LMFDB - Embedded newform 6912.2.a.bc.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(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.64575$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{11} +5.29150 q^{13} +5.29150 q^{17} -5.29150 q^{19} -5.29150 q^{23} -4.00000 q^{25} +6.00000 q^{29} -7.00000 q^{31} +1.00000 q^{35} -5.29150 q^{37} +5.29150 q^{41} +10.5830 q^{43} -6.00000 q^{49} +9.00000 q^{53} +3.00000 q^{55} +4.00000 q^{59} -5.29150 q^{65} +15.8745 q^{71} -3.00000 q^{73} +3.00000 q^{77} +4.00000 q^{79} -7.00000 q^{83} -5.29150 q^{85} -10.5830 q^{89} -5.29150 q^{91} +5.29150 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} - 6 q^{11} - 8 q^{25} + 12 q^{29} - 14 q^{31} + 2 q^{35} - 12 q^{49} + 18 q^{53} + 6 q^{55} + 8 q^{59} - 6 q^{73} + 6 q^{77} + 8 q^{79} - 14 q^{83} + 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^11 - 8 * q^25 + 12 * q^29 - 14 * q^31 + 2 * q^35 - 12 * q^49 + 18 * q^53 + 6 * q^55 + 8 * q^59 - 6 * q^73 + 6 * q^77 + 8 * q^79 - 14 * q^83 + 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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.29150	1.28338	0.641689	−	0.766965i	$-0.278234\pi$
$17$	5.29150	1.28338	0.641689	+	0.766965i	$0.278234\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.29150	−1.10335	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	−5.29150	−1.10335	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−5.29150	−0.869918	−0.434959	−	0.900450i	$-0.643237\pi$
$37$	−5.29150	−0.869918	−0.434959	+	0.900450i	$0.643237\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.29150	0.826394	0.413197	−	0.910642i	$-0.364412\pi$
$41$	5.29150	0.826394	0.413197	+	0.910642i	$0.364412\pi$
$42$	0	0
$43$	10.5830	1.61389	0.806947	−	0.590624i	$-0.201119\pi$
$43$	10.5830	1.61389	0.806947	+	0.590624i	$0.201119\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.29150	−0.656330
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.8745	1.88396	0.941979	−	0.335673i	$-0.108964\pi$
$71$	15.8745	1.88396	0.941979	+	0.335673i	$0.108964\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	−5.29150	−0.573944
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.5830	−1.12180	−0.560898	−	0.827885i	$-0.689544\pi$
$89$	−10.5830	−1.12180	−0.560898	+	0.827885i	$0.689544\pi$
$90$	0	0
$91$	−5.29150	−0.554700
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.29150	0.542897
$96$	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
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.8745	−1.49335	−0.746674	−	0.665190i	$-0.768350\pi$
$113$	−15.8745	−1.49335	−0.746674	+	0.665190i	$0.768350\pi$
$114$	0	0
$115$	5.29150	0.493435
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.29150	−0.485071
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	5.29150	0.458831
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−10.5830	−0.897639	−0.448819	−	0.893622i	$-0.648155\pi$
$139$	−10.5830	−0.897639	−0.448819	+	0.893622i	$0.648155\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.8745	−1.32749
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.00000	0.562254
$156$	0	0
$157$	−10.5830	−0.844616	−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830	−0.844616	−0.422308	+	0.906452i	$0.638780\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.29150	0.417029
$162$	0	0
$163$	5.29150	0.414462	0.207231	−	0.978292i	$-0.433555\pi$
$163$	5.29150	0.414462	0.207231	+	0.978292i	$0.433555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.5830	−0.818938	−0.409469	−	0.912324i	$-0.634286\pi$
$167$	−10.5830	−0.818938	−0.409469	+	0.912324i	$0.634286\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.0000	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$179$	23.0000	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$180$	0	0
$181$	21.1660	1.57326	0.786629	−	0.617426i	$-0.211825\pi$
$181$	21.1660	1.57326	0.786629	+	0.617426i	$0.211825\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.29150	0.389039
$186$	0	0
$187$	−15.8745	−1.16086
