LMFDB - Embedded newform 6336.2.a.da.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(1,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.5932147207$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 3168)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	6336.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{5} -3.46410 q^{7} -1.00000 q^{11} +2.89898 q^{13} +0.635674 q^{17} +0.635674 q^{19} -4.89898 q^{23} +3.00000 q^{25} +6.29253 q^{29} -5.65685 q^{31} -9.79796 q^{35} -11.7980 q^{37} -6.29253 q^{41} -6.29253 q^{43} +0.898979 q^{47} +5.00000 q^{49} +4.09978 q^{53} -2.82843 q^{55} +5.79796 q^{59} -10.8990 q^{61} +8.19955 q^{65} -8.89898 q^{71} +6.00000 q^{73} +3.46410 q^{77} +4.73545 q^{79} +5.79796 q^{83} +1.79796 q^{85} +11.3137 q^{89} -10.0424 q^{91} +1.79796 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} - 8 q^{13} + 12 q^{25} - 8 q^{37} - 16 q^{47} + 20 q^{49} - 16 q^{59} - 24 q^{61} - 16 q^{71} + 24 q^{73} - 16 q^{83} - 32 q^{85} - 32 q^{95} + 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^11 - 8 * q^13 + 12 * q^25 - 8 * q^37 - 16 * q^47 + 20 * q^49 - 16 * q^59 - 24 * q^61 - 16 * q^71 + 24 * q^73 - 16 * q^83 - 32 * q^85 - 32 * q^95 + 40 * q^97

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.89898	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$13$	2.89898	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.635674	0.154174	0.0770869	−	0.997024i	$-0.475438\pi$
$17$	0.635674	0.154174	0.0770869	+	0.997024i	$0.475438\pi$
$18$	0	0
$19$	0.635674	0.145834	0.0729169	−	0.997338i	$-0.476769\pi$
$19$	0.635674	0.145834	0.0729169	+	0.997338i	$0.476769\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.89898	−1.02151	−0.510754	−	0.859727i	$-0.670634\pi$
$23$	−4.89898	−1.02151	−0.510754	+	0.859727i	$0.670634\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.29253	1.16849	0.584247	−	0.811576i	$-0.301390\pi$
$29$	6.29253	1.16849	0.584247	+	0.811576i	$0.301390\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.79796	−1.65616
$36$	0	0
$37$	−11.7980	−1.93957	−0.969786	−	0.243956i	$-0.921555\pi$
$37$	−11.7980	−1.93957	−0.969786	+	0.243956i	$0.921555\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.29253	−0.982728	−0.491364	−	0.870954i	$-0.663501\pi$
$41$	−6.29253	−0.982728	−0.491364	+	0.870954i	$0.663501\pi$
$42$	0	0
$43$	−6.29253	−0.959602	−0.479801	−	0.877377i	$-0.659291\pi$
$43$	−6.29253	−0.959602	−0.479801	+	0.877377i	$0.659291\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.898979	0.131130	0.0655648	−	0.997848i	$-0.479115\pi$
$47$	0.898979	0.131130	0.0655648	+	0.997848i	$0.479115\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.09978	0.563148	0.281574	−	0.959540i	$-0.409144\pi$
$53$	4.09978	0.563148	0.281574	+	0.959540i	$0.409144\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.79796	0.754830	0.377415	−	0.926044i	$-0.376813\pi$
$59$	5.79796	0.754830	0.377415	+	0.926044i	$0.376813\pi$
$60$	0	0
$61$	−10.8990	−1.39547	−0.697736	−	0.716355i	$-0.745809\pi$
$61$	−10.8990	−1.39547	−0.697736	+	0.716355i	$0.745809\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.19955	1.01703
$66$	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
$71$	−8.89898	−1.05611	−0.528057	−	0.849209i	$-0.677079\pi$
$71$	−8.89898	−1.05611	−0.528057	+	0.849209i	$0.677079\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	4.73545	0.532780	0.266390	−	0.963865i	$-0.414169\pi$
$79$	4.73545	0.532780	0.266390	+	0.963865i	$0.414169\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.79796	0.636409	0.318204	−	0.948022i	$-0.396920\pi$
$83$	5.79796	0.636409	0.318204	+	0.948022i	$0.396920\pi$
$84$	0	0
$85$	1.79796	0.195016
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.3137	1.19925	0.599625	−	0.800281i	$-0.295316\pi$
$89$	11.3137	1.19925	0.599625	+	0.800281i	$0.295316\pi$
$90$	0	0
$91$	−10.0424	−1.05273
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.79796	0.184467
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.2207	−1.31551	−0.657756	−	0.753231i	$-0.728494\pi$
