LMFDB - Embedded newform 5184.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(1,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([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("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.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:	$41.3944484078$
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 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5184.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{5} -2.00000 q^{7} +O(q^{10})$	$q+1.73205 q^{5} -2.00000 q^{7} -3.46410 q^{11} +1.00000 q^{13} +5.19615 q^{17} +2.00000 q^{19} -3.46410 q^{23} -2.00000 q^{25} -1.73205 q^{29} -8.00000 q^{31} -3.46410 q^{35} +7.00000 q^{37} -6.92820 q^{41} +2.00000 q^{43} -6.92820 q^{47} -3.00000 q^{49} -6.00000 q^{55} +13.8564 q^{59} +7.00000 q^{61} +1.73205 q^{65} -10.0000 q^{67} +10.3923 q^{71} -7.00000 q^{73} +6.92820 q^{77} -2.00000 q^{79} -13.8564 q^{83} +9.00000 q^{85} -5.19615 q^{89} -2.00000 q^{91} +3.46410 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} + 2 q^{13} + 4 q^{19} - 4 q^{25} - 16 q^{31} + 14 q^{37} + 4 q^{43} - 6 q^{49} - 12 q^{55} + 14 q^{61} - 20 q^{67} - 14 q^{73} - 4 q^{79} + 18 q^{85} - 4 q^{91} + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 + 2 * q^13 + 4 * q^19 - 4 * q^25 - 16 * q^31 + 14 * q^37 + 4 * q^43 - 6 * q^49 - 12 * q^55 + 14 * q^61 - 20 * q^67 - 14 * q^73 - 4 * q^79 + 18 * q^85 - 4 * q^91 + 4 * 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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.19615	1.26025	0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	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
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	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
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.73205	0.214834
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.92820	0.789542
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.8564	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.92820	−0.689382	−0.344691	−	0.938716i	$-0.612016\pi$
$101$	−6.92820	−0.689382	−0.344691	+	0.938716i	$0.612016\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.73205	−0.162938	−0.0814688	−	0.996676i	$-0.525961\pi$
$113$	−1.73205	−0.162938	−0.0814688	+	0.996676i	$0.525961\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.3923	−0.952661
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.73205	0.147979	0.0739895	−	0.997259i	$-0.476427\pi$
$137$	1.73205	0.147979	0.0739895	+	0.997259i	$0.476427\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.66025	−0.709476	−0.354738	−	0.934966i	$-0.615430\pi$
$149$	−8.66025	−0.709476	−0.354738	+	0.934966i	$0.615430\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.8564	−1.11297
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.92820	0.546019
$162$	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
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.0526	1.44854	0.724270	−	0.689517i	$-0.242177\pi$
$173$	19.0526	1.44854	0.724270	+	0.689517i	$0.242177\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.1244	0.891400
$186$	0	0
$187$	−18.0000	−1.31629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.3205	−1.25327	−0.626634	−	0.779314i	$-0.715568\pi$
$191$	−17.3205	−1.25327	−0.626634	+	0.779314i	$0.715568\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.19615	0.370211	0.185105	−	0.982719i	$-0.440737\pi$
$197$	5.19615	0.370211	0.185105	+	0.982719i	$0.440737\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.46410	0.243132
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.46410	0.236250
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.19615	0.349531
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.9808	−1.70206	−0.851028	−	0.525120i	$-0.824020\pi$
$233$	−25.9808	−1.70206	−0.851028	+	0.525120i	$0.824020\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.7128	1.79259	0.896296	−	0.443455i	$-0.146248\pi$
$239$	27.7128	1.79259	0.896296	+	0.443455i	$0.146248\pi$
$240$	0	0
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.19615	−0.331970
