LMFDB - Embedded newform 8112.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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:	$ 2 $
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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{11} +6.00000 q^{17} -3.46410 q^{19} -3.46410 q^{21} -5.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +3.46410 q^{31} -3.46410 q^{33} -6.92820 q^{37} -6.92820 q^{41} -4.00000 q^{43} +3.46410 q^{47} +5.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} -3.46410 q^{57} +10.3923 q^{59} -2.00000 q^{61} -3.46410 q^{63} +10.3923 q^{67} -3.46410 q^{71} -5.00000 q^{75} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} +6.00000 q^{87} +6.92820 q^{89} +3.46410 q^{93} -13.8564 q^{97} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 10 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{43} + 10 q^{49} + 12 q^{51} + 12 q^{53} - 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 10 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^43 + 10 * q^49 + 12 * q^51 + 12 * q^53 - 4 * q^61 - 10 * q^75 + 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	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
$9$	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
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	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
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	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
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.3923	1.26962	0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923	1.26962	0.634811	+	0.772667i	$0.281078\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	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	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.46410	0.359211
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.8564	−1.40690	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564	−1.40690	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	−3.46410	−0.348155
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	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
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	−6.92820	−0.657596
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.7846	−1.90532
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.92820	−0.624695
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.7846	1.77575	0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846	1.77575	0.887875	+	0.460086i	$0.152181\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.00000	0.412393
$148$	0	0
$149$	13.8564	1.13516	0.567581	−	0.823318i	$-0.307880\pi$
$149$	13.8564	1.13516	0.567581	+	0.823318i	$0.307880\pi$
$150$	0	0
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.46410	0.271329	0.135665	−	0.990755i	$-0.456683\pi$
$163$	3.46410	0.271329	0.135665	+	0.990755i	$0.456683\pi$
$164$	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
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	17.3205	1.30931
$176$	0	0
$177$	10.3923	0.781133
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.7846	−1.51992
$188$	0	0
$189$	−3.46410	−0.251976
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	10.3923	0.733017
$202$	0	0
$203$	−20.7846	−1.45879
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	−3.46410	−0.237356
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.46410	0.231973	0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410	0.231973	0.115987	+	0.993251i	$0.462997\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	17.3205	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$228$	0	0
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	−13.8564	−0.892570	−0.446285	−	0.894891i	$-0.647253\pi$
$241$	−13.8564	−0.892570	−0.446285	+	0.894891i	$0.647253\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.92820	0.423999
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	10.3923	0.631288	0.315644	−	0.948878i	$-0.397780\pi$
$271$	10.3923	0.631288	0.315644	+	0.948878i	$0.397780\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	17.3205	1.04447
$276$	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
$279$	3.46410	0.207390
$280$	0	0
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−13.8564	−0.812277
$292$	0	0
$293$	−27.7128	−1.61900	−0.809500	−	0.587120i	$-0.800262\pi$
$293$	−27.7128	−1.61900	−0.809500	+	0.587120i	$0.800262\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.46410	−0.201008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.8564	0.798670
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.3923	−0.593120	−0.296560	−	0.955014i	$-0.595840\pi$
$307$	−10.3923	−0.593120	−0.296560	+	0.955014i	$0.595840\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.8564	0.778253	0.389127	−	0.921184i	$-0.372777\pi$
