LMFDB - Embedded newform 8112.2.a.bu.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{9} -3.46410 q^{11} -3.46410 q^{15} +3.46410 q^{19} +1.73205 q^{21} -6.00000 q^{23} +7.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +1.73205 q^{31} -3.46410 q^{33} -6.00000 q^{35} -6.92820 q^{41} -1.00000 q^{43} -3.46410 q^{45} -3.46410 q^{47} -4.00000 q^{49} +12.0000 q^{53} +12.0000 q^{55} +3.46410 q^{57} +3.46410 q^{59} +1.00000 q^{61} +1.73205 q^{63} -8.66025 q^{67} -6.00000 q^{69} -10.3923 q^{71} +1.73205 q^{73} +7.00000 q^{75} -6.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -13.8564 q^{83} +6.00000 q^{87} -6.92820 q^{89} +1.73205 q^{93} -12.0000 q^{95} +5.19615 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} - 12 q^{23} + 14 q^{25} + 2 q^{27} + 12 q^{29} - 12 q^{35} - 2 q^{43} - 8 q^{49} + 24 q^{53} + 24 q^{55} + 2 q^{61} - 12 q^{69} + 14 q^{75} - 12 q^{77} + 22 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots - 24 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^23 + 14 * q^25 + 2 * q^27 + 12 * q^29 - 12 * q^35 - 2 * q^43 - 8 * q^49 + 24 * q^53 + 24 * q^55 + 2 * q^61 - 12 * q^69 + 14 * q^75 - 12 * q^77 + 22 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^95

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	1.73205	0.377964
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	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
$31$	1.73205	0.311086	0.155543	−	0.987829i	$-0.450287\pi$
$31$	1.73205	0.311086	0.155543	+	0.987829i	$0.450287\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	−6.00000	−1.01419
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	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
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	1.73205	0.218218
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.66025	−1.05802	−0.529009	−	0.848616i	$-0.677436\pi$
$67$	−8.66025	−1.05802	−0.529009	+	0.848616i	$0.677436\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	1.73205	0.202721	0.101361	−	0.994850i	$-0.467680\pi$
$73$	1.73205	0.202721	0.101361	+	0.994850i	$0.467680\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	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
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.73205	0.179605
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	5.19615	0.527589	0.263795	−	0.964579i	$-0.415026\pi$
$97$	5.19615	0.527589	0.263795	+	0.964579i	$0.415026\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	15.5885	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$109$	15.5885	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	20.7846	1.93817
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.92820	−0.624695
$124$	0	0
$125$	−6.92820	−0.619677
$126$	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
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−3.46410	−0.298142
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−20.7846	−1.72607
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	6.92820	0.567581	0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820	0.567581	0.283790	+	0.958886i	$0.408408\pi$
$150$	0	0
$151$	3.46410	0.281905	0.140952	−	0.990016i	$-0.454984\pi$
$151$	3.46410	0.281905	0.140952	+	0.990016i	$0.454984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−10.3923	−0.819028
$162$	0	0
$163$	19.0526	1.49231	0.746156	−	0.665771i	$-0.231897\pi$
$163$	19.0526	1.49231	0.746156	+	0.665771i	$0.231897\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	3.46410	0.264906
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	12.1244	0.916515
$176$	0	0
$177$	3.46410	0.260378
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.73205	0.125988
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	15.5885	1.12208	0.561041	−	0.827788i	$-0.310401\pi$
$193$	15.5885	1.12208	0.561041	+	0.827788i	$0.310401\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	−8.66025	−0.610847
