LMFDB - Embedded newform 6840.2.a.bk.1.1

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:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.60975 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.60975 q^{7} +1.60975 q^{11} +0.640023 q^{13} -5.28005 q^{17} +1.00000 q^{19} +8.24977 q^{23} +1.00000 q^{25} +4.88979 q^{29} -7.28005 q^{31} -3.60975 q^{35} -9.92007 q^{37} -4.57947 q^{41} +7.92007 q^{43} -0.249771 q^{47} +6.03028 q^{49} -1.75023 q^{53} +1.60975 q^{55} -12.3103 q^{59} +11.5298 q^{61} +0.640023 q^{65} +12.4995 q^{67} -13.7796 q^{71} +3.93945 q^{73} -5.81078 q^{77} -3.03028 q^{79} +10.4995 q^{83} -5.28005 q^{85} -15.9201 q^{89} -2.31032 q^{91} +1.00000 q^{95} -9.92007 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 2 * q^7	$ 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * 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
$7$	−3.60975	−1.36436	−0.682178	−	0.731186i	$-0.738967\pi$
$7$	−3.60975	−1.36436	−0.682178	+	0.731186i	$0.738967\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.60975	0.485357	0.242679	−	0.970107i	$-0.421974\pi$
$11$	1.60975	0.485357	0.242679	+	0.970107i	$0.421974\pi$
$12$	0	0
$13$	0.640023	0.177511	0.0887553	−	0.996053i	$-0.471711\pi$
$13$	0.640023	0.177511	0.0887553	+	0.996053i	$0.471711\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.28005	−1.28060	−0.640300	−	0.768125i	$-0.721190\pi$
$17$	−5.28005	−1.28060	−0.640300	+	0.768125i	$0.721190\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.24977	1.72020	0.860098	−	0.510129i	$-0.170402\pi$
$23$	8.24977	1.72020	0.860098	+	0.510129i	$0.170402\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.88979	0.908012	0.454006	−	0.890999i	$-0.349994\pi$
$29$	4.88979	0.908012	0.454006	+	0.890999i	$0.349994\pi$
$30$	0	0
$31$	−7.28005	−1.30754	−0.653768	−	0.756695i	$-0.726813\pi$
$31$	−7.28005	−1.30754	−0.653768	+	0.756695i	$0.726813\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.60975	−0.610159
$36$	0	0
$37$	−9.92007	−1.63085	−0.815425	−	0.578863i	$-0.803497\pi$
$37$	−9.92007	−1.63085	−0.815425	+	0.578863i	$0.803497\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.57947	−0.715193	−0.357597	−	0.933876i	$-0.616404\pi$
$41$	−4.57947	−0.715193	−0.357597	+	0.933876i	$0.616404\pi$
$42$	0	0
$43$	7.92007	1.20780	0.603900	−	0.797060i	$-0.293613\pi$
$43$	7.92007	1.20780	0.603900	+	0.797060i	$0.293613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.249771	−0.0364328	−0.0182164	−	0.999834i	$-0.505799\pi$
$47$	−0.249771	−0.0364328	−0.0182164	+	0.999834i	$0.505799\pi$
$48$	0	0
$49$	6.03028	0.861468
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.75023	−0.240412	−0.120206	−	0.992749i	$-0.538356\pi$
$53$	−1.75023	−0.240412	−0.120206	+	0.992749i	$0.538356\pi$
$54$	0	0
$55$	1.60975	0.217058
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.3103	−1.60267	−0.801334	−	0.598218i	$-0.795876\pi$
$59$	−12.3103	−1.60267	−0.801334	+	0.598218i	$0.795876\pi$
$60$	0	0
$61$	11.5298	1.47624	0.738121	−	0.674668i	$-0.235713\pi$
$61$	11.5298	1.47624	0.738121	+	0.674668i	$0.235713\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.640023	0.0793851
$66$	0	0
$67$	12.4995	1.52706	0.763531	−	0.645771i	$-0.223464\pi$
$67$	12.4995	1.52706	0.763531	+	0.645771i	$0.223464\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.7796	−1.63534	−0.817668	−	0.575690i	$-0.804734\pi$
$71$	−13.7796	−1.63534	−0.817668	+	0.575690i	$0.804734\pi$
$72$	0	0
$73$	3.93945	0.461077	0.230539	−	0.973063i	$-0.425951\pi$
$73$	3.93945	0.461077	0.230539	+	0.973063i	$0.425951\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.81078	−0.662200
$78$	0	0
$79$	−3.03028	−0.340933	−0.170466	−	0.985363i	$-0.554527\pi$
$79$	−3.03028	−0.340933	−0.170466	+	0.985363i	$0.554527\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.4995	1.15247	0.576237	−	0.817282i	$-0.304520\pi$
$83$	10.4995	1.15247	0.576237	+	0.817282i	$0.304520\pi$
$84$	0	0
$85$	−5.28005	−0.572701
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.9201	−1.68752	−0.843762	−	0.536717i	$-0.819664\pi$
$89$	−15.9201	−1.68752	−0.843762	+	0.536717i	$0.819664\pi$
$90$	0	0
$91$	−2.31032	−0.242188
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−9.92007	−1.00723	−0.503615	−	0.863928i	$-0.667997\pi$
