LMFDB - Embedded newform 6840.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +3.58774 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.58774 q^{7} +1.35793 q^{13} -5.58774 q^{17} +1.00000 q^{19} +4.87189 q^{23} +1.00000 q^{25} -9.58774 q^{29} -7.17548 q^{31} -3.58774 q^{35} -0.945668 q^{37} -10.4596 q^{41} -2.71585 q^{43} +5.89134 q^{47} +5.87189 q^{49} -9.81756 q^{53} -10.1560 q^{59} +3.28415 q^{61} -1.35793 q^{65} +10.3859 q^{67} -14.3510 q^{71} -4.15604 q^{73} +1.28415 q^{79} +11.1755 q^{83} +5.58774 q^{85} +6.45963 q^{89} +4.87189 q^{91} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - q^7	$ 3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - q^7 + 5 * q^13 - 5 * q^17 + 3 * q^19 + q^23 + 3 * q^25 - 17 * q^29 + 2 * q^31 + q^35 + 8 * q^37 - 6 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^49 - 5 * q^53 - 15 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 + 4 * q^71 + 3 * q^73 + 2 * q^79 + 10 * q^83 + 5 * q^85 - 6 * q^89 + q^91 - 3 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.58774	1.35604	0.678019	−	0.735044i	$-0.262839\pi$
$7$	3.58774	1.35604	0.678019	+	0.735044i	$0.262839\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.35793	0.376621	0.188311	−	0.982110i	$-0.439699\pi$
$13$	1.35793	0.376621	0.188311	+	0.982110i	$0.439699\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.58774	−1.35523	−0.677613	−	0.735419i	$-0.736986\pi$
$17$	−5.58774	−1.35523	−0.677613	+	0.735419i	$0.736986\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.87189	1.01586	0.507930	−	0.861399i	$-0.330411\pi$
$23$	4.87189	1.01586	0.507930	+	0.861399i	$0.330411\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.58774	−1.78040	−0.890199	−	0.455571i	$-0.849435\pi$
$29$	−9.58774	−1.78040	−0.890199	+	0.455571i	$0.849435\pi$
$30$	0	0
$31$	−7.17548	−1.28875	−0.644377	−	0.764708i	$-0.722883\pi$
$31$	−7.17548	−1.28875	−0.644377	+	0.764708i	$0.722883\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.58774	−0.606439
$36$	0	0
$37$	−0.945668	−0.155467	−0.0777334	−	0.996974i	$-0.524768\pi$
$37$	−0.945668	−0.155467	−0.0777334	+	0.996974i	$0.524768\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.4596	−1.63352	−0.816760	−	0.576978i	$-0.804232\pi$
$41$	−10.4596	−1.63352	−0.816760	+	0.576978i	$0.804232\pi$
$42$	0	0
$43$	−2.71585	−0.414164	−0.207082	−	0.978324i	$-0.566397\pi$
$43$	−2.71585	−0.414164	−0.207082	+	0.978324i	$0.566397\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.89134	0.859340	0.429670	−	0.902986i	$-0.358630\pi$
$47$	5.89134	0.859340	0.429670	+	0.902986i	$0.358630\pi$
$48$	0	0
$49$	5.87189	0.838841
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.81756	−1.34855	−0.674273	−	0.738483i	$-0.735543\pi$
$53$	−9.81756	−1.34855	−0.674273	+	0.738483i	$0.735543\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.1560	−1.32220	−0.661102	−	0.750296i	$-0.729911\pi$
$59$	−10.1560	−1.32220	−0.661102	+	0.750296i	$0.729911\pi$
$60$	0	0
$61$	3.28415	0.420492	0.210246	−	0.977649i	$-0.432574\pi$
$61$	3.28415	0.420492	0.210246	+	0.977649i	$0.432574\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.35793	−0.168430
$66$	0	0
$67$	10.3859	1.26883	0.634417	−	0.772991i	$-0.281240\pi$
$67$	10.3859	1.26883	0.634417	+	0.772991i	$0.281240\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.3510	−1.70315	−0.851573	−	0.524236i	$-0.824351\pi$
$71$	−14.3510	−1.70315	−0.851573	+	0.524236i	$0.824351\pi$
$72$	0	0
$73$	−4.15604	−0.486427	−0.243214	−	0.969973i	$-0.578202\pi$
$73$	−4.15604	−0.486427	−0.243214	+	0.969973i	$0.578202\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.28415	0.144478	0.0722389	−	0.997387i	$-0.476986\pi$
$79$	1.28415	0.144478	0.0722389	+	0.997387i	$0.476986\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.1755	1.22667	0.613334	−	0.789823i	$-0.289828\pi$
$83$	11.1755	1.22667	0.613334	+	0.789823i	$0.289828\pi$
$84$	0	0
$85$	5.58774	0.606076
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.45963	0.684719	0.342360	−	0.939569i	$-0.388774\pi$
$89$	6.45963	0.684719	0.342360	+	0.939569i	$0.388774\pi$
$90$	0	0
$91$	4.87189	0.510713
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−13.4053	−1.36110	−0.680551	−	0.732701i	$-0.738259\pi$