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.29150	0.382880	0.191440	−	0.981504i	$-0.438684\pi$
$191$	5.29150	0.382880	0.191440	+	0.981504i	$0.438684\pi$
$192$	0	0
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.0000	0.926212	0.463106	−	0.886303i	$-0.346735\pi$
$197$	13.0000	0.926212	0.463106	+	0.886303i	$0.346735\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−5.29150	−0.369575
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.8745	1.09806
$210$	0	0
$211$	5.29150	0.364282	0.182141	−	0.983272i	$-0.441697\pi$
$211$	5.29150	0.364282	0.182141	+	0.983272i	$0.441697\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.5830	−0.721755
$216$	0	0
$217$	7.00000	0.475191
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28.0000	1.88348
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	21.1660	1.39869	0.699345	−	0.714785i	$-0.253475\pi$
$229$	21.1660	1.39869	0.699345	+	0.714785i	$0.253475\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.5830	−0.693316	−0.346658	−	0.937992i	$-0.612684\pi$
$233$	−10.5830	−0.693316	−0.346658	+	0.937992i	$0.612684\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8745	1.02684	0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745	1.02684	0.513418	+	0.858138i	$0.328379\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−28.0000	−1.78160
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	15.8745	0.998022
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.8745	0.990225	0.495112	−	0.868829i	$-0.335127\pi$
$257$	15.8745	0.990225	0.495112	+	0.868829i	$0.335127\pi$
$258$	0	0
$259$	5.29150	0.328798
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.29150	0.326288	0.163144	−	0.986602i	$-0.447836\pi$
$263$	5.29150	0.326288	0.163144	+	0.986602i	$0.447836\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−10.5830	−0.635871	−0.317936	−	0.948112i	$-0.602990\pi$
$277$	−10.5830	−0.635871	−0.317936	+	0.948112i	$0.602990\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.8745	0.946994	0.473497	−	0.880795i	$-0.342992\pi$
$281$	15.8745	0.946994	0.473497	+	0.880795i	$0.342992\pi$
$282$	0	0
$283$	26.4575	1.57274	0.786368	−	0.617758i	$-0.211959\pi$
$283$	26.4575	1.57274	0.786368	+	0.617758i	$0.211959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.29150	−0.312348
$288$	0	0
$289$	11.0000	0.647059
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−28.0000	−1.61928
$300$	0	0
$301$	−10.5830	−0.609994
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.8745	0.906006	0.453003	−	0.891509i	$-0.350353\pi$
$307$	15.8745	0.906006	0.453003	+	0.891509i	$0.350353\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.4575	1.50027	0.750134	−	0.661286i	$-0.229989\pi$
$311$	26.4575	1.50027	0.750134	+	0.661286i	$0.229989\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.0000	−1.55796
$324$	0	0
$325$	−21.1660	−1.17408
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.1660	−1.16339	−0.581695	−	0.813407i	$-0.697610\pi$
$331$	−21.1660	−1.16339	−0.581695	+	0.813407i	$0.697610\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.0000	1.13721
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	0	0
$349$	5.29150	0.283248	0.141624	−	0.989921i	$-0.454768\pi$
$349$	5.29150	0.283248	0.141624	+	0.989921i	$0.454768\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.8745	−0.844915	−0.422457	−	0.906383i	$-0.638832\pi$
$353$	−15.8745	−0.844915	−0.422457	+	0.906383i	$0.638832\pi$
$354$	0	0
$355$	−15.8745	−0.842531
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.1660	−1.11710	−0.558550	−	0.829471i	$-0.688642\pi$
$359$	−21.1660	−1.11710	−0.558550	+	0.829471i	$0.688642\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.00000	0.157027
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.00000	−0.467257
$372$	0	0
$373$	−26.4575	−1.36992	−0.684959	−	0.728582i	$-0.740180\pi$
$373$	−26.4575	−1.36992	−0.684959	+	0.728582i	$0.740180\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.7490	1.63516
$378$	0	0
$379$	21.1660	1.08722	0.543612	−	0.839336i	$-0.317056\pi$
