LMFDB - Embedded newform 8028.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 38x^3 - 46x^2 - 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 892)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.33527$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.389930 q^{5} -3.58741 q^{7} +O(q^{10})$	$q+0.389930 q^{5} -3.58741 q^{7} +4.32233 q^{11} +1.54291 q^{13} -7.64375 q^{17} -5.47783 q^{19} +1.01294 q^{23} -4.84795 q^{25} +8.39160 q^{29} -8.16759 q^{31} -1.39884 q^{35} +1.03997 q^{37} -2.25214 q^{41} +11.2916 q^{43} -3.18575 q^{47} +5.86954 q^{49} +1.14518 q^{53} +1.68541 q^{55} +5.08941 q^{59} -6.18542 q^{61} +0.601625 q^{65} +0.217047 q^{67} +9.19757 q^{71} +12.6410 q^{73} -15.5060 q^{77} +13.3146 q^{79} +9.03233 q^{83} -2.98053 q^{85} +4.28576 q^{89} -5.53504 q^{91} -2.13597 q^{95} +1.23551 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} - q^{7}+O(q^{10}) $ 7 * q + 7 * q^5 - q^7	$ 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.389930	0.174382	0.0871910	−	0.996192i	$-0.472211\pi$
$5$	0.389930	0.174382	0.0871910	+	0.996192i	$0.472211\pi$
$6$	0	0
$7$	−3.58741	−1.35591	−0.677957	−	0.735101i	$-0.737135\pi$
$7$	−3.58741	−1.35591	−0.677957	+	0.735101i	$0.737135\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.32233	1.30323	0.651616	−	0.758549i	$-0.274091\pi$
$11$	4.32233	1.30323	0.651616	+	0.758549i	$0.274091\pi$
$12$	0	0
$13$	1.54291	0.427925	0.213962	−	0.976842i	$-0.431363\pi$
$13$	1.54291	0.427925	0.213962	+	0.976842i	$0.431363\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.64375	−1.85388	−0.926940	−	0.375209i	$-0.877571\pi$
$17$	−7.64375	−1.85388	−0.926940	+	0.375209i	$0.877571\pi$
$18$	0	0
$19$	−5.47783	−1.25670	−0.628350	−	0.777931i	$-0.716269\pi$
$19$	−5.47783	−1.25670	−0.628350	+	0.777931i	$0.716269\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.01294	0.211213	0.105607	−	0.994408i	$-0.466322\pi$
$23$	1.01294	0.211213	0.105607	+	0.994408i	$0.466322\pi$
$24$	0	0
$25$	−4.84795	−0.969591
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.39160	1.55828	0.779141	−	0.626849i	$-0.215656\pi$
$29$	8.39160	1.55828	0.779141	+	0.626849i	$0.215656\pi$
$30$	0	0
$31$	−8.16759	−1.46694	−0.733471	−	0.679720i	$-0.762101\pi$
$31$	−8.16759	−1.46694	−0.733471	+	0.679720i	$0.762101\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.39884	−0.236447
$36$	0	0
$37$	1.03997	0.170969	0.0854847	−	0.996339i	$-0.472756\pi$
$37$	1.03997	0.170969	0.0854847	+	0.996339i	$0.472756\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.25214	−0.351725	−0.175863	−	0.984415i	$-0.556271\pi$
$41$	−2.25214	−0.351725	−0.175863	+	0.984415i	$0.556271\pi$
$42$	0	0
$43$	11.2916	1.72196	0.860979	−	0.508640i	$-0.169852\pi$
$43$	11.2916	1.72196	0.860979	+	0.508640i	$0.169852\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.18575	−0.464689	−0.232345	−	0.972634i	$-0.574640\pi$
$47$	−3.18575	−0.464689	−0.232345	+	0.972634i	$0.574640\pi$
$48$	0	0
$49$	5.86954	0.838505
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.14518	0.157303	0.0786515	−	0.996902i	$-0.474939\pi$
$53$	1.14518	0.157303	0.0786515	+	0.996902i	$0.474939\pi$
$54$	0	0
$55$	1.68541	0.227260
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.08941	0.662584	0.331292	−	0.943528i	$-0.392515\pi$
$59$	5.08941	0.662584	0.331292	+	0.943528i	$0.392515\pi$
$60$	0	0
$61$	−6.18542	−0.791962	−0.395981	−	0.918259i	$-0.629595\pi$
$61$	−6.18542	−0.791962	−0.395981	+	0.918259i	$0.629595\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.601625	0.0746224
$66$	0	0
$67$	0.217047	0.0265165	0.0132583	−	0.999912i	$-0.495780\pi$
$67$	0.217047	0.0265165	0.0132583	+	0.999912i	$0.495780\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.19757	1.09155	0.545776	−	0.837931i	$-0.316235\pi$
$71$	9.19757	1.09155	0.545776	+	0.837931i	$0.316235\pi$
$72$	0	0
$73$	12.6410	1.47952	0.739760	−	0.672871i	$-0.234939\pi$
$73$	12.6410	1.47952	0.739760	+	0.672871i	$0.234939\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.5060	−1.76707
$78$	0	0
$79$	13.3146	1.49801	0.749006	−	0.662563i	$-0.230531\pi$
$79$	13.3146	1.49801	0.749006	+	0.662563i	$0.230531\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.03233	0.991427	0.495714	−	0.868486i	$-0.334907\pi$
$83$	9.03233	0.991427	0.495714	+	0.868486i	$0.334907\pi$
$84$	0	0
$85$	−2.98053	−0.323283