$202$	0	0
$203$	10.3923	0.729397
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−10.3923	−0.712069
$214$	0	0
$215$	3.46410	0.236250
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	1.73205	0.117041
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−17.3205	−1.15987	−0.579934	−	0.814664i	$-0.696921\pi$
$223$	−17.3205	−1.15987	−0.579934	+	0.814664i	$0.696921\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	20.7846	1.37952	0.689761	−	0.724037i	$-0.257715\pi$
$227$	20.7846	1.37952	0.689761	+	0.724037i	$0.257715\pi$
$228$	0	0
$229$	27.7128	1.83131	0.915657	−	0.401960i	$-0.131671\pi$
$229$	27.7128	1.83131	0.915657	+	0.401960i	$0.131671\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	11.0000	0.714527
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	20.7846	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$241$	20.7846	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	13.8564	0.885253
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−13.8564	−0.878114
$250$	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
$253$	20.7846	1.30672
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−41.5692	−2.55358
$266$	0	0
$267$	−6.92820	−0.423999
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−5.19615	−0.315644	−0.157822	−	0.987468i	$-0.550447\pi$
$271$	−5.19615	−0.315644	−0.157822	+	0.987468i	$0.550447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.2487	−1.46225
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	1.73205	0.103695
$280$	0	0
$281$	−24.2487	−1.44656	−0.723278	−	0.690557i	$-0.757366\pi$
$281$	−24.2487	−1.44656	−0.723278	+	0.690557i	$0.757366\pi$
$282$	0	0
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	5.19615	0.304604
$292$	0	0
$293$	−17.3205	−1.01187	−0.505937	−	0.862570i	$-0.668853\pi$
$293$	−17.3205	−1.01187	−0.505937	+	0.862570i	$0.668853\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−3.46410	−0.201008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.73205	−0.0998337
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	−3.46410	−0.198354
$306$	0	0
$307$	1.73205	0.0988534	0.0494267	−	0.998778i	$-0.484261\pi$
$307$	1.73205	0.0988534	0.0494267	+	0.998778i	$0.484261\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	0	0
$315$	−6.00000	−0.338062
$316$	0	0
$317$	−6.92820	−0.389127	−0.194563	−	0.980890i	$-0.562329\pi$
$317$	−6.92820	−0.389127	−0.194563	+	0.980890i	$0.562329\pi$
$318$	0	0
$319$	−20.7846	−1.16371
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.5885	0.862044
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	5.19615	0.285606	0.142803	−	0.989751i	$-0.454388\pi$
$331$	5.19615	0.285606	0.142803	+	0.989751i	$0.454388\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	30.0000	1.63908
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	20.7846	1.11901
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	19.0526	1.01986	0.509930	−	0.860216i	$-0.329671\pi$
$349$	19.0526	1.01986	0.509930	+	0.860216i	$0.329671\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3923	0.553127	0.276563	−	0.960996i	$-0.410804\pi$
$353$	10.3923	0.553127	0.276563	+	0.960996i	$0.410804\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.92820	−0.365657	−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820	−0.365657	−0.182828	+	0.983145i	$0.558525\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	−6.92820	−0.360668
$370$	0	0
$371$	20.7846	1.07908
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	0	0
$378$	0	0
$379$	22.5167	1.15660	0.578302	−	0.815823i	$-0.303716\pi$
$379$	22.5167	1.15660	0.578302	+	0.815823i	$0.303716\pi$
$380$	0	0
$381$	13.0000	0.666010
$382$	0	0
$383$	27.7128	1.41606	0.708029	−	0.706183i	$-0.249584\pi$
$383$	27.7128	1.41606	0.708029	+	0.706183i	$0.249584\pi$
$384$	0	0
$385$	20.7846	1.05928
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	−38.1051	−1.91728
$396$	0	0
$397$	15.5885	0.782362	0.391181	−	0.920314i	$-0.372067\pi$