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.66025	0.540212	0.270106	−	0.962831i	$-0.412941\pi$
$257$	8.66025	0.540212	0.270106	+	0.962831i	$0.412941\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.92820	−0.427211	−0.213606	−	0.976920i	$-0.568521\pi$
$263$	−6.92820	−0.427211	−0.213606	+	0.976920i	$0.568521\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.5885	0.950445	0.475223	−	0.879866i	$-0.342368\pi$
$269$	15.5885	0.950445	0.475223	+	0.879866i	$0.342368\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.1244	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$281$	12.1244	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.8564	0.817918
$288$	0	0
$289$	10.0000	0.588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.46410	−0.200334
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.1244	0.694239
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.92820	0.392862	0.196431	−	0.980518i	$-0.437065\pi$
$311$	6.92820	0.392862	0.196431	+	0.980518i	$0.437065\pi$
$312$	0	0
$313$	−25.0000	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$313$	−25.0000	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.66025	0.486408	0.243204	−	0.969975i	$-0.421801\pi$
$317$	8.66025	0.486408	0.243204	+	0.969975i	$0.421801\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.8564	0.763928
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.3205	−0.946320
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	27.7128	1.50073
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.46410	0.185963	0.0929814	−	0.995668i	$-0.470360\pi$
$347$	3.46410	0.185963	0.0929814	+	0.995668i	$0.470360\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.8564	−0.737502	−0.368751	−	0.929528i	$-0.620215\pi$
$353$	−13.8564	−0.737502	−0.368751	+	0.929528i	$0.620215\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.3923	−0.548485	−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923	−0.548485	−0.274242	+	0.961661i	$0.588427\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.1244	−0.634618
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.73205	−0.0892052
$378$	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
$383$	17.3205	0.885037	0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205	0.885037	0.442518	+	0.896759i	$0.354085\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−27.7128	−1.40510	−0.702548	−	0.711637i	$-0.747954\pi$
$389$	−27.7128	−1.40510	−0.702548	+	0.711637i	$0.747954\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.46410	−0.174298
$396$	0	0
$397$	−29.0000	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.1244	−0.605461	−0.302731	−	0.953076i	$-0.597898\pi$
$401$	−12.1244	−0.605461	−0.302731	+	0.953076i	$0.597898\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.2487	−1.20196
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−27.7128	−1.36366
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.92820	−0.338465	−0.169232	−	0.985576i	$-0.554129\pi$
$419$	−6.92820	−0.338465	−0.169232	+	0.985576i	$0.554129\pi$
$420$	0	0
$421$	25.0000	1.21843	0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000	1.21843	0.609213	+	0.793007i	$0.291486\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.3923	−0.504101
$426$	0	0
$427$	−14.0000	−0.677507
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.6410	−1.64584	−0.822922	−	0.568154i	$-0.807658\pi$
$443$	−34.6410	−1.64584	−0.822922	+	0.568154i	$0.807658\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.7846	0.980886	0.490443	−	0.871473i	$-0.336835\pi$
$449$	20.7846	0.980886	0.490443	+	0.871473i	$0.336835\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.46410	−0.162400
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.7846	−0.961797	−0.480899	−	0.876776i	$-0.659689\pi$
$467$	−20.7846	−0.961797	−0.480899	+	0.876776i	$0.659689\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.92820	−0.318559
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.46410	0.157297