$101$	−13.2207	−1.31551	−0.657756	+	0.753231i	$0.728494\pi$
$102$	0	0
$103$	12.5851	1.24004	0.620021	−	0.784585i	$-0.287124\pi$
$103$	12.5851	1.24004	0.620021	+	0.784585i	$0.287124\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−5.10102	−0.488589	−0.244295	−	0.969701i	$-0.578556\pi$
$109$	−5.10102	−0.488589	−0.244295	+	0.969701i	$0.578556\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.27135	−0.119598	−0.0597992	−	0.998210i	$-0.519046\pi$
$113$	−1.27135	−0.119598	−0.0597992	+	0.998210i	$0.519046\pi$
$114$	0	0
$115$	−13.8564	−1.29212
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.20204	−0.201861
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−20.4347	−1.81328	−0.906642	−	0.421902i	$-0.861363\pi$
$127$	−20.4347	−1.81328	−0.906642	+	0.421902i	$0.861363\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.7980	−1.20553	−0.602767	−	0.797918i	$-0.705935\pi$
$131$	−13.7980	−1.20553	−0.602767	+	0.797918i	$0.705935\pi$
$132$	0	0
$133$	−2.20204	−0.190941
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	−1.90702	−0.161752	−0.0808758	−	0.996724i	$-0.525772\pi$
$139$	−1.90702	−0.161752	−0.0808758	+	0.996724i	$0.525772\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.89898	−0.242425
$144$	0	0
$145$	17.7980	1.47804
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.1489	−1.65067	−0.825333	−	0.564647i	$-0.809012\pi$
$149$	−20.1489	−1.65067	−0.825333	+	0.564647i	$0.809012\pi$
$150$	0	0
$151$	13.5065	1.09914	0.549570	−	0.835448i	$-0.314792\pi$
$151$	13.5065	1.09914	0.549570	+	0.835448i	$0.314792\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−15.7980	−1.26081	−0.630407	−	0.776265i	$-0.717112\pi$
$157$	−15.7980	−1.26081	−0.630407	+	0.776265i	$0.717112\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.9706	1.33747
$162$	0	0
$163$	23.8988	1.87190	0.935948	−	0.352138i	$-0.114545\pi$
$163$	23.8988	1.87190	0.935948	+	0.352138i	$0.114545\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−4.59592	−0.353532
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.83523	0.671730	0.335865	−	0.941910i	$-0.390971\pi$
$173$	8.83523	0.671730	0.335865	+	0.941910i	$0.390971\pi$
$174$	0	0
$175$	−10.3923	−0.785584
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.79796	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$180$	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
$185$	−33.3697	−2.45339
$186$	0	0
$187$	−0.635674	−0.0464851
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.6969	−1.93172	−0.965861	−	0.259060i	$-0.916587\pi$
$191$	−26.6969	−1.93172	−0.965861	+	0.259060i	$0.916587\pi$
$192$	0	0
$193$	−7.79796	−0.561309	−0.280655	−	0.959809i	$-0.590552\pi$
$193$	−7.79796	−0.561309	−0.280655	+	0.959809i	$0.590552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.5344	−1.74801	−0.874003	−	0.485920i	$-0.838485\pi$
$197$	−24.5344	−1.74801	−0.874003	+	0.485920i	$0.838485\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.7980	−1.52992
$204$	0	0
$205$	−17.7980	−1.24306
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.635674	−0.0439705
$210$	0	0
$211$	20.1489	1.38711	0.693555	−	0.720404i	$-0.256043\pi$
$211$	20.1489	1.38711	0.693555	+	0.720404i	$0.256043\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−17.7980	−1.21381
$216$	0	0
$217$	19.5959	1.33026
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.84281	0.123961
$222$	0	0
$223$	23.8988	1.60038	0.800190	−	0.599747i	$-0.204732\pi$
$223$	23.8988	1.60038	0.800190	+	0.599747i	$0.204732\pi$
$224$	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
$229$	−7.79796	−0.515304	−0.257652	−	0.966238i	$-0.582949\pi$
$229$	−7.79796	−0.515304	−0.257652	+	0.966238i	$0.582949\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.56388	0.495526	0.247763	−	0.968821i	$-0.420305\pi$
$233$	7.56388	0.495526	0.247763	+	0.968821i	$0.420305\pi$
$234$	0	0
$235$	2.54270	0.165867
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	15.7980	1.01764	0.508818	−	0.860874i	$-0.330083\pi$