$97$	−9.92007	−1.00723	−0.503615	+	0.863928i	$0.667997\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.2195	−1.71340	−0.856702	−	0.515812i	$-0.827490\pi$
$101$	−17.2195	−1.71340	−0.856702	+	0.515812i	$0.827490\pi$
$102$	0	0
$103$	12.4995	1.23162	0.615808	−	0.787896i	$-0.288830\pi$
$103$	12.4995	1.23162	0.615808	+	0.787896i	$0.288830\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.5601	1.02088	0.510441	−	0.859913i	$-0.329482\pi$
$107$	10.5601	1.02088	0.510441	+	0.859913i	$0.329482\pi$
$108$	0	0
$109$	10.3103	0.987550	0.493775	−	0.869590i	$-0.335617\pi$
$109$	10.3103	0.987550	0.493775	+	0.869590i	$0.335617\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.470182	−0.0442310	−0.0221155	−	0.999755i	$-0.507040\pi$
$113$	−0.470182	−0.0442310	−0.0221155	+	0.999755i	$0.507040\pi$
$114$	0	0
$115$	8.24977	0.769295
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.0596	1.74719
$120$	0	0
$121$	−8.40871	−0.764428
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.4995	−1.10915	−0.554577	−	0.832132i	$-0.687120\pi$
$127$	−12.4995	−1.10915	−0.554577	+	0.832132i	$0.687120\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.32970	−0.378288	−0.189144	−	0.981949i	$-0.560571\pi$
$131$	−4.32970	−0.378288	−0.189144	+	0.981949i	$0.560571\pi$
$132$	0	0
$133$	−3.60975	−0.313005
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.03028	−0.429765	−0.214883	−	0.976640i	$-0.568937\pi$
$137$	−5.03028	−0.429765	−0.214883	+	0.976640i	$0.568937\pi$
$138$	0	0
$139$	−9.77959	−0.829494	−0.414747	−	0.909937i	$-0.636130\pi$
$139$	−9.77959	−0.829494	−0.414747	+	0.909937i	$0.636130\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.03028	0.0861560
$144$	0	0
$145$	4.88979	0.406075
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	1.34060	0.109096	0.0545482	−	0.998511i	$-0.482628\pi$
$151$	1.34060	0.109096	0.0545482	+	0.998511i	$0.482628\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.28005	−0.584747
$156$	0	0
$157$	−10.1892	−0.813188	−0.406594	−	0.913609i	$-0.633284\pi$
$157$	−10.1892	−0.813188	−0.406594	+	0.913609i	$0.633284\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−29.7796	−2.34696
$162$	0	0
$163$	15.7990	1.23747	0.618735	−	0.785600i	$-0.287645\pi$
$163$	15.7990	1.23747	0.618735	+	0.785600i	$0.287645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.52982	−0.737439	−0.368720	−	0.929541i	$-0.620204\pi$
$167$	−9.52982	−0.737439	−0.368720	+	0.929541i	$0.620204\pi$
$168$	0	0
$169$	−12.5904	−0.968490
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.4693	−1.02405	−0.512025	−	0.858971i	$-0.671104\pi$
$173$	−13.4693	−1.02405	−0.512025	+	0.858971i	$0.671104\pi$
$174$	0	0
$175$	−3.60975	−0.272871
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.1892	−1.50901	−0.754507	−	0.656293i	$-0.772124\pi$
$179$	−20.1892	−1.50901	−0.754507	+	0.656293i	$0.772124\pi$
$180$	0	0
$181$	−7.75023	−0.576070	−0.288035	−	0.957620i	$-0.593002\pi$
$181$	−7.75023	−0.576070	−0.288035	+	0.957620i	$0.593002\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.92007	−0.729338
$186$	0	0
$187$	−8.49954	−0.621548
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.39025	0.172953	0.0864763	−	0.996254i	$-0.472439\pi$
$191$	2.39025	0.172953	0.0864763	+	0.996254i	$0.472439\pi$
$192$	0	0
$193$	−0.640023	−0.0460699	−0.0230349	−	0.999735i	$-0.507333\pi$
$193$	−0.640023	−0.0460699	−0.0230349	+	0.999735i	$0.507333\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.7493	1.05085	0.525423	−	0.850841i	$-0.323907\pi$
$197$	14.7493	1.05085	0.525423	+	0.850841i	$0.323907\pi$
$198$	0	0
$199$	19.8401	1.40643	0.703215	−	0.710977i	$-0.251747\pi$
$199$	19.8401	1.40643	0.703215	+	0.710977i	$0.251747\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−17.6509	−1.23885
$204$	0	0
$205$	−4.57947	−0.319844
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.60975	0.111349
$210$	0	0
$211$	−2.06055	−0.141854	−0.0709271	−	0.997481i	$-0.522596\pi$
$211$	−2.06055	−0.141854	−0.0709271	+	0.997481i	$0.522596\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.92007	0.540144
$216$	0	0
$217$	26.2791	1.78394
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.37935	−0.227320
$222$	0	0
$223$	−26.5601	−1.77860	−0.889298	−	0.457329i	$-0.848806\pi$