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.28576	0.454290	0.227145	−	0.973861i	$-0.427061\pi$
$89$	4.28576	0.454290	0.227145	+	0.973861i	$0.427061\pi$
$90$	0	0
$91$	−5.53504	−0.580230
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.13597	−0.219146
$96$	0	0
$97$	1.23551	0.125447	0.0627233	−	0.998031i	$-0.480021\pi$
$97$	1.23551	0.125447	0.0627233	+	0.998031i	$0.480021\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.84255	−0.282844	−0.141422	−	0.989949i	$-0.545167\pi$
$101$	−2.84255	−0.282844	−0.141422	+	0.989949i	$0.545167\pi$
$102$	0	0
$103$	12.5444	1.23603	0.618017	−	0.786164i	$-0.287936\pi$
$103$	12.5444	1.23603	0.618017	+	0.786164i	$0.287936\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.66206	−0.934067	−0.467033	−	0.884240i	$-0.654677\pi$
$107$	−9.66206	−0.934067	−0.467033	+	0.884240i	$0.654677\pi$
$108$	0	0
$109$	−0.831490	−0.0796423	−0.0398212	−	0.999207i	$-0.512679\pi$
$109$	−0.831490	−0.0796423	−0.0398212	+	0.999207i	$0.512679\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.57883	0.618884	0.309442	−	0.950918i	$-0.399858\pi$
$113$	6.57883	0.618884	0.309442	+	0.950918i	$0.399858\pi$
$114$	0	0
$115$	0.394977	0.0368318
$116$	0	0
$117$	0	0
$118$	0	0
$119$	27.4213	2.51370
$120$	0	0
$121$	7.68254	0.698413
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.84001	−0.343461
$126$	0	0
$127$	−11.7112	−1.03920	−0.519601	−	0.854409i	$-0.673919\pi$
$127$	−11.7112	−1.03920	−0.519601	+	0.854409i	$0.673919\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.878977	−0.0767966	−0.0383983	−	0.999263i	$-0.512226\pi$
$131$	−0.878977	−0.0767966	−0.0383983	+	0.999263i	$0.512226\pi$
$132$	0	0
$133$	19.6512	1.70398
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.69955	−0.743253	−0.371626	−	0.928382i	$-0.621200\pi$
$137$	−8.69955	−0.743253	−0.371626	+	0.928382i	$0.621200\pi$
$138$	0	0
$139$	−8.86606	−0.752010	−0.376005	−	0.926618i	$-0.622702\pi$
$139$	−8.86606	−0.752010	−0.376005	+	0.926618i	$0.622702\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.66895	0.557685
$144$	0	0
$145$	3.27214	0.271736
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.9106	−1.46729	−0.733647	−	0.679531i	$-0.762183\pi$
$149$	−17.9106	−1.46729	−0.733647	+	0.679531i	$0.762183\pi$
$150$	0	0
$151$	−21.1847	−1.72398	−0.861992	−	0.506923i	$-0.830783\pi$
$151$	−21.1847	−1.72398	−0.861992	+	0.506923i	$0.830783\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.18479	−0.255808
$156$	0	0
$157$	−15.3853	−1.22788	−0.613942	−	0.789351i	$-0.710417\pi$
$157$	−15.3853	−1.22788	−0.613942	+	0.789351i	$0.710417\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.63384	−0.286387
$162$	0	0
$163$	19.0937	1.49554	0.747768	−	0.663960i	$-0.231125\pi$
$163$	19.0937	1.49554	0.747768	+	0.663960i	$0.231125\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.0543	1.08755	0.543775	−	0.839231i	$-0.316994\pi$
$167$	14.0543	1.08755	0.543775	+	0.839231i	$0.316994\pi$
$168$	0	0
$169$	−10.6194	−0.816880
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.0494	1.22022	0.610108	−	0.792318i	$-0.291126\pi$
$173$	16.0494	1.22022	0.610108	+	0.792318i	$0.291126\pi$
$174$	0	0
$175$	17.3916	1.31468
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.3222	−1.74319	−0.871593	−	0.490230i	$-0.836913\pi$
$179$	−23.3222	−1.74319	−0.871593	+	0.490230i	$0.836913\pi$
$180$	0	0
$181$	15.2738	1.13529	0.567647	−	0.823272i	$-0.307854\pi$
$181$	15.2738	1.13529	0.567647	+	0.823272i	$0.307854\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.405514	0.0298140
$186$	0	0
$187$	−33.0388	−2.41604
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.3894	1.47533	0.737663	−	0.675169i	$-0.235929\pi$
$191$	20.3894	1.47533	0.737663	+	0.675169i	$0.235929\pi$
$192$	0	0
$193$	20.3734	1.46651	0.733255	−	0.679953i	$-0.238000\pi$
$193$	20.3734	1.46651	0.733255	+	0.679953i	$0.238000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4674	0.817021	0.408510	−	0.912754i	$-0.366048\pi$
$197$	11.4674	0.817021	0.408510	+	0.912754i	$0.366048\pi$
$198$	0	0
$199$	3.87408	0.274626	0.137313	−	0.990528i	$-0.456153\pi$
$199$	3.87408	0.274626	0.137313	+	0.990528i	$0.456153\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−30.1042	−2.11290
$204$	0	0
$205$	−0.878178	−0.0613346
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−23.6770	−1.63777
$210$	0	0
$211$	14.5809	1.00379	0.501894	−	0.864929i	$-0.332637\pi$