$241$	15.7980	1.01764	0.508818	+	0.860874i	$0.330083\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.1421	0.903508
$246$	0	0
$247$	1.84281	0.117255
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.79796	−0.618442	−0.309221	−	0.950990i	$-0.600068\pi$
$251$	−9.79796	−0.618442	−0.309221	+	0.950990i	$0.600068\pi$
$252$	0	0
$253$	4.89898	0.307996
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.54270	0.158609	0.0793046	−	0.996850i	$-0.474730\pi$
$257$	2.54270	0.158609	0.0793046	+	0.996850i	$0.474730\pi$
$258$	0	0
$259$	40.8693	2.53950
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.20204	−0.382434	−0.191217	−	0.981548i	$-0.561243\pi$
$263$	−6.20204	−0.382434	−0.191217	+	0.981548i	$0.561243\pi$
$264$	0	0
$265$	11.5959	0.712332
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.21393	−0.439841	−0.219921	−	0.975518i	$-0.570580\pi$
$269$	−7.21393	−0.439841	−0.219921	+	0.975518i	$0.570580\pi$
$270$	0	0
$271$	−29.9056	−1.81663	−0.908317	−	0.418283i	$-0.862632\pi$
$271$	−29.9056	−1.81663	−0.908317	+	0.418283i	$0.862632\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−6.89898	−0.414520	−0.207260	−	0.978286i	$-0.566455\pi$
$277$	−6.89898	−0.414520	−0.207260	+	0.978286i	$0.566455\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.9494	−0.712840	−0.356420	−	0.934326i	$-0.616003\pi$
$281$	−11.9494	−0.712840	−0.356420	+	0.934326i	$0.616003\pi$
$282$	0	0
$283$	−17.6062	−1.04658	−0.523291	−	0.852154i	$-0.675296\pi$
$283$	−17.6062	−1.04658	−0.523291	+	0.852154i	$0.675296\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.7980	1.28669
$288$	0	0
$289$	−16.5959	−0.976230
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.6780	0.623817	0.311909	−	0.950112i	$-0.399032\pi$
$293$	10.6780	0.623817	0.311909	+	0.950112i	$0.399032\pi$
$294$	0	0
$295$	16.3991	0.954793
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.2020	−0.821325
$300$	0	0
$301$	21.7980	1.25641
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−30.8270	−1.76515
$306$	0	0
$307$	−1.90702	−0.108840	−0.0544198	−	0.998518i	$-0.517331\pi$
$307$	−1.90702	−0.108840	−0.0544198	+	0.998518i	$0.517331\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.8990	−1.63871	−0.819355	−	0.573286i	$-0.805668\pi$
$311$	−28.8990	−1.63871	−0.819355	+	0.573286i	$0.805668\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.3417	1.25483	0.627417	−	0.778683i	$-0.284112\pi$
$317$	22.3417	1.25483	0.627417	+	0.778683i	$0.284112\pi$
$318$	0	0
$319$	−6.29253	−0.352314
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.404082	0.0224837
$324$	0	0
$325$	8.69694	0.482419
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.11416	−0.171689
$330$	0	0
$331$	23.8988	1.31359	0.656797	−	0.754067i	$-0.271911\pi$
$331$	23.8988	1.31359	0.656797	+	0.754067i	$0.271911\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−29.5959	−1.61219	−0.806096	−	0.591785i	$-0.798424\pi$
$337$	−29.5959	−1.61219	−0.806096	+	0.591785i	$0.798424\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.7980	−1.17018	−0.585088	−	0.810970i	$-0.698940\pi$
$347$	−21.7980	−1.17018	−0.585088	+	0.810970i	$0.698940\pi$
$348$	0	0
$349$	−9.10102	−0.487166	−0.243583	−	0.969880i	$-0.578323\pi$
$349$	−9.10102	−0.487166	−0.243583	+	0.969880i	$0.578323\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.1701	1.33967	0.669835	−	0.742510i	$-0.266365\pi$
$353$	25.1701	1.33967	0.669835	+	0.742510i	$0.266365\pi$
$354$	0	0
$355$	−25.1701	−1.33589
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.7980	−1.78379	−0.891894	−	0.452244i	$-0.850623\pi$
$359$	−33.7980	−1.78379	−0.891894	+	0.452244i	$0.850623\pi$
$360$	0	0
$361$	−18.5959	−0.978733
$362$	0	0
$363$	0	0
$364$	0	0
$365$	16.9706	0.888280
$366$	0	0
$367$	23.8988	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$367$	23.8988	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.2020	−0.737333
$372$	0	0
$373$	3.30306	0.171026	0.0855130	−	0.996337i	$-0.472747\pi$
$373$	3.30306	0.171026	0.0855130	+	0.996337i	$0.472747\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.2419	0.939506