$97$	−13.4053	−1.36110	−0.680551	+	0.732701i	$0.738259\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.0668	1.89722	0.948610	−	0.316449i	$-0.102490\pi$
$101$	19.0668	1.89722	0.948610	+	0.316449i	$0.102490\pi$
$102$	0	0
$103$	−9.05433	−0.892150	−0.446075	−	0.894996i	$-0.647179\pi$
$103$	−9.05433	−0.892150	−0.446075	+	0.894996i	$0.647179\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.24926	0.507465	0.253733	−	0.967274i	$-0.418342\pi$
$107$	5.24926	0.507465	0.253733	+	0.967274i	$0.418342\pi$
$108$	0	0
$109$	−6.04737	−0.579233	−0.289617	−	0.957143i	$-0.593528\pi$
$109$	−6.04737	−0.579233	−0.289617	+	0.957143i	$0.593528\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.22982	0.586052	0.293026	−	0.956105i	$-0.405338\pi$
$113$	6.22982	0.586052	0.293026	+	0.956105i	$0.405338\pi$
$114$	0	0
$115$	−4.87189	−0.454306
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.0474	−1.83774
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.1212	−1.25305	−0.626525	−	0.779402i	$-0.715523\pi$
$127$	−14.1212	−1.25305	−0.626525	+	0.779402i	$0.715523\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.3510	1.60333	0.801666	−	0.597773i	$-0.203947\pi$
$131$	18.3510	1.60333	0.801666	+	0.597773i	$0.203947\pi$
$132$	0	0
$133$	3.58774	0.311097
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.01945	−0.257969	−0.128984	−	0.991647i	$-0.541172\pi$
$137$	−3.01945	−0.257969	−0.128984	+	0.991647i	$0.541172\pi$
$138$	0	0
$139$	11.7827	0.999393	0.499697	−	0.866201i	$-0.333445\pi$
$139$	11.7827	0.999393	0.499697	+	0.866201i	$0.333445\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.58774	0.796219
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.4985	−1.67930	−0.839652	−	0.543124i	$-0.817241\pi$
$149$	−20.4985	−1.67930	−0.839652	+	0.543124i	$0.817241\pi$
$150$	0	0
$151$	−3.85244	−0.313507	−0.156754	−	0.987638i	$-0.550103\pi$
$151$	−3.85244	−0.313507	−0.156754	+	0.987638i	$0.550103\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.17548	0.576349
$156$	0	0
$157$	3.89134	0.310562	0.155281	−	0.987870i	$-0.450372\pi$
$157$	3.89134	0.310562	0.155281	+	0.987870i	$0.450372\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	17.4791	1.37754
$162$	0	0
$163$	−18.7159	−1.46594	−0.732969	−	0.680262i	$-0.761866\pi$
$163$	−18.7159	−1.46594	−0.732969	+	0.680262i	$0.761866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.09323	0.239361	0.119681	−	0.992812i	$-0.461813\pi$
$167$	3.09323	0.239361	0.119681	+	0.992812i	$0.461813\pi$
$168$	0	0
$169$	−11.1560	−0.858157
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.51396	−0.571276	−0.285638	−	0.958338i	$-0.592205\pi$
$173$	−7.51396	−0.571276	−0.285638	+	0.958338i	$0.592205\pi$
$174$	0	0
$175$	3.58774	0.271208
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.45963	0.333328	0.166664	−	0.986014i	$-0.446700\pi$
$179$	4.45963	0.333328	0.166664	+	0.986014i	$0.446700\pi$
$180$	0	0
$181$	8.56829	0.636876	0.318438	−	0.947944i	$-0.396842\pi$
$181$	8.56829	0.636876	0.318438	+	0.947944i	$0.396842\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.945668	0.0695269
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.73530	−0.559707	−0.279853	−	0.960043i	$-0.590286\pi$
$191$	−7.73530	−0.559707	−0.279853	+	0.960043i	$0.590286\pi$
$192$	0	0
$193$	6.08226	0.437810	0.218905	−	0.975746i	$-0.429751\pi$
$193$	6.08226	0.437810	0.218905	+	0.975746i	$0.429751\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−6.91078	−0.489892	−0.244946	−	0.969537i	$-0.578770\pi$
$199$	−6.91078	−0.489892	−0.244946	+	0.969537i	$0.578770\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−34.3983	−2.41429
$204$	0	0
$205$	10.4596	0.730532
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.1949	−0.839534	−0.419767	−	0.907632i	$-0.637888\pi$
$211$	−12.1949	−0.839534	−0.419767	+	0.907632i	$0.637888\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.71585	0.185220
$216$	0	0
$217$	−25.7438	−1.74760
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.58774	−0.510407
$222$	0	0
$223$	−1.87885	−0.125817	−0.0629085	−	0.998019i	$-0.520038\pi$
$223$	−1.87885	−0.125817	−0.0629085	+	0.998019i	$0.520038\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.35793	−0.488363	−0.244181	−	0.969730i	$-0.578519\pi$