$223$	−26.5601	−1.77860	−0.889298	+	0.457329i	$0.848806\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.03028	0.333871	0.166936	−	0.985968i	$-0.446613\pi$
$227$	5.03028	0.333871	0.166936	+	0.985968i	$0.446613\pi$
$228$	0	0
$229$	23.5298	1.55489	0.777447	−	0.628948i	$-0.216514\pi$
$229$	23.5298	1.55489	0.777447	+	0.628948i	$0.216514\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.9688	−1.04615	−0.523076	−	0.852286i	$-0.675215\pi$
$233$	−15.9688	−1.04615	−0.523076	+	0.852286i	$0.675215\pi$
$234$	0	0
$235$	−0.249771	−0.0162933
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.6703	−1.01363	−0.506814	−	0.862056i	$-0.669177\pi$
$239$	−15.6703	−1.01363	−0.506814	+	0.862056i	$0.669177\pi$
$240$	0	0
$241$	−28.4390	−1.83192	−0.915958	−	0.401274i	$-0.868568\pi$
$241$	−28.4390	−1.83192	−0.915958	+	0.401274i	$0.868568\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.03028	0.385260
$246$	0	0
$247$	0.640023	0.0407237
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.8292	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$251$	24.8292	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$252$	0	0
$253$	13.2800	0.834909
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.46927	0.590677	0.295338	−	0.955393i	$-0.404568\pi$
$257$	9.46927	0.590677	0.295338	+	0.955393i	$0.404568\pi$
$258$	0	0
$259$	35.8089	2.22506
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.5298	0.834284	0.417142	−	0.908841i	$-0.363032\pi$
$263$	13.5298	0.834284	0.417142	+	0.908841i	$0.363032\pi$
$264$	0	0
$265$	−1.75023	−0.107516
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.8889	−1.57847	−0.789236	−	0.614090i	$-0.789523\pi$
$269$	−25.8889	−1.57847	−0.789236	+	0.614090i	$0.789523\pi$
$270$	0	0
$271$	−29.6585	−1.80162	−0.900812	−	0.434209i	$-0.857028\pi$
$271$	−29.6585	−1.80162	−0.900812	+	0.434209i	$0.857028\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.60975	0.0970714
$276$	0	0
$277$	1.03028	0.0619033	0.0309516	−	0.999521i	$-0.490146\pi$
$277$	1.03028	0.0619033	0.0309516	+	0.999521i	$0.490146\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0799	−0.720628	−0.360314	−	0.932831i	$-0.617330\pi$
$281$	−12.0799	−0.720628	−0.360314	+	0.932831i	$0.617330\pi$
$282$	0	0
$283$	−13.2001	−0.784666	−0.392333	−	0.919823i	$-0.628332\pi$
$283$	−13.2001	−0.784666	−0.392333	+	0.919823i	$0.628332\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.5307	0.975778
$288$	0	0
$289$	10.8789	0.639935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.0303	−0.644396	−0.322198	−	0.946672i	$-0.604422\pi$
$293$	−11.0303	−0.644396	−0.322198	+	0.946672i	$0.604422\pi$
$294$	0	0
$295$	−12.3103	−0.716735
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.28005	0.305353
$300$	0	0
$301$	−28.5895	−1.64787
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.5298	0.660195
$306$	0	0
$307$	17.7796	1.01473	0.507367	−	0.861730i	$-0.330619\pi$
$307$	17.7796	1.01473	0.507367	+	0.861730i	$0.330619\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.88979	−0.163865	−0.0819326	−	0.996638i	$-0.526109\pi$
$311$	−2.88979	−0.163865	−0.0819326	+	0.996638i	$0.526109\pi$
$312$	0	0
$313$	−3.40115	−0.192244	−0.0961222	−	0.995370i	$-0.530644\pi$
$313$	−3.40115	−0.192244	−0.0961222	+	0.995370i	$0.530644\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.18922	−0.235290	−0.117645	−	0.993056i	$-0.537534\pi$
$317$	−4.18922	−0.235290	−0.117645	+	0.993056i	$0.537534\pi$
$318$	0	0
$319$	7.87133	0.440710
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.28005	−0.293790
$324$	0	0
$325$	0.640023	0.0355021
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.901610	0.0497073
$330$	0	0
$331$	10.3103	0.566707	0.283353	−	0.959016i	$-0.408553\pi$
$331$	10.3103	0.566707	0.283353	+	0.959016i	$0.408553\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.4995	0.682923
$336$	0	0
$337$	−16.6400	−0.906440	−0.453220	−	0.891399i	$-0.649725\pi$
$337$	−16.6400	−0.906440	−0.453220	+	0.891399i	$0.649725\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.7190	−0.634621
$342$	0	0
$343$	3.50046	0.189007
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0606	0.862176	0.431088	−	0.902310i	$-0.358130\pi$
$347$	16.0606	0.862176	0.431088	+	0.902310i	$0.358130\pi$
$348$	0	0
$349$	−12.0606	−0.645587	−0.322793	−	0.946469i	$-0.604622\pi$
