LMFDB - Embedded newform 6912.2.a.bq.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:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
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)

$f(q)$	$=$	$q+3.46410 q^{5} +1.73205 q^{7} +O(q^{10})$	$q+3.46410 q^{5} +1.73205 q^{7} +6.00000 q^{11} +5.19615 q^{13} -6.00000 q^{17} -5.00000 q^{19} -3.46410 q^{23} +7.00000 q^{25} +6.92820 q^{29} -3.46410 q^{31} +6.00000 q^{35} -1.73205 q^{37} +4.00000 q^{43} +3.46410 q^{47} -4.00000 q^{49} +6.92820 q^{53} +20.7846 q^{55} -6.00000 q^{59} +12.1244 q^{61} +18.0000 q^{65} +5.00000 q^{67} +13.8564 q^{71} +7.00000 q^{73} +10.3923 q^{77} -5.19615 q^{79} -20.7846 q^{85} -6.00000 q^{89} +9.00000 q^{91} -17.3205 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 12 q^{11} - 12 q^{17} - 10 q^{19} + 14 q^{25} + 12 q^{35} + 8 q^{43} - 8 q^{49} - 12 q^{59} + 36 q^{65} + 10 q^{67} + 14 q^{73} - 12 q^{89} + 18 q^{91} + 14 q^{97}+O(q^{100}) $ 2 * q + 12 * q^11 - 12 * q^17 - 10 * q^19 + 14 * q^25 + 12 * q^35 + 8 * q^43 - 8 * q^49 - 12 * q^59 + 36 * q^65 + 10 * q^67 + 14 * q^73 - 12 * q^89 + 18 * q^91 + 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	5.19615	1.44115	0.720577	−	0.693375i	$-0.243877\pi$
$13$	5.19615	1.44115	0.720577	+	0.693375i	$0.243877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−1.73205	−0.284747	−0.142374	−	0.989813i	$-0.545473\pi$
$37$	−1.73205	−0.284747	−0.142374	+	0.989813i	$0.545473\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	20.7846	2.80260
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	12.1244	1.55236	0.776182	−	0.630509i	$-0.217154\pi$
$61$	12.1244	1.55236	0.776182	+	0.630509i	$0.217154\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.0000	2.23263
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	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
$77$	10.3923	1.18431
$78$	0	0
$79$	−5.19615	−0.584613	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	−5.19615	−0.584613	−0.292306	+	0.956325i	$0.594423\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−20.7846	−2.25441
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−17.3205	−1.77705
$96$	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
$101$	6.92820	0.689382	0.344691	−	0.938716i	$-0.387984\pi$
$101$	6.92820	0.689382	0.344691	+	0.938716i	$0.387984\pi$
$102$	0	0
$103$	−8.66025	−0.853320	−0.426660	−	0.904412i	$-0.640310\pi$
$103$	−8.66025	−0.853320	−0.426660	+	0.904412i	$0.640310\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.3923	−0.952661
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−17.3205	−1.53695	−0.768473	−	0.639882i	$-0.778983\pi$
$127$	−17.3205	−1.53695	−0.768473	+	0.639882i	$0.778983\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−8.66025	−0.750939
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31.1769	2.60714
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	1.73205	0.140952	0.0704761	−	0.997513i	$-0.477548\pi$
$151$	1.73205	0.140952	0.0704761	+	0.997513i	$0.477548\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−12.0000	−0.963863
$156$	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
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	14.0000	1.07692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.8564	−1.05348	−0.526742	−	0.850026i	$-0.676586\pi$
$173$	−13.8564	−1.05348	−0.526742	+	0.850026i	$0.676586\pi$
$174$	0	0
$175$	12.1244	0.916515
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	12.1244	0.901196	0.450598	−	0.892727i	$-0.351211\pi$
$181$	12.1244	0.901196	0.450598	+	0.892727i	$0.351211\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−36.0000	−2.63258
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.2487	−1.75458	−0.877288	−	0.479965i	$-0.840649\pi$
$191$	−24.2487	−1.75458	−0.877288	+	0.479965i	$0.840649\pi$
$192$	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
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	5.19615	0.368345	0.184173	−	0.982894i	$-0.441039\pi$
$199$	5.19615	0.368345	0.184173	+	0.982894i	$0.441039\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.8564	0.944999
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−31.1769	−2.09719
