LMFDB - Embedded newform 6864.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1620.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 12x - 14 $ x^3 - 12*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.55247$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.55247 q^{5} -3.62008 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.55247 q^{5} -3.62008 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.55247 q^{15} +3.06760 q^{17} +3.62008 q^{19} -3.62008 q^{21} -3.62008 q^{23} +1.51513 q^{25} +1.00000 q^{27} -6.17255 q^{29} -4.55247 q^{31} +1.00000 q^{33} +9.24015 q^{35} +4.00000 q^{37} -1.00000 q^{39} -1.62008 q^{41} -6.17255 q^{43} -2.55247 q^{45} -4.13520 q^{47} +6.10495 q^{49} +3.06760 q^{51} -0.135202 q^{53} -2.55247 q^{55} +3.62008 q^{57} +1.10495 q^{59} +1.86480 q^{61} -3.62008 q^{63} +2.55247 q^{65} -11.7926 q^{67} -3.62008 q^{69} +16.4803 q^{71} +16.7250 q^{73} +1.51513 q^{75} -3.62008 q^{77} -9.06760 q^{79} +1.00000 q^{81} -8.34510 q^{83} -7.82997 q^{85} -6.17255 q^{87} +9.65742 q^{89} +3.62008 q^{91} -4.55247 q^{93} -9.24015 q^{95} +8.00000 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{9} + 3 q^{11} - 3 q^{13} + 6 q^{17} + 9 q^{25} + 3 q^{27} - 6 q^{31} + 3 q^{33} + 6 q^{35} + 12 q^{37} - 3 q^{39} + 6 q^{41} - 6 q^{47} + 3 q^{49} + 6 q^{51} + 6 q^{53} - 12 q^{59} + 12 q^{61} - 6 q^{67} + 6 q^{71} + 24 q^{73} + 9 q^{75} - 24 q^{79} + 3 q^{81} + 12 q^{83} + 18 q^{85} + 6 q^{89} - 6 q^{93} - 6 q^{95} + 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^11 - 3 * q^13 + 6 * q^17 + 9 * q^25 + 3 * q^27 - 6 * q^31 + 3 * q^33 + 6 * q^35 + 12 * q^37 - 3 * q^39 + 6 * q^41 - 6 * q^47 + 3 * q^49 + 6 * q^51 + 6 * q^53 - 12 * q^59 + 12 * q^61 - 6 * q^67 + 6 * q^71 + 24 * q^73 + 9 * q^75 - 24 * q^79 + 3 * q^81 + 12 * q^83 + 18 * q^85 + 6 * q^89 - 6 * q^93 - 6 * q^95 + 24 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.55247	−1.14150	−0.570751	−	0.821123i	$-0.693348\pi$
$5$	−2.55247	−1.14150	−0.570751	+	0.821123i	$0.693348\pi$
$6$	0	0
$7$	−3.62008	−1.36826	−0.684130	−	0.729360i	$-0.739818\pi$
$7$	−3.62008	−1.36826	−0.684130	+	0.729360i	$0.739818\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.55247	−0.659046
$16$	0	0
$17$	3.06760	0.744003	0.372001	−	0.928232i	$-0.378672\pi$
$17$	3.06760	0.744003	0.372001	+	0.928232i	$0.378672\pi$
$18$	0	0
$19$	3.62008	0.830502	0.415251	−	0.909707i	$-0.363694\pi$
$19$	3.62008	0.830502	0.415251	+	0.909707i	$0.363694\pi$
$20$	0	0
$21$	−3.62008	−0.789965
$22$	0	0
$23$	−3.62008	−0.754838	−0.377419	−	0.926043i	$-0.623188\pi$
$23$	−3.62008	−0.754838	−0.377419	+	0.926043i	$0.623188\pi$
$24$	0	0
$25$	1.51513	0.303025
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.17255	−1.14621	−0.573107	−	0.819481i	$-0.694262\pi$
$29$	−6.17255	−1.14621	−0.573107	+	0.819481i	$0.694262\pi$
$30$	0	0
$31$	−4.55247	−0.817649	−0.408824	−	0.912613i	$-0.634061\pi$
$31$	−4.55247	−0.817649	−0.408824	+	0.912613i	$0.634061\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	9.24015	1.56187
