -4.00000 q^{49} +8.66025 q^{53} +5.19615 q^{55} -13.8564 q^{61} +4.00000 q^{67} +13.8564 q^{71} +7.00000 q^{73} -5.19615 q^{77} +10.3923 q^{79} -9.00000 q^{83} -12.0000 q^{89} -6.92820 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 6 q^{11} - 8 q^{19} - 4 q^{25} - 6 q^{35} - 24 q^{41} - 8 q^{43} - 8 q^{49} + 8 q^{67} + 14 q^{73} - 18 q^{83} - 24 q^{89} + 14 q^{97}+O(q^{100}) $ 2 * q + 6 * q^11 - 8 * q^19 - 4 * q^25 - 6 * q^35 - 24 * q^41 - 8 * q^43 - 8 * q^49 + 8 * q^67 + 14 * q^73 - 18 * q^83 - 24 * q^89 + 14 * q^97

Display 100 coefficients 10 coefficients

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
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	−1.73205	−0.311086	−0.155543	−	0.987829i	$-0.549713\pi$
$31$	−1.73205	−0.311086	−0.155543	+	0.987829i	$0.549713\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.66025	1.18958	0.594789	−	0.803882i	$-0.297236\pi$
$53$	8.66025	1.18958	0.594789	+	0.803882i	$0.297236\pi$
$54$	0	0
$55$	5.19615	0.700649
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.8564	−1.77413	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	−13.8564	−1.77413	−0.887066	+	0.461644i	$0.847260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.19615	−0.592157
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.92820	−0.710819
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.66025	0.861727	0.430864	−	0.902417i	$-0.358209\pi$
$101$	8.66025	0.861727	0.430864	+	0.902417i	$0.358209\pi$
$102$	0	0
$103$	3.46410	0.341328	0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−19.0526	−1.69064	−0.845321	−	0.534259i	$-0.820591\pi$
$127$	−19.0526	−1.69064	−0.845321	+	0.534259i	$0.820591\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	6.92820	0.600751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.19615	0.425685	0.212843	−	0.977086i	$-0.431728\pi$
$149$	5.19615	0.425685	0.212843	+	0.977086i	$0.431728\pi$
$150$	0	0
$151$	−22.5167	−1.83238	−0.916190	−	0.400744i	$-0.868752\pi$
$151$	−22.5167	−1.83238	−0.916190	+	0.400744i	$0.868752\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	13.8564	1.10586	0.552931	−	0.833227i	$-0.313509\pi$
$157$	13.8564	1.10586	0.552931	+	0.833227i	$0.313509\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.66025	0.658427	0.329213	−	0.944256i	$-0.393216\pi$
$173$	8.66025	0.658427	0.329213	+	0.944256i	$0.393216\pi$
$174$	0	0
$175$	3.46410	0.261861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.8564	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.5885	−1.11063	−0.555316	−	0.831640i	$-0.687403\pi$
$197$	−15.5885	−1.11063	−0.555316	+	0.831640i	$0.687403\pi$
$198$	0	0
$199$	15.5885	1.10504	0.552518	−	0.833501i	$-0.313667\pi$
$199$	15.5885	1.10504	0.552518	+	0.833501i	$0.313667\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−20.7846	−1.45166
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.3205	1.15987	0.579934	−	0.814664i	$-0.303079\pi$
$223$	17.3205	1.15987	0.579934	+	0.814664i	$0.303079\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.92820	−0.442627
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	20.7846	1.30672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.7846	−1.28163	−0.640817	−	0.767694i	$-0.721404\pi$
$263$	−20.7846	−1.28163	−0.640817	+	0.767694i	$0.721404\pi$
$264$	0	0
$265$	15.0000	0.921443
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.7846	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$269$	20.7846	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$270$	0	0
$271$	1.73205	0.105215	0.0526073	−	0.998615i	$-0.483247\pi$
$271$	1.73205	0.105215	0.0526073	+	0.998615i	$0.483247\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	27.7128	1.66510	0.832551	−	0.553949i	$-0.186880\pi$
$277$	27.7128	1.66510	0.832551	+	0.553949i	$0.186880\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.7846	1.22688
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.92820	−0.404750	−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820	−0.404750	−0.202375	+	0.979308i	$0.564866\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.92820	−0.392862	−0.196431	−	0.980518i	$-0.562935\pi$
$311$	−6.92820	−0.392862	−0.196431	+	0.980518i	$0.562935\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5885	0.875535	0.437767	−	0.899088i	$-0.355769\pi$
$317$	15.5885	0.875535	0.437767	+	0.899088i	$0.355769\pi$