$317$	13.8564	0.778253	0.389127	+	0.921184i	$0.372777\pi$
$318$	0	0
$319$	−20.7846	−1.16371
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.92820	0.383131
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−3.46410	−0.190404	−0.0952021	−	0.995458i	$-0.530350\pi$
$331$	−3.46410	−0.190404	−0.0952021	+	0.995458i	$0.530350\pi$
$332$	0	0
$333$	−6.92820	−0.379663
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	6.92820	0.370858	0.185429	−	0.982658i	$-0.440632\pi$
$349$	6.92820	0.370858	0.185429	+	0.982658i	$0.440632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	34.6410	1.84376	0.921878	−	0.387481i	$-0.126655\pi$
$353$	34.6410	1.84376	0.921878	+	0.387481i	$0.126655\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−20.7846	−1.10004
$358$	0	0
$359$	17.3205	0.914141	0.457071	−	0.889430i	$-0.348899\pi$
$359$	17.3205	0.914141	0.457071	+	0.889430i	$0.348899\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−6.92820	−0.360668
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−17.3205	−0.889695	−0.444847	−	0.895606i	$-0.646742\pi$
$379$	−17.3205	−0.889695	−0.444847	+	0.895606i	$0.646742\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.6410	1.73858	0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410	1.73858	0.869291	+	0.494300i	$0.164576\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	−6.92820	−0.345978	−0.172989	−	0.984924i	$-0.555343\pi$
$401$	−6.92820	−0.345978	−0.172989	+	0.984924i	$0.555343\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	27.7128	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$409$	27.7128	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$410$	0	0
$411$	20.7846	1.02523
$412$	0	0
$413$	−36.0000	−1.77144
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	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
$421$	−34.6410	−1.68830	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	−34.6410	−1.68830	−0.844150	+	0.536107i	$0.819894\pi$
$422$	0	0
$423$	3.46410	0.168430
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	6.92820	0.335279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.8564	0.655386
$448$	0	0
$449$	−6.92820	−0.326962	−0.163481	−	0.986546i	$-0.552272\pi$
$449$	−6.92820	−0.326962	−0.163481	+	0.986546i	$0.552272\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	10.3923	0.488273
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.7128	−1.29635	−0.648175	−	0.761491i	$-0.724468\pi$
$457$	−27.7128	−1.29635	−0.648175	+	0.761491i	$0.724468\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	−17.3205	−0.804952	−0.402476	−	0.915430i	$-0.631850\pi$
$463$	−17.3205	−0.804952	−0.402476	+	0.915430i	$0.631850\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	13.8564	0.637118
$474$	0	0
$475$	17.3205	0.794719
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−38.1051	−1.72671	−0.863354	−	0.504599i	$-0.831640\pi$
$487$	−38.1051	−1.72671	−0.863354	+	0.504599i	$0.831640\pi$
$488$	0	0
$489$	3.46410	0.156652
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	10.3923	0.465223	0.232612	−	0.972570i	$-0.425273\pi$
$499$	10.3923	0.465223	0.232612	+	0.972570i	$0.425273\pi$
$500$	0	0
$501$	17.3205	0.773823
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−41.5692	−1.84252	−0.921262	−	0.388943i	$-0.872840\pi$
$509$	−41.5692	−1.84252	−0.921262	+	0.388943i	$0.872840\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.46410	−0.152944
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	17.3205	0.755929
$526$	0	0
$527$	20.7846	0.905392
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	10.3923	0.450988
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−17.3205	−0.746047
$540$	0	0
$541$	−6.92820	−0.297867	−0.148933	−	0.988847i	$-0.547584\pi$
$541$	−6.92820	−0.297867	−0.148933	+	0.988847i	$0.547584\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−20.7846	−0.885454
$552$	0	0
$553$	−27.7128	−1.17847
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.8564	−0.587115	−0.293557	−	0.955941i	$-0.594839\pi$
$557$	−13.8564	−0.587115	−0.293557	+	0.955941i	$0.594839\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−20.7846	−0.877527
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.46410	−0.145479
$568$	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
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−20.7846	−0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.3923	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.92820	−0.284507	−0.142254	−	0.989830i	$-0.545435\pi$
$593$	−6.92820	−0.284507	−0.142254	+	0.989830i	$0.545435\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	10.3923	0.423207
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	−20.7846	−0.842235
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−20.7846	−0.839482	−0.419741	−	0.907644i	$-0.637879\pi$