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.62008	−0.253013	−0.126507	−	0.991966i	$-0.540377\pi$
$41$	−1.62008	−0.253013	−0.126507	+	0.991966i	$0.540377\pi$
$42$	0	0
$43$	−6.17255	−0.941305	−0.470653	−	0.882319i	$-0.655981\pi$
$43$	−6.17255	−0.941305	−0.470653	+	0.882319i	$0.655981\pi$
$44$	0	0
$45$	−2.55247	−0.380500
$46$	0	0
$47$	−4.13520	−0.603181	−0.301591	−	0.953438i	$-0.597518\pi$
$47$	−4.13520	−0.603181	−0.301591	+	0.953438i	$0.597518\pi$
$48$	0	0
$49$	6.10495	0.872136
$50$	0	0
$51$	3.06760	0.429550
$52$	0	0
$53$	−0.135202	−0.0185715	−0.00928575	−	0.999957i	$-0.502956\pi$
$53$	−0.135202	−0.0185715	−0.00928575	+	0.999957i	$0.502956\pi$
$54$	0	0
$55$	−2.55247	−0.344176
$56$	0	0
$57$	3.62008	0.479491
$58$	0	0
$59$	1.10495	0.143852	0.0719261	−	0.997410i	$-0.477085\pi$
$59$	1.10495	0.143852	0.0719261	+	0.997410i	$0.477085\pi$
$60$	0	0
$61$	1.86480	0.238763	0.119381	−	0.992848i	$-0.461909\pi$
$61$	1.86480	0.238763	0.119381	+	0.992848i	$0.461909\pi$
$62$	0	0
$63$	−3.62008	−0.456087
$64$	0	0
$65$	2.55247	0.316596
$66$	0	0
$67$	−11.7926	−1.44070	−0.720349	−	0.693611i	$-0.756018\pi$
$67$	−11.7926	−1.44070	−0.720349	+	0.693611i	$0.756018\pi$
$68$	0	0
$69$	−3.62008	−0.435806
$70$	0	0
$71$	16.4803	1.95585	0.977926	−	0.208951i	$-0.0670049\pi$
$71$	16.4803	1.95585	0.977926	+	0.208951i	$0.0670049\pi$
$72$	0	0
$73$	16.7250	1.95752	0.978758	−	0.205019i	$-0.0657255\pi$
$73$	16.7250	1.95752	0.978758	+	0.205019i	$0.0657255\pi$
$74$	0	0
$75$	1.51513	0.174952
$76$	0	0
$77$	−3.62008	−0.412546
$78$	0	0
$79$	−9.06760	−1.02018	−0.510092	−	0.860120i	$-0.670389\pi$
$79$	−9.06760	−1.02018	−0.510092	+	0.860120i	$0.670389\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.34510	−0.915994	−0.457997	−	0.888954i	$-0.651433\pi$
$83$	−8.34510	−0.915994	−0.457997	+	0.888954i	$0.651433\pi$
$84$	0	0
$85$	−7.82997	−0.849280
$86$	0	0
$87$	−6.17255	−0.661767
$88$	0	0
$89$	9.65742	1.02368	0.511842	−	0.859079i	$-0.328963\pi$
$89$	9.65742	1.02368	0.511842	+	0.859079i	$0.328963\pi$
$90$	0	0
$91$	3.62008	0.379487
$92$	0	0
$93$	−4.55247	−0.472070
$94$	0	0
$95$	−9.24015	−0.948020
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.17255	0.415184	0.207592	−	0.978215i	$-0.433437\pi$
$101$	4.17255	0.415184	0.207592	+	0.978215i	$0.433437\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	9.24015	0.901746
$106$	0	0
$107$	13.1049	1.26690	0.633452	−	0.773782i	$-0.281638\pi$
$107$	13.1049	1.26690	0.633452	+	0.773782i	$0.281638\pi$
$108$	0	0
$109$	17.7553	1.70065	0.850324	−	0.526260i	$-0.176406\pi$
$109$	17.7553	1.70065	0.850324	+	0.526260i	$0.176406\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	0.895051	0.0841993	0.0420996	−	0.999113i	$-0.486595\pi$
$113$	0.895051	0.0841993	0.0420996	+	0.999113i	$0.486595\pi$
$114$	0	0
$115$	9.24015	0.861649
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−11.1049	−1.01799
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.62008	−0.146077
$124$	0	0
$125$	8.89505	0.795598
$126$	0	0
$127$	−6.03735	−0.535728	−0.267864	−	0.963457i	$-0.586318\pi$
$127$	−6.03735	−0.535728	−0.267864	+	0.963457i	$0.586318\pi$
$128$	0	0