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.3205	−0.781664	−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205	−0.781664	−0.390832	+	0.920462i	$0.627813\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.7846	−0.932317
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.7846	−0.926740	−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846	−0.926740	−0.463370	+	0.886165i	$0.653360\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.8564	−0.610586
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.5692	−1.81078
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.92820	−0.300094
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.3923	0.447628
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.0526	−0.816122
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.3731	−1.54118	−0.770588	−	0.637333i	$-0.780037\pi$
$557$	−36.3731	−1.54118	−0.770588	+	0.637333i	$0.780037\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	−3.00000	−0.126211
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.9090	1.37962	0.689808	−	0.723993i	$-0.257695\pi$
$569$	32.9090	1.37962	0.689808	+	0.723993i	$0.257695\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.92820	0.288926
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	27.7128	1.14972
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.1051	1.57277	0.786383	−	0.617739i	$-0.211951\pi$
$587$	38.1051	1.57277	0.786383	+	0.617739i	$0.211951\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.5885	0.640141	0.320071	−	0.947394i	$-0.396293\pi$
$593$	15.5885	0.640141	0.320071	+	0.947394i	$0.396293\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.73205	0.0704179
$606$	0	0
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.1244	−0.488108	−0.244054	−	0.969762i	$-0.578477\pi$
$617$	−12.1244	−0.488108	−0.244054	+	0.969762i	$0.578477\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	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
$629$	36.3731	1.45029
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.46410	−0.137469
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22.5167	0.889355	0.444677	−	0.895691i	$-0.353318\pi$
$641$	22.5167	0.889355	0.444677	+	0.895691i	$0.353318\pi$
$642$	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
$647$	31.1769	1.22569	0.612845	−	0.790203i	$-0.290025\pi$
$647$	31.1769	1.22569	0.612845	+	0.790203i	$0.290025\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.8564	−0.542243	−0.271122	−	0.962545i	$-0.587395\pi$
$653$	−13.8564	−0.542243	−0.271122	+	0.962545i	$0.587395\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.46410	−0.134942	−0.0674711	−	0.997721i	$-0.521493\pi$
$659$	−3.46410	−0.134942	−0.0674711	+	0.997721i	$0.521493\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.92820	−0.268664
$666$	0	0
$667$	6.00000	0.232321
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.2487	−0.936111
$672$	0	0
$673$	−25.0000	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$673$	−25.0000	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.8564	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$677$	13.8564	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.7846	−0.795301	−0.397650	−	0.917537i	$-0.630174\pi$
$683$	−20.7846	−0.795301	−0.397650	+	0.917537i	$0.630174\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.8564	0.525603
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−46.7654	−1.76630	−0.883152	−	0.469087i	$-0.844583\pi$
$701$	−46.7654	−1.76630	−0.883152	+	0.469087i	$0.844583\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8564	0.521124
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.7128	1.03785
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.3923	0.384373
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.6410	1.27602
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.92820	0.254171	0.127086	−	0.991892i	$-0.459438\pi$
$743$	6.92820	0.254171	0.127086	+	0.991892i	$0.459438\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−34.6410	−1.26072
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.4449	1.06738	0.533688	−	0.845682i	$-0.320806\pi$
$761$	29.4449	1.06738	0.533688	+	0.845682i	$0.320806\pi$