$378$	0	0
$379$	−35.2125	−1.80874	−0.904372	−	0.426746i	$-0.859660\pi$
$379$	−35.2125	−1.80874	−0.904372	+	0.426746i	$0.859660\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.4949	−1.66041	−0.830206	−	0.557457i	$-0.811777\pi$
$383$	−32.4949	−1.66041	−0.830206	+	0.557457i	$0.811777\pi$
$384$	0	0
$385$	9.79796	0.499350
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.9985	1.41958	0.709791	−	0.704412i	$-0.248789\pi$
$389$	27.9985	1.41958	0.709791	+	0.704412i	$0.248789\pi$
$390$	0	0
$391$	−3.11416	−0.157490
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.3939	0.673919
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.0983	−1.60291	−0.801457	−	0.598053i	$-0.795941\pi$
$401$	−32.0983	−1.60291	−0.801457	+	0.598053i	$0.795941\pi$
$402$	0	0
$403$	−16.3991	−0.816898
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.7980	0.584803
$408$	0	0
$409$	−11.7980	−0.583372	−0.291686	−	0.956514i	$-0.594216\pi$
$409$	−11.7980	−0.583372	−0.291686	+	0.956514i	$0.594216\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−20.0847	−0.988304
$414$	0	0
$415$	16.3991	0.805000
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.2020	−0.693815	−0.346908	−	0.937899i	$-0.612768\pi$
$419$	−14.2020	−0.693815	−0.346908	+	0.937899i	$0.612768\pi$
$420$	0	0
$421$	7.79796	0.380049	0.190025	−	0.981779i	$-0.439143\pi$
$421$	7.79796	0.380049	0.190025	+	0.981779i	$0.439143\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.90702	0.0925042
$426$	0	0
$427$	37.7552	1.82710
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.7980	1.24264	0.621322	−	0.783555i	$-0.286596\pi$
$431$	25.7980	1.24264	0.621322	+	0.783555i	$0.286596\pi$
$432$	0	0
$433$	17.5959	0.845606	0.422803	−	0.906222i	$-0.361046\pi$
$433$	17.5959	0.845606	0.422803	+	0.906222i	$0.361046\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.11416	−0.148970
$438$	0	0
$439$	12.2351	0.583950	0.291975	−	0.956426i	$-0.405688\pi$
$439$	12.2351	0.583950	0.291975	+	0.956426i	$0.405688\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.3939	−1.20650	−0.603250	−	0.797552i	$-0.706128\pi$
$443$	−25.3939	−1.20650	−0.603250	+	0.797552i	$0.706128\pi$
$444$	0	0
$445$	32.0000	1.51695
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.5133	0.920888	0.460444	−	0.887689i	$-0.347690\pi$
$449$	19.5133	0.920888	0.460444	+	0.887689i	$0.347690\pi$
$450$	0	0
$451$	6.29253	0.296304
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−28.4041	−1.33160
$456$	0	0
$457$	19.3939	0.907207	0.453604	−	0.891204i	$-0.350138\pi$
$457$	19.3939	0.907207	0.453604	+	0.891204i	$0.350138\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.2631	−1.08347	−0.541735	−	0.840549i	$-0.682232\pi$
$461$	−23.2631	−1.08347	−0.541735	+	0.840549i	$0.682232\pi$
$462$	0	0
$463$	15.1278	0.703046	0.351523	−	0.936179i	$-0.385664\pi$
$463$	15.1278	0.703046	0.351523	+	0.936179i	$0.385664\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.5959	1.09189	0.545944	−	0.837821i	$-0.316171\pi$
$467$	23.5959	1.09189	0.545944	+	0.837821i	$0.316171\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.29253	0.289331
$474$	0	0
$475$	1.90702	0.0875002
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.3939	0.977511	0.488756	−	0.872421i	$-0.337451\pi$
$479$	21.3939	0.977511	0.488756	+	0.872421i	$0.337451\pi$
$480$	0	0
$481$	−34.2020	−1.55948
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.2843	1.28432
$486$	0	0
$487$	3.11416	0.141116	0.0705579	−	0.997508i	$-0.477522\pi$
$487$	3.11416	0.141116	0.0705579	+	0.997508i	$0.477522\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.8270	1.38278
$498$	0	0
$499$	−11.3137	−0.506471	−0.253236	−	0.967405i	$-0.581495\pi$
$499$	−11.3137	−0.506471	−0.253236	+	0.967405i	$0.581495\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	43.5959	1.94385	0.971923	−	0.235299i	$-0.0756070\pi$
$503$	43.5959	1.94385	0.971923	+	0.235299i	$0.0756070\pi$
$504$	0	0
$505$	−37.3939	−1.66401