$613$	−20.7846	−0.839482	−0.419741	+	0.907644i	$0.637879\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	31.1769	1.25311	0.626553	−	0.779379i	$-0.284465\pi$
$619$	31.1769	1.25311	0.626553	+	0.779379i	$0.284465\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	−41.5692	−1.65747
$630$	0	0
$631$	38.1051	1.51694	0.758470	−	0.651707i	$-0.225947\pi$
$631$	38.1051	1.51694	0.758470	+	0.651707i	$0.225947\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.46410	−0.137038
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	10.3923	0.409832	0.204916	−	0.978780i	$-0.434308\pi$
$643$	10.3923	0.409832	0.204916	+	0.978780i	$0.434308\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	20.7846	0.808428	0.404214	−	0.914665i	$-0.367545\pi$
$661$	20.7846	0.808428	0.404214	+	0.914665i	$0.367545\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	3.46410	0.133930
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	48.0000	1.84207
$680$	0	0
$681$	17.3205	0.663723
$682$	0	0
$683$	−31.1769	−1.19295	−0.596476	−	0.802631i	$-0.703433\pi$
$683$	−31.1769	−1.19295	−0.596476	+	0.802631i	$0.703433\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.92820	0.264327
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−45.0333	−1.71315	−0.856574	−	0.516024i	$-0.827412\pi$
$691$	−45.0333	−1.71315	−0.856574	+	0.516024i	$0.827412\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−41.5692	−1.57455
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	6.92820	0.260194	0.130097	−	0.991501i	$-0.458471\pi$
$709$	6.92820	0.260194	0.130097	+	0.991501i	$0.458471\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.3923	0.388108
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−27.7128	−1.03208
$722$	0	0
$723$	−13.8564	−0.515325
$724$	0	0
$725$	−30.0000	−1.11417
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	34.6410	1.27950	0.639748	−	0.768585i	$-0.279039\pi$
$733$	34.6410	1.27950	0.639748	+	0.768585i	$0.279039\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	−38.1051	−1.40172	−0.700860	−	0.713299i	$-0.747200\pi$
$739$	−38.1051	−1.40172	−0.700860	+	0.713299i	$0.747200\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.46410	−0.127086	−0.0635428	−	0.997979i	$-0.520240\pi$
$743$	−3.46410	−0.127086	−0.0635428	+	0.997979i	$0.520240\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.46410	0.126745
$748$	0	0
$749$	41.5692	1.51891
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.4974	1.75803	0.879015	−	0.476794i	$-0.158201\pi$
$761$	48.4974	1.75803	0.879015	+	0.476794i	$0.158201\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	27.7128	0.999350	0.499675	−	0.866213i	$-0.333453\pi$
$769$	27.7128	0.999350	0.499675	+	0.866213i	$0.333453\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−13.8564	−0.498380	−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564	−0.498380	−0.249190	+	0.968455i	$0.580164\pi$
$774$	0	0
$775$	−17.3205	−0.622171
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−10.3923	−0.370446	−0.185223	−	0.982697i	$-0.559301\pi$
$787$	−10.3923	−0.370446	−0.185223	+	0.982697i	$0.559301\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	0	0
$801$	6.92820	0.244796
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	38.1051	1.33805	0.669026	−	0.743239i	$-0.266712\pi$
$811$	38.1051	1.33805	0.669026	+	0.743239i	$0.266712\pi$
$812$	0	0
$813$	10.3923	0.364474
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.8564	0.483592	0.241796	−	0.970327i	$-0.422264\pi$
$821$	13.8564	0.483592	0.241796	+	0.970327i	$0.422264\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	17.3205	0.603023
$826$	0	0
$827$	24.2487	0.843210	0.421605	−	0.906780i	$-0.361467\pi$
$827$	24.2487	0.843210	0.421605	+	0.906780i	$0.361467\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.46410	0.119737
$838$	0	0
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−6.92820	−0.238620
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	20.7846	0.711651	0.355826	−	0.934552i	$-0.384200\pi$
$853$	20.7846	0.711651	0.355826	+	0.934552i	$0.384200\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	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
$865$	0	0
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	−27.7128	−0.940093
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−13.8564	−0.468968
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.4974	1.63764	0.818821	−	0.574049i	$-0.194628\pi$
$877$	48.4974	1.63764	0.818821	+	0.574049i	$0.194628\pi$
$878$	0	0
$879$	−27.7128	−0.934730
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	27.7128	0.929458
$890$	0	0
$891$	−3.46410	−0.116052
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.7846	0.693206
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	13.8564	0.461112
$904$	0	0
$905$	0	0
$906$	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