$227$	−7.35793	−0.488363	−0.244181	+	0.969730i	$0.578519\pi$
$228$	0	0
$229$	13.0279	0.860909	0.430455	−	0.902612i	$-0.358353\pi$
$229$	13.0279	0.860909	0.430455	+	0.902612i	$0.358353\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.7019	1.74930	0.874651	−	0.484753i	$-0.161091\pi$
$233$	26.7019	1.74930	0.874651	+	0.484753i	$0.161091\pi$
$234$	0	0
$235$	−5.89134	−0.384308
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.47908	0.613151	0.306575	−	0.951846i	$-0.400817\pi$
$239$	9.47908	0.613151	0.306575	+	0.951846i	$0.400817\pi$
$240$	0	0
$241$	−24.2034	−1.55908	−0.779539	−	0.626353i	$-0.784547\pi$
$241$	−24.2034	−1.55908	−0.779539	+	0.626353i	$0.784547\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.87189	−0.375141
$246$	0	0
$247$	1.35793	0.0864028
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.43171	−0.595324	−0.297662	−	0.954671i	$-0.596207\pi$
$251$	−9.43171	−0.595324	−0.297662	+	0.954671i	$0.596207\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.1102	−1.56633	−0.783165	−	0.621814i	$-0.786396\pi$
$257$	−25.1102	−1.56633	−0.783165	+	0.621814i	$0.786396\pi$
$258$	0	0
$259$	−3.39281	−0.210819
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.7019	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$263$	28.7019	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$264$	0	0
$265$	9.81756	0.603088
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.350966	0.0213988	0.0106994	−	0.999943i	$-0.496594\pi$
$269$	0.350966	0.0213988	0.0106994	+	0.999943i	$0.496594\pi$
$270$	0	0
$271$	16.8719	1.02489	0.512447	−	0.858719i	$-0.328739\pi$
$271$	16.8719	1.02489	0.512447	+	0.858719i	$0.328739\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.14756	−0.369371	−0.184685	−	0.982798i	$-0.559127\pi$
$277$	−6.14756	−0.369371	−0.184685	+	0.982798i	$0.559127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.0279	1.01580	0.507900	−	0.861416i	$-0.330422\pi$
$281$	17.0279	1.01580	0.507900	+	0.861416i	$0.330422\pi$
$282$	0	0
$283$	5.74378	0.341432	0.170716	−	0.985320i	$-0.445392\pi$
$283$	5.74378	0.341432	0.170716	+	0.985320i	$0.445392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−37.5264	−2.21512
$288$	0	0
$289$	14.2229	0.836639
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.2772	0.834082	0.417041	−	0.908888i	$-0.363067\pi$
$293$	14.2772	0.834082	0.417041	+	0.908888i	$0.363067\pi$
$294$	0	0
$295$	10.1560	0.591307
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.61567	0.382594
$300$	0	0
$301$	−9.74378	−0.561622
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.28415	−0.188050
$306$	0	0
$307$	8.83700	0.504354	0.252177	−	0.967681i	$-0.418853\pi$
$307$	8.83700	0.504354	0.252177	+	0.967681i	$0.418853\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.2229	−1.31685	−0.658424	−	0.752648i	$-0.728776\pi$
$311$	−23.2229	−1.31685	−0.658424	+	0.752648i	$0.728776\pi$
$312$	0	0
$313$	30.1421	1.70373	0.851867	−	0.523759i	$-0.175471\pi$
$313$	30.1421	1.70373	0.851867	+	0.523759i	$0.175471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.96511	−0.335034	−0.167517	−	0.985869i	$-0.553575\pi$
$317$	−5.96511	−0.335034	−0.167517	+	0.985869i	$0.553575\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.58774	−0.310910
$324$	0	0
$325$	1.35793	0.0753242
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.1366	1.16530
$330$	0	0
$331$	−2.15604	−0.118506	−0.0592532	−	0.998243i	$-0.518872\pi$
$331$	−2.15604	−0.118506	−0.0592532	+	0.998243i	$0.518872\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.3859	−0.567440
$336$	0	0
$337$	0.945668	0.0515138	0.0257569	−	0.999668i	$-0.491800\pi$
$337$	0.945668	0.0515138	0.0257569	+	0.999668i	$0.491800\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.04737	−0.218538
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.2562	−0.550583	−0.275291	−	0.961361i	$-0.588774\pi$
$347$	−10.2562	−0.550583	−0.275291	+	0.961361i	$0.588774\pi$
$348$	0	0
$349$	−24.0558	−1.28768	−0.643840	−	0.765160i	$-0.722660\pi$
$349$	−24.0558	−1.28768	−0.643840	+	0.765160i	$0.722660\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.3315	−0.816014	−0.408007	−	0.912979i	$-0.633776\pi$
$353$	−15.3315	−0.816014	−0.408007	+	0.912979i	$0.633776\pi$
$354$	0	0
$355$	14.3510	0.761670