$129$	−6.17255	−0.543463
$130$	0	0
$131$	−15.5106	−1.35516	−0.677582	−	0.735447i	$-0.736972\pi$
$131$	−15.5106	−1.35516	−0.677582	+	0.735447i	$0.736972\pi$
$132$	0	0
$133$	−13.1049	−1.13634
$134$	0	0
$135$	−2.55247	−0.219682
$136$	0	0
$137$	−0.552475	−0.0472011	−0.0236005	−	0.999721i	$-0.507513\pi$
$137$	−0.552475	−0.0472011	−0.0236005	+	0.999721i	$0.507513\pi$
$138$	0	0
$139$	−2.79720	−0.237255	−0.118628	−	0.992939i	$-0.537849\pi$
$139$	−2.79720	−0.237255	−0.118628	+	0.992939i	$0.537849\pi$
$140$	0	0
$141$	−4.13520	−0.348247
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	15.7553	1.30840
$146$	0	0
$147$	6.10495	0.503528
$148$	0	0
$149$	19.0701	1.56229	0.781143	−	0.624352i	$-0.214637\pi$
$149$	19.0701	1.56229	0.781143	+	0.624352i	$0.214637\pi$
$150$	0	0
$151$	18.1004	1.47299	0.736494	−	0.676444i	$-0.236480\pi$
$151$	18.1004	1.47299	0.736494	+	0.676444i	$0.236480\pi$
$152$	0	0
$153$	3.06760	0.248001
$154$	0	0
$155$	11.6201	0.933347
$156$	0	0
$157$	−17.2053	−1.37313	−0.686567	−	0.727066i	$-0.740883\pi$
$157$	−17.2053	−1.37313	−0.686567	+	0.727066i	$0.740883\pi$
$158$	0	0
$159$	−0.135202	−0.0107223
$160$	0	0
$161$	13.1049	1.03281
$162$	0	0
$163$	20.0025	1.56672	0.783359	−	0.621569i	$-0.213505\pi$
$163$	20.0025	1.56672	0.783359	+	0.621569i	$0.213505\pi$
$164$	0	0
$165$	−2.55247	−0.198710
$166$	0	0
$167$	−9.45005	−0.731267	−0.365633	−	0.930759i	$-0.619148\pi$
$167$	−9.45005	−0.731267	−0.365633	+	0.930759i	$0.619148\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.62008	0.276834
$172$	0	0
$173$	7.41270	0.563577	0.281789	−	0.959476i	$-0.409072\pi$
$173$	7.41270	0.563577	0.281789	+	0.959476i	$0.409072\pi$
$174$	0	0
$175$	−5.48487	−0.414617
$176$	0	0
$177$	1.10495	0.0830531
$178$	0	0
$179$	−1.75528	−0.131196	−0.0655978	−	0.997846i	$-0.520895\pi$
$179$	−1.75528	−0.131196	−0.0655978	+	0.997846i	$0.520895\pi$
$180$	0	0
$181$	10.9954	0.817284	0.408642	−	0.912695i	$-0.366002\pi$
$181$	10.9954	0.817284	0.408642	+	0.912695i	$0.366002\pi$
$182$	0	0
$183$	1.86480	0.137850
$184$	0	0
$185$	−10.2099	−0.750647
$186$	0	0
$187$	3.06760	0.224325
$188$	0	0
$189$	−3.62008	−0.263322
$190$	0	0
$191$	8.34510	0.603830	0.301915	−	0.953335i	$-0.402374\pi$
$191$	8.34510	0.603830	0.301915	+	0.953335i	$0.402374\pi$
$192$	0	0
$193$	−2.51513	−0.181043	−0.0905214	−	0.995895i	$-0.528853\pi$
$193$	−2.51513	−0.181043	−0.0905214	+	0.995895i	$0.528853\pi$
$194$	0	0
$195$	2.55247	0.182787
$196$	0	0
$197$	0.244722	0.0174357	0.00871785	−	0.999962i	$-0.497225\pi$
$197$	0.244722	0.0174357	0.00871785	+	0.999962i	$0.497225\pi$
$198$	0	0
$199$	10.2099	0.723761	0.361880	−	0.932225i	$-0.382135\pi$
$199$	10.2099	0.723761	0.361880	+	0.932225i	$0.382135\pi$
$200$	0	0
$201$	−11.7926	−0.831788
$202$	0	0
$203$	22.3451	1.56832
$204$	0	0
$205$	4.13520	0.288815
$206$	0	0
$207$	−3.62008	−0.251613
$208$	0	0
$209$	3.62008	0.250406
$210$	0	0
$211$	−0.307753	−0.0211866	−0.0105933	−	0.999944i	$-0.503372\pi$
$211$	−0.307753	−0.0211866	−0.0105933	+	0.999944i	$0.503372\pi$
$212$	0	0
$213$	16.4803	1.12921
$214$	0	0
$215$	15.7553	1.07450