$379$	21.1660	1.08722	0.543612	+	0.839336i	$0.317056\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.8745	−0.811149	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	−15.8745	−0.811149	−0.405575	+	0.914062i	$0.632929\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−5.29150	−0.265573	−0.132786	−	0.991145i	$-0.542392\pi$
$397$	−5.29150	−0.265573	−0.132786	+	0.991145i	$0.542392\pi$
$398$	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
$403$	−37.0405	−1.84512
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.8745	0.786870
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	7.00000	0.343616
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−10.5830	−0.515784	−0.257892	−	0.966174i	$-0.583028\pi$
$421$	−10.5830	−0.515784	−0.257892	+	0.966174i	$0.583028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−21.1660	−1.02670
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.7490	1.52930	0.764648	−	0.644448i	$-0.222913\pi$
$431$	31.7490	1.52930	0.764648	+	0.644448i	$0.222913\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.0000	1.33942
$438$	0	0
$439$	21.0000	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$439$	21.0000	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	10.5830	0.501683
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.8745	−0.749164	−0.374582	−	0.927194i	$-0.622214\pi$
$449$	−15.8745	−0.749164	−0.374582	+	0.927194i	$0.622214\pi$
$450$	0	0
$451$	−15.8745	−0.747501
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.29150	0.248069
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.00000	0.0462745	0.0231372	−	0.999732i	$-0.492635\pi$
$467$	1.00000	0.0462745	0.0231372	+	0.999732i	$0.492635\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−31.7490	−1.45982
$474$	0	0
$475$	21.1660	0.971163
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.1660	−0.967100	−0.483550	−	0.875317i	$-0.660653\pi$
$479$	−21.1660	−0.967100	−0.483550	+	0.875317i	$0.660653\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.0000	−1.30875	−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000	−1.30875	−0.654376	+	0.756169i	$0.727069\pi$
$492$	0	0
$493$	31.7490	1.42990
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.8745	−0.712069
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.8745	0.707809	0.353905	−	0.935282i	$-0.384854\pi$
$503$	15.8745	0.707809	0.353905	+	0.935282i	$0.384854\pi$
$504$	0	0
$505$	−17.0000	−0.756490
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.8745	0.695475	0.347737	−	0.937592i	$-0.386950\pi$
$521$	15.8745	0.695475	0.347737	+	0.937592i	$0.386950\pi$
$522$	0	0
$523$	15.8745	0.694144	0.347072	−	0.937839i	$-0.387176\pi$
$523$	15.8745	0.694144	0.347072	+	0.937839i	$0.387176\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.0405	−1.61351
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.0000	1.21281
$534$	0	0
$535$	3.00000	0.129701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−10.5830	−0.454999	−0.227499	−	0.973778i	$-0.573055\pi$
$541$	−10.5830	−0.454999	−0.227499	+	0.973778i	$0.573055\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.29150	0.226663
$546$	0	0
$547$	−5.29150	−0.226248	−0.113124	−	0.993581i	$-0.536086\pi$
$547$	−5.29150	−0.226248	−0.113124	+	0.993581i	$0.536086\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.7490	−1.35255
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	0	0
$559$	56.0000	2.36855
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	15.8745	0.667846
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.7490	−1.33099	−0.665494	−	0.746403i	$-0.731779\pi$
$569$	−31.7490	−1.33099	−0.665494	+	0.746403i	$0.731779\pi$
$570$	0	0
$571$	−26.4575	−1.10721	−0.553606	−	0.832779i	$-0.686749\pi$
$571$	−26.4575	−1.10721	−0.553606	+	0.832779i	$0.686749\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.1660	0.882684
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.00000	0.290409
$582$	0	0
$583$	−27.0000	−1.11823