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.1281	−0.798431	−0.399216	−	0.916857i	$-0.630718\pi$
$359$	−15.1281	−0.798431	−0.399216	+	0.916857i	$0.630718\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.15604	0.217537
$366$	0	0
$367$	−33.6740	−1.75777	−0.878884	−	0.477035i	$-0.841712\pi$
$367$	−33.6740	−1.75777	−0.878884	+	0.477035i	$0.841712\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−35.2229	−1.82868
$372$	0	0
$373$	16.0738	0.832269	0.416134	−	0.909303i	$-0.363385\pi$
$373$	16.0738	0.832269	0.416134	+	0.909303i	$0.363385\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.0194	−0.670536
$378$	0	0
$379$	−6.00848	−0.308635	−0.154317	−	0.988021i	$-0.549318\pi$
$379$	−6.00848	−0.308635	−0.154317	+	0.988021i	$0.549318\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.8649	0.810660	0.405330	−	0.914170i	$-0.367157\pi$
$383$	15.8649	0.810660	0.405330	+	0.914170i	$0.367157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.7438	−0.798241	−0.399121	−	0.916898i	$-0.630685\pi$
$389$	−15.7438	−0.798241	−0.399121	+	0.916898i	$0.630685\pi$
$390$	0	0
$391$	−27.2229	−1.37672
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.28415	−0.0646125
$396$	0	0
$397$	−23.9861	−1.20383	−0.601913	−	0.798561i	$-0.705595\pi$
$397$	−23.9861	−1.20383	−0.601913	+	0.798561i	$0.705595\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.5683	1.22688	0.613441	−	0.789741i	$-0.289785\pi$
$401$	24.5683	1.22688	0.613441	+	0.789741i	$0.289785\pi$
$402$	0	0
$403$	−9.74378	−0.485372
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.9193	−0.737710	−0.368855	−	0.929487i	$-0.620250\pi$
$409$	−14.9193	−0.737710	−0.368855	+	0.929487i	$0.620250\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−36.4372	−1.79296
$414$	0	0
$415$	−11.1755	−0.548583
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.0389	−0.685845	−0.342922	−	0.939364i	$-0.611417\pi$
$419$	−14.0389	−0.685845	−0.342922	+	0.939364i	$0.611417\pi$
$420$	0	0
$421$	39.9387	1.94649	0.973247	−	0.229762i	$-0.0737949\pi$
$421$	39.9387	1.94649	0.973247	+	0.229762i	$0.0737949\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.58774	−0.271045
$426$	0	0
$427$	11.7827	0.570203
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.0279	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$431$	−27.0279	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$432$	0	0
$433$	−40.0683	−1.92556	−0.962781	−	0.270284i	$-0.912882\pi$
$433$	−40.0683	−1.92556	−0.962781	+	0.270284i	$0.912882\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.87189	0.233054
$438$	0	0
$439$	6.42074	0.306445	0.153223	−	0.988192i	$-0.451035\pi$
$439$	6.42074	0.306445	0.153223	+	0.988192i	$0.451035\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.2702	1.67574	0.837870	−	0.545871i	$-0.183801\pi$
$443$	35.2702	1.67574	0.837870	+	0.545871i	$0.183801\pi$
$444$	0	0
$445$	−6.45963	−0.306216
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.78267	−0.272901	−0.136451	−	0.990647i	$-0.543569\pi$
$449$	−5.78267	−0.272901	−0.136451	+	0.990647i	$0.543569\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.87189	−0.228398
$456$	0	0
$457$	−9.12811	−0.426995	−0.213498	−	0.976944i	$-0.568486\pi$
$457$	−9.12811	−0.426995	−0.213498	+	0.976944i	$0.568486\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.86341	−0.412810	−0.206405	−	0.978467i	$-0.566176\pi$
$461$	−8.86341	−0.412810	−0.206405	+	0.978467i	$0.566176\pi$
$462$	0	0
$463$	−21.1366	−0.982301	−0.491150	−	0.871075i	$-0.663423\pi$
$463$	−21.1366	−0.982301	−0.491150	+	0.871075i	$0.663423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.78267	−0.360139	−0.180070	−	0.983654i	$-0.557632\pi$
$467$	−7.78267	−0.360139	−0.180070	+	0.983654i	$0.557632\pi$
$468$	0	0
$469$	37.2617	1.72059
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.4068	−1.84623	−0.923117	−	0.384519i	$-0.874367\pi$
$479$	−40.4068	−1.84623	−0.923117	+	0.384519i	$0.874367\pi$
$480$	0	0
$481$	−1.28415	−0.0585521
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.4053	0.608703
$486$	0	0
$487$	5.82603	0.264003	0.132001	−	0.991250i	$-0.457860\pi$
$487$	5.82603	0.264003	0.132001	+	0.991250i	$0.457860\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.6630	−1.02277	−0.511384	−	0.859352i	$-0.670867\pi$