$349$	−12.0606	−0.645587	−0.322793	+	0.946469i	$0.604622\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.5298	−0.720120	−0.360060	−	0.932929i	$-0.617244\pi$
$353$	−13.5298	−0.720120	−0.360060	+	0.932929i	$0.617244\pi$
$354$	0	0
$355$	−13.7796	−0.731345
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.6097	−1.56274	−0.781371	−	0.624066i	$-0.785479\pi$
$359$	−29.6097	−1.56274	−0.781371	+	0.624066i	$0.785479\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.93945	0.206200
$366$	0	0
$367$	−11.4886	−0.599702	−0.299851	−	0.953986i	$-0.596937\pi$
$367$	−11.4886	−0.599702	−0.299851	+	0.953986i	$0.596937\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.31789	0.328008
$372$	0	0
$373$	7.20012	0.372808	0.186404	−	0.982473i	$-0.440317\pi$
$373$	7.20012	0.372808	0.186404	+	0.982473i	$0.440317\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.12958	0.161182
$378$	0	0
$379$	9.68968	0.497725	0.248863	−	0.968539i	$-0.419943\pi$
$379$	9.68968	0.497725	0.248863	+	0.968539i	$0.419943\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.9385	−1.17210	−0.586052	−	0.810273i	$-0.699319\pi$
$383$	−22.9385	−1.17210	−0.586052	+	0.810273i	$0.699319\pi$
$384$	0	0
$385$	−5.81078	−0.296145
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.5592	1.09309	0.546547	−	0.837429i	$-0.315942\pi$
$389$	21.5592	1.09309	0.546547	+	0.837429i	$0.315942\pi$
$390$	0	0
$391$	−43.5592	−2.20288
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.03028	−0.152470
$396$	0	0
$397$	22.0294	1.10562	0.552811	−	0.833307i	$-0.313555\pi$
$397$	22.0294	1.10562	0.552811	+	0.833307i	$0.313555\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.0790	0.852885	0.426443	−	0.904515i	$-0.359767\pi$
$401$	17.0790	0.852885	0.426443	+	0.904515i	$0.359767\pi$
$402$	0	0
$403$	−4.65940	−0.232101
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.9688	−0.791544
$408$	0	0
$409$	−23.6585	−1.16984	−0.584918	−	0.811092i	$-0.698873\pi$
$409$	−23.6585	−1.16984	−0.584918	+	0.811092i	$0.698873\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	44.4372	2.18661
$414$	0	0
$415$	10.4995	0.515402
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.7687	−1.11232	−0.556162	−	0.831074i	$-0.687726\pi$
$419$	−22.7687	−1.11232	−0.556162	+	0.831074i	$0.687726\pi$
$420$	0	0
$421$	−2.47018	−0.120389	−0.0601947	−	0.998187i	$-0.519172\pi$
$421$	−2.47018	−0.120389	−0.0601947	+	0.998187i	$0.519172\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.28005	−0.256120
$426$	0	0
$427$	−41.6197	−2.01412
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.40115	0.452838	0.226419	−	0.974030i	$-0.427298\pi$
$431$	9.40115	0.452838	0.226419	+	0.974030i	$0.427298\pi$
$432$	0	0
$433$	36.4802	1.75312	0.876562	−	0.481288i	$-0.159831\pi$
$433$	36.4802	1.75312	0.876562	+	0.481288i	$0.159831\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.24977	0.394640
$438$	0	0
$439$	−30.5895	−1.45995	−0.729977	−	0.683471i	$-0.760469\pi$
$439$	−30.5895	−1.45995	−0.729977	+	0.683471i	$0.760469\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.9612	−1.04341	−0.521705	−	0.853126i	$-0.674704\pi$
$443$	−21.9612	−1.04341	−0.521705	+	0.853126i	$0.674704\pi$
$444$	0	0
$445$	−15.9201	−0.754684
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.3212	−0.628667	−0.314334	−	0.949313i	$-0.601781\pi$
$449$	−13.3212	−0.628667	−0.314334	+	0.949313i	$0.601781\pi$
$450$	0	0
$451$	−7.37179	−0.347124
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.31032	−0.108310
$456$	0	0
$457$	−17.9612	−0.840192	−0.420096	−	0.907480i	$-0.638004\pi$
$457$	−17.9612	−0.840192	−0.420096	+	0.907480i	$0.638004\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.5592	−1.37671	−0.688354	−	0.725375i	$-0.741666\pi$
$461$	−29.5592	−1.37671	−0.688354	+	0.725375i	$0.741666\pi$
$462$	0	0
$463$	11.3288	0.526493	0.263247	−	0.964729i	$-0.415207\pi$
$463$	11.3288	0.526493	0.263247	+	0.964729i	$0.415207\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.84014	0.0851516	0.0425758	−	0.999093i	$-0.486444\pi$
$467$	1.84014	0.0851516	0.0425758	+	0.999093i	$0.486444\pi$
$468$	0	0
$469$	−45.1202	−2.08346
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.7493	0.586214
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.51044	−0.343161	−0.171580	−	0.985170i	$-0.554887\pi$