$762$	0	0
$763$	22.0000	0.796453
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8564	0.500326
$768$	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
$773$	25.9808	0.934463	0.467232	−	0.884135i	$-0.345251\pi$
$773$	25.9808	0.934463	0.467232	+	0.884135i	$0.345251\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.8564	−0.496457
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29.4449	−1.05093
$786$	0	0
$787$	26.0000	0.926800	0.463400	−	0.886149i	$-0.346629\pi$
$787$	26.0000	0.926800	0.463400	+	0.886149i	$0.346629\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.46410	0.123169
$792$	0	0
$793$	7.00000	0.248577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.6936	1.90192	0.950962	−	0.309308i	$-0.100097\pi$
$797$	53.6936	1.90192	0.950962	+	0.309308i	$0.100097\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24.2487	0.855718
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−46.7654	−1.64418	−0.822091	−	0.569355i	$-0.807193\pi$
$809$	−46.7654	−1.64418	−0.822091	+	0.569355i	$0.807193\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−27.7128	−0.970737
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.1244	−0.423143	−0.211571	−	0.977363i	$-0.567858\pi$
$821$	−12.1244	−0.423143	−0.211571	+	0.977363i	$0.567858\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.5885	−0.540108
$834$	0	0
$835$	30.0000	1.03819
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−20.7846	−0.715012
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.2487	−0.831235
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.5167	0.769154	0.384577	−	0.923093i	$-0.374347\pi$
$857$	22.5167	0.769154	0.384577	+	0.923093i	$0.374347\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.1769	−1.06127	−0.530637	−	0.847599i	$-0.678047\pi$
$863$	−31.1769	−1.06127	−0.530637	+	0.847599i	$0.678047\pi$
$864$	0	0
$865$	33.0000	1.12203
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.92820	0.235023
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24.2487	0.819756
$876$	0	0
$877$	−53.0000	−1.78968	−0.894841	−	0.446384i	$-0.852711\pi$
$877$	−53.0000	−1.78968	−0.894841	+	0.446384i	$0.852711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.7846	0.700251	0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846	0.700251	0.350126	+	0.936703i	$0.386139\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.46410	−0.116313	−0.0581566	−	0.998307i	$-0.518522\pi$
$887$	−3.46410	−0.116313	−0.0581566	+	0.998307i	$0.518522\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.8564	−0.463687
$894$	0	0
$895$	36.0000	1.20335
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.8564	0.462137
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.46410	−0.115151
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.2487	0.803396	0.401698	−	0.915772i	$-0.368420\pi$
$911$	24.2487	0.803396	0.401698	+	0.915772i	$0.368420\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.92820	−0.228789
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.3923	0.342067
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.2295	−1.64798	−0.823988	−	0.566608i	$-0.808256\pi$
$929$	−50.2295	−1.64798	−0.823988	+	0.566608i	$0.808256\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−31.1769	−1.01959
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50.2295	−1.63743	−0.818717	−	0.574197i	$-0.805314\pi$
$941$	−50.2295	−1.63743	−0.818717	+	0.574197i	$0.805314\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.3205	0.562841	0.281420	−	0.959585i	$-0.409194\pi$
$947$	17.3205	0.562841	0.281420	+	0.959585i	$0.409194\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.19615	−0.168320	−0.0841599	−	0.996452i	$-0.526821\pi$
$953$	−5.19615	−0.168320	−0.0841599	+	0.996452i	$0.526821\pi$
$954$	0	0
$955$	−30.0000	−0.970777
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.46410	−0.111862
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.73205	−0.0557567
$966$	0	0
$967$	46.0000	1.47926	0.739630	−	0.673014i	$-0.235000\pi$