$491$	−22.6630	−1.02277	−0.511384	+	0.859352i	$0.670867\pi$
$492$	0	0
$493$	53.5738	2.41284
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−51.4876	−2.30953
$498$	0	0
$499$	−25.7438	−1.15245	−0.576225	−	0.817291i	$-0.695475\pi$
$499$	−25.7438	−1.15245	−0.576225	+	0.817291i	$0.695475\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.4791	−1.13606	−0.568028	−	0.823009i	$-0.692293\pi$
$503$	−25.4791	−1.13606	−0.568028	+	0.823009i	$0.692293\pi$
$504$	0	0
$505$	−19.0668	−0.848462
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.54037	0.422869	0.211435	−	0.977392i	$-0.432186\pi$
$509$	9.54037	0.422869	0.211435	+	0.977392i	$0.432186\pi$
$510$	0	0
$511$	−14.9108	−0.659614
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.05433	0.398982
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.31207	−0.276537	−0.138268	−	0.990395i	$-0.544154\pi$
$521$	−6.31207	−0.276537	−0.138268	+	0.990395i	$0.544154\pi$
$522$	0	0
$523$	8.93719	0.390796	0.195398	−	0.980724i	$-0.437400\pi$
$523$	8.93719	0.390796	0.195398	+	0.980724i	$0.437400\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.0947	1.74655
$528$	0	0
$529$	0.735300	0.0319696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14.2034	−0.615218
$534$	0	0
$535$	−5.24926	−0.226945
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.71585	0.374724	0.187362	−	0.982291i	$-0.440006\pi$
$541$	8.71585	0.374724	0.187362	+	0.982291i	$0.440006\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.04737	0.259041
$546$	0	0
$547$	−26.9457	−1.15211	−0.576057	−	0.817410i	$-0.695409\pi$
$547$	−26.9457	−1.15211	−0.576057	+	0.817410i	$0.695409\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.58774	−0.408452
$552$	0	0
$553$	4.60719	0.195918
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.3619	0.989877	0.494938	−	0.868928i	$-0.335191\pi$
$557$	23.3619	0.989877	0.494938	+	0.868928i	$0.335191\pi$
$558$	0	0
$559$	−3.68793	−0.155983
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.6476	−1.33379	−0.666894	−	0.745153i	$-0.732376\pi$
$563$	−31.6476	−1.33379	−0.666894	+	0.745153i	$0.732376\pi$
$564$	0	0
$565$	−6.22982	−0.262090
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.8804	0.875350	0.437675	−	0.899133i	$-0.355802\pi$
$569$	20.8804	0.875350	0.437675	+	0.899133i	$0.355802\pi$
$570$	0	0
$571$	42.6630	1.78539	0.892696	−	0.450659i	$-0.148811\pi$
$571$	42.6630	1.78539	0.892696	+	0.450659i	$0.148811\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.87189	0.203172
$576$	0	0
$577$	−8.59871	−0.357969	−0.178985	−	0.983852i	$-0.557281\pi$
$577$	−8.59871	−0.357969	−0.178985	+	0.983852i	$0.557281\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	40.0947	1.66341
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.7438	−1.22766	−0.613829	−	0.789439i	$-0.710371\pi$
$587$	−29.7438	−1.22766	−0.613829	+	0.789439i	$0.710371\pi$
$588$	0	0
$589$	−7.17548	−0.295661
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.56829	−0.187597	−0.0937987	−	0.995591i	$-0.529901\pi$
$593$	−4.56829	−0.187597	−0.0937987	+	0.995591i	$0.529901\pi$
$594$	0	0
$595$	20.0474	0.821862
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.7159	−0.601273	−0.300637	−	0.953739i	$-0.597199\pi$
$599$	−14.7159	−0.601273	−0.300637	+	0.953739i	$0.597199\pi$
$600$	0	0
$601$	37.3230	1.52244	0.761219	−	0.648495i	$-0.224601\pi$
$601$	37.3230	1.52244	0.761219	+	0.648495i	$0.224601\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	23.0404	0.935181	0.467591	−	0.883945i	$-0.345122\pi$
$607$	23.0404	0.935181	0.467591	+	0.883945i	$0.345122\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	32.4985	1.31260	0.656302	−	0.754499i	$-0.272120\pi$
$613$	32.4985	1.31260	0.656302	+	0.754499i	$0.272120\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.2982	1.30027	0.650137	−	0.759817i	$-0.274711\pi$
$617$	32.2982	1.30027	0.650137	+	0.759817i	$0.274711\pi$
$618$	0	0
$619$	−34.5683	−1.38942	−0.694709	−	0.719291i	$-0.744467\pi$
$619$	−34.5683	−1.38942	−0.694709	+	0.719291i	$0.744467\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	23.1755	0.928506
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.28415	0.210693
$630$	0	0
$631$	−19.2702	−0.767136	−0.383568	−	0.923513i	$-0.625305\pi$