$211$	14.5809	1.00379	0.501894	+	0.864929i	$0.332637\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.40295	0.300279
$216$	0	0
$217$	29.3005	1.98905
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.7936	−0.793322
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.9085	1.58686	0.793432	−	0.608659i	$-0.208292\pi$
$227$	23.9085	1.58686	0.793432	+	0.608659i	$0.208292\pi$
$228$	0	0
$229$	−12.0112	−0.793721	−0.396860	−	0.917879i	$-0.629900\pi$
$229$	−12.0112	−0.793721	−0.396860	+	0.917879i	$0.629900\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.89596	−0.124209	−0.0621043	−	0.998070i	$-0.519781\pi$
$233$	−1.89596	−0.124209	−0.0621043	+	0.998070i	$0.519781\pi$
$234$	0	0
$235$	−1.24222	−0.0810335
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.87044	−0.444412	−0.222206	−	0.975000i	$-0.571326\pi$
$239$	−6.87044	−0.444412	−0.222206	+	0.975000i	$0.571326\pi$
$240$	0	0
$241$	−1.74457	−0.112378	−0.0561889	−	0.998420i	$-0.517895\pi$
$241$	−1.74457	−0.112378	−0.0561889	+	0.998420i	$0.517895\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.28871	0.146220
$246$	0	0
$247$	−8.45177	−0.537773
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.3289	1.91434	0.957170	−	0.289526i	$-0.0934978\pi$
$251$	30.3289	1.91434	0.957170	+	0.289526i	$0.0934978\pi$
$252$	0	0
$253$	4.37827	0.275259
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.668752	0.0417156	0.0208578	−	0.999782i	$-0.493360\pi$
$257$	0.668752	0.0417156	0.0208578	+	0.999782i	$0.493360\pi$
$258$	0	0
$259$	−3.73079	−0.231820
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.6100	1.02422	0.512108	−	0.858921i	$-0.328865\pi$
$263$	16.6100	1.02422	0.512108	+	0.858921i	$0.328865\pi$
$264$	0	0
$265$	0.446541	0.0274308
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−27.0110	−1.64689	−0.823444	−	0.567398i	$-0.807950\pi$
$269$	−27.0110	−1.64689	−0.823444	+	0.567398i	$0.807950\pi$
$270$	0	0
$271$	−5.02572	−0.305291	−0.152645	−	0.988281i	$-0.548779\pi$
$271$	−5.02572	−0.305291	−0.152645	+	0.988281i	$0.548779\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.9545	−1.26360
$276$	0	0
$277$	−2.96644	−0.178236	−0.0891180	−	0.996021i	$-0.528405\pi$
$277$	−2.96644	−0.178236	−0.0891180	+	0.996021i	$0.528405\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.0753	1.85380	0.926900	−	0.375309i	$-0.122463\pi$
$281$	31.0753	1.85380	0.926900	+	0.375309i	$0.122463\pi$
$282$	0	0
$283$	−4.93449	−0.293325	−0.146662	−	0.989187i	$-0.546853\pi$
$283$	−4.93449	−0.293325	−0.146662	+	0.989187i	$0.546853\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.07936	0.476910
$288$	0	0
$289$	41.4268	2.43687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.4410	1.07734	0.538668	−	0.842518i	$-0.318928\pi$
$293$	18.4410	1.07734	0.538668	+	0.842518i	$0.318928\pi$
$294$	0	0
$295$	1.98451	0.115543
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.56287	0.0903833
$300$	0	0
$301$	−40.5078	−2.33483
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.41188	−0.138104
$306$	0	0
$307$	−24.3092	−1.38740	−0.693699	−	0.720265i	$-0.744020\pi$
$307$	−24.3092	−1.38740	−0.693699	+	0.720265i	$0.744020\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.8462	0.841850	0.420925	−	0.907095i	$-0.361706\pi$
$311$	14.8462	0.841850	0.420925	+	0.907095i	$0.361706\pi$
$312$	0	0
$313$	21.0008	1.18704	0.593519	−	0.804820i	$-0.297738\pi$
$313$	21.0008	1.18704	0.593519	+	0.804820i	$0.297738\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7284	1.05189	0.525945	−	0.850518i	$-0.323712\pi$
$317$	18.7284	1.05189	0.525945	+	0.850518i	$0.323712\pi$
$318$	0	0
$319$	36.2713	2.03080
$320$	0	0
$321$	0	0
$322$	0	0
$323$	41.8711	2.32977
$324$	0	0
$325$	−7.47993	−0.414912
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.4286	0.630079
$330$	0	0
$331$	28.2629	1.55347	0.776734	−	0.629829i	$-0.216875\pi$
$331$	28.2629	1.55347	0.776734	+	0.629829i	$0.216875\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.0846332	0.00462401
$336$	0	0
$337$	25.9669	1.41451	0.707254	−	0.706960i	$-0.249934\pi$
$337$	25.9669	1.41451	0.707254	+	0.706960i	$0.249934\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−35.3030	−1.91177
$342$	0	0
$343$	4.05544	0.218973
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.3412	0.555145	0.277573	−	0.960705i	$-0.410470\pi$
$347$	10.3412	0.555145	0.277573	+	0.960705i	$0.410470\pi$