$479$	−7.51044	−0.343161	−0.171580	+	0.985170i	$0.554887\pi$
$480$	0	0
$481$	−6.34908	−0.289493
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.92007	−0.450447
$486$	0	0
$487$	18.2791	0.828306	0.414153	−	0.910207i	$-0.364078\pi$
$487$	18.2791	0.828306	0.414153	+	0.910207i	$0.364078\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.7678	1.61418	0.807089	−	0.590430i	$-0.201042\pi$
$491$	35.7678	1.61418	0.807089	+	0.590430i	$0.201042\pi$
$492$	0	0
$493$	−25.8183	−1.16280
$494$	0	0
$495$	0	0
$496$	0	0
$497$	49.7408	2.23118
$498$	0	0
$499$	−5.65848	−0.253309	−0.126654	−	0.991947i	$-0.540424\pi$
$499$	−5.65848	−0.253309	−0.126654	+	0.991947i	$0.540424\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.4078	−1.71252	−0.856260	−	0.516546i	$-0.827218\pi$
$503$	−38.4078	−1.71252	−0.856260	+	0.516546i	$0.827218\pi$
$504$	0	0
$505$	−17.2195	−0.766257
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.1093	0.714032	0.357016	−	0.934098i	$-0.383794\pi$
$509$	16.1093	0.714032	0.357016	+	0.934098i	$0.383794\pi$
$510$	0	0
$511$	−14.2204	−0.629074
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.4995	0.550796
$516$	0	0
$517$	−0.402068	−0.0176829
$518$	0	0
$519$	0	0
$520$	0	0
$521$	41.9182	1.83647	0.918236	−	0.396034i	$-0.129614\pi$
$521$	41.9182	1.83647	0.918236	+	0.396034i	$0.129614\pi$
$522$	0	0
$523$	27.5979	1.20677	0.603387	−	0.797449i	$-0.293818\pi$
$523$	27.5979	1.20677	0.603387	+	0.797449i	$0.293818\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.4390	1.67443
$528$	0	0
$529$	45.0587	1.95907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.93097	−0.126954
$534$	0	0
$535$	10.5601	0.456553
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.70722	0.418120
$540$	0	0
$541$	38.9991	1.67670	0.838351	−	0.545131i	$-0.183520\pi$
$541$	38.9991	1.67670	0.838351	+	0.545131i	$0.183520\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.3103	0.441646
$546$	0	0
$547$	−7.21949	−0.308683	−0.154342	−	0.988018i	$-0.549326\pi$
$547$	−7.21949	−0.308683	−0.154342	+	0.988018i	$0.549326\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.88979	0.208312
$552$	0	0
$553$	10.9385	0.465154
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.3179	−0.521926	−0.260963	−	0.965349i	$-0.584040\pi$
$557$	−12.3179	−0.521926	−0.260963	+	0.965349i	$0.584040\pi$
$558$	0	0
$559$	5.06903	0.214397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.2110	1.35753	0.678766	−	0.734354i	$-0.262515\pi$
$563$	32.2110	1.35753	0.678766	+	0.734354i	$0.262515\pi$
$564$	0	0
$565$	−0.470182	−0.0197807
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.9201	1.17047	0.585235	−	0.810864i	$-0.301002\pi$
$569$	27.9201	1.17047	0.585235	+	0.810864i	$0.301002\pi$
$570$	0	0
$571$	40.9603	1.71414	0.857068	−	0.515203i	$-0.172283\pi$
$571$	40.9603	1.71414	0.857068	+	0.515203i	$0.172283\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.24977	0.344039
$576$	0	0
$577$	−39.6585	−1.65100	−0.825502	−	0.564399i	$-0.809108\pi$
$577$	−39.6585	−1.65100	−0.825502	+	0.564399i	$0.809108\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−37.9007	−1.57239
$582$	0	0
$583$	−2.81743	−0.116686
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.3388	−1.78878	−0.894391	−	0.447286i	$-0.852391\pi$
$587$	−43.3388	−1.78878	−0.894391	+	0.447286i	$0.852391\pi$
$588$	0	0
$589$	−7.28005	−0.299969
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.5895	1.33829	0.669144	−	0.743133i	$-0.266661\pi$
$593$	32.5895	1.33829	0.669144	+	0.743133i	$0.266661\pi$
$594$	0	0
$595$	19.0596	0.781369
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.1807	1.11057	0.555287	−	0.831658i	$-0.312608\pi$
$599$	27.1807	1.11057	0.555287	+	0.831658i	$0.312608\pi$
$600$	0	0
$601$	−9.71904	−0.396448	−0.198224	−	0.980157i	$-0.563517\pi$
$601$	−9.71904	−0.396448	−0.198224	+	0.980157i	$0.563517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.40871	−0.341863
$606$	0	0
$607$	−17.7796	−0.721651	−0.360826	−	0.932633i	$-0.617505\pi$
$607$	−17.7796	−0.721651	−0.360826	+	0.932633i	$0.617505\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.159859	−0.00646721
$612$	0	0
$613$	−16.9915	−0.686281	−0.343141	−	0.939284i	$-0.611491\pi$