$318$	0	0
$319$	−20.7846	−1.16371
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.92820	0.378528
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.19615	−0.281387
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	6.92820	0.370858	0.185429	−	0.982658i	$-0.440632\pi$
$349$	6.92820	0.370858	0.185429	+	0.982658i	$0.440632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.92820	−0.365657	−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820	−0.365657	−0.182828	+	0.983145i	$0.558525\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.1244	0.634618
$366$	0	0
$367$	12.1244	0.632886	0.316443	−	0.948611i	$-0.397511\pi$
$367$	12.1244	0.632886	0.316443	+	0.948611i	$0.397511\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.0000	−0.778761
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	27.7128	1.41606	0.708029	−	0.706183i	$-0.249584\pi$
$383$	27.7128	1.41606	0.708029	+	0.706183i	$0.249584\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.5167	−1.14164	−0.570820	−	0.821075i	$-0.693375\pi$
$389$	−22.5167	−1.14164	−0.570820	+	0.821075i	$0.693375\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.0000	0.905678
$396$	0	0
$397$	13.8564	0.695433	0.347717	−	0.937600i	$-0.386957\pi$
$397$	13.8564	0.695433	0.347717	+	0.937600i	$0.386957\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.7846	−1.03025
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−15.5885	−0.765207
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	13.8564	0.675320	0.337660	−	0.941268i	$-0.390365\pi$
$421$	13.8564	0.675320	0.337660	+	0.941268i	$0.390365\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.0000	1.16144
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	0	0
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.7128	−1.32568
$438$	0	0
$439$	−25.9808	−1.23999	−0.619997	−	0.784604i	$-0.712866\pi$
$439$	−25.9808	−1.23999	−0.619997	+	0.784604i	$0.712866\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−20.7846	−0.985285
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.5885	−0.726027	−0.363013	−	0.931784i	$-0.618252\pi$
$461$	−15.5885	−0.726027	−0.363013	+	0.931784i	$0.618252\pi$
$462$	0	0
$463$	15.5885	0.724457	0.362229	−	0.932089i	$-0.382016\pi$
$463$	15.5885	0.724457	0.362229	+	0.932089i	$0.382016\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	−6.92820	−0.319915
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.1244	0.550539
$486$	0	0
$487$	−31.1769	−1.41276	−0.706380	−	0.707832i	$-0.749673\pi$
$487$	−31.1769	−1.41276	−0.706380	+	0.707832i	$0.749673\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.92820	0.308913	0.154457	−	0.988000i	$-0.450637\pi$
$503$	6.92820	0.308913	0.154457	+	0.988000i	$0.450637\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.9090	1.45866	0.729332	−	0.684160i	$-0.239831\pi$
$509$	32.9090	1.45866	0.729332	+	0.684160i	$0.239831\pi$
$510$	0	0
$511$	−12.1244	−0.536350
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	−20.7846	−0.914106
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	15.5885	0.673948
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−13.8564	−0.595733	−0.297867	−	0.954607i	$-0.596275\pi$
$541$	−13.8564	−0.595733	−0.297867	+	0.954607i	$0.596275\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	27.7128	1.18061
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.9808	1.10084	0.550420	−	0.834888i	$-0.314468\pi$
$557$	25.9808	1.10084	0.550420	+	0.834888i	$0.314468\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	0	0
$565$	−20.7846	−0.874415
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.8564	−0.577852
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.5885	0.646718
$582$	0	0
$583$	25.9808	1.07601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.92820	0.283079	0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820	0.283079	0.141539	+	0.989933i	$0.454795\pi$
$600$	0	0
$601$	1.00000	0.0407909	0.0203954	−	0.999792i	$-0.493507\pi$
$601$	1.00000	0.0407909	0.0203954	+	0.999792i	$0.493507\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.46410	−0.140836
$606$	0	0
$607$	17.3205	0.703018	0.351509	−	0.936185i	$-0.385669\pi$
$607$	17.3205	0.703018	0.351509	+	0.936185i	$0.385669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.92820	−0.279827	−0.139914	−	0.990164i	$-0.544683\pi$