$397$	15.5885	0.782362	0.391181	+	0.920314i	$0.372067\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.46410	−0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8.66025	−0.428222	−0.214111	−	0.976809i	$-0.568685\pi$
$409$	−8.66025	−0.428222	−0.214111	+	0.976809i	$0.568685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	12.1244	0.590905	0.295452	−	0.955357i	$-0.404530\pi$
$421$	12.1244	0.590905	0.295452	+	0.955357i	$0.404530\pi$
$422$	0	0
$423$	−3.46410	−0.168430
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.73205	0.0838198
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.7846	−1.00116	−0.500580	−	0.865690i	$-0.666880\pi$
$431$	−20.7846	−1.00116	−0.500580	+	0.865690i	$0.666880\pi$
$432$	0	0
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	0	0
$435$	−20.7846	−0.996546
$436$	0	0
$437$	−20.7846	−0.994263
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	6.92820	0.327693
$448$	0	0
$449$	38.1051	1.79829	0.899146	−	0.437649i	$-0.144189\pi$
$449$	38.1051	1.79829	0.899146	+	0.437649i	$0.144189\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	3.46410	0.162758
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.3731	−1.70146	−0.850730	−	0.525603i	$-0.823840\pi$
$457$	−36.3731	−1.70146	−0.850730	+	0.525603i	$0.823840\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.5692	1.93607	0.968036	−	0.250812i	$-0.0806976\pi$
$461$	41.5692	1.93607	0.968036	+	0.250812i	$0.0806976\pi$
$462$	0	0
$463$	36.3731	1.69040	0.845200	−	0.534450i	$-0.179481\pi$
$463$	36.3731	1.69040	0.845200	+	0.534450i	$0.179481\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	11.0000	0.506853
$472$	0	0
$473$	3.46410	0.159280
$474$	0	0
$475$	24.2487	1.11261
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	34.6410	1.58279	0.791394	−	0.611306i	$-0.209356\pi$
$479$	34.6410	1.58279	0.791394	+	0.611306i	$0.209356\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−10.3923	−0.472866
$484$	0	0
$485$	−18.0000	−0.817338
$486$	0	0
$487$	−24.2487	−1.09881	−0.549407	−	0.835555i	$-0.685146\pi$
$487$	−24.2487	−1.09881	−0.549407	+	0.835555i	$0.685146\pi$
$488$	0	0
$489$	19.0526	0.861586
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	−18.0000	−0.807410
$498$	0	0
$499$	31.1769	1.39567	0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769	1.39567	0.697835	+	0.716258i	$0.254147\pi$
$500$	0	0
$501$	−6.92820	−0.309529
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	−62.3538	−2.77471
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.3205	−0.767718	−0.383859	−	0.923392i	$-0.625405\pi$
$509$	−17.3205	−0.767718	−0.383859	+	0.923392i	$0.625405\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	3.46410	0.152944
$514$	0	0
$515$	3.46410	0.152647
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	12.1244	0.529150
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	3.46410	0.150329
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−20.7846	−0.898597
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	13.8564	0.596838
$540$	0	0
$541$	−29.4449	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$541$	−29.4449	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−54.0000	−2.31311
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	19.0526	0.810197
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.7128	1.17423	0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128	1.17423	0.587115	+	0.809504i	$0.300264\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	0	0
$565$	−20.7846	−0.874415
$566$	0	0
$567$	1.73205	0.0727393
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	−42.0000	−1.75152
$576$	0	0
$577$	34.6410	1.44212	0.721062	−	0.692870i	$-0.243654\pi$
$577$	34.6410	1.44212	0.721062	+	0.692870i	$0.243654\pi$
$578$	0	0
$579$	15.5885	0.647834
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−41.5692	−1.72162