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.0000	−0.949312	−0.474656	−	0.880172i	$-0.657427\pi$
$587$	−23.0000	−0.949312	−0.474656	+	0.880172i	$0.657427\pi$
$588$	0	0
$589$	37.0405	1.52623
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.7490	−1.30378	−0.651888	−	0.758315i	$-0.726023\pi$
$593$	−31.7490	−1.30378	−0.651888	+	0.758315i	$0.726023\pi$
$594$	0	0
$595$	5.29150	0.216930
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.1660	−0.864820	−0.432410	−	0.901677i	$-0.642337\pi$
$599$	−21.1660	−0.864820	−0.432410	+	0.901677i	$0.642337\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−10.5830	−0.427444	−0.213722	−	0.976895i	$-0.568559\pi$
$613$	−10.5830	−0.427444	−0.213722	+	0.976895i	$0.568559\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.29150	−0.213028	−0.106514	−	0.994311i	$-0.533969\pi$
$617$	−5.29150	−0.213028	−0.106514	+	0.994311i	$0.533969\pi$
$618$	0	0
$619$	15.8745	0.638050	0.319025	−	0.947746i	$-0.396645\pi$
$619$	15.8745	0.638050	0.319025	+	0.947746i	$0.396645\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.5830	0.423999
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.0000	−1.11643
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.0000	−0.515889
$636$	0	0
$637$	−31.7490	−1.25794
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.0405	1.46301	0.731506	−	0.681835i	$-0.238818\pi$
$641$	37.0405	1.46301	0.731506	+	0.681835i	$0.238818\pi$
$642$	0	0
$643$	21.1660	0.834706	0.417353	−	0.908744i	$-0.362958\pi$
$643$	21.1660	0.834706	0.417353	+	0.908744i	$0.362958\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.7490	−1.24818	−0.624091	−	0.781351i	$-0.714531\pi$
$647$	−31.7490	−1.24818	−0.624091	+	0.781351i	$0.714531\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	7.00000	0.273513
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	26.4575	1.02908	0.514539	−	0.857467i	$-0.327963\pi$
$661$	26.4575	1.02908	0.514539	+	0.857467i	$0.327963\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.29150	−0.205196
$666$	0	0
$667$	−31.7490	−1.22933
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	47.6235	1.81431
$690$	0	0
$691$	42.3320	1.61039	0.805193	−	0.593013i	$-0.202062\pi$
$691$	42.3320	1.61039	0.805193	+	0.593013i	$0.202062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.5830	0.401436
$696$	0	0
$697$	28.0000	1.06058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.00000	0.0377695	0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000	0.0377695	0.0188847	+	0.999822i	$0.493988\pi$
$702$	0	0
$703$	28.0000	1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.0000	−0.639351
$708$	0	0
$709$	21.1660	0.794906	0.397453	−	0.917622i	$-0.369894\pi$
$709$	21.1660	0.794906	0.397453	+	0.917622i	$0.369894\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37.0405	1.38718
$714$	0	0
$715$	15.8745	0.593673
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.1660	0.789359	0.394679	−	0.918819i	$-0.370856\pi$
$719$	21.1660	0.789359	0.394679	+	0.918819i	$0.370856\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−33.0000	−1.22390	−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000	−1.22390	−0.611951	+	0.790896i	$0.709615\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	56.0000	2.07123
$732$	0	0
$733$	5.29150	0.195446	0.0977231	−	0.995214i	$-0.468844\pi$
$733$	5.29150	0.195446	0.0977231	+	0.995214i	$0.468844\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	5.29150	0.194651	0.0973255	−	0.995253i	$-0.468971\pi$
$739$	5.29150	0.194651	0.0973255	+	0.995253i	$0.468971\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.0405	1.35888	0.679442	−	0.733729i	$-0.262222\pi$
$743$	37.0405	1.35888	0.679442	+	0.733729i	$0.262222\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.00000	0.109618
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.00000	0.109181
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.5830	0.383634	0.191817	−	0.981431i	$-0.438562\pi$
$761$	10.5830	0.383634	0.191817	+	0.981431i	$0.438562\pi$
$762$	0	0