$631$	−19.2702	−0.767136	−0.383568	+	0.923513i	$0.625305\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.1212	0.560381
$636$	0	0
$637$	7.97359	0.315925
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.06682	0.121132	0.0605660	−	0.998164i	$-0.480709\pi$
$641$	3.06682	0.121132	0.0605660	+	0.998164i	$0.480709\pi$
$642$	0	0
$643$	−25.5264	−1.00666	−0.503332	−	0.864093i	$-0.667893\pi$
$643$	−25.5264	−1.00666	−0.503332	+	0.864093i	$0.667893\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.4791	−1.00169	−0.500843	−	0.865538i	$-0.666977\pi$
$647$	−25.4791	−1.00169	−0.500843	+	0.865538i	$0.666977\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.2034	1.41675	0.708374	−	0.705837i	$-0.249429\pi$
$653$	36.2034	1.41675	0.708374	+	0.705837i	$0.249429\pi$
$654$	0	0
$655$	−18.3510	−0.717032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.6406	1.66104	0.830522	−	0.556986i	$-0.188042\pi$
$659$	42.6406	1.66104	0.830522	+	0.556986i	$0.188042\pi$
$660$	0	0
$661$	24.4681	0.951699	0.475850	−	0.879527i	$-0.342141\pi$
$661$	24.4681	0.951699	0.475850	+	0.879527i	$0.342141\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.58774	−0.139127
$666$	0	0
$667$	−46.7104	−1.80863
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−26.7672	−1.03180	−0.515901	−	0.856649i	$-0.672543\pi$
$673$	−26.7672	−1.03180	−0.515901	+	0.856649i	$0.672543\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.65304	0.217264	0.108632	−	0.994082i	$-0.465353\pi$
$677$	5.65304	0.217264	0.108632	+	0.994082i	$0.465353\pi$
$678$	0	0
$679$	−48.0947	−1.84571
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.1102	0.425119	0.212560	−	0.977148i	$-0.431820\pi$
$683$	11.1102	0.425119	0.212560	+	0.977148i	$0.431820\pi$
$684$	0	0
$685$	3.01945	0.115367
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.3315	−0.507890
$690$	0	0
$691$	−5.96111	−0.226771	−0.113386	−	0.993551i	$-0.536170\pi$
$691$	−5.96111	−0.226771	−0.113386	+	0.993551i	$0.536170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.7827	−0.446942
$696$	0	0
$697$	58.4457	2.21379
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.8524	0.976433	0.488217	−	0.872722i	$-0.337648\pi$
$701$	25.8524	0.976433	0.488217	+	0.872722i	$0.337648\pi$
$702$	0	0
$703$	−0.945668	−0.0356665
$704$	0	0
$705$	0	0
$706$	0	0
$707$	68.4068	2.57270
$708$	0	0
$709$	−23.4317	−0.879996	−0.439998	−	0.897999i	$-0.645021\pi$
$709$	−23.4317	−0.879996	−0.439998	+	0.897999i	$0.645021\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−34.9582	−1.30919
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.20885	0.231551	0.115776	−	0.993275i	$-0.463065\pi$
$719$	6.20885	0.231551	0.115776	+	0.993275i	$0.463065\pi$
$720$	0	0
$721$	−32.4846	−1.20979
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.58774	−0.356080
$726$	0	0
$727$	29.7214	1.10230	0.551152	−	0.834405i	$-0.314188\pi$
$727$	29.7214	1.10230	0.551152	+	0.834405i	$0.314188\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.1755	0.561286
$732$	0	0
$733$	−7.52645	−0.277996	−0.138998	−	0.990293i	$-0.544388\pi$
$733$	−7.52645	−0.277996	−0.138998	+	0.990293i	$0.544388\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22.9582	0.844529	0.422265	−	0.906473i	$-0.361235\pi$
$739$	22.9582	0.844529	0.422265	+	0.906473i	$0.361235\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.524931	−0.0192579	−0.00962893	−	0.999954i	$-0.503065\pi$
$743$	−0.524931	−0.0192579	−0.00962893	+	0.999954i	$0.503065\pi$
$744$	0	0
$745$	20.4985	0.751008
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.8330	0.688143
$750$	0	0
$751$	21.3619	0.779508	0.389754	−	0.920919i	$-0.372560\pi$
$751$	21.3619	0.779508	0.389754	+	0.920919i	$0.372560\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.85244	0.140205
$756$	0	0
$757$	50.4068	1.83207	0.916033	−	0.401102i	$-0.131373\pi$
$757$	50.4068	1.83207	0.916033	+	0.401102i	$0.131373\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.2338	0.733476	0.366738	−	0.930324i	$-0.380475\pi$
$761$	20.2338	0.733476	0.366738	+	0.930324i	$0.380475\pi$
$762$	0	0
$763$	−21.6964	−0.785463
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.7911	−0.497970
$768$	0	0
$769$	−27.1142	−0.977763	−0.488881	−	0.872350i	$-0.662595\pi$