$222$	0	0
$223$	3.46410	0.231973	0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410	0.231973	0.115987	+	0.993251i	$0.462997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.8564	−0.885253
$246$	0	0
$247$	−25.9808	−1.65312
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−20.7846	−1.30672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.7846	1.28163	0.640817	−	0.767694i	$-0.278596\pi$
$263$	20.7846	1.28163	0.640817	+	0.767694i	$0.278596\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	−22.5167	−1.36779	−0.683895	−	0.729581i	$-0.739715\pi$
$271$	−22.5167	−1.36779	−0.683895	+	0.729581i	$0.739715\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	42.0000	2.53270
$276$	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
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.2487	−1.41662	−0.708312	−	0.705899i	$-0.750543\pi$
$293$	−24.2487	−1.41662	−0.708312	+	0.705899i	$0.750543\pi$
$294$	0	0
$295$	−20.7846	−1.21013
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	42.0000	2.40491
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.3205	−0.982156	−0.491078	−	0.871116i	$-0.663397\pi$
$311$	−17.3205	−0.982156	−0.491078	+	0.871116i	$0.663397\pi$
$312$	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
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	41.5692	2.32743
$320$	0	0
$321$	0	0
$322$	0	0
$323$	30.0000	1.66924
$324$	0	0
$325$	36.3731	2.01761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	17.3205	0.946320
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.7846	−1.12555
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	1.73205	0.0927146	0.0463573	−	0.998925i	$-0.485239\pi$
$349$	1.73205	0.0927146	0.0463573	+	0.998925i	$0.485239\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	48.0000	2.54758
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.46410	0.182828	0.0914141	−	0.995813i	$-0.470861\pi$
$359$	3.46410	0.182828	0.0914141	+	0.995813i	$0.470861\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	24.2487	1.26924
$366$	0	0
$367$	29.4449	1.53701	0.768505	−	0.639844i	$-0.221001\pi$
$367$	29.4449	1.53701	0.768505	+	0.639844i	$0.221001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	−15.5885	−0.807140	−0.403570	−	0.914949i	$-0.632231\pi$
$373$	−15.5885	−0.807140	−0.403570	+	0.914949i	$0.632231\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.0000	1.85409
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.6410	−1.77007	−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410	−1.77007	−0.885037	+	0.465521i	$0.845867\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.46410	−0.175637	−0.0878185	−	0.996136i	$-0.527990\pi$
$389$	−3.46410	−0.175637	−0.0878185	+	0.996136i	$0.527990\pi$
$390$	0	0
$391$	20.7846	1.05112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.0000	−0.905678
$396$	0	0
$397$	−27.7128	−1.39087	−0.695433	−	0.718591i	$-0.744787\pi$
$397$	−27.7128	−1.39087	−0.695433	+	0.718591i	$0.744787\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	−18.0000	−0.896644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.3923	−0.515127
$408$	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
$413$	−10.3923	−0.511372
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−12.1244	−0.590905	−0.295452	−	0.955357i	$-0.595470\pi$
$421$	−12.1244	−0.590905	−0.295452	+	0.955357i	$0.595470\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−42.0000	−2.03730
$426$	0	0
$427$	21.0000	1.01626
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.46410	−0.166860	−0.0834300	−	0.996514i	$-0.526587\pi$
$431$	−3.46410	−0.166860	−0.0834300	+	0.996514i	$0.526587\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.3205	0.828552
$438$	0	0
$439$	−10.3923	−0.495998	−0.247999	−	0.968760i	$-0.579773\pi$
$439$	−10.3923	−0.495998	−0.247999	+	0.968760i	$0.579773\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	−20.7846	−0.985285
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	31.1769	1.46160
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.1769	−1.45205	−0.726027	−	0.687666i	$-0.758635\pi$
$461$	−31.1769	−1.45205	−0.726027	+	0.687666i	$0.758635\pi$