$613$	−6.92820	−0.279827	−0.139914	+	0.990164i	$0.544683\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	20.7846	0.832718
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	12.1244	0.482663	0.241331	−	0.970443i	$-0.422416\pi$
$631$	12.1244	0.482663	0.241331	+	0.970443i	$0.422416\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−33.0000	−1.30957
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.7128	−1.08950	−0.544752	−	0.838597i	$-0.683376\pi$
$647$	−27.7128	−1.08950	−0.544752	+	0.838597i	$0.683376\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.7654	−1.83007	−0.915035	−	0.403374i	$-0.867837\pi$
$653$	−46.7654	−1.83007	−0.915035	+	0.403374i	$0.867837\pi$
$654$	0	0
$655$	−5.19615	−0.203030
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	−6.92820	−0.269476	−0.134738	−	0.990881i	$-0.543019\pi$
$661$	−6.92820	−0.269476	−0.134738	+	0.990881i	$0.543019\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−48.0000	−1.85857
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−41.5692	−1.60476
$672$	0	0
$673$	−47.0000	−1.81172	−0.905858	−	0.423581i	$-0.860773\pi$
$673$	−47.0000	−1.81172	−0.905858	+	0.423581i	$0.860773\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.92820	0.266272	0.133136	−	0.991098i	$-0.457495\pi$
$677$	6.92820	0.266272	0.133136	+	0.991098i	$0.457495\pi$
$678$	0	0
$679$	−12.1244	−0.465290
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	−20.7846	−0.794139
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.7128	1.05121
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.3731	−1.37379	−0.686896	−	0.726756i	$-0.741027\pi$
$701$	−36.3731	−1.37379	−0.686896	+	0.726756i	$0.741027\pi$
$702$	0	0
$703$	27.7128	1.04521
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.0000	−0.564133
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.8564	0.514614
$726$	0	0
$727$	25.9808	0.963573	0.481787	−	0.876289i	$-0.339988\pi$
$727$	25.9808	0.963573	0.481787	+	0.876289i	$0.339988\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−34.6410	−1.27950	−0.639748	−	0.768585i	$-0.720961\pi$
$733$	−34.6410	−1.27950	−0.639748	+	0.768585i	$0.720961\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.8564	0.508342	0.254171	−	0.967159i	$-0.418197\pi$
$743$	13.8564	0.508342	0.254171	+	0.967159i	$0.418197\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15.5885	−0.569590
$750$	0	0
$751$	−29.4449	−1.07446	−0.537229	−	0.843436i	$-0.680529\pi$
$751$	−29.4449	−1.07446	−0.537229	+	0.843436i	$0.680529\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−39.0000	−1.41936
$756$	0	0
$757$	48.4974	1.76267	0.881334	−	0.472493i	$-0.156646\pi$
$757$	48.4974	1.76267	0.881334	+	0.472493i	$0.156646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.92820	−0.249190	−0.124595	−	0.992208i	$-0.539763\pi$
$773$	−6.92820	−0.249190	−0.124595	+	0.992208i	$0.539763\pi$
$774$	0	0
$775$	3.46410	0.124434
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48.0000	1.71978
$780$	0	0
$781$	41.5692	1.48746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.66025	0.306762	0.153381	−	0.988167i	$-0.450984\pi$
$797$	8.66025	0.306762	0.153381	+	0.988167i	$0.450984\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	21.0000	0.741074
$804$	0	0
$805$	−20.7846	−0.732561
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−27.7128	−0.970737
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.4974	1.69257	0.846286	−	0.532729i	$-0.178834\pi$
$821$	48.4974	1.69257	0.846286	+	0.532729i	$0.178834\pi$
$822$	0	0
$823$	−36.3731	−1.26789	−0.633943	−	0.773380i	$-0.718565\pi$
$823$	−36.3731	−1.26789	−0.633943	+	0.773380i	$0.718565\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	−20.7846	−0.721879	−0.360940	−	0.932589i	$-0.617544\pi$
$829$	−20.7846	−0.721879	−0.360940	+	0.932589i	$0.617544\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.7128	0.956753	0.478376	−	0.878155i	$-0.341226\pi$
$839$	27.7128	0.956753	0.478376	+	0.878155i	$0.341226\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.5167	−0.774597
$846$	0	0
$847$	3.46410	0.119028
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−34.6410	−1.18609	−0.593043	−	0.805171i	$-0.702074\pi$
$853$	−34.6410	−1.18609	−0.593043	+	0.805171i	$0.702074\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−55.4256	−1.88671	−0.943355	−	0.331785i	$-0.892349\pi$