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.1769	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	13.8564	0.569976
$592$	0	0
$593$	3.46410	0.142254	0.0711268	−	0.997467i	$-0.477341\pi$
$593$	3.46410	0.142254	0.0711268	+	0.997467i	$0.477341\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.00000	0.286491
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−8.66025	−0.352673
$604$	0	0
$605$	−3.46410	−0.140836
$606$	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
$609$	10.3923	0.421117
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.66025	0.349784	0.174892	−	0.984588i	$-0.444042\pi$
$613$	8.66025	0.349784	0.174892	+	0.984588i	$0.444042\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	10.3923	0.418378	0.209189	−	0.977875i	$-0.432918\pi$
$617$	10.3923	0.418378	0.209189	+	0.977875i	$0.432918\pi$
$618$	0	0
$619$	−25.9808	−1.04425	−0.522127	−	0.852867i	$-0.674861\pi$
$619$	−25.9808	−1.04425	−0.522127	+	0.852867i	$0.674861\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1.73205	−0.0689519	−0.0344759	−	0.999406i	$-0.510976\pi$
$631$	−1.73205	−0.0689519	−0.0344759	+	0.999406i	$0.510976\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	−45.0333	−1.78709
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.3923	−0.411113
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	19.0526	0.751360	0.375680	−	0.926750i	$-0.377409\pi$
$643$	19.0526	0.751360	0.375680	+	0.926750i	$0.377409\pi$
$644$	0	0
$645$	3.46410	0.136399
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	3.00000	0.117579
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	20.7846	0.812122
$656$	0	0
$657$	1.73205	0.0675737
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	25.9808	1.01053	0.505267	−	0.862963i	$-0.331394\pi$
$661$	25.9808	1.01053	0.505267	+	0.862963i	$0.331394\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.7846	−0.805993
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	−17.3205	−0.669650
$670$	0	0
$671$	−3.46410	−0.133730
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	0	0
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	9.00000	0.345388
$680$	0	0
$681$	20.7846	0.796468
$682$	0	0
$683$	−24.2487	−0.927851	−0.463926	−	0.885874i	$-0.653559\pi$
$683$	−24.2487	−0.927851	−0.463926	+	0.885874i	$0.653559\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	27.7128	1.05731
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−43.3013	−1.64726	−0.823629	−	0.567129i	$-0.808054\pi$
$691$	−43.3013	−1.64726	−0.823629	+	0.567129i	$0.808054\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	17.3205	0.657004
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	31.1769	1.17253
$708$	0	0
$709$	−19.0526	−0.715534	−0.357767	−	0.933811i	$-0.616462\pi$
$709$	−19.0526	−0.715534	−0.357767	+	0.933811i	$0.616462\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−10.3923	−0.389195
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−1.73205	−0.0645049
$722$	0	0
$723$	20.7846	0.772988
$724$	0	0
$725$	42.0000	1.55984
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−39.8372	−1.47142	−0.735710	−	0.677297i	$-0.763151\pi$
$733$	−39.8372	−1.47142	−0.735710	+	0.677297i	$0.763151\pi$
$734$	0	0
$735$	13.8564	0.511101
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	45.0333	1.65658	0.828289	−	0.560301i	$-0.189315\pi$
$739$	45.0333	1.65658	0.828289	+	0.560301i	$0.189315\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.3923	0.381257	0.190628	−	0.981662i	$-0.438947\pi$
$743$	10.3923	0.381257	0.190628	+	0.981662i	$0.438947\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	−13.8564	−0.506979
$748$	0	0
$749$	10.3923	0.379727
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	20.7846	0.754434
$760$	0	0
$761$	−20.7846	−0.753442	−0.376721	−	0.926327i	$-0.622948\pi$
$761$	−20.7846	−0.753442	−0.376721	+	0.926327i	$0.622948\pi$
$762$	0	0
$763$	27.0000	0.977466
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	51.9615	1.86893	0.934463	−	0.356060i	$-0.115880\pi$