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.55708	0.0690163	0.0345081	−	0.999404i	$-0.489014\pi$
$509$	1.55708	0.0690163	0.0345081	+	0.999404i	$0.489014\pi$
$510$	0	0
$511$	−20.7846	−0.919457
$512$	0	0
$513$	0	0
$514$	0	0
$515$	35.5959	1.56854
$516$	0	0
$517$	−0.898979	−0.0395371
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27.0129	1.18346	0.591729	−	0.806137i	$-0.298446\pi$
$521$	27.0129	1.18346	0.591729	+	0.806137i	$0.298446\pi$
$522$	0	0
$523$	−22.6916	−0.992236	−0.496118	−	0.868255i	$-0.665242\pi$
$523$	−22.6916	−0.992236	−0.496118	+	0.868255i	$0.665242\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.59592	−0.156641
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18.2419	−0.790145
$534$	0	0
$535$	−33.9411	−1.46740
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	34.8990	1.50042	0.750212	−	0.661197i	$-0.229951\pi$
$541$	34.8990	1.50042	0.750212	+	0.661197i	$0.229951\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.4279	−0.618022
$546$	0	0
$547$	−11.9494	−0.510919	−0.255459	−	0.966820i	$-0.582227\pi$
$547$	−11.9494	−0.510919	−0.255459	+	0.966820i	$0.582227\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	−16.4041	−0.697573
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.9917	0.931820	0.465910	−	0.884832i	$-0.345727\pi$
$557$	21.9917	0.931820	0.465910	+	0.884832i	$0.345727\pi$
$558$	0	0
$559$	−18.2419	−0.771551
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.59592	0.320130	0.160065	−	0.987106i	$-0.448830\pi$
$563$	7.59592	0.320130	0.160065	+	0.987106i	$0.448830\pi$
$564$	0	0
$565$	−3.59592	−0.151281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.9494	0.500944	0.250472	−	0.968124i	$-0.419414\pi$
$569$	11.9494	0.500944	0.250472	+	0.968124i	$0.419414\pi$
$570$	0	0
$571$	−11.9494	−0.500066	−0.250033	−	0.968237i	$-0.580441\pi$
$571$	−11.9494	−0.500066	−0.250033	+	0.968237i	$0.580441\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.6969	−0.612905
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.0847	−0.833255
$582$	0	0
$583$	−4.09978	−0.169795
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.2020	0.421083	0.210542	−	0.977585i	$-0.432477\pi$
$587$	10.2020	0.421083	0.210542	+	0.977585i	$0.432477\pi$
$588$	0	0
$589$	−3.59592	−0.148167
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.3485	−1.16413	−0.582066	−	0.813141i	$-0.697756\pi$
$593$	−28.3485	−1.16413	−0.582066	+	0.813141i	$0.697756\pi$
$594$	0	0
$595$	−6.22831	−0.255336
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.30306	−0.0532417	−0.0266208	−	0.999646i	$-0.508475\pi$
$599$	−1.30306	−0.0532417	−0.0266208	+	0.999646i	$0.508475\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.82843	0.114992
$606$	0	0
$607$	2.19275	0.0890011	0.0445005	−	0.999009i	$-0.485830\pi$
$607$	2.19275	0.0890011	0.0445005	+	0.999009i	$0.485830\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.60612	0.105432
$612$	0	0
$613$	−18.8990	−0.763323	−0.381661	−	0.924302i	$-0.624648\pi$
$613$	−18.8990	−0.763323	−0.381661	+	0.924302i	$0.624648\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.5262	1.87307	0.936537	−	0.350569i	$-0.114012\pi$
$617$	46.5262	1.87307	0.936537	+	0.350569i	$0.114012\pi$
$618$	0	0
$619$	23.8988	0.960573	0.480286	−	0.877112i	$-0.340533\pi$
$619$	23.8988	0.960573	0.480286	+	0.877112i	$0.340533\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.1918	−1.57019
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.49966	−0.299031
$630$	0	0
$631$	−16.9706	−0.675587	−0.337794	−	0.941220i	$-0.609681\pi$
$631$	−16.9706	−0.675587	−0.337794	+	0.941220i	$0.609681\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−57.7980	−2.29364
$636$	0	0
$637$	14.4949	0.574309
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.65685	0.223432	0.111716	−	0.993740i	$-0.464365\pi$
$641$	5.65685	0.223432	0.111716	+	0.993740i	$0.464365\pi$
$642$	0	0
$643$	−12.5851	−0.496306	−0.248153	−	0.968721i	$-0.579824\pi$
$643$	−12.5851	−0.496306	−0.248153	+	0.968721i	$0.579824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.8990	−1.13614	−0.568068	−	0.822982i	$-0.692309\pi$