$763$	5.29150	0.191565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.1660	0.764260
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.0000	−1.00320
$780$	0	0
$781$	−47.6235	−1.70410
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.5830	0.377724
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.8745	0.564433
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.0000	−1.73567	−0.867835	−	0.496853i	$-0.834489\pi$
$797$	−49.0000	−1.73567	−0.867835	+	0.496853i	$0.834489\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.00000	0.317603
$804$	0	0
$805$	−5.29150	−0.186501
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.7490	1.11624	0.558118	−	0.829762i	$-0.311524\pi$
$809$	31.7490	1.11624	0.558118	+	0.829762i	$0.311524\pi$
$810$	0	0
$811$	−37.0405	−1.30067	−0.650334	−	0.759648i	$-0.725371\pi$
$811$	−37.0405	−1.30067	−0.650334	+	0.759648i	$0.725371\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.29150	−0.185353
$816$	0	0
$817$	−56.0000	−1.95919
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−21.1660	−0.735126	−0.367563	−	0.929999i	$-0.619808\pi$
$829$	−21.1660	−0.735126	−0.367563	+	0.929999i	$0.619808\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−31.7490	−1.10004
$834$	0	0
$835$	10.5830	0.366240
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.0405	1.27878	0.639390	−	0.768882i	$-0.279187\pi$
$839$	37.0405	1.27878	0.639390	+	0.768882i	$0.279187\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15.0000	−0.516016
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.0000	0.959828
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.8745	0.542263	0.271131	−	0.962542i	$-0.412602\pi$
$857$	15.8745	0.542263	0.271131	+	0.962542i	$0.412602\pi$
$858$	0	0
$859$	−52.9150	−1.80544	−0.902719	−	0.430231i	$-0.858432\pi$
$859$	−52.9150	−1.80544	−0.902719	+	0.430231i	$0.858432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−52.9150	−1.78681	−0.893407	−	0.449249i	$-0.851692\pi$
$877$	−52.9150	−1.78681	−0.893407	+	0.449249i	$0.851692\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.4575	0.891376	0.445688	−	0.895188i	$-0.352959\pi$
$881$	26.4575	0.891376	0.445688	+	0.895188i	$0.352959\pi$
$882$	0	0
$883$	−58.2065	−1.95881	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	−58.2065	−1.95881	−0.979403	+	0.201916i	$0.935283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.8745	−0.533014	−0.266507	−	0.963833i	$-0.585870\pi$
$887$	−15.8745	−0.533014	−0.266507	+	0.963833i	$0.585870\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−23.0000	−0.768805
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−42.0000	−1.40078
$900$	0	0
$901$	47.6235	1.58657
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1660	−0.703582
$906$	0	0
$907$	5.29150	0.175701	0.0878507	−	0.996134i	$-0.472000\pi$
$907$	5.29150	0.175701	0.0878507	+	0.996134i	$0.472000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.7490	−1.05189	−0.525946	−	0.850518i	$-0.676289\pi$
$911$	−31.7490	−1.05189	−0.525946	+	0.850518i	$0.676289\pi$
$912$	0	0
$913$	21.0000	0.694999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.00000	0.231160
$918$	0	0
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	84.0000	2.76489
$924$	0	0
$925$	21.1660	0.695934
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.5830	−0.347217	−0.173609	−	0.984815i	$-0.555543\pi$
$929$	−10.5830	−0.347217	−0.173609	+	0.984815i	$0.555543\pi$
$930$	0	0
$931$	31.7490	1.04053
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.8745	0.519152
$936$	0	0
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	0	0
$943$	−28.0000	−0.911805
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.0000	1.26733	0.633665	−	0.773608i	$-0.281550\pi$
$947$	39.0000	1.26733	0.633665	+	0.773608i	$0.281550\pi$
$948$	0	0
$949$	−15.8745	−0.515308
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.7490	−1.02845	−0.514226	−	0.857655i	$-0.671921\pi$
$953$	−31.7490	−1.02845	−0.514226	+	0.857655i	$0.671921\pi$
$954$	0	0