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−41.5692	−1.37274
$918$	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
$921$	−10.3923	−0.342438
$922$	0	0
$923$	0	0
$924$	0	0
$925$	34.6410	1.13899
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−20.7846	−0.681921	−0.340960	−	0.940078i	$-0.610752\pi$
$929$	−20.7846	−0.681921	−0.340960	+	0.940078i	$0.610752\pi$
$930$	0	0
$931$	−17.3205	−0.567657
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.9615	1.68852	0.844261	−	0.535932i	$-0.180040\pi$
$947$	51.9615	1.68852	0.844261	+	0.535932i	$0.180040\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	13.8564	0.449325
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−20.7846	−0.671871
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−10.3923	−0.334194	−0.167097	−	0.985940i	$-0.553439\pi$
$967$	−10.3923	−0.334194	−0.167097	+	0.985940i	$0.553439\pi$
$968$	0	0
$969$	−20.7846	−0.667698
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−13.8564	−0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.4974	1.55157	0.775785	−	0.630997i	$-0.217354\pi$
$977$	48.4974	1.55157	0.775785	+	0.630997i	$0.217354\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	6.92820	0.221201
$982$	0	0
$983$	−51.9615	−1.65732	−0.828658	−	0.559756i	$-0.810895\pi$
$983$	−51.9615	−1.65732	−0.828658	+	0.559756i	$0.810895\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.0000	−0.381964
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	−3.46410	−0.109930
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	−6.92820	−0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bv.1.1		2
4.3	odd	2		507.2.a.f.1.2		2
12.11	even	2		1521.2.a.l.1.1		2
13.5	odd	4		624.2.c.e.337.1		2
13.8	odd	4		624.2.c.e.337.2		2
13.12	even	2	inner	8112.2.a.bv.1.2		2
39.5	even	4		1872.2.c.e.1585.1		2
39.8	even	4		1872.2.c.e.1585.2		2
52.3	odd	6		507.2.e.e.22.1		4
52.7	even	12		507.2.j.c.361.1		2
52.11	even	12		507.2.j.a.316.1		2
52.15	even	12		507.2.j.c.316.1		2
52.19	even	12		507.2.j.a.361.1		2
52.23	odd	6		507.2.e.e.22.2		4
52.31	even	4		39.2.b.a.25.1	✓	2
52.35	odd	6		507.2.e.e.484.1		4
52.43	odd	6		507.2.e.e.484.2		4
52.47	even	4		39.2.b.a.25.2	yes	2
52.51	odd	2		507.2.a.f.1.1		2
104.5	odd	4		2496.2.c.d.961.1		2
104.21	odd	4		2496.2.c.d.961.2		2
104.83	even	4		2496.2.c.k.961.2		2
104.99	even	4		2496.2.c.k.961.1		2
156.47	odd	4		117.2.b.a.64.1		2
156.83	odd	4		117.2.b.a.64.2		2
156.155	even	2		1521.2.a.l.1.2		2
260.47	odd	4		975.2.h.f.649.2		4
260.83	odd	4		975.2.h.f.649.1		4
260.99	even	4		975.2.b.d.376.1		2
260.187	odd	4		975.2.h.f.649.4		4
260.203	odd	4		975.2.h.f.649.3		4
260.239	even	4		975.2.b.d.376.2		2
364.83	odd	4		1911.2.c.d.883.1		2
364.307	odd	4		1911.2.c.d.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.b.a.25.1	✓	2	52.31	even	4
39.2.b.a.25.2	yes	2	52.47	even	4
117.2.b.a.64.1		2	156.47	odd	4
117.2.b.a.64.2		2	156.83	odd	4
507.2.a.f.1.1		2	52.51	odd	2
507.2.a.f.1.2		2	4.3	odd	2
507.2.e.e.22.1		4	52.3	odd	6
507.2.e.e.22.2		4	52.23	odd	6
507.2.e.e.484.1		4	52.35	odd	6
507.2.e.e.484.2		4	52.43	odd	6
507.2.j.a.316.1		2	52.11	even	12
507.2.j.a.361.1		2	52.19	even	12
507.2.j.c.316.1		2	52.15	even	12
507.2.j.c.361.1		2	52.7	even	12
624.2.c.e.337.1		2	13.5	odd	4
624.2.c.e.337.2		2	13.8	odd	4
975.2.b.d.376.1		2	260.99	even	4
975.2.b.d.376.2		2	260.239	even	4
975.2.h.f.649.1		4	260.83	odd	4
975.2.h.f.649.2		4	260.47	odd	4
975.2.h.f.649.3		4	260.203	odd	4
975.2.h.f.649.4		4	260.187	odd	4
1521.2.a.l.1.1		2	12.11	even	2
1521.2.a.l.1.2		2	156.155	even	2
1872.2.c.e.1585.1		2	39.5	even	4
1872.2.c.e.1585.2		2	39.8	even	4
1911.2.c.d.883.1		2	364.83	odd	4
1911.2.c.d.883.2		2	364.307	odd	4
2496.2.c.d.961.1		2	104.5	odd	4
2496.2.c.d.961.2		2	104.21	odd	4
2496.2.c.k.961.1		2	104.99	even	4
2496.2.c.k.961.2		2	104.83	even	4
8112.2.a.bv.1.1		2	1.1	even	1	trivial
8112.2.a.bv.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{11} +6.00000 q^{17} -3.46410 q^{19} -3.46410 q^{21} -5.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +3.46410 q^{31} -3.46410 q^{33} -6.92820 q^{37} -6.92820 q^{41} -4.00000 q^{43} +3.46410 q^{47} +5.00000 q^{49} +6.00000 q^{51} +6.00000 q^{53} -3.46410 q^{57} +10.3923 q^{59} -2.00000 q^{61} -3.46410 q^{63} +10.3923 q^{67} -3.46410 q^{71} -5.00000 q^{75} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} +6.00000 q^{87} +6.92820 q^{89} +3.46410 q^{93} -13.8564 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 10 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{43} + 10 q^{49} + 12 q^{51} + 12 q^{53} - 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 10 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^43 + 10 * q^49 + 12 * q^51 + 12 * q^53 - 4 * q^61 - 10 * q^75 + 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87

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