$769$	−27.1142	−0.977763	−0.488881	+	0.872350i	$0.662595\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	12.1685	0.437671	0.218836	−	0.975762i	$-0.429774\pi$
$773$	12.1685	0.437671	0.218836	+	0.975762i	$0.429774\pi$
$774$	0	0
$775$	−7.17548	−0.257751
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.4596	−0.374755
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.89134	−0.138888
$786$	0	0
$787$	−36.1296	−1.28788	−0.643941	−	0.765075i	$-0.722702\pi$
$787$	−36.1296	−1.28788	−0.643941	+	0.765075i	$0.722702\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	22.3510	0.794709
$792$	0	0
$793$	4.45963	0.158366
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.4417	−1.50336	−0.751681	−	0.659527i	$-0.770757\pi$
$797$	−42.4417	−1.50336	−0.751681	+	0.659527i	$0.770757\pi$
$798$	0	0
$799$	−32.9193	−1.16460
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−17.4791	−0.616057
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.3006	1.31142	0.655710	−	0.755013i	$-0.272369\pi$
$809$	37.3006	1.31142	0.655710	+	0.755013i	$0.272369\pi$
$810$	0	0
$811$	27.4402	0.963555	0.481778	−	0.876294i	$-0.339991\pi$
$811$	27.4402	0.963555	0.481778	+	0.876294i	$0.339991\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.7159	0.655588
$816$	0	0
$817$	−2.71585	−0.0950157
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.0140	−1.64080	−0.820400	−	0.571790i	$-0.806249\pi$
$821$	−47.0140	−1.64080	−0.820400	+	0.571790i	$0.806249\pi$
$822$	0	0
$823$	35.9776	1.25410	0.627050	−	0.778979i	$-0.284262\pi$
$823$	35.9776	1.25410	0.627050	+	0.778979i	$0.284262\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.6032	−0.507802	−0.253901	−	0.967230i	$-0.581714\pi$
$827$	−14.6032	−0.507802	−0.253901	+	0.967230i	$0.581714\pi$
$828$	0	0
$829$	−6.96663	−0.241961	−0.120981	−	0.992655i	$-0.538604\pi$
$829$	−6.96663	−0.241961	−0.120981	+	0.992655i	$0.538604\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32.8106	−1.13682
$834$	0	0
$835$	−3.09323	−0.107046
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.459630	0.0158682	0.00793410	−	0.999969i	$-0.497474\pi$
$839$	0.459630	0.0158682	0.00793410	+	0.999969i	$0.497474\pi$
$840$	0	0
$841$	62.9248	2.16982
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.1560	0.383779
$846$	0	0
$847$	−39.4652	−1.35604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.60719	−0.157932
$852$	0	0
$853$	−5.48755	−0.187890	−0.0939451	−	0.995577i	$-0.529948\pi$
$853$	−5.48755	−0.187890	−0.0939451	+	0.995577i	$0.529948\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.6586	−1.79878	−0.899391	−	0.437145i	$-0.855990\pi$
$857$	−52.6586	−1.79878	−0.899391	+	0.437145i	$0.855990\pi$
$858$	0	0
$859$	−4.29512	−0.146547	−0.0732737	−	0.997312i	$-0.523345\pi$
$859$	−4.29512	−0.146547	−0.0732737	+	0.997312i	$0.523345\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.6506	0.498711	0.249355	−	0.968412i	$-0.419781\pi$
$863$	14.6506	0.498711	0.249355	+	0.968412i	$0.419781\pi$
$864$	0	0
$865$	7.51396	0.255482
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	14.1032	0.477869
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.58774	−0.121288
$876$	0	0
$877$	−19.7089	−0.665522	−0.332761	−	0.943011i	$-0.607980\pi$
$877$	−19.7089	−0.665522	−0.332761	+	0.943011i	$0.607980\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.32304	−0.0445744	−0.0222872	−	0.999752i	$-0.507095\pi$
$881$	−1.32304	−0.0445744	−0.0222872	+	0.999752i	$0.507095\pi$
$882$	0	0
$883$	−25.0668	−0.843566	−0.421783	−	0.906697i	$-0.638596\pi$
$883$	−25.0668	−0.843566	−0.421783	+	0.906697i	$0.638596\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.5669	1.22779	0.613897	−	0.789386i	$-0.289601\pi$
$887$	36.5669	1.22779	0.613897	+	0.789386i	$0.289601\pi$
$888$	0	0
$889$	−50.6630	−1.69918
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.89134	0.197146
$894$	0	0
$895$	−4.45963	−0.149069
$896$	0	0
$897$	0	0
$898$	0	0
$899$	68.7967	2.29450
$900$	0	0
$901$	54.8580	1.82758
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.56829	−0.284820
$906$	0	0
$907$	−40.7368	−1.35264	−0.676322	−	0.736606i	$-0.736427\pi$
$907$	−40.7368	−1.35264	−0.676322	+	0.736606i	$0.736427\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	58.2982	1.93150	0.965752	−	0.259467i	$-0.0835469\pi$