$462$	0	0
$463$	5.19615	0.241486	0.120743	−	0.992684i	$-0.461472\pi$
$463$	5.19615	0.241486	0.120743	+	0.992684i	$0.461472\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	8.66025	0.399893
$470$	0	0
$471$	0	0
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	−35.0000	−1.60591
$476$	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
$481$	−9.00000	−0.410365
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.2487	1.10108
$486$	0	0
$487$	−5.19615	−0.235460	−0.117730	−	0.993046i	$-0.537562\pi$
$487$	−5.19615	−0.235460	−0.117730	+	0.993046i	$0.537562\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	−41.5692	−1.87218
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.2487	−1.08120	−0.540598	−	0.841281i	$-0.681802\pi$
$503$	−24.2487	−1.08120	−0.540598	+	0.841281i	$0.681802\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.3205	−0.767718	−0.383859	−	0.923392i	$-0.625405\pi$
$509$	−17.3205	−0.767718	−0.383859	+	0.923392i	$0.625405\pi$
$510$	0	0
$511$	12.1244	0.536350
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−30.0000	−1.32196
$516$	0	0
$517$	20.7846	0.914106
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.7846	0.905392
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−20.7846	−0.898597
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24.0000	−1.03375
$540$	0	0
$541$	12.1244	0.521267	0.260633	−	0.965438i	$-0.416069\pi$
$541$	12.1244	0.521267	0.260633	+	0.965438i	$0.416069\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	1.00000	0.0427569	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000	0.0427569	0.0213785	+	0.999771i	$0.493195\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.6410	−1.47576
$552$	0	0
$553$	−9.00000	−0.382719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.3923	−0.440336	−0.220168	−	0.975462i	$-0.570661\pi$
$557$	−10.3923	−0.440336	−0.220168	+	0.975462i	$0.570661\pi$
$558$	0	0
$559$	20.7846	0.879095
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−20.7846	−0.874415
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.2487	−1.01124
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	41.5692	1.72162
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	17.3205	0.713679
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	−36.0000	−1.47586
$596$	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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	86.6025	3.52089
$606$	0	0
$607$	−1.73205	−0.0703018	−0.0351509	−	0.999382i	$-0.511191\pi$
$607$	−1.73205	−0.0703018	−0.0351509	+	0.999382i	$0.511191\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.0000	0.728202
$612$	0	0
$613$	−32.9090	−1.32918	−0.664590	−	0.747208i	$-0.731394\pi$
$613$	−32.9090	−1.32918	−0.664590	+	0.747208i	$0.731394\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.3923	−0.416359
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.3923	0.414368
$630$	0	0
$631$	29.4449	1.17218	0.586091	−	0.810245i	$-0.300666\pi$
$631$	29.4449	1.17218	0.586091	+	0.810245i	$0.300666\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−60.0000	−2.38103
$636$	0	0
$637$	−20.7846	−0.823516
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.8564	0.544752	0.272376	−	0.962191i	$-0.412191\pi$
$647$	13.8564	0.544752	0.272376	+	0.962191i	$0.412191\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.5692	1.62673	0.813365	−	0.581754i	$-0.197633\pi$
$653$	41.5692	1.62673	0.813365	+	0.581754i	$0.197633\pi$
$654$	0	0
$655$	−41.5692	−1.62424
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	8.66025	0.336845	0.168422	−	0.985715i	$-0.446133\pi$
$661$	8.66025	0.336845	0.168422	+	0.985715i	$0.446133\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−30.0000	−1.16335
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	72.7461	2.80833
$672$	0	0
$673$	25.0000	0.963679	0.481840	−	0.876259i	$-0.339969\pi$
$673$	25.0000	0.963679	0.481840	+	0.876259i	$0.339969\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.2487	0.931954	0.465977	−	0.884797i	$-0.345703\pi$
$677$	24.2487	0.931954	0.465977	+	0.884797i	$0.345703\pi$
$678$	0	0
$679$	12.1244	0.465290
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	62.3538	2.38242