$613$	−16.9915	−0.686281	−0.343141	+	0.939284i	$0.611491\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.3406	−1.10069	−0.550346	−	0.834937i	$-0.685504\pi$
$617$	−27.3406	−1.10069	−0.550346	+	0.834937i	$0.685504\pi$
$618$	0	0
$619$	41.2800	1.65919	0.829593	−	0.558369i	$-0.188573\pi$
$619$	41.2800	1.65919	0.829593	+	0.558369i	$0.188573\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	57.4674	2.30238
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	52.3784	2.08847
$630$	0	0
$631$	−26.5601	−1.05734	−0.528670	−	0.848827i	$-0.677309\pi$
$631$	−26.5601	−1.05734	−0.528670	+	0.848827i	$0.677309\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.4995	−0.496029
$636$	0	0
$637$	3.85952	0.152920
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.01846	0.119222	0.0596110	−	0.998222i	$-0.481014\pi$
$641$	3.01846	0.119222	0.0596110	+	0.998222i	$0.481014\pi$
$642$	0	0
$643$	−11.7990	−0.465306	−0.232653	−	0.972560i	$-0.574741\pi$
$643$	−11.7990	−0.465306	−0.232653	+	0.972560i	$0.574741\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.46927	0.293647	0.146824	−	0.989163i	$-0.453095\pi$
$647$	7.46927	0.293647	0.146824	+	0.989163i	$0.453095\pi$
$648$	0	0
$649$	−19.8165	−0.777866
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−41.8089	−1.63611	−0.818055	−	0.575140i	$-0.804948\pi$
$653$	−41.8089	−1.63611	−0.818055	+	0.575140i	$0.804948\pi$
$654$	0	0
$655$	−4.32970	−0.169175
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.8477	−1.16270	−0.581351	−	0.813653i	$-0.697476\pi$
$659$	−29.8477	−1.16270	−0.581351	+	0.813653i	$0.697476\pi$
$660$	0	0
$661$	18.5677	0.722198	0.361099	−	0.932527i	$-0.382402\pi$
$661$	18.5677	0.722198	0.361099	+	0.932527i	$0.382402\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.60975	−0.139980
$666$	0	0
$667$	40.3397	1.56196
$668$	0	0
$669$	0	0
$670$	0	0
$671$	18.5601	0.716504
$672$	0	0
$673$	−4.64002	−0.178860	−0.0894299	−	0.995993i	$-0.528504\pi$
$673$	−4.64002	−0.178860	−0.0894299	+	0.995993i	$0.528504\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.3709	−0.706050	−0.353025	−	0.935614i	$-0.614847\pi$
$677$	−18.3709	−0.706050	−0.353025	+	0.935614i	$0.614847\pi$
$678$	0	0
$679$	35.8089	1.37422
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.9385	−0.724663	−0.362331	−	0.932049i	$-0.618019\pi$
$683$	−18.9385	−0.724663	−0.362331	+	0.932049i	$0.618019\pi$
$684$	0	0
$685$	−5.03028	−0.192197
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.12019	−0.0426758
$690$	0	0
$691$	38.1580	1.45160	0.725800	−	0.687906i	$-0.241470\pi$
$691$	38.1580	1.45160	0.725800	+	0.687906i	$0.241470\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.77959	−0.370961
$696$	0	0
$697$	24.1798	0.915876
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.9394	−0.904180	−0.452090	−	0.891972i	$-0.649322\pi$
$701$	−23.9394	−0.904180	−0.452090	+	0.891972i	$0.649322\pi$
$702$	0	0
$703$	−9.92007	−0.374143
$704$	0	0
$705$	0	0
$706$	0	0
$707$	62.1580	2.33769
$708$	0	0
$709$	−13.0908	−0.491636	−0.245818	−	0.969316i	$-0.579057\pi$
$709$	−13.0908	−0.491636	−0.245818	+	0.969316i	$0.579057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−60.0587	−2.24922
$714$	0	0
$715$	1.03028	0.0385301
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.6088	−0.843167	−0.421584	−	0.906790i	$-0.638526\pi$
$719$	−22.6088	−0.843167	−0.421584	+	0.906790i	$0.638526\pi$
$720$	0	0
$721$	−45.1202	−1.68036
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.88979	0.181602
$726$	0	0
$727$	4.10929	0.152405	0.0762025	−	0.997092i	$-0.475720\pi$
$727$	4.10929	0.152405	0.0762025	+	0.997092i	$0.475720\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−41.8183	−1.54671
$732$	0	0
$733$	−6.52890	−0.241150	−0.120575	−	0.992704i	$-0.538474\pi$
$733$	−6.52890	−0.241150	−0.120575	+	0.992704i	$0.538474\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.1211	0.741170
$738$	0	0
$739$	32.8780	1.20944	0.604718	−	0.796440i	$-0.293286\pi$
$739$	32.8780	1.20944	0.604718	+	0.796440i	$0.293286\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.06055	−0.0755943	−0.0377972	−	0.999285i	$-0.512034\pi$
$743$	−2.06055	−0.0755943	−0.0377972	+	0.999285i	$0.512034\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−38.1193	−1.39285
$750$	0	0
$751$	12.7787	0.466300	0.233150	−	0.972441i	$-0.425097\pi$