$216$	0	0
$217$	16.4803	1.11876
$218$	0	0
$219$	16.7250	1.13017
$220$	0	0
$221$	−3.06760	−0.206349
$222$	0	0
$223$	0.822880	0.0551041	0.0275520	−	0.999620i	$-0.491229\pi$
$223$	0.822880	0.0551041	0.0275520	+	0.999620i	$0.491229\pi$
$224$	0	0
$225$	1.51513	0.101008
$226$	0	0
$227$	23.8300	1.58165	0.790825	−	0.612042i	$-0.209652\pi$
$227$	23.8300	1.58165	0.790825	+	0.612042i	$0.209652\pi$
$228$	0	0
$229$	16.0747	1.06225	0.531123	−	0.847295i	$-0.321770\pi$
$229$	16.0747	1.06225	0.531123	+	0.847295i	$0.321770\pi$
$230$	0	0
$231$	−3.62008	−0.238184
$232$	0	0
$233$	−27.5479	−1.80472	−0.902362	−	0.430980i	$-0.858168\pi$
$233$	−27.5479	−1.80472	−0.902362	+	0.430980i	$0.858168\pi$
$234$	0	0
$235$	10.5550	0.688532
$236$	0	0
$237$	−9.06760	−0.589004
$238$	0	0
$239$	10.6503	0.688913	0.344456	−	0.938802i	$-0.388063\pi$
$239$	10.6503	0.688913	0.344456	+	0.938802i	$0.388063\pi$
$240$	0	0
$241$	−17.5853	−1.13277	−0.566383	−	0.824142i	$-0.691658\pi$
$241$	−17.5853	−1.13277	−0.566383	+	0.824142i	$0.691658\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−15.5827	−0.995544
$246$	0	0
$247$	−3.62008	−0.230340
$248$	0	0
$249$	−8.34510	−0.528849
$250$	0	0
$251$	6.40561	0.404318	0.202159	−	0.979353i	$-0.435204\pi$
$251$	6.40561	0.404318	0.202159	+	0.979353i	$0.435204\pi$
$252$	0	0
$253$	−3.62008	−0.227592
$254$	0	0
$255$	−7.82997	−0.490332
$256$	0	0
$257$	23.4501	1.46277	0.731387	−	0.681963i	$-0.238873\pi$
$257$	23.4501	1.46277	0.731387	+	0.681963i	$0.238873\pi$
$258$	0	0
$259$	−14.4803	−0.899762
$260$	0	0
$261$	−6.17255	−0.382071
$262$	0	0
$263$	21.1049	1.30139	0.650693	−	0.759341i	$-0.274478\pi$
$263$	21.1049	1.30139	0.650693	+	0.759341i	$0.274478\pi$
$264$	0	0
$265$	0.345101	0.0211994
$266$	0	0
$267$	9.65742	0.591025
$268$	0	0
$269$	8.13520	0.496012	0.248006	−	0.968758i	$-0.420225\pi$
$269$	8.13520	0.496012	0.248006	+	0.968758i	$0.420225\pi$
$270$	0	0
$271$	14.0257	0.852000	0.426000	−	0.904723i	$-0.359922\pi$
$271$	14.0257	0.852000	0.426000	+	0.904723i	$0.359922\pi$
$272$	0	0
$273$	3.62008	0.219097
$274$	0	0
$275$	1.51513	0.0913656
$276$	0	0
$277$	−4.96975	−0.298603	−0.149302	−	0.988792i	$-0.547703\pi$
$277$	−4.96975	−0.298603	−0.149302	+	0.988792i	$0.547703\pi$
$278$	0	0
$279$	−4.55247	−0.272550
$280$	0	0
$281$	7.75528	0.462641	0.231321	−	0.972878i	$-0.425695\pi$
$281$	7.75528	0.462641	0.231321	+	0.972878i	$0.425695\pi$
$282$	0	0
$283$	7.82745	0.465294	0.232647	−	0.972561i	$-0.425261\pi$
$283$	7.82745	0.465294	0.232647	+	0.972561i	$0.425261\pi$
$284$	0	0
$285$	−9.24015	−0.547339
$286$	0	0
$287$	5.86480	0.346188
$288$	0	0
$289$	−7.58982	−0.446460
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	28.9349	1.69040	0.845198	−	0.534453i	$-0.179482\pi$
$293$	28.9349	1.69040	0.845198	+	0.534453i	$0.179482\pi$
$294$	0	0
$295$	−2.82035	−0.164207
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	3.62008	0.209354
$300$	0	0
$301$	22.3451	1.28795
$302$	0	0
$303$	4.17255	0.239707
$304$	0	0
$305$	−4.75985	−0.272548
$306$	0	0
$307$	−22.1004	−1.26134	−0.630668	−	0.776053i	$-0.717219\pi$