$967$	46.0000	1.47926	0.739630	+	0.673014i	$0.235000\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1769	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.4974	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.6410	1.10488	0.552438	−	0.833554i	$-0.313697\pi$
$983$	34.6410	1.10488	0.552438	+	0.833554i	$0.313697\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.92820	−0.220304
$990$	0	0
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−34.6410	−1.09819
$996$	0	0
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.a.bq.1.2		2
3.2	odd	2	inner	5184.2.a.bq.1.1		2
4.3	odd	2		5184.2.a.br.1.2		2
8.3	odd	2		81.2.a.a.1.2	yes	2
8.5	even	2		1296.2.a.o.1.1		2
12.11	even	2		5184.2.a.br.1.1		2
24.5	odd	2		1296.2.a.o.1.2		2
24.11	even	2		81.2.a.a.1.1	✓	2
40.3	even	4		2025.2.b.k.649.1		4
40.19	odd	2		2025.2.a.j.1.1		2
40.27	even	4		2025.2.b.k.649.4		4
56.27	even	2		3969.2.a.i.1.2		2
72.5	odd	6		1296.2.i.s.865.1		4
72.11	even	6		81.2.c.b.28.2		4
72.13	even	6		1296.2.i.s.865.2		4
72.29	odd	6		1296.2.i.s.433.1		4
72.43	odd	6		81.2.c.b.28.1		4
72.59	even	6		81.2.c.b.55.2		4
72.61	even	6		1296.2.i.s.433.2		4
72.67	odd	6		81.2.c.b.55.1		4
88.43	even	2		9801.2.a.v.1.1		2
120.59	even	2		2025.2.a.j.1.2		2
120.83	odd	4		2025.2.b.k.649.3		4
120.107	odd	4		2025.2.b.k.649.2		4
168.83	odd	2		3969.2.a.i.1.1		2
216.11	even	18		729.2.e.o.82.1		12
216.43	odd	18		729.2.e.o.82.2		12
216.59	even	18		729.2.e.o.649.1		12
216.67	odd	18		729.2.e.o.163.1		12
216.83	even	18		729.2.e.o.568.2		12
216.115	odd	18		729.2.e.o.325.2		12
216.131	even	18		729.2.e.o.406.1		12
216.139	odd	18		729.2.e.o.406.2		12
216.155	even	18		729.2.e.o.325.1		12
216.187	odd	18		729.2.e.o.568.1		12
216.203	even	18		729.2.e.o.163.2		12
216.211	odd	18		729.2.e.o.649.2		12
264.131	odd	2		9801.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.a.a.1.1	✓	2	24.11	even	2
81.2.a.a.1.2	yes	2	8.3	odd	2
81.2.c.b.28.1		4	72.43	odd	6
81.2.c.b.28.2		4	72.11	even	6
81.2.c.b.55.1		4	72.67	odd	6
81.2.c.b.55.2		4	72.59	even	6
729.2.e.o.82.1		12	216.11	even	18
729.2.e.o.82.2		12	216.43	odd	18
729.2.e.o.163.1		12	216.67	odd	18
729.2.e.o.163.2		12	216.203	even	18
729.2.e.o.325.1		12	216.155	even	18
729.2.e.o.325.2		12	216.115	odd	18
729.2.e.o.406.1		12	216.131	even	18
729.2.e.o.406.2		12	216.139	odd	18
729.2.e.o.568.1		12	216.187	odd	18
729.2.e.o.568.2		12	216.83	even	18
729.2.e.o.649.1		12	216.59	even	18
729.2.e.o.649.2		12	216.211	odd	18
1296.2.a.o.1.1		2	8.5	even	2
1296.2.a.o.1.2		2	24.5	odd	2
1296.2.i.s.433.1		4	72.29	odd	6
1296.2.i.s.433.2		4	72.61	even	6
1296.2.i.s.865.1		4	72.5	odd	6
1296.2.i.s.865.2		4	72.13	even	6
2025.2.a.j.1.1		2	40.19	odd	2
2025.2.a.j.1.2		2	120.59	even	2
2025.2.b.k.649.1		4	40.3	even	4
2025.2.b.k.649.2		4	120.107	odd	4
2025.2.b.k.649.3		4	120.83	odd	4
2025.2.b.k.649.4		4	40.27	even	4
3969.2.a.i.1.1		2	168.83	odd	2
3969.2.a.i.1.2		2	56.27	even	2
5184.2.a.bq.1.1		2	3.2	odd	2	inner
5184.2.a.bq.1.2		2	1.1	even	1	trivial
5184.2.a.br.1.1		2	12.11	even	2
5184.2.a.br.1.2		2	4.3	odd	2
9801.2.a.v.1.1		2	88.43	even	2
9801.2.a.v.1.2		2	264.131	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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q+1.73205 q^{5} -2.00000 q^{7} -3.46410 q^{11} +1.00000 q^{13} +5.19615 q^{17} +2.00000 q^{19} -3.46410 q^{23} -2.00000 q^{25} -1.73205 q^{29} -8.00000 q^{31} -3.46410 q^{35} +7.00000 q^{37} -6.92820 q^{41} +2.00000 q^{43} -6.92820 q^{47} -3.00000 q^{49} -6.00000 q^{55} +13.8564 q^{59} +7.00000 q^{61} +1.73205 q^{65} -10.0000 q^{67} +10.3923 q^{71} -7.00000 q^{73} +6.92820 q^{77} -2.00000 q^{79} -13.8564 q^{83} +9.00000 q^{85} -5.19615 q^{89} -2.00000 q^{91} +3.46410 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} + 2 q^{13} + 4 q^{19} - 4 q^{25} - 16 q^{31} + 14 q^{37} + 4 q^{43} - 6 q^{49} - 12 q^{55} + 14 q^{61} - 20 q^{67} - 14 q^{73} - 4 q^{79} + 18 q^{85} - 4 q^{91} + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 + 2 * q^13 + 4 * q^19 - 4 * q^25 - 16 * q^31 + 14 * q^37 + 4 * q^43 - 6 * q^49 - 12 * q^55 + 14 * q^61 - 20 * q^67 - 14 * q^73 - 4 * q^79 + 18 * q^85 - 4 * q^91 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more