$647$	−28.8990	−1.13614	−0.568068	+	0.822982i	$0.692309\pi$
$648$	0	0
$649$	−5.79796	−0.227590
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.1263	−1.68766	−0.843831	−	0.536609i	$-0.819705\pi$
$653$	−43.1263	−1.68766	−0.843831	+	0.536609i	$0.819705\pi$
$654$	0	0
$655$	−39.0265	−1.52489
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.5959	−0.607531	−0.303765	−	0.952747i	$-0.598244\pi$
$659$	−15.5959	−0.607531	−0.303765	+	0.952747i	$0.598244\pi$
$660$	0	0
$661$	43.7980	1.70354	0.851772	−	0.523913i	$-0.175528\pi$
$661$	43.7980	1.70354	0.851772	+	0.523913i	$0.175528\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.22831	−0.241524
$666$	0	0
$667$	−30.8270	−1.19363
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.8990	0.420750
$672$	0	0
$673$	8.20204	0.316165	0.158083	−	0.987426i	$-0.449469\pi$
$673$	8.20204	0.316165	0.158083	+	0.987426i	$0.449469\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.17837	0.122155	0.0610774	−	0.998133i	$-0.480546\pi$
$677$	3.17837	0.122155	0.0610774	+	0.998133i	$0.480546\pi$
$678$	0	0
$679$	−34.6410	−1.32940
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33.7980	−1.29324	−0.646621	−	0.762811i	$-0.723819\pi$
$683$	−33.7980	−1.29324	−0.646621	+	0.762811i	$0.723819\pi$
$684$	0	0
$685$	−19.5959	−0.748722
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.8852	0.452789
$690$	0	0
$691$	28.9842	1.10261	0.551305	−	0.834304i	$-0.314130\pi$
$691$	28.9842	1.10261	0.551305	+	0.834304i	$0.314130\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.39388	−0.204601
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−42.2049	−1.59406	−0.797028	−	0.603942i	$-0.793596\pi$
$701$	−42.2049	−1.59406	−0.797028	+	0.603942i	$0.793596\pi$
$702$	0	0
$703$	−7.49966	−0.282855
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.7980	1.72241
$708$	0	0
$709$	27.7980	1.04397	0.521987	−	0.852953i	$-0.325191\pi$
$709$	27.7980	1.04397	0.521987	+	0.852953i	$0.325191\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.7128	1.03785
$714$	0	0
$715$	−8.19955	−0.306646
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.2929	0.831383	0.415692	−	0.909506i	$-0.363540\pi$
$719$	22.2929	0.831383	0.415692	+	0.909506i	$0.363540\pi$
$720$	0	0
$721$	−43.5959	−1.62360
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.8776	0.701096
$726$	0	0
$727$	−3.11416	−0.115498	−0.0577488	−	0.998331i	$-0.518392\pi$
$727$	−3.11416	−0.115498	−0.0577488	+	0.998331i	$0.518392\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	18.8990	0.698050	0.349025	−	0.937113i	$-0.386513\pi$
$733$	18.8990	0.698050	0.349025	+	0.937113i	$0.386513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−6.29253	−0.231474	−0.115737	−	0.993280i	$-0.536923\pi$
$739$	−6.29253	−0.231474	−0.115737	+	0.993280i	$0.536923\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.7980	−1.23993	−0.619963	−	0.784631i	$-0.712853\pi$
$743$	−33.7980	−1.23993	−0.619963	+	0.784631i	$0.712853\pi$
$744$	0	0
$745$	−56.9898	−2.08794
$746$	0	0
$747$	0	0
$748$	0	0
$749$	41.5692	1.51891
$750$	0	0
$751$	19.5133	0.712049	0.356024	−	0.934477i	$-0.384132\pi$
$751$	19.5133	0.712049	0.356024	+	0.934477i	$0.384132\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	38.2020	1.39031
$756$	0	0
$757$	7.79796	0.283422	0.141711	−	0.989908i	$-0.454740\pi$
$757$	7.79796	0.283422	0.141711	+	0.989908i	$0.454740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−52.2473	−1.89396	−0.946981	−	0.321290i	$-0.895883\pi$
$761$	−52.2473	−1.89396	−0.946981	+	0.321290i	$0.895883\pi$
$762$	0	0
$763$	17.6705	0.639713
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.8082	0.606908
$768$	0	0
$769$	15.3939	0.555117	0.277559	−	0.960709i	$-0.410475\pi$
$769$	15.3939	0.555117	0.277559	+	0.960709i	$0.410475\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17.9562	0.645839	0.322919	−	0.946426i	$-0.395336\pi$
$773$	17.9562	0.645839	0.322919	+	0.946426i	$0.395336\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	8.89898	0.318431