$307$	−22.1004	−1.26134	−0.630668	+	0.776053i	$0.717219\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−6.86023	−0.389008	−0.194504	−	0.980902i	$-0.562310\pi$
$311$	−6.86023	−0.389008	−0.194504	+	0.980902i	$0.562310\pi$
$312$	0	0
$313$	0.589823	0.0333387	0.0166694	−	0.999861i	$-0.494694\pi$
$313$	0.589823	0.0333387	0.0166694	+	0.999861i	$0.494694\pi$
$314$	0	0
$315$	9.24015	0.520624
$316$	0	0
$317$	26.3476	1.47983	0.739915	−	0.672700i	$-0.234866\pi$
$317$	26.3476	1.47983	0.739915	+	0.672700i	$0.234866\pi$
$318$	0	0
$319$	−6.17255	−0.345596
$320$	0	0
$321$	13.1049	0.731447
$322$	0	0
$323$	11.1049	0.617896
$324$	0	0
$325$	−1.51513	−0.0840441
$326$	0	0
$327$	17.7553	0.981869
$328$	0	0
$329$	14.9697	0.825309
$330$	0	0
$331$	24.7624	1.36106	0.680532	−	0.732719i	$-0.261749\pi$
$331$	24.7624	1.36106	0.680532	+	0.732719i	$0.261749\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	30.1004	1.64456
$336$	0	0
$337$	35.7952	1.94989	0.974943	−	0.222455i	$-0.0714069\pi$
$337$	35.7952	1.94989	0.974943	+	0.222455i	$0.0714069\pi$
$338$	0	0
$339$	0.895051	0.0486125
$340$	0	0
$341$	−4.55247	−0.246530
$342$	0	0
$343$	3.24015	0.174952
$344$	0	0
$345$	9.24015	0.497473
$346$	0	0
$347$	−0.209898	−0.0112679	−0.00563397	−	0.999984i	$-0.501793\pi$
$347$	−0.209898	−0.0112679	−0.00563397	+	0.999984i	$0.501793\pi$
$348$	0	0
$349$	−6.83454	−0.365845	−0.182922	−	0.983127i	$-0.558556\pi$
$349$	−6.83454	−0.365845	−0.182922	+	0.983127i	$0.558556\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−1.44753	−0.0770440	−0.0385220	−	0.999258i	$-0.512265\pi$
$353$	−1.44753	−0.0770440	−0.0385220	+	0.999258i	$0.512265\pi$
$354$	0	0
$355$	−42.0656	−2.23261
$356$	0	0
$357$	−11.1049	−0.587736
$358$	0	0
$359$	−8.86023	−0.467625	−0.233812	−	0.972282i	$-0.575120\pi$
$359$	−8.86023	−0.467625	−0.233812	+	0.972282i	$0.575120\pi$
$360$	0	0
$361$	−5.89505	−0.310266
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−42.6902	−2.23451
$366$	0	0
$367$	−8.34510	−0.435611	−0.217805	−	0.975992i	$-0.569890\pi$
$367$	−8.34510	−0.435611	−0.217805	+	0.975992i	$0.569890\pi$
$368$	0	0
$369$	−1.62008	−0.0843378
$370$	0	0
$371$	0.489443	0.0254106
$372$	0	0
$373$	17.3148	0.896529	0.448264	−	0.893901i	$-0.352042\pi$
$373$	17.3148	0.896529	0.448264	+	0.893901i	$0.352042\pi$
$374$	0	0
$375$	8.89505	0.459338
$376$	0	0
$377$	6.17255	0.317903
$378$	0	0
$379$	−30.0630	−1.54423	−0.772117	−	0.635480i	$-0.780802\pi$
$379$	−30.0630	−1.54423	−0.772117	+	0.635480i	$0.780802\pi$
$380$	0	0
$381$	−6.03735	−0.309303
$382$	0	0
$383$	−26.8254	−1.37071	−0.685357	−	0.728207i	$-0.740354\pi$
$383$	−26.8254	−1.37071	−0.685357	+	0.728207i	$0.740354\pi$
$384$	0	0
$385$	9.24015	0.470922
$386$	0	0
$387$	−6.17255	−0.313768
$388$	0	0
$389$	−1.16546	−0.0590910	−0.0295455	−	0.999563i	$-0.509406\pi$
$389$	−1.16546	−0.0590910	−0.0295455	+	0.999563i	$0.509406\pi$
$390$	0	0
$391$	−11.1049	−0.561601
$392$	0	0
$393$	−15.5106	−0.782404
$394$	0	0
$395$	23.1448	1.16454
$396$	0	0
$397$	−38.7507	−1.94484	−0.972421	−	0.233232i	$-0.925070\pi$
$397$	−38.7507	−1.94484	−0.972421	+	0.233232i	$0.925070\pi$