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	38.1051	1.44541
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.1769	1.17754	0.588768	−	0.808302i	$-0.299613\pi$
$701$	31.1769	1.17754	0.588768	+	0.808302i	$0.299613\pi$
$702$	0	0
$703$	8.66025	0.326628
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000	0.451306
$708$	0	0
$709$	−5.19615	−0.195146	−0.0975728	−	0.995228i	$-0.531108\pi$
$709$	−5.19615	−0.195146	−0.0975728	+	0.995228i	$0.531108\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	108.000	4.03897
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.5692	−1.55027	−0.775135	−	0.631795i	$-0.782318\pi$
$719$	−41.5692	−1.55027	−0.775135	+	0.631795i	$0.782318\pi$
$720$	0	0
$721$	−15.0000	−0.558629
$722$	0	0
$723$	0	0
$724$	0	0
$725$	48.4974	1.80115
$726$	0	0
$727$	−31.1769	−1.15629	−0.578144	−	0.815935i	$-0.696223\pi$
$727$	−31.1769	−1.15629	−0.578144	+	0.815935i	$0.696223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−13.8564	−0.511798	−0.255899	−	0.966704i	$-0.582371\pi$
$733$	−13.8564	−0.511798	−0.255899	+	0.966704i	$0.582371\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.46410	0.127086	0.0635428	−	0.997979i	$-0.479760\pi$
$743$	3.46410	0.127086	0.0635428	+	0.997979i	$0.479760\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.3923	−0.379727
$750$	0	0
$751$	−53.6936	−1.95931	−0.979653	−	0.200698i	$-0.935679\pi$
$751$	−53.6936	−1.95931	−0.979653	+	0.200698i	$0.935679\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.00000	0.218362
$756$	0	0
$757$	22.5167	0.818382	0.409191	−	0.912449i	$-0.365811\pi$
$757$	22.5167	0.818382	0.409191	+	0.912449i	$0.365811\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.1769	−1.12573
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.7128	0.996761	0.498380	−	0.866959i	$-0.333928\pi$
$773$	27.7128	0.996761	0.498380	+	0.866959i	$0.333928\pi$
$774$	0	0
$775$	−24.2487	−0.871039
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	83.1384	2.97493
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.0000	1.71319
$786$	0	0
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	63.0000	2.23720
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34.6410	−1.22705	−0.613524	−	0.789676i	$-0.710249\pi$
$797$	−34.6410	−1.22705	−0.613524	+	0.789676i	$0.710249\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	0	0
$801$	0	0
$802$	0	0
$803$	42.0000	1.48215
$804$	0	0
$805$	−20.7846	−0.732561
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−38.1051	−1.33476
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.3205	−0.604490	−0.302245	−	0.953230i	$-0.597736\pi$
$821$	−17.3205	−0.604490	−0.302245	+	0.953230i	$0.597736\pi$
$822$	0	0
$823$	36.3731	1.26789	0.633943	−	0.773380i	$-0.281435\pi$
$823$	36.3731	1.26789	0.633943	+	0.773380i	$0.281435\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	0	0
$829$	36.3731	1.26329	0.631644	−	0.775258i	$-0.282380\pi$
$829$	36.3731	1.26329	0.631644	+	0.775258i	$0.282380\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	−60.0000	−2.07639
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.92820	0.239188	0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820	0.239188	0.119594	+	0.992823i	$0.461841\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	48.4974	1.66836
$846$	0	0
$847$	43.3013	1.48785
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−8.66025	−0.296521	−0.148261	−	0.988948i	$-0.547367\pi$
$853$	−8.66025	−0.296521	−0.148261	+	0.988948i	$0.547367\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.3205	0.589597	0.294798	−	0.955559i	$-0.404747\pi$
$863$	17.3205	0.589597	0.294798	+	0.955559i	$0.404747\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−31.1769	−1.05760
$870$	0	0
$871$	25.9808	0.880325
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−19.0526	−0.643359	−0.321680	−	0.946849i	$-0.604247\pi$
$877$	−19.0526	−0.643359	−0.321680	+	0.946849i	$0.604247\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.5692	1.39576	0.697879	−	0.716216i	$-0.254127\pi$
$887$	41.5692	1.39576	0.697879	+	0.716216i	$0.254127\pi$
$888$	0	0