$751$	12.7787	0.466300	0.233150	+	0.972441i	$0.425097\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.34060	0.0487894
$756$	0	0
$757$	−11.3094	−0.411047	−0.205524	−	0.978652i	$-0.565890\pi$
$757$	−11.3094	−0.411047	−0.205524	+	0.978652i	$0.565890\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−37.2177	−1.34737
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.87890	−0.284490
$768$	0	0
$769$	10.6206	0.382990	0.191495	−	0.981494i	$-0.438666\pi$
$769$	10.6206	0.382990	0.191495	+	0.981494i	$0.438666\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.8695	1.00240	0.501198	−	0.865333i	$-0.332893\pi$
$773$	27.8695	1.00240	0.501198	+	0.865333i	$0.332893\pi$
$774$	0	0
$775$	−7.28005	−0.261507
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.57947	−0.164077
$780$	0	0
$781$	−22.1817	−0.793722
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.1892	−0.363669
$786$	0	0
$787$	−43.2782	−1.54270	−0.771351	−	0.636410i	$-0.780419\pi$
$787$	−43.2782	−1.54270	−0.771351	+	0.636410i	$0.780419\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.69724	0.0603469
$792$	0	0
$793$	7.37935	0.262049
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.53073	−0.0896432	−0.0448216	−	0.998995i	$-0.514272\pi$
$797$	−2.53073	−0.0896432	−0.0448216	+	0.998995i	$0.514272\pi$
$798$	0	0
$799$	1.31880	0.0466559
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.34152	0.223787
$804$	0	0
$805$	−29.7796	−1.04959
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.99908	0.105442	0.0527211	−	0.998609i	$-0.483211\pi$
$809$	2.99908	0.105442	0.0527211	+	0.998609i	$0.483211\pi$
$810$	0	0
$811$	36.3784	1.27742	0.638710	−	0.769448i	$-0.279468\pi$
$811$	36.3784	1.27742	0.638710	+	0.769448i	$0.279468\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.7990	0.553414
$816$	0	0
$817$	7.92007	0.277088
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.7181	−0.932469	−0.466234	−	0.884661i	$-0.654390\pi$
$821$	−26.7181	−0.932469	−0.466234	+	0.884661i	$0.654390\pi$
$822$	0	0
$823$	10.8292	0.377484	0.188742	−	0.982027i	$-0.439559\pi$
$823$	10.8292	0.377484	0.188742	+	0.982027i	$0.439559\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17.0303	0.592201	0.296100	−	0.955157i	$-0.404314\pi$
$827$	17.0303	0.592201	0.296100	+	0.955157i	$0.404314\pi$
$828$	0	0
$829$	−53.8695	−1.87097	−0.935483	−	0.353373i	$-0.885035\pi$
$829$	−53.8695	−1.87097	−0.935483	+	0.353373i	$0.885035\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−31.8401	−1.10320
$834$	0	0
$835$	−9.52982	−0.329793
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.4608	−1.25877	−0.629383	−	0.777095i	$-0.716692\pi$
$839$	−36.4608	−1.25877	−0.629383	+	0.777095i	$0.716692\pi$
$840$	0	0
$841$	−5.08991	−0.175514
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.5904	−0.433122
$846$	0	0
$847$	30.3533	1.04295
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−81.8383	−2.80538
$852$	0	0
$853$	−34.6888	−1.18772	−0.593860	−	0.804568i	$-0.702397\pi$
$853$	−34.6888	−1.18772	−0.593860	+	0.804568i	$0.702397\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.2498	1.57986	0.789931	−	0.613196i	$-0.210116\pi$
$857$	46.2498	1.57986	0.789931	+	0.613196i	$0.210116\pi$
$858$	0	0
$859$	−23.0596	−0.786785	−0.393392	−	0.919371i	$-0.628699\pi$
$859$	−23.0596	−0.786785	−0.393392	+	0.919371i	$0.628699\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.9385	−0.780837	−0.390418	−	0.920638i	$-0.627670\pi$
$863$	−22.9385	−0.780837	−0.390418	+	0.920638i	$0.627670\pi$
$864$	0	0
$865$	−13.4693	−0.457969
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.87798	−0.165474
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.60975	−0.122032
$876$	0	0
$877$	1.42053	0.0479678	0.0239839	−	0.999712i	$-0.492365\pi$
$877$	1.42053	0.0479678	0.0239839	+	0.999712i	$0.492365\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.90161	0.232521	0.116261	−	0.993219i	$-0.462909\pi$
$881$	6.90161	0.232521	0.116261	+	0.993219i	$0.462909\pi$
$882$	0	0
$883$	1.98062	0.0666533	0.0333266	−	0.999445i	$-0.489390\pi$
$883$	1.98062	0.0666533	0.0333266	+	0.999445i	$0.489390\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.56101	0.321027	0.160514	−	0.987034i	$-0.448685\pi$
$887$	9.56101	0.321027	0.160514	+	0.987034i	$0.448685\pi$
$888$	0	0