$398$	0	0
$399$	−13.1049	−0.656068
$400$	0	0
$401$	−18.8976	−0.943700	−0.471850	−	0.881679i	$-0.656414\pi$
$401$	−18.8976	−0.943700	−0.471850	+	0.881679i	$0.656414\pi$
$402$	0	0
$403$	4.55247	0.226775
$404$	0	0
$405$	−2.55247	−0.126833
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−13.7553	−0.680155	−0.340077	−	0.940397i	$-0.610453\pi$
$409$	−13.7553	−0.680155	−0.340077	+	0.940397i	$0.610453\pi$
$410$	0	0
$411$	−0.552475	−0.0272516
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	21.3007	1.04561
$416$	0	0
$417$	−2.79720	−0.136979
$418$	0	0
$419$	2.24472	0.109662	0.0548309	−	0.998496i	$-0.482538\pi$
$419$	2.24472	0.109662	0.0548309	+	0.998496i	$0.482538\pi$
$420$	0	0
$421$	−19.7205	−0.961116	−0.480558	−	0.876963i	$-0.659566\pi$
$421$	−19.7205	−0.961116	−0.480558	+	0.876963i	$0.659566\pi$
$422$	0	0
$423$	−4.13520	−0.201060
$424$	0	0
$425$	4.64780	0.225452
$426$	0	0
$427$	−6.75071	−0.326690
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−2.96975	−0.143048	−0.0715238	−	0.997439i	$-0.522786\pi$
$431$	−2.96975	−0.143048	−0.0715238	+	0.997439i	$0.522786\pi$
$432$	0	0
$433$	−28.9001	−1.38885	−0.694425	−	0.719565i	$-0.744341\pi$
$433$	−28.9001	−1.38885	−0.694425	+	0.719565i	$0.744341\pi$
$434$	0	0
$435$	15.7553	0.755408
$436$	0	0
$437$	−13.1049	−0.626895
$438$	0	0
$439$	12.3078	0.587417	0.293708	−	0.955895i	$-0.405111\pi$
$439$	12.3078	0.587417	0.293708	+	0.955895i	$0.405111\pi$
$440$	0	0
$441$	6.10495	0.290712
$442$	0	0
$443$	24.2704	1.15312	0.576561	−	0.817054i	$-0.304394\pi$
$443$	24.2704	1.15312	0.576561	+	0.817054i	$0.304394\pi$
$444$	0	0
$445$	−24.6503	−1.16854
$446$	0	0
$447$	19.0701	0.901986
$448$	0	0
$449$	36.6877	1.73140	0.865699	−	0.500564i	$-0.166874\pi$
$449$	36.6877	1.73140	0.865699	+	0.500564i	$0.166874\pi$
$450$	0	0
$451$	−1.62008	−0.0762864
$452$	0	0
$453$	18.1004	0.850430
$454$	0	0
$455$	−9.24015	−0.433185
$456$	0	0
$457$	7.03025	0.328861	0.164431	−	0.986389i	$-0.447421\pi$
$457$	7.03025	0.328861	0.164431	+	0.986389i	$0.447421\pi$
$458$	0	0
$459$	3.06760	0.143183
$460$	0	0
$461$	0.319418	0.0148768	0.00743838	−	0.999972i	$-0.497632\pi$
$461$	0.319418	0.0148768	0.00743838	+	0.999972i	$0.497632\pi$
$462$	0	0
$463$	37.7230	1.75314	0.876568	−	0.481278i	$-0.159827\pi$
$463$	37.7230	1.75314	0.876568	+	0.481278i	$0.159827\pi$
$464$	0	0
$465$	11.6201	0.538868
$466$	0	0
$467$	25.0701	1.16011	0.580054	−	0.814578i	$-0.303032\pi$
$467$	25.0701	1.16011	0.580054	+	0.814578i	$0.303032\pi$
$468$	0	0
$469$	42.6902	1.97125
$470$	0	0
$471$	−17.2053	−0.792780
$472$	0	0
$473$	−6.17255	−0.283814
$474$	0	0
$475$	5.48487	0.251663
$476$	0	0
$477$	−0.135202	−0.00619050
$478$	0	0
$479$	30.0399	1.37256	0.686278	−	0.727339i	$-0.259243\pi$
$479$	30.0399	1.37256	0.686278	+	0.727339i	$0.259243\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	13.1049	0.596296
$484$	0	0
$485$	−20.4198	−0.927215
$486$	0	0
$487$	−30.0630	−1.36229	−0.681143	−	0.732150i	$-0.738517\pi$
$487$	−30.0630	−1.36229	−0.681143	+	0.732150i	$0.738517\pi$
$488$	0	0
$489$	20.0025	0.904545
$490$	0	0