$773$	51.9615	1.86893	0.934463	+	0.356060i	$0.115880\pi$
$774$	0	0
$775$	12.1244	0.435520
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−38.1051	−1.36003
$786$	0	0
$787$	−32.9090	−1.17308	−0.586539	−	0.809921i	$-0.699510\pi$
$787$	−32.9090	−1.17308	−0.586539	+	0.809921i	$0.699510\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	10.3923	0.369508
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−41.5692	−1.47431
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.92820	−0.244796
$802$	0	0
$803$	−6.00000	−0.211735
$804$	0	0
$805$	36.0000	1.26883
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	−25.9808	−0.912308	−0.456154	−	0.889901i	$-0.650773\pi$
$811$	−25.9808	−0.912308	−0.456154	+	0.889901i	$0.650773\pi$
$812$	0	0
$813$	−5.19615	−0.182237
$814$	0	0
$815$	−66.0000	−2.31188
$816$	0	0
$817$	−3.46410	−0.121194
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.2487	0.846286	0.423143	−	0.906063i	$-0.360927\pi$
$821$	24.2487	0.846286	0.423143	+	0.906063i	$0.360927\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	−24.2487	−0.844232
$826$	0	0
$827$	−48.4974	−1.68642	−0.843210	−	0.537584i	$-0.819337\pi$
$827$	−48.4974	−1.68642	−0.843210	+	0.537584i	$0.819337\pi$
$828$	0	0
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	1.73205	0.0598684
$838$	0	0
$839$	−31.1769	−1.07635	−0.538173	−	0.842834i	$-0.680885\pi$
$839$	−31.1769	−1.07635	−0.538173	+	0.842834i	$0.680885\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−24.2487	−0.835170
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.73205	0.0595140
$848$	0	0
$849$	−11.0000	−0.377519
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−25.9808	−0.889564	−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808	−0.889564	−0.444782	+	0.895639i	$0.646719\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	−17.3205	−0.589597	−0.294798	−	0.955559i	$-0.595253\pi$
$863$	−17.3205	−0.589597	−0.294798	+	0.955559i	$0.595253\pi$
$864$	0	0
$865$	−20.7846	−0.706698
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−38.1051	−1.29263
$870$	0	0
$871$	0	0
$872$	0	0
$873$	5.19615	0.175863
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−41.5692	−1.40369	−0.701846	−	0.712328i	$-0.747641\pi$
$877$	−41.5692	−1.40369	−0.701846	+	0.712328i	$0.747641\pi$
$878$	0	0
$879$	−17.3205	−0.584206
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	−5.00000	−0.168263	−0.0841317	−	0.996455i	$-0.526812\pi$
$883$	−5.00000	−0.168263	−0.0841317	+	0.996455i	$0.526812\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	22.5167	0.755185
$890$	0	0
$891$	−3.46410	−0.116052
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−41.5692	−1.38951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.3923	0.346603
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1.73205	−0.0576390
$904$	0	0
$905$	48.4974	1.61211
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	−3.46410	−0.114520
$916$	0	0
$917$	−10.3923	−0.343184
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	1.73205	0.0570730
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.00000	−0.0328443
$928$	0	0
$929$	20.7846	0.681921	0.340960	−	0.940078i	$-0.389248\pi$
$929$	20.7846	0.681921	0.340960	+	0.940078i	$0.389248\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	13.0000	0.424239
$940$	0	0
$941$	10.3923	0.338779	0.169390	−	0.985549i	$-0.445820\pi$
$941$	10.3923	0.338779	0.169390	+	0.985549i	$0.445820\pi$
$942$	0	0
$943$	41.5692	1.35368
$944$	0	0
$945$	−6.00000	−0.195180
$946$	0	0
$947$	−48.4974	−1.57595	−0.787977	−	0.615704i	$-0.788872\pi$
$947$	−48.4974	−1.57595	−0.787977	+	0.615704i	$0.788872\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.92820	−0.224662
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−62.3538	−2.01772
$956$	0	0
$957$	−20.7846	−0.671871
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.0000	−0.903226
$962$	0	0
$963$	6.00000	0.193347
$964$	0	0