$911$	58.2982	1.93150	0.965752	+	0.259467i	$0.0835469\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	65.8385	2.17418
$918$	0	0
$919$	−14.5209	−0.479001	−0.239501	−	0.970896i	$-0.576984\pi$
$919$	−14.5209	−0.479001	−0.239501	+	0.970896i	$0.576984\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−19.4876	−0.641441
$924$	0	0
$925$	−0.945668	−0.0310934
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.7523	−0.451197	−0.225598	−	0.974220i	$-0.572434\pi$
$929$	−13.7523	−0.451197	−0.225598	+	0.974220i	$0.572434\pi$
$930$	0	0
$931$	5.87189	0.192443
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.4512	−0.406761	−0.203381	−	0.979100i	$-0.565193\pi$
$937$	−12.4512	−0.406761	−0.203381	+	0.979100i	$0.565193\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34.3594	1.12009	0.560043	−	0.828464i	$-0.310785\pi$
$941$	34.3594	1.12009	0.560043	+	0.828464i	$0.310785\pi$
$942$	0	0
$943$	−50.9582	−1.65943
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.4876	−0.763243	−0.381621	−	0.924319i	$-0.624634\pi$
$947$	−23.4876	−0.763243	−0.381621	+	0.924319i	$0.624634\pi$
$948$	0	0
$949$	−5.64359	−0.183199
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.5808	−1.57369	−0.786843	−	0.617153i	$-0.788286\pi$
$953$	−48.5808	−1.57369	−0.786843	+	0.617153i	$0.788286\pi$
$954$	0	0
$955$	7.73530	0.250308
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.8330	−0.349816
$960$	0	0
$961$	20.4876	0.660889
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.08226	−0.195795
$966$	0	0
$967$	53.3261	1.71485	0.857425	−	0.514608i	$-0.172063\pi$
$967$	53.3261	1.71485	0.857425	+	0.514608i	$0.172063\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.1366	1.06340	0.531702	−	0.846932i	$-0.321553\pi$
$971$	33.1366	1.06340	0.531702	+	0.846932i	$0.321553\pi$
$972$	0	0
$973$	42.2732	1.35522
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.62263	−0.179884	−0.0899419	−	0.995947i	$-0.528668\pi$
$977$	−5.62263	−0.179884	−0.0899419	+	0.995947i	$0.528668\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.3664	−0.681482	−0.340741	−	0.940157i	$-0.610678\pi$
$983$	−21.3664	−0.681482	−0.340741	+	0.940157i	$0.610678\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.2313	−0.420732
$990$	0	0
$991$	41.1446	1.30700	0.653501	−	0.756926i	$-0.273300\pi$
$991$	41.1446	1.30700	0.653501	+	0.756926i	$0.273300\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.91078	0.219087
$996$	0	0
$997$	−48.5155	−1.53650	−0.768250	−	0.640150i	$-0.778872\pi$
$997$	−48.5155	−1.53650	−0.768250	+	0.640150i	$0.778872\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bg.1.3		3
3.2	odd	2		760.2.a.j.1.2	✓	3
12.11	even	2		1520.2.a.s.1.2		3
15.2	even	4		3800.2.d.l.3649.4		6
15.8	even	4		3800.2.d.l.3649.3		6
15.14	odd	2		3800.2.a.x.1.2		3
24.5	odd	2		6080.2.a.bv.1.2		3
24.11	even	2		6080.2.a.bq.1.2		3
60.59	even	2		7600.2.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.2	✓	3	3.2	odd	2
1520.2.a.s.1.2		3	12.11	even	2
3800.2.a.x.1.2		3	15.14	odd	2
3800.2.d.l.3649.3		6	15.8	even	4
3800.2.d.l.3649.4		6	15.2	even	4
6080.2.a.bq.1.2		3	24.11	even	2
6080.2.a.bv.1.2		3	24.5	odd	2
6840.2.a.bg.1.3		3	1.1	even	1	trivial
7600.2.a.bq.1.2		3	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.58774 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.58774 q^{7} +1.35793 q^{13} -5.58774 q^{17} +1.00000 q^{19} +4.87189 q^{23} +1.00000 q^{25} -9.58774 q^{29} -7.17548 q^{31} -3.58774 q^{35} -0.945668 q^{37} -10.4596 q^{41} -2.71585 q^{43} +5.89134 q^{47} +5.87189 q^{49} -9.81756 q^{53} -10.1560 q^{59} +3.28415 q^{61} -1.35793 q^{65} +10.3859 q^{67} -14.3510 q^{71} -4.15604 q^{73} +1.28415 q^{79} +11.1755 q^{83} +5.58774 q^{85} +6.45963 q^{89} +4.87189 q^{91} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - q^7	\( 3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - q^7 + 5 * q^13 - 5 * q^17 + 3 * q^19 + q^23 + 3 * q^25 - 17 * q^29 + 2 * q^31 + q^35 + 8 * q^37 - 6 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^49 - 5 * q^53 - 15 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 + 4 * q^71 + 3 * q^73 + 2 * q^79 + 10 * q^83 + 5 * q^85 - 6 * q^89 + q^91 - 3 * q^95 - 4 * q^97

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