$491$	33.9304	1.53126	0.765628	−	0.643284i	$-0.222429\pi$
$491$	33.9304	1.53126	0.765628	+	0.643284i	$0.222429\pi$
$492$	0	0
$493$	−18.9349	−0.852786
$494$	0	0
$495$	−2.55247	−0.114725
$496$	0	0
$497$	−59.6599	−2.67611
$498$	0	0
$499$	−20.5525	−0.920055	−0.460028	−	0.887905i	$-0.652160\pi$
$499$	−20.5525	−0.920055	−0.460028	+	0.887905i	$0.652160\pi$
$500$	0	0
$501$	−9.45005	−0.422197
$502$	0	0
$503$	−19.1655	−0.854545	−0.427273	−	0.904123i	$-0.640526\pi$
$503$	−19.1655	−0.854545	−0.427273	+	0.904123i	$0.640526\pi$
$504$	0	0
$505$	−10.6503	−0.473933
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	25.0933	1.11224	0.556120	−	0.831102i	$-0.312289\pi$
$509$	25.0933	1.11224	0.556120	+	0.831102i	$0.312289\pi$
$510$	0	0
$511$	−60.5459	−2.67839
$512$	0	0
$513$	3.62008	0.159830
$514$	0	0
$515$	20.4198	0.899804
$516$	0	0
$517$	−4.13520	−0.181866
$518$	0	0
$519$	7.41270	0.325382
$520$	0	0
$521$	9.72960	0.426261	0.213131	−	0.977024i	$-0.431634\pi$
$521$	9.72960	0.426261	0.213131	+	0.977024i	$0.431634\pi$
$522$	0	0
$523$	−0.0373480	−0.00163311	−0.000816556	−	1.00000i	$-0.500260\pi$
$523$	−0.0373480	−0.00163311	−0.000816556		1.00000i	$0.500260\pi$
$524$	0	0
$525$	−5.48487	−0.239379
$526$	0	0
$527$	−13.9652	−0.608333
$528$	0	0
$529$	−9.89505	−0.430220
$530$	0	0
$531$	1.10495	0.0479507
$532$	0	0
$533$	1.62008	0.0701733
$534$	0	0
$535$	−33.4501	−1.44617
$536$	0	0
$537$	−1.75528	−0.0757459
$538$	0	0
$539$	6.10495	0.262959
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	10.9954	0.471859
$544$	0	0
$545$	−45.3199	−1.94129
$546$	0	0
$547$	−0.932399	−0.0398665	−0.0199332	−	0.999801i	$-0.506345\pi$
$547$	−0.932399	−0.0398665	−0.0199332	+	0.999801i	$0.506345\pi$
$548$	0	0
$549$	1.86480	0.0795876
$550$	0	0
$551$	−22.3451	−0.951933
$552$	0	0
$553$	32.8254	1.39588
$554$	0	0
$555$	−10.2099	−0.433386
$556$	0	0
$557$	16.8602	0.714391	0.357195	−	0.934030i	$-0.383733\pi$
$557$	16.8602	0.714391	0.357195	+	0.934030i	$0.383733\pi$
$558$	0	0
$559$	6.17255	0.261071
$560$	0	0
$561$	3.06760	0.129514
$562$	0	0
$563$	9.30066	0.391976	0.195988	−	0.980606i	$-0.437209\pi$
$563$	9.30066	0.391976	0.195988	+	0.980606i	$0.437209\pi$
$564$	0	0
$565$	−2.28459	−0.0961136
$566$	0	0
$567$	−3.62008	−0.152029
$568$	0	0
$569$	24.1029	1.01045	0.505223	−	0.862989i	$-0.331410\pi$
$569$	24.1029	1.01045	0.505223	+	0.862989i	$0.331410\pi$
$570$	0	0
$571$	18.9324	0.792296	0.396148	−	0.918187i	$-0.370347\pi$
$571$	18.9324	0.792296	0.396148	+	0.918187i	$0.370347\pi$
$572$	0	0
$573$	8.34510	0.348622
$574$	0	0
$575$	−5.48487	−0.228735
$576$	0	0
$577$	−44.2613	−1.84262	−0.921310	−	0.388828i	$-0.872880\pi$
$577$	−44.2613	−1.84262	−0.921310	+	0.388828i	$0.872880\pi$
$578$	0	0
$579$	−2.51513	−0.104525
$580$	0	0
$581$	30.2099	1.25332
$582$	0	0
$583$	−0.135202	−0.00559952
$584$	0	0
$585$	2.55247	0.105532
$586$	0	0
$587$	−0.564139	−0.0232845	−0.0116423	−	0.999932i	$-0.503706\pi$
$587$	−0.564139	−0.0232845	−0.0116423	+	0.999932i	$0.503706\pi$
$588$	0	0
$589$	−16.4803	−0.679059
$590$	0	0
$591$	0.244722	0.0100665
$592$	0	0
$593$	10.0348	0.412081	0.206040	−	0.978543i	$-0.433942\pi$