$965$	−54.0000	−1.73832
$966$	0	0
$967$	−24.2487	−0.779786	−0.389893	−	0.920860i	$-0.627488\pi$
$967$	−24.2487	−0.779786	−0.389893	+	0.920860i	$0.627488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	−8.66025	−0.277635
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.7128	−0.886611	−0.443306	−	0.896370i	$-0.646194\pi$
$977$	−27.7128	−0.886611	−0.443306	+	0.896370i	$0.646194\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	15.5885	0.497701
$982$	0	0
$983$	10.3923	0.331463	0.165732	−	0.986171i	$-0.447001\pi$
$983$	10.3923	0.331463	0.165732	+	0.986171i	$0.447001\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	5.19615	0.164895
$994$	0	0
$995$	−24.2487	−0.768736
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bu.1.1		2
4.3	odd	2		507.2.a.e.1.1		2
12.11	even	2		1521.2.a.h.1.2		2
13.2	odd	12		624.2.bv.b.433.1		2
13.7	odd	12		624.2.bv.b.49.1		2
13.12	even	2	inner	8112.2.a.bu.1.2		2
39.2	even	12		1872.2.by.f.433.1		2
39.20	even	12		1872.2.by.f.1297.1		2
52.3	odd	6		507.2.e.f.22.1		4
52.7	even	12		39.2.j.a.10.1	yes	2
52.11	even	12		507.2.j.b.316.1		2
52.15	even	12		39.2.j.a.4.1	✓	2
52.19	even	12		507.2.j.b.361.1		2
52.23	odd	6		507.2.e.f.22.2		4
52.31	even	4		507.2.b.c.337.2		2
52.35	odd	6		507.2.e.f.484.1		4
52.43	odd	6		507.2.e.f.484.2		4
52.47	even	4		507.2.b.c.337.1		2
52.51	odd	2		507.2.a.e.1.2		2
156.47	odd	4		1521.2.b.f.1351.2		2
156.59	odd	12		117.2.q.a.10.1		2
156.83	odd	4		1521.2.b.f.1351.1		2
156.119	odd	12		117.2.q.a.82.1		2
156.155	even	2		1521.2.a.h.1.1		2
260.7	odd	12		975.2.w.d.49.2		4
260.59	even	12		975.2.bc.c.751.1		2
260.67	odd	12		975.2.w.d.199.1		4
260.119	even	12		975.2.bc.c.901.1		2
260.163	odd	12		975.2.w.d.49.1		4
260.223	odd	12		975.2.w.d.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.j.a.4.1	✓	2	52.15	even	12
39.2.j.a.10.1	yes	2	52.7	even	12
117.2.q.a.10.1		2	156.59	odd	12
117.2.q.a.82.1		2	156.119	odd	12
507.2.a.e.1.1		2	4.3	odd	2
507.2.a.e.1.2		2	52.51	odd	2
507.2.b.c.337.1		2	52.47	even	4
507.2.b.c.337.2		2	52.31	even	4
507.2.e.f.22.1		4	52.3	odd	6
507.2.e.f.22.2		4	52.23	odd	6
507.2.e.f.484.1		4	52.35	odd	6
507.2.e.f.484.2		4	52.43	odd	6
507.2.j.b.316.1		2	52.11	even	12
507.2.j.b.361.1		2	52.19	even	12
624.2.bv.b.49.1		2	13.7	odd	12
624.2.bv.b.433.1		2	13.2	odd	12
975.2.w.d.49.1		4	260.163	odd	12
975.2.w.d.49.2		4	260.7	odd	12
975.2.w.d.199.1		4	260.67	odd	12
975.2.w.d.199.2		4	260.223	odd	12
975.2.bc.c.751.1		2	260.59	even	12
975.2.bc.c.901.1		2	260.119	even	12
1521.2.a.h.1.1		2	156.155	even	2
1521.2.a.h.1.2		2	12.11	even	2
1521.2.b.f.1351.1		2	156.83	odd	4
1521.2.b.f.1351.2		2	156.47	odd	4
1872.2.by.f.433.1		2	39.2	even	12
1872.2.by.f.1297.1		2	39.20	even	12
8112.2.a.bu.1.1		2	1.1	even	1	trivial
8112.2.a.bu.1.2		2	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{9} -3.46410 q^{11} -3.46410 q^{15} +3.46410 q^{19} +1.73205 q^{21} -6.00000 q^{23} +7.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +1.73205 q^{31} -3.46410 q^{33} -6.00000 q^{35} -6.92820 q^{41} -1.00000 q^{43} -3.46410 q^{45} -3.46410 q^{47} -4.00000 q^{49} +12.0000 q^{53} +12.0000 q^{55} +3.46410 q^{57} +3.46410 q^{59} +1.00000 q^{61} +1.73205 q^{63} -8.66025 q^{67} -6.00000 q^{69} -10.3923 q^{71} +1.73205 q^{73} +7.00000 q^{75} -6.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} -13.8564 q^{83} +6.00000 q^{87} -6.92820 q^{89} +1.73205 q^{93} -12.0000 q^{95} +5.19615 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} - 12 q^{23} + 14 q^{25} + 2 q^{27} + 12 q^{29} - 12 q^{35} - 2 q^{43} - 8 q^{49} + 24 q^{53} + 24 q^{55} + 2 q^{61} - 12 q^{69} + 14 q^{75} - 12 q^{77} + 22 q^{79} + 2 q^{81} + 12 q^{87}+ \cdots - 24 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^23 + 14 * q^25 + 2 * q^27 + 12 * q^29 - 12 * q^35 - 2 * q^43 - 8 * q^49 + 24 * q^53 + 24 * q^55 + 2 * q^61 - 12 * q^69 + 14 * q^75 - 12 * q^77 + 22 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more