$348$	0	0
$349$	3.85006	0.206089	0.103044	−	0.994677i	$-0.467142\pi$
$349$	3.85006	0.206089	0.103044	+	0.994677i	$0.467142\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.1404	−0.805840	−0.402920	−	0.915235i	$-0.632005\pi$
$353$	−15.1404	−0.805840	−0.402920	+	0.915235i	$0.632005\pi$
$354$	0	0
$355$	3.58641	0.190347
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.9679	1.73998	0.869988	−	0.493072i	$-0.164126\pi$
$359$	32.9679	1.73998	0.869988	+	0.493072i	$0.164126\pi$
$360$	0	0
$361$	11.0066	0.579295
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.92912	0.258002
$366$	0	0
$367$	12.3166	0.642922	0.321461	−	0.946923i	$-0.395826\pi$
$367$	12.3166	0.642922	0.321461	+	0.946923i	$0.395826\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.10825	−0.213289
$372$	0	0
$373$	−35.3116	−1.82837	−0.914184	−	0.405300i	$-0.867167\pi$
$373$	−35.3116	−1.82837	−0.914184	+	0.405300i	$0.867167\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.9474	0.666828
$378$	0	0
$379$	−1.82856	−0.0939267	−0.0469633	−	0.998897i	$-0.514954\pi$
$379$	−1.82856	−0.0939267	−0.0469633	+	0.998897i	$0.514954\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.1140	0.721193	0.360596	−	0.932722i	$-0.382573\pi$
$383$	14.1140	0.721193	0.360596	+	0.932722i	$0.382573\pi$
$384$	0	0
$385$	−6.04625	−0.308146
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.8191	0.903465	0.451733	−	0.892153i	$-0.350806\pi$
$389$	17.8191	0.903465	0.451733	+	0.892153i	$0.350806\pi$
$390$	0	0
$391$	−7.74267	−0.391564
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.19177	0.261226
$396$	0	0
$397$	3.10692	0.155932	0.0779660	−	0.996956i	$-0.475157\pi$
$397$	3.10692	0.155932	0.0779660	+	0.996956i	$0.475157\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.4916	−1.22305	−0.611526	−	0.791224i	$-0.709444\pi$
$401$	−24.4916	−1.22305	−0.611526	+	0.791224i	$0.709444\pi$
$402$	0	0
$403$	−12.6018	−0.627741
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.49508	0.222813
$408$	0	0
$409$	−4.14549	−0.204981	−0.102491	−	0.994734i	$-0.532681\pi$
$409$	−4.14549	−0.204981	−0.102491	+	0.994734i	$0.532681\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.2578	−0.898408
$414$	0	0
$415$	3.52198	0.172887
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.01297	0.293753	0.146876	−	0.989155i	$-0.453078\pi$
$419$	6.01297	0.293753	0.146876	+	0.989155i	$0.453078\pi$
$420$	0	0
$421$	−31.1891	−1.52006	−0.760031	−	0.649886i	$-0.774816\pi$
$421$	−31.1891	−1.52006	−0.760031	+	0.649886i	$0.774816\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37.0565	1.79751
$426$	0	0
$427$	22.1897	1.07383
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.53070	−0.218236	−0.109118	−	0.994029i	$-0.534803\pi$
$431$	−4.53070	−0.218236	−0.109118	+	0.994029i	$0.534803\pi$
$432$	0	0
$433$	14.9566	0.718767	0.359383	−	0.933190i	$-0.382987\pi$
$433$	14.9566	0.718767	0.359383	+	0.933190i	$0.382987\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.54872	−0.265431
$438$	0	0
$439$	1.62302	0.0774624	0.0387312	−	0.999250i	$-0.487668\pi$
$439$	1.62302	0.0774624	0.0387312	+	0.999250i	$0.487668\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.10664	0.100089	0.0500447	−	0.998747i	$-0.484064\pi$
$443$	2.10664	0.100089	0.0500447	+	0.998747i	$0.484064\pi$
$444$	0	0
$445$	1.67115	0.0792200
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.84409	0.134221	0.0671105	−	0.997746i	$-0.478622\pi$
$449$	2.84409	0.134221	0.0671105	+	0.997746i	$0.478622\pi$
$450$	0	0
$451$	−9.73450	−0.458380
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.15828	−0.101182
$456$	0	0
$457$	−15.9121	−0.744338	−0.372169	−	0.928165i	$-0.621386\pi$
$457$	−15.9121	−0.744338	−0.372169	+	0.928165i	$0.621386\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.0803	1.30783	0.653914	−	0.756568i	$-0.273126\pi$
$461$	28.0803	1.30783	0.653914	+	0.756568i	$0.273126\pi$
$462$	0	0
$463$	22.6247	1.05146	0.525731	−	0.850651i	$-0.323792\pi$
$463$	22.6247	1.05146	0.525731	+	0.850651i	$0.323792\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.60228	0.305517	0.152758	−	0.988264i	$-0.451184\pi$
$467$	6.60228	0.305517	0.152758	+	0.988264i	$0.451184\pi$
$468$	0	0
$469$	−0.778638	−0.0359541
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.8062	2.24411
$474$	0	0
$475$	26.5563	1.21848