$863$	−55.4256	−1.88671	−0.943355	+	0.331785i	$0.892349\pi$
$864$	0	0
$865$	15.0000	0.510015
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.1769	1.05760
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21.0000	0.709930
$876$	0	0
$877$	−34.6410	−1.16974	−0.584872	−	0.811126i	$-0.698855\pi$
$877$	−34.6410	−1.16974	−0.584872	+	0.811126i	$0.698855\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.5692	−1.39576	−0.697879	−	0.716216i	$-0.745873\pi$
$887$	−41.5692	−1.39576	−0.697879	+	0.716216i	$0.745873\pi$
$888$	0	0
$889$	33.0000	1.10678
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.7128	0.927374
$894$	0	0
$895$	5.19615	0.173688
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.4974	−1.60679	−0.803396	−	0.595446i	$-0.796976\pi$
$911$	−48.4974	−1.60679	−0.803396	+	0.595446i	$0.796976\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.19615	0.171592
$918$	0	0
$919$	−36.3731	−1.19984	−0.599918	−	0.800061i	$-0.704800\pi$
$919$	−36.3731	−1.19984	−0.599918	+	0.800061i	$0.704800\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	13.8564	0.455596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	16.0000	0.524379
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.3013	−1.41158	−0.705791	−	0.708421i	$-0.749408\pi$
$941$	−43.3013	−1.41158	−0.705791	+	0.708421i	$0.749408\pi$
$942$	0	0
$943$	−83.1384	−2.70736
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.7846	0.671170
$960$	0	0
$961$	−28.0000	−0.903226
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.0526	0.613324
$966$	0	0
$967$	22.5167	0.724087	0.362043	−	0.932161i	$-0.382079\pi$
$967$	22.5167	0.724087	0.362043	+	0.932161i	$0.382079\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	0	0
$973$	−27.7128	−0.888432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27.7128	−0.881216
$990$	0	0
$991$	32.9090	1.04539	0.522694	−	0.852520i	$-0.324927\pi$
$991$	32.9090	1.04539	0.522694	+	0.852520i	$0.324927\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.0000	0.855958
$996$	0	0
$997$	−41.5692	−1.31651	−0.658255	−	0.752795i	$-0.728705\pi$
$997$	−41.5692	−1.31651	−0.658255	+	0.752795i	$0.728705\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6912.2.a.bo.1.2		2
3.2	odd	2		6912.2.a.bi.1.1		2
4.3	odd	2		6912.2.a.bj.1.2		2
8.3	odd	2	inner	6912.2.a.bo.1.1		2
8.5	even	2		6912.2.a.bj.1.1		2
12.11	even	2		6912.2.a.bp.1.1		2
16.3	odd	4		1728.2.d.g.865.1	yes	4
16.5	even	4		1728.2.d.g.865.4	yes	4
16.11	odd	4		1728.2.d.g.865.3	yes	4
16.13	even	4		1728.2.d.g.865.2	yes	4
24.5	odd	2		6912.2.a.bp.1.2		2
24.11	even	2		6912.2.a.bi.1.2		2
48.5	odd	4		1728.2.d.f.865.2	yes	4
48.11	even	4		1728.2.d.f.865.1	✓	4
48.29	odd	4		1728.2.d.f.865.4	yes	4
48.35	even	4		1728.2.d.f.865.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.2.d.f.865.1	✓	4	48.11	even	4
1728.2.d.f.865.2	yes	4	48.5	odd	4
1728.2.d.f.865.3	yes	4	48.35	even	4
1728.2.d.f.865.4	yes	4	48.29	odd	4
1728.2.d.g.865.1	yes	4	16.3	odd	4
1728.2.d.g.865.2	yes	4	16.13	even	4
1728.2.d.g.865.3	yes	4	16.11	odd	4
1728.2.d.g.865.4	yes	4	16.5	even	4
6912.2.a.bi.1.1		2	3.2	odd	2
6912.2.a.bi.1.2		2	24.11	even	2
6912.2.a.bj.1.1		2	8.5	even	2
6912.2.a.bj.1.2		2	4.3	odd	2
6912.2.a.bo.1.1		2	8.3	odd	2	inner
6912.2.a.bo.1.2		2	1.1	even	1	trivial
6912.2.a.bp.1.1		2	12.11	even	2
6912.2.a.bp.1.2		2	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6912 = 2^{8} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6912.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{5} -1.73205 q^{7} +O(q^{10})\)	\(q+1.73205 q^{5} -1.73205 q^{7} +3.00000 q^{11} -4.00000 q^{19} +6.92820 q^{23} -2.00000 q^{25} -6.92820 q^{29} -1.73205 q^{31} -3.00000 q^{35} -6.92820 q^{37} -12.0000 q^{41} -4.00000 q^{43} -6.92820 q^{47} -4.00000 q^{49} +8.66025 q^{53} +5.19615 q^{55} -13.8564 q^{61} +4.00000 q^{67} +13.8564 q^{71} +7.00000 q^{73} -5.19615 q^{77} +10.3923 q^{79} -9.00000 q^{83} -12.0000 q^{89} -6.92820 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 6 q^{11} - 8 q^{19} - 4 q^{25} - 6 q^{35} - 24 q^{41} - 8 q^{43} - 8 q^{49} + 8 q^{67} + 14 q^{73} - 18 q^{83} - 24 q^{89} + 14 q^{97}+O(q^{100}) \) 2 * q + 6 * q^11 - 8 * q^19 - 4 * q^25 - 6 * q^35 - 24 * q^41 - 8 * q^43 - 8 * q^49 + 8 * q^67 + 14 * q^73 - 18 * q^83 - 24 * q^89 + 14 * q^97

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