$889$	−30.0000	−1.00617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17.3205	−0.579609
$894$	0	0
$895$	−83.1384	−2.77901
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−41.5692	−1.38487
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.0000	1.39613
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.8564	0.459083	0.229542	−	0.973299i	$-0.426277\pi$
$911$	13.8564	0.459083	0.229542	+	0.973299i	$0.426277\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.7846	−0.686368
$918$	0	0
$919$	51.9615	1.71405	0.857026	−	0.515273i	$-0.172309\pi$
$919$	51.9615	1.71405	0.857026	+	0.515273i	$0.172309\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	72.0000	2.36991
$924$	0	0
$925$	−12.1244	−0.398646
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	0	0
$931$	20.0000	0.655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−124.708	−4.07838
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.1051	1.24219	0.621096	−	0.783735i	$-0.286688\pi$
$941$	38.1051	1.24219	0.621096	+	0.783735i	$0.286688\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	36.3731	1.18072
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	−84.0000	−2.71818
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.1769	1.00676
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	38.1051	1.22665
$966$	0	0
$967$	−22.5167	−0.724087	−0.362043	−	0.932161i	$-0.617921\pi$
$967$	−22.5167	−0.724087	−0.362043	+	0.932161i	$0.617921\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	19.0526	0.610797
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	10.3923	0.331463	0.165732	−	0.986171i	$-0.447001\pi$
$983$	10.3923	0.331463	0.165732	+	0.986171i	$0.447001\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.8564	−0.440608
$990$	0	0
$991$	−12.1244	−0.385143	−0.192571	−	0.981283i	$-0.561683\pi$
$991$	−12.1244	−0.385143	−0.192571	+	0.981283i	$0.561683\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.0000	0.570638
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6912.2.a.bq.1.2		2
3.2	odd	2		6912.2.a.bh.1.1		2
4.3	odd	2		6912.2.a.bg.1.2		2
8.3	odd	2	inner	6912.2.a.bq.1.1		2
8.5	even	2		6912.2.a.bg.1.1		2
12.11	even	2		6912.2.a.br.1.1		2
16.3	odd	4		1728.2.d.e.865.2	yes	4
16.5	even	4		1728.2.d.e.865.3	yes	4
16.11	odd	4		1728.2.d.e.865.4	yes	4
16.13	even	4		1728.2.d.e.865.1	✓	4
24.5	odd	2		6912.2.a.br.1.2		2
24.11	even	2		6912.2.a.bh.1.2		2
48.5	odd	4		1728.2.d.j.865.1	yes	4
48.11	even	4		1728.2.d.j.865.2	yes	4
48.29	odd	4		1728.2.d.j.865.3	yes	4
48.35	even	4		1728.2.d.j.865.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.2.d.e.865.1	✓	4	16.13	even	4
1728.2.d.e.865.2	yes	4	16.3	odd	4
1728.2.d.e.865.3	yes	4	16.5	even	4
1728.2.d.e.865.4	yes	4	16.11	odd	4
1728.2.d.j.865.1	yes	4	48.5	odd	4
1728.2.d.j.865.2	yes	4	48.11	even	4
1728.2.d.j.865.3	yes	4	48.29	odd	4
1728.2.d.j.865.4	yes	4	48.35	even	4
6912.2.a.bg.1.1		2	8.5	even	2
6912.2.a.bg.1.2		2	4.3	odd	2
6912.2.a.bh.1.1		2	3.2	odd	2
6912.2.a.bh.1.2		2	24.11	even	2
6912.2.a.bq.1.1		2	8.3	odd	2	inner
6912.2.a.bq.1.2		2	1.1	even	1	trivial
6912.2.a.br.1.1		2	12.11	even	2
6912.2.a.br.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)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} +1.73205 q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} +1.73205 q^{7} +6.00000 q^{11} +5.19615 q^{13} -6.00000 q^{17} -5.00000 q^{19} -3.46410 q^{23} +7.00000 q^{25} +6.92820 q^{29} -3.46410 q^{31} +6.00000 q^{35} -1.73205 q^{37} +4.00000 q^{43} +3.46410 q^{47} -4.00000 q^{49} +6.92820 q^{53} +20.7846 q^{55} -6.00000 q^{59} +12.1244 q^{61} +18.0000 q^{65} +5.00000 q^{67} +13.8564 q^{71} +7.00000 q^{73} +10.3923 q^{77} -5.19615 q^{79} -20.7846 q^{85} -6.00000 q^{89} +9.00000 q^{91} -17.3205 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 12 q^{11} - 12 q^{17} - 10 q^{19} + 14 q^{25} + 12 q^{35} + 8 q^{43} - 8 q^{49} - 12 q^{59} + 36 q^{65} + 10 q^{67} + 14 q^{73} - 12 q^{89} + 18 q^{91} + 14 q^{97}+O(q^{100}) \) 2 * q + 12 * q^11 - 12 * q^17 - 10 * q^19 + 14 * q^25 + 12 * q^35 + 8 * q^43 - 8 * q^49 - 12 * q^59 + 36 * q^65 + 10 * q^67 + 14 * q^73 - 12 * q^89 + 18 * q^91 + 14 * q^97

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