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.23341	−0.421885	−0.210943	−	0.977498i	$-0.567653\pi$
$479$	−9.23341	−0.421885	−0.210943	+	0.977498i	$0.567653\pi$
$480$	0	0
$481$	1.60457	0.0731621
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.481761	0.0218756
$486$	0	0
$487$	42.5714	1.92910	0.964548	−	0.263908i	$-0.0850115\pi$
$487$	42.5714	1.92910	0.964548	+	0.263908i	$0.0850115\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.2013	1.45323	0.726613	−	0.687047i	$-0.241093\pi$
$491$	32.2013	1.45323	0.726613	+	0.687047i	$0.241093\pi$
$492$	0	0
$493$	−64.1433	−2.88887
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−32.9955	−1.48005
$498$	0	0
$499$	28.8594	1.29193	0.645963	−	0.763369i	$-0.276456\pi$
$499$	28.8594	1.29193	0.645963	+	0.763369i	$0.276456\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.49961	−0.334391	−0.167196	−	0.985924i	$-0.553471\pi$
$503$	−7.49961	−0.334391	−0.167196	+	0.985924i	$0.553471\pi$
$504$	0	0
$505$	−1.10840	−0.0493230
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.3283	1.12266	0.561329	−	0.827592i	$-0.310290\pi$
$509$	25.3283	1.12266	0.561329	+	0.827592i	$0.310290\pi$
$510$	0	0
$511$	−45.3486	−2.00610
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.89143	0.215542
$516$	0	0
$517$	−13.7699	−0.605598
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.71708	0.381902	0.190951	−	0.981600i	$-0.438843\pi$
$521$	8.71708	0.381902	0.190951	+	0.981600i	$0.438843\pi$
$522$	0	0
$523$	−1.44794	−0.0633140	−0.0316570	−	0.999499i	$-0.510078\pi$
$523$	−1.44794	−0.0633140	−0.0316570	+	0.999499i	$0.510078\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	62.4310	2.71954
$528$	0	0
$529$	−21.9739	−0.955389
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.47484	−0.150512
$534$	0	0
$535$	−3.76753	−0.162884
$536$	0	0
$537$	0	0
$538$	0	0
$539$	25.3701	1.09277
$540$	0	0
$541$	2.54291	0.109328	0.0546641	−	0.998505i	$-0.482591\pi$
$541$	2.54291	0.109328	0.0546641	+	0.998505i	$0.482591\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.324223	−0.0138882
$546$	0	0
$547$	−7.10112	−0.303622	−0.151811	−	0.988410i	$-0.548510\pi$
$547$	−7.10112	−0.303622	−0.151811	+	0.988410i	$0.548510\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−45.9678	−1.95829
$552$	0	0
$553$	−47.7651	−2.03118
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.26108	−0.0958048	−0.0479024	−	0.998852i	$-0.515254\pi$
$557$	−2.26108	−0.0958048	−0.0479024	+	0.998852i	$0.515254\pi$
$558$	0	0
$559$	17.4219	0.736869
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.6854	−0.576769	−0.288384	−	0.957515i	$-0.593118\pi$
$563$	−13.6854	−0.576769	−0.288384	+	0.957515i	$0.593118\pi$
$564$	0	0
$565$	2.56528	0.107922
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.1016	1.01039	0.505196	−	0.863005i	$-0.331420\pi$
$569$	24.1016	1.01039	0.505196	+	0.863005i	$0.331420\pi$
$570$	0	0
$571$	−11.1018	−0.464594	−0.232297	−	0.972645i	$-0.574624\pi$
$571$	−11.1018	−0.464594	−0.232297	+	0.972645i	$0.574624\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.91070	−0.204790
$576$	0	0
$577$	−40.4906	−1.68565	−0.842823	−	0.538192i	$-0.819108\pi$
$577$	−40.4906	−1.68565	−0.842823	+	0.538192i	$0.819108\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−32.4027	−1.34429
$582$	0	0
$583$	4.94986	0.205002
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.1667	−0.832368	−0.416184	−	0.909280i	$-0.636633\pi$
$587$	−20.1667	−0.832368	−0.416184	+	0.909280i	$0.636633\pi$
$588$	0	0
$589$	44.7407	1.84351
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.5959	−1.25642	−0.628211	−	0.778043i	$-0.716212\pi$
$593$	−30.5959	−1.25642	−0.628211	+	0.778043i	$0.716212\pi$
$594$	0	0
$595$	10.6924	0.438345
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.8807	−1.22089	−0.610446	−	0.792058i	$-0.709010\pi$
$599$	−29.8807	−1.22089	−0.610446	+	0.792058i	$0.709010\pi$
$600$	0	0
$601$	−23.0503	−0.940240	−0.470120	−	0.882603i	$-0.655789\pi$
$601$	−23.0503	−0.940240	−0.470120	+	0.882603i	$0.655789\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.99565	0.121791
$606$	0	0
$607$	30.2867	1.22930	0.614649	−	0.788801i	$-0.289298\pi$
$607$	30.2867	1.22930	0.614649	+	0.788801i	$0.289298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.91531	−0.198852
$612$	0	0
$613$	−2.13121	−0.0860787	−0.0430394	−	0.999073i	$-0.513704\pi$