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−44.6834	−1.59482
$786$	0	0
$787$	−1.20713	−0.0430296	−0.0215148	−	0.999769i	$-0.506849\pi$
$787$	−1.20713	−0.0430296	−0.0215148	+	0.999769i	$0.506849\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.40408	0.156591
$792$	0	0
$793$	−31.5959	−1.12200
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.1127	−1.10207	−0.551034	−	0.834483i	$-0.685767\pi$
$797$	−31.1127	−1.10207	−0.551034	+	0.834483i	$0.685767\pi$
$798$	0	0
$799$	0.571458	0.0202167
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	48.0000	1.69178
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.1913	−1.06147	−0.530735	−	0.847538i	$-0.678084\pi$
$809$	−30.1913	−1.06147	−0.530735	+	0.847538i	$0.678084\pi$
$810$	0	0
$811$	42.7764	1.50208	0.751040	−	0.660256i	$-0.229552\pi$
$811$	42.7764	1.50208	0.751040	+	0.660256i	$0.229552\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	67.5959	2.36778
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.7634	−0.550147	−0.275074	−	0.961423i	$-0.588702\pi$
$821$	−15.7634	−0.550147	−0.275074	+	0.961423i	$0.588702\pi$
$822$	0	0
$823$	28.9842	1.01032	0.505162	−	0.863024i	$-0.331433\pi$
$823$	28.9842	1.01032	0.505162	+	0.863024i	$0.331433\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.20204	−0.0765725	−0.0382862	−	0.999267i	$-0.512190\pi$
$827$	−2.20204	−0.0765725	−0.0382862	+	0.999267i	$0.512190\pi$
$828$	0	0
$829$	21.5959	0.750057	0.375029	−	0.927013i	$-0.377633\pi$
$829$	21.5959	0.750057	0.375029	+	0.927013i	$0.377633\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.17837	0.110124
$834$	0	0
$835$	−22.6274	−0.783054
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.1010	−1.21182	−0.605911	−	0.795533i	$-0.707191\pi$
$839$	−35.1010	−1.21182	−0.605911	+	0.795533i	$0.707191\pi$
$840$	0	0
$841$	10.5959	0.365376
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.9992	−0.447187
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.7980	1.98129
$852$	0	0
$853$	−7.30306	−0.250052	−0.125026	−	0.992153i	$-0.539901\pi$
$853$	−7.30306	−0.250052	−0.125026	+	0.992153i	$0.539901\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.5050	−1.41778	−0.708892	−	0.705317i	$-0.750805\pi$
$857$	−41.5050	−1.41778	−0.708892	+	0.705317i	$0.750805\pi$
$858$	0	0
$859$	−15.1278	−0.516152	−0.258076	−	0.966125i	$-0.583089\pi$
$859$	−15.1278	−0.516152	−0.258076	+	0.966125i	$0.583089\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.4949	1.65079	0.825393	−	0.564559i	$-0.190954\pi$
$863$	48.4949	1.65079	0.825393	+	0.564559i	$0.190954\pi$
$864$	0	0
$865$	24.9898	0.849679
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.73545	−0.160639
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	19.5959	0.662463
$876$	0	0
$877$	6.89898	0.232962	0.116481	−	0.993193i	$-0.462839\pi$
$877$	6.89898	0.232962	0.116481	+	0.993193i	$0.462839\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−28.9842	−0.976501	−0.488251	−	0.872703i	$-0.662365\pi$
$881$	−28.9842	−0.976501	−0.488251	+	0.872703i	$0.662365\pi$
$882$	0	0
$883$	−27.7128	−0.932610	−0.466305	−	0.884624i	$-0.654415\pi$
$883$	−27.7128	−0.932610	−0.466305	+	0.884624i	$0.654415\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.39388	0.181109	0.0905543	−	0.995892i	$-0.471136\pi$
$887$	5.39388	0.181109	0.0905543	+	0.995892i	$0.471136\pi$
$888$	0	0
$889$	70.7878	2.37415
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.571458	0.0191231
$894$	0	0
$895$	27.7128	0.926337
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.5959	−1.18719
$900$	0	0
$901$	2.60612	0.0868225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.65685	−0.188040
$906$	0	0
$907$	46.5262	1.54488	0.772438	−	0.635090i	$-0.219037\pi$
$907$	46.5262	1.54488	0.772438	+	0.635090i	$0.219037\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.2929	1.00365	0.501824	−	0.864970i	$-0.332663\pi$
$911$	30.2929	1.00365	0.501824	+	0.864970i	$0.332663\pi$
$912$	0	0
$913$	−5.79796	−0.191884