$889$	45.1202	1.51328
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.249771	−0.00835826
$894$	0	0
$895$	−20.1892	−0.674851
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−35.5979	−1.18726
$900$	0	0
$901$	9.24129	0.307872
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.75023	−0.257626
$906$	0	0
$907$	−15.2195	−0.505355	−0.252678	−	0.967551i	$-0.581311\pi$
$907$	−15.2195	−0.505355	−0.252678	+	0.967551i	$0.581311\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44.9991	1.49089	0.745443	−	0.666569i	$-0.232238\pi$
$911$	44.9991	1.49089	0.745443	+	0.666569i	$0.232238\pi$
$912$	0	0
$913$	16.9016	0.559362
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.6291	0.516119
$918$	0	0
$919$	45.3775	1.49687	0.748433	−	0.663210i	$-0.230806\pi$
$919$	45.3775	1.49687	0.748433	+	0.663210i	$0.230806\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.81926	−0.290289
$924$	0	0
$925$	−9.92007	−0.326170
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.3406	0.700162	0.350081	−	0.936719i	$-0.386154\pi$
$929$	21.3406	0.700162	0.350081	+	0.936719i	$0.386154\pi$
$930$	0	0
$931$	6.03028	0.197634
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.49954	−0.277965
$936$	0	0
$937$	19.9007	0.650127	0.325064	−	0.945692i	$-0.394614\pi$
$937$	19.9007	0.650127	0.325064	+	0.945692i	$0.394614\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.70905	−0.186110	−0.0930549	−	0.995661i	$-0.529663\pi$
$941$	−5.70905	−0.186110	−0.0930549	+	0.995661i	$0.529663\pi$
$942$	0	0
$943$	−37.7796	−1.23027
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.3179	0.920208	0.460104	−	0.887865i	$-0.347812\pi$
$947$	28.3179	0.920208	0.460104	+	0.887865i	$0.347812\pi$
$948$	0	0
$949$	2.52134	0.0818461
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.97064	−0.128622	−0.0643108	−	0.997930i	$-0.520485\pi$
$953$	−3.97064	−0.128622	−0.0643108	+	0.997930i	$0.520485\pi$
$954$	0	0
$955$	2.39025	0.0773468
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.1580	0.586353
$960$	0	0
$961$	21.9991	0.709648
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.640023	−0.0206031
$966$	0	0
$967$	4.55011	0.146322	0.0731609	−	0.997320i	$-0.476691\pi$
$967$	4.55011	0.146322	0.0731609	+	0.997320i	$0.476691\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.5298	−0.370009	−0.185005	−	0.982738i	$-0.559230\pi$
$971$	−11.5298	−0.370009	−0.185005	+	0.982738i	$0.559230\pi$
$972$	0	0
$973$	35.3018	1.13173
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.4920	−0.335668	−0.167834	−	0.985815i	$-0.553677\pi$
$977$	−10.4920	−0.335668	−0.167834	+	0.985815i	$0.553677\pi$
$978$	0	0
$979$	−25.6273	−0.819052
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.40871	0.172511	0.0862556	−	0.996273i	$-0.472510\pi$
$983$	5.40871	0.172511	0.0862556	+	0.996273i	$0.472510\pi$
$984$	0	0
$985$	14.7493	0.469952
$986$	0	0
$987$	0	0
$988$	0	0
$989$	65.3388	2.07765
$990$	0	0
$991$	−27.0890	−0.860510	−0.430255	−	0.902707i	$-0.641576\pi$
$991$	−27.0890	−0.860510	−0.430255	+	0.902707i	$0.641576\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19.8401	0.628975
$996$	0	0
$997$	22.8099	0.722396	0.361198	−	0.932489i	$-0.382368\pi$
$997$	22.8099	0.722396	0.361198	+	0.932489i	$0.382368\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bk.1.1		3
3.2	odd	2		2280.2.a.r.1.1	✓	3
12.11	even	2		4560.2.a.bt.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.r.1.1	✓	3	3.2	odd	2
4560.2.a.bt.1.3		3	12.11	even	2
6840.2.a.bk.1.1		3	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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.60975 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.60975 q^{7} +1.60975 q^{11} +0.640023 q^{13} -5.28005 q^{17} +1.00000 q^{19} +8.24977 q^{23} +1.00000 q^{25} +4.88979 q^{29} -7.28005 q^{31} -3.60975 q^{35} -9.92007 q^{37} -4.57947 q^{41} +7.92007 q^{43} -0.249771 q^{47} +6.03028 q^{49} -1.75023 q^{53} +1.60975 q^{55} -12.3103 q^{59} +11.5298 q^{61} +0.640023 q^{65} +12.4995 q^{67} -13.7796 q^{71} +3.93945 q^{73} -5.81078 q^{77} -3.03028 q^{79} +10.4995 q^{83} -5.28005 q^{85} -15.9201 q^{89} -2.31032 q^{91} +1.00000 q^{95} -9.92007 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 2 * q^7	\( 3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more