$613$	−2.13121	−0.0860787	−0.0430394	+	0.999073i	$0.513704\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.9338	−0.923281	−0.461641	−	0.887067i	$-0.652739\pi$
$617$	−22.9338	−0.923281	−0.461641	+	0.887067i	$0.652739\pi$
$618$	0	0
$619$	−31.5530	−1.26822	−0.634111	−	0.773242i	$-0.718634\pi$
$619$	−31.5530	−1.26822	−0.634111	+	0.773242i	$0.718634\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.3748	−0.615978
$624$	0	0
$625$	22.7424	0.909697
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.94924	−0.316957
$630$	0	0
$631$	−17.0052	−0.676967	−0.338484	−	0.940972i	$-0.609914\pi$
$631$	−17.0052	−0.676967	−0.338484	+	0.940972i	$0.609914\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.56656	−0.181218
$636$	0	0
$637$	9.05614	0.358817
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.1595	−1.58620	−0.793102	−	0.609089i	$-0.791535\pi$
$641$	−40.1595	−1.58620	−0.793102	+	0.609089i	$0.791535\pi$
$642$	0	0
$643$	−16.6117	−0.655103	−0.327551	−	0.944833i	$-0.606223\pi$
$643$	−16.6117	−0.655103	−0.327551	+	0.944833i	$0.606223\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.3967	−0.801878	−0.400939	−	0.916105i	$-0.631316\pi$
$647$	−20.3967	−0.801878	−0.400939	+	0.916105i	$0.631316\pi$
$648$	0	0
$649$	21.9981	0.863501
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.9761	1.48612	0.743059	−	0.669226i	$-0.233374\pi$
$653$	37.9761	1.48612	0.743059	+	0.669226i	$0.233374\pi$
$654$	0	0
$655$	−0.342740	−0.0133919
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.2346	−0.632410	−0.316205	−	0.948691i	$-0.602409\pi$
$659$	−16.2346	−0.632410	−0.316205	+	0.948691i	$0.602409\pi$
$660$	0	0
$661$	−4.73158	−0.184037	−0.0920186	−	0.995757i	$-0.529332\pi$
$661$	−4.73158	−0.184037	−0.0920186	+	0.995757i	$0.529332\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.66261	0.297143
$666$	0	0
$667$	8.50021	0.329129
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.7354	−1.03211
$672$	0	0
$673$	−24.7488	−0.953997	−0.476999	−	0.878904i	$-0.658275\pi$
$673$	−24.7488	−0.953997	−0.476999	+	0.878904i	$0.658275\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.3513	−1.24336	−0.621681	−	0.783270i	$-0.713550\pi$
$677$	−32.3513	−1.24336	−0.621681	+	0.783270i	$0.713550\pi$
$678$	0	0
$679$	−4.43227	−0.170095
$680$	0	0
$681$	0	0
$682$	0	0
$683$	45.5859	1.74430	0.872149	−	0.489241i	$-0.162726\pi$
$683$	45.5859	1.74430	0.872149	+	0.489241i	$0.162726\pi$
$684$	0	0
$685$	−3.39222	−0.129610
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.76691	0.0673139
$690$	0	0
$691$	−43.1132	−1.64010	−0.820052	−	0.572289i	$-0.806055\pi$
$691$	−43.1132	−1.64010	−0.820052	+	0.572289i	$0.806055\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.45715	−0.131137
$696$	0	0
$697$	17.2148	0.652057
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.3402	1.44809	0.724045	−	0.689753i	$-0.242281\pi$
$701$	38.3402	1.44809	0.724045	+	0.689753i	$0.242281\pi$
$702$	0	0
$703$	−5.69676	−0.214857
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.1974	0.383513
$708$	0	0
$709$	−6.26163	−0.235161	−0.117580	−	0.993063i	$-0.537514\pi$
$709$	−6.26163	−0.235161	−0.117580	+	0.993063i	$0.537514\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.27330	−0.309837
$714$	0	0
$715$	2.60042	0.0972503
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.0851526	−0.00317566	−0.00158783	−	0.999999i	$-0.500505\pi$
$719$	−0.0851526	−0.00317566	−0.00158783	+	0.999999i	$0.500505\pi$
$720$	0	0
$721$	−45.0019	−1.67596
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−40.6821	−1.51090
$726$	0	0
$727$	9.14608	0.339209	0.169605	−	0.985512i	$-0.445751\pi$
$727$	9.14608	0.339209	0.169605	+	0.985512i	$0.445751\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−86.3104	−3.19230
$732$	0	0
$733$	14.6329	0.540480	0.270240	−	0.962793i	$-0.412897\pi$
$733$	14.6329	0.540480	0.270240	+	0.962793i	$0.412897\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.938149	0.0345572
$738$	0	0
$739$	−36.9047	−1.35756	−0.678780	−	0.734342i	$-0.737491\pi$
$739$	−36.9047	−1.35756	−0.678780	+	0.734342i	$0.737491\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.0507	1.17582	0.587912	−	0.808925i	$-0.299950\pi$
$743$	32.0507	1.17582	0.587912	+	0.808925i	$0.299950\pi$
$744$	0	0
$745$	−6.98388	−0.255870
$746$	0	0
$747$	0	0
$748$	0	0
$749$	34.6618	1.26651
$750$	0	0
$751$	−3.12590	−0.114066	−0.0570329	−	0.998372i	$-0.518164\pi$