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47.7975	1.57841
$918$	0	0
$919$	45.6048	1.50436	0.752181	−	0.658956i	$-0.229002\pi$
$919$	45.6048	1.50436	0.752181	+	0.658956i	$0.229002\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−25.7980	−0.849150
$924$	0	0
$925$	−35.3939	−1.16374
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.47090	0.310730	0.155365	−	0.987857i	$-0.450345\pi$
$929$	9.47090	0.310730	0.155365	+	0.987857i	$0.450345\pi$
$930$	0	0
$931$	3.17837	0.104167
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.79796	−0.0587995
$936$	0	0
$937$	53.5959	1.75090	0.875451	−	0.483307i	$-0.160564\pi$
$937$	53.5959	1.75090	0.875451	+	0.483307i	$0.160564\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.3908	−1.25151	−0.625753	−	0.780021i	$-0.715208\pi$
$941$	−38.3908	−1.25151	−0.625753	+	0.780021i	$0.715208\pi$
$942$	0	0
$943$	30.8270	1.00386
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.1918	−1.27356	−0.636782	−	0.771044i	$-0.719735\pi$
$947$	−39.1918	−1.27356	−0.636782	+	0.771044i	$0.719735\pi$
$948$	0	0
$949$	17.3939	0.564629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17.6062	0.570322	0.285161	−	0.958480i	$-0.407953\pi$
$953$	17.6062	0.570322	0.285161	+	0.958480i	$0.407953\pi$
$954$	0	0
$955$	−75.5103	−2.44346
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.0560	−0.710006
$966$	0	0
$967$	−34.2911	−1.10273	−0.551363	−	0.834265i	$-0.685892\pi$
$967$	−34.2911	−1.10273	−0.551363	+	0.834265i	$0.685892\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	60.9898	1.95726	0.978628	−	0.205639i	$-0.0659274\pi$
$971$	60.9898	1.95726	0.978628	+	0.205639i	$0.0659274\pi$
$972$	0	0
$973$	6.60612	0.211783
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.38551	−0.140305	−0.0701524	−	0.997536i	$-0.522349\pi$
$977$	−4.38551	−0.140305	−0.0701524	+	0.997536i	$0.522349\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.4949	−1.54675	−0.773374	−	0.633951i	$-0.781432\pi$
$983$	−48.4949	−1.54675	−0.773374	+	0.633951i	$0.781432\pi$
$984$	0	0
$985$	−69.3939	−2.21107
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.8270	0.980241
$990$	0	0
$991$	16.9706	0.539088	0.269544	−	0.962988i	$-0.413127\pi$
$991$	16.9706	0.539088	0.269544	+	0.962988i	$0.413127\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−41.1010	−1.30168	−0.650841	−	0.759214i	$-0.725584\pi$
$997$	−41.1010	−1.30168	−0.650841	+	0.759214i	$0.725584\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.a.da.1.3		4
3.2	odd	2		6336.2.a.db.1.1		4
4.3	odd	2		6336.2.a.db.1.4		4
8.3	odd	2		3168.2.a.bi.1.2	✓	4
8.5	even	2		3168.2.a.bj.1.1	yes	4
12.11	even	2	inner	6336.2.a.da.1.2		4
24.5	odd	2		3168.2.a.bi.1.3	yes	4
24.11	even	2		3168.2.a.bj.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3168.2.a.bi.1.2	✓	4	8.3	odd	2
3168.2.a.bi.1.3	yes	4	24.5	odd	2
3168.2.a.bj.1.1	yes	4	8.5	even	2
3168.2.a.bj.1.4	yes	4	24.11	even	2
6336.2.a.da.1.2		4	12.11	even	2	inner
6336.2.a.da.1.3		4	1.1	even	1	trivial
6336.2.a.db.1.1		4	3.2	odd	2
6336.2.a.db.1.4		4	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} -3.46410 q^{7} -1.00000 q^{11} +2.89898 q^{13} +0.635674 q^{17} +0.635674 q^{19} -4.89898 q^{23} +3.00000 q^{25} +6.29253 q^{29} -5.65685 q^{31} -9.79796 q^{35} -11.7980 q^{37} -6.29253 q^{41} -6.29253 q^{43} +0.898979 q^{47} +5.00000 q^{49} +4.09978 q^{53} -2.82843 q^{55} +5.79796 q^{59} -10.8990 q^{61} +8.19955 q^{65} -8.89898 q^{71} +6.00000 q^{73} +3.46410 q^{77} +4.73545 q^{79} +5.79796 q^{83} +1.79796 q^{85} +11.3137 q^{89} -10.0424 q^{91} +1.79796 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} - 8 q^{13} + 12 q^{25} - 8 q^{37} - 16 q^{47} + 20 q^{49} - 16 q^{59} - 24 q^{61} - 16 q^{71} + 24 q^{73} - 16 q^{83} - 32 q^{85} - 32 q^{95} + 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^11 - 8 * q^13 + 12 * q^25 - 8 * q^37 - 16 * q^47 + 20 * q^49 - 16 * q^59 - 24 * q^61 - 16 * q^71 + 24 * q^73 - 16 * q^83 - 32 * q^85 - 32 * q^95 + 40 * q^97

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