$955$	−5.29150	−0.171229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.00000	0.0965734
$966$	0	0
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	10.5830	0.339276
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.5830	−0.338580	−0.169290	−	0.985566i	$-0.554148\pi$
$977$	−10.5830	−0.338580	−0.169290	+	0.985566i	$0.554148\pi$
$978$	0	0
$979$	31.7490	1.01470
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.3320	−1.35018	−0.675091	−	0.737735i	$-0.735896\pi$
$983$	−42.3320	−1.35018	−0.675091	+	0.737735i	$0.735896\pi$
$984$	0	0
$985$	−13.0000	−0.414214
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−56.0000	−1.78070
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.0000	−0.538936
$996$	0	0
$997$	−15.8745	−0.502751	−0.251375	−	0.967890i	$-0.580883\pi$
$997$	−15.8745	−0.502751	−0.251375	+	0.967890i	$0.580883\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6912.2.a.bc.1.2		2
3.2	odd	2		6912.2.a.bu.1.2		2
4.3	odd	2		6912.2.a.bd.1.2		2
8.3	odd	2		6912.2.a.bv.1.1		2
8.5	even	2		6912.2.a.bu.1.1		2
12.11	even	2		6912.2.a.bv.1.2		2
16.3	odd	4		864.2.d.a.433.4		4
16.5	even	4		216.2.d.b.109.2	yes	4
16.11	odd	4		864.2.d.a.433.1		4
16.13	even	4		216.2.d.b.109.1	✓	4
24.5	odd	2	inner	6912.2.a.bc.1.1		2
24.11	even	2		6912.2.a.bd.1.1		2
48.5	odd	4		216.2.d.b.109.3	yes	4
48.11	even	4		864.2.d.a.433.3		4
48.29	odd	4		216.2.d.b.109.4	yes	4
48.35	even	4		864.2.d.a.433.2		4
144.5	odd	12		648.2.n.n.541.2		8
144.11	even	12		2592.2.r.p.433.2		8
144.13	even	12		648.2.n.n.541.4		8
144.29	odd	12		648.2.n.n.109.2		8
144.43	odd	12		2592.2.r.p.433.4		8
144.59	even	12		2592.2.r.p.2161.3		8
144.61	even	12		648.2.n.n.109.3		8
144.67	odd	12		2592.2.r.p.2161.4		8
144.77	odd	12		648.2.n.n.541.1		8
144.83	even	12		2592.2.r.p.433.3		8
144.85	even	12		648.2.n.n.541.3		8
144.101	odd	12		648.2.n.n.109.1		8
144.115	odd	12		2592.2.r.p.433.1		8
144.131	even	12		2592.2.r.p.2161.2		8
144.133	even	12		648.2.n.n.109.4		8
144.139	odd	12		2592.2.r.p.2161.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.b.109.1	✓	4	16.13	even	4
216.2.d.b.109.2	yes	4	16.5	even	4
216.2.d.b.109.3	yes	4	48.5	odd	4
216.2.d.b.109.4	yes	4	48.29	odd	4
648.2.n.n.109.1		8	144.101	odd	12
648.2.n.n.109.2		8	144.29	odd	12
648.2.n.n.109.3		8	144.61	even	12
648.2.n.n.109.4		8	144.133	even	12
648.2.n.n.541.1		8	144.77	odd	12
648.2.n.n.541.2		8	144.5	odd	12
648.2.n.n.541.3		8	144.85	even	12
648.2.n.n.541.4		8	144.13	even	12
864.2.d.a.433.1		4	16.11	odd	4
864.2.d.a.433.2		4	48.35	even	4
864.2.d.a.433.3		4	48.11	even	4
864.2.d.a.433.4		4	16.3	odd	4
2592.2.r.p.433.1		8	144.115	odd	12
2592.2.r.p.433.2		8	144.11	even	12
2592.2.r.p.433.3		8	144.83	even	12
2592.2.r.p.433.4		8	144.43	odd	12
2592.2.r.p.2161.1		8	144.139	odd	12
2592.2.r.p.2161.2		8	144.131	even	12
2592.2.r.p.2161.3		8	144.59	even	12
2592.2.r.p.2161.4		8	144.67	odd	12
6912.2.a.bc.1.1		2	24.5	odd	2	inner
6912.2.a.bc.1.2		2	1.1	even	1	trivial
6912.2.a.bd.1.1		2	24.11	even	2
6912.2.a.bd.1.2		2	4.3	odd	2
6912.2.a.bu.1.1		2	8.5	even	2
6912.2.a.bu.1.2		2	3.2	odd	2
6912.2.a.bv.1.1		2	8.3	odd	2
6912.2.a.bv.1.2		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{11} +5.29150 q^{13} +5.29150 q^{17} -5.29150 q^{19} -5.29150 q^{23} -4.00000 q^{25} +6.00000 q^{29} -7.00000 q^{31} +1.00000 q^{35} -5.29150 q^{37} +5.29150 q^{41} +10.5830 q^{43} -6.00000 q^{49} +9.00000 q^{53} +3.00000 q^{55} +4.00000 q^{59} -5.29150 q^{65} +15.8745 q^{71} -3.00000 q^{73} +3.00000 q^{77} +4.00000 q^{79} -7.00000 q^{83} -5.29150 q^{85} -10.5830 q^{89} -5.29150 q^{91} +5.29150 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} - 6 q^{11} - 8 q^{25} + 12 q^{29} - 14 q^{31} + 2 q^{35} - 12 q^{49} + 18 q^{53} + 6 q^{55} + 8 q^{59} - 6 q^{73} + 6 q^{77} + 8 q^{79} - 14 q^{83} + 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 - 6 * q^11 - 8 * q^25 + 12 * q^29 - 14 * q^31 + 2 * q^35 - 12 * q^49 + 18 * q^53 + 6 * q^55 + 8 * q^59 - 6 * q^73 + 6 * q^77 + 8 * q^79 - 14 * q^83 + 14 * q^97

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