$751$	−3.12590	−0.114066	−0.0570329	+	0.998372i	$0.518164\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.26054	−0.300632
$756$	0	0
$757$	−13.6273	−0.495294	−0.247647	−	0.968850i	$-0.579657\pi$
$757$	−13.6273	−0.495294	−0.247647	+	0.968850i	$0.579657\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.6677	−0.459202	−0.229601	−	0.973285i	$-0.573742\pi$
$761$	−12.6677	−0.459202	−0.229601	+	0.973285i	$0.573742\pi$
$762$	0	0
$763$	2.98290	0.107988
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.85247	0.283536
$768$	0	0
$769$	−27.0312	−0.974772	−0.487386	−	0.873187i	$-0.662049\pi$
$769$	−27.0312	−0.974772	−0.487386	+	0.873187i	$0.662049\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.11802	−0.112147	−0.0560737	−	0.998427i	$-0.517858\pi$
$773$	−3.11802	−0.112147	−0.0560737	+	0.998427i	$0.517858\pi$
$774$	0	0
$775$	39.5961	1.42233
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.3368	0.442013
$780$	0	0
$781$	39.7550	1.42254
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.99921	−0.214121
$786$	0	0
$787$	16.8719	0.601419	0.300710	−	0.953716i	$-0.402776\pi$
$787$	16.8719	0.601419	0.300710	+	0.953716i	$0.402776\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.6010	−0.839154
$792$	0	0
$793$	−9.54351	−0.338900
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.0001	−0.673018	−0.336509	−	0.941680i	$-0.609246\pi$
$797$	−19.0001	−0.673018	−0.336509	+	0.941680i	$0.609246\pi$
$798$	0	0
$799$	24.3511	0.861479
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.6387	1.92816
$804$	0	0
$805$	−1.41694	−0.0499407
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.46595	0.262489	0.131244	−	0.991350i	$-0.458103\pi$
$809$	7.46595	0.262489	0.131244	+	0.991350i	$0.458103\pi$
$810$	0	0
$811$	32.1188	1.12784	0.563922	−	0.825828i	$-0.309292\pi$
$811$	32.1188	1.12784	0.563922	+	0.825828i	$0.309292\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.44522	0.260795
$816$	0	0
$817$	−61.8536	−2.16398
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.73949	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$821$	7.73949	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$822$	0	0
$823$	45.5180	1.58666	0.793329	−	0.608793i	$-0.208346\pi$
$823$	45.5180	1.58666	0.793329	+	0.608793i	$0.208346\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.4893	0.955897	0.477949	−	0.878388i	$-0.341381\pi$
$827$	27.4893	0.955897	0.477949	+	0.878388i	$0.341381\pi$
$828$	0	0
$829$	−18.6939	−0.649267	−0.324634	−	0.945840i	$-0.605241\pi$
$829$	−18.6939	−0.649267	−0.324634	+	0.945840i	$0.605241\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−44.8652	−1.55449
$834$	0	0
$835$	5.48018	0.189649
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.62418	0.125121	0.0625603	−	0.998041i	$-0.480073\pi$
$839$	3.62418	0.125121	0.0625603	+	0.998041i	$0.480073\pi$
$840$	0	0
$841$	41.4190	1.42824
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.14084	−0.142449
$846$	0	0
$847$	−27.5605	−0.946988
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.05343	0.0361110
$852$	0	0
$853$	49.4726	1.69391	0.846955	−	0.531665i	$-0.178433\pi$
$853$	49.4726	1.69391	0.846955	+	0.531665i	$0.178433\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.5678	−1.31745	−0.658725	−	0.752383i	$-0.728904\pi$
$857$	−38.5678	−1.31745	−0.658725	+	0.752383i	$0.728904\pi$
$858$	0	0
$859$	1.86954	0.0637878	0.0318939	−	0.999491i	$-0.489846\pi$
$859$	1.86954	0.0637878	0.0318939	+	0.999491i	$0.489846\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.2357	−0.654792	−0.327396	−	0.944887i	$-0.606171\pi$
$863$	−19.2357	−0.654792	−0.327396	+	0.944887i	$0.606171\pi$
$864$	0	0
$865$	6.25816	0.212784
$866$	0	0
$867$	0	0
$868$	0	0
$869$	57.5502	1.95226
$870$	0	0
$871$	0.334883	0.0113471
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.7757	0.465704
$876$	0	0
$877$	18.2556	0.616447	0.308224	−	0.951314i	$-0.400265\pi$
$877$	18.2556	0.616447	0.308224	+	0.951314i	$0.400265\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.0448	1.01223	0.506117	−	0.862465i	$-0.331080\pi$
$881$	30.0448	1.01223	0.506117	+	0.862465i	$0.331080\pi$
$882$	0	0
$883$	24.9091	0.838257	0.419128	−	0.907927i	$-0.362336\pi$
$883$	24.9091	0.838257	0.419128	+	0.907927i	$0.362336\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.2260	0.544815	0.272407	−	0.962182i	$-0.412180\pi$
$887$	16.2260	0.544815	0.272407	+	0.962182i	$0.412180\pi$
$888$	0	0
$889$	42.0130	1.40907
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.4510	0.583975
$894$	0	0
$895$	−9.09405	−0.303980
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−68.5392	−2.28591
$900$	0	0
$901$	−8.75349	−0.291621
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.95572	0.197975
$906$	0	0
$907$	−12.3850	−0.411237	−0.205619	−	0.978632i	$-0.565921\pi$
$907$	−12.3850	−0.411237	−0.205619	+	0.978632i	$0.565921\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.3022	0.407589	0.203795	−	0.979014i	$-0.434673\pi$
$911$	12.3022	0.407589	0.203795	+	0.979014i	$0.434673\pi$
$912$	0	0
$913$	39.0407	1.29206
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.15325	0.104130
$918$	0	0
$919$	24.6451	0.812968	0.406484	−	0.913658i	$-0.366755\pi$
$919$	24.6451	0.812968	0.406484	+	0.913658i	$0.366755\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.1910	0.467102
$924$	0	0
$925$	−5.04171	−0.165770
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.3913	−1.68609	−0.843046	−	0.537841i	$-0.819240\pi$
$929$	−51.3913	−1.68609	−0.843046	+	0.537841i	$0.819240\pi$
$930$	0	0
$931$	−32.1523	−1.05375
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.8828	−0.421313
$936$	0	0
$937$	−33.9876	−1.11033	−0.555163	−	0.831742i	$-0.687344\pi$
$937$	−33.9876	−1.11033	−0.555163	+	0.831742i	$0.687344\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.2065	1.31069	0.655347	−	0.755328i	$-0.272523\pi$
$941$	40.2065	1.31069	0.655347	+	0.755328i	$0.272523\pi$
$942$	0	0
$943$	−2.28129	−0.0742890
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.8855	0.516208	0.258104	−	0.966117i	$-0.416902\pi$
$947$	15.8855	0.516208	0.258104	+	0.966117i	$0.416902\pi$
$948$	0	0
$949$	19.5039	0.633123
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.0436	0.422524	0.211262	−	0.977430i	$-0.432243\pi$
$953$	13.0436	0.422524	0.211262	+	0.977430i	$0.432243\pi$
$954$	0	0
$955$	7.95044	0.257270
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.2089	1.00779
$960$	0	0
$961$	35.7095	1.15192
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.94421	0.255733
$966$	0	0
$967$	7.39524	0.237815	0.118907	−	0.992905i	$-0.462061\pi$
$967$	7.39524	0.237815	0.118907	+	0.992905i	$0.462061\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.4046	0.398083	0.199041	−	0.979991i	$-0.436217\pi$
$971$	12.4046	0.398083	0.199041	+	0.979991i	$0.436217\pi$
$972$	0	0
$973$	31.8062	1.01966
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.6120	−0.339508	−0.169754	−	0.985486i	$-0.554297\pi$
$977$	−10.6120	−0.339508	−0.169754	+	0.985486i	$0.554297\pi$
$978$	0	0
$979$	18.5245	0.592045
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.45116	−0.205760	−0.102880	−	0.994694i	$-0.532806\pi$
$983$	−6.45116	−0.205760	−0.102880	+	0.994694i	$0.532806\pi$
$984$	0	0
$985$	4.47150	0.142474
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.4378	0.363700
$990$	0	0
$991$	−9.68730	−0.307727	−0.153864	−	0.988092i	$-0.549172\pi$
$991$	−9.68730	−0.307727	−0.153864	+	0.988092i	$0.549172\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.51062	0.0478899
$996$	0	0
$997$	43.8759	1.38956	0.694782	−	0.719221i	$-0.255501\pi$
$997$	43.8759	1.38956	0.694782	+	0.719221i	$0.255501\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.j.1.3		7
3.2	odd	2		892.2.a.d.1.6	✓	7
12.11	even	2		3568.2.a.m.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
892.2.a.d.1.6	✓	7	3.2	odd	2
3568.2.a.m.1.2		7	12.11	even	2
8028.2.a.j.1.3		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.389930 q^{5} -3.58741 q^{7} +O(q^{10})\)	\(q+0.389930 q^{5} -3.58741 q^{7} +4.32233 q^{11} +1.54291 q^{13} -7.64375 q^{17} -5.47783 q^{19} +1.01294 q^{23} -4.84795 q^{25} +8.39160 q^{29} -8.16759 q^{31} -1.39884 q^{35} +1.03997 q^{37} -2.25214 q^{41} +11.2916 q^{43} -3.18575 q^{47} +5.86954 q^{49} +1.14518 q^{53} +1.68541 q^{55} +5.08941 q^{59} -6.18542 q^{61} +0.601625 q^{65} +0.217047 q^{67} +9.19757 q^{71} +12.6410 q^{73} -15.5060 q^{77} +13.3146 q^{79} +9.03233 q^{83} -2.98053 q^{85} +4.28576 q^{89} -5.53504 q^{91} -2.13597 q^{95} +1.23551 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - q^{7}+O(q^{10}) \) 7 * q + 7 * q^5 - q^7	\( 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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