$593$	10.0348	0.412081	0.206040	+	0.978543i	$0.433942\pi$
$594$	0	0
$595$	28.3451	1.16204
$596$	0	0
$597$	10.2099	0.417863
$598$	0	0
$599$	−14.1352	−0.577549	−0.288774	−	0.957397i	$-0.593248\pi$
$599$	−14.1352	−0.577549	−0.288774	+	0.957397i	$0.593248\pi$
$600$	0	0
$601$	−20.4803	−0.835409	−0.417705	−	0.908583i	$-0.637165\pi$
$601$	−20.4803	−0.835409	−0.417705	+	0.908583i	$0.637165\pi$
$602$	0	0
$603$	−11.7926	−0.480233
$604$	0	0
$605$	−2.55247	−0.103773
$606$	0	0
$607$	0.442955	0.0179790	0.00898950	−	0.999960i	$-0.497139\pi$
$607$	0.442955	0.0179790	0.00898950	+	0.999960i	$0.497139\pi$
$608$	0	0
$609$	22.3451	0.905469
$610$	0	0
$611$	4.13520	0.167292
$612$	0	0
$613$	−5.48487	−0.221532	−0.110766	−	0.993847i	$-0.535330\pi$
$613$	−5.48487	−0.221532	−0.110766	+	0.993847i	$0.535330\pi$
$614$	0	0
$615$	4.13520	0.166747
$616$	0	0
$617$	−5.10242	−0.205416	−0.102708	−	0.994712i	$-0.532751\pi$
$617$	−5.10242	−0.205416	−0.102708	+	0.994712i	$0.532751\pi$
$618$	0	0
$619$	−42.9723	−1.72720	−0.863601	−	0.504176i	$-0.831796\pi$
$619$	−42.9723	−1.72720	−0.863601	+	0.504176i	$0.831796\pi$
$620$	0	0
$621$	−3.62008	−0.145269
$622$	0	0
$623$	−34.9606	−1.40067
$624$	0	0
$625$	−30.2800	−1.21120
$626$	0	0
$627$	3.62008	0.144572
$628$	0	0
$629$	12.2704	0.489253
$630$	0	0
$631$	−23.9278	−0.952552	−0.476276	−	0.879296i	$-0.658014\pi$
$631$	−23.9278	−0.952552	−0.476276	+	0.879296i	$0.658014\pi$
$632$	0	0
$633$	−0.307753	−0.0122321
$634$	0	0
$635$	15.4102	0.611534
$636$	0	0
$637$	−6.10495	−0.241887
$638$	0	0
$639$	16.4803	0.651951
$640$	0	0
$641$	−33.9304	−1.34017	−0.670084	−	0.742285i	$-0.733742\pi$
$641$	−33.9304	−1.34017	−0.670084	+	0.742285i	$0.733742\pi$
$642$	0	0
$643$	−12.3568	−0.487303	−0.243652	−	0.969863i	$-0.578345\pi$
$643$	−12.3568	−0.487303	−0.243652	+	0.969863i	$0.578345\pi$
$644$	0	0
$645$	15.7553	0.620363
$646$	0	0
$647$	30.8254	1.21187	0.605936	−	0.795514i	$-0.292799\pi$
$647$	30.8254	1.21187	0.605936	+	0.795514i	$0.292799\pi$
$648$	0	0
$649$	1.10495	0.0433731
$650$	0	0
$651$	16.4803	0.645914
$652$	0	0
$653$	1.24015	0.0485309	0.0242654	−	0.999706i	$-0.492275\pi$
$653$	1.24015	0.0485309	0.0242654	+	0.999706i	$0.492275\pi$
$654$	0	0
$655$	39.5903	1.54692
$656$	0	0
$657$	16.7250	0.652505
$658$	0	0
$659$	2.61551	0.101886	0.0509428	−	0.998702i	$-0.483777\pi$
$659$	2.61551	0.101886	0.0509428	+	0.998702i	$0.483777\pi$
$660$	0	0
$661$	27.3057	1.06207	0.531034	−	0.847350i	$-0.321803\pi$
$661$	27.3057	1.06207	0.531034	+	0.847350i	$0.321803\pi$
$662$	0	0
$663$	−3.06760	−0.119136
$664$	0	0
$665$	33.4501	1.29714
$666$	0	0
$667$	22.3451	0.865206
$668$	0	0
$669$	0.822880	0.0318144
$670$	0	0
$671$	1.86480	0.0719897
$672$	0	0
$673$	−38.6297	−1.48907	−0.744533	−	0.667586i	$-0.767328\pi$
$673$	−38.6297	−1.48907	−0.744533	+	0.667586i	$0.767328\pi$
$674$	0	0
$675$	1.51513	0.0583173
$676$	0	0
$677$	−7.61755	−0.292766	−0.146383	−	0.989228i	$-0.546763\pi$
$677$	−7.61755	−0.292766	−0.146383	+	0.989228i	$0.546763\pi$
$678$	0	0
$679$	−28.9606	−1.11141
$680$	0	0
$681$	23.8300	0.913167
$682$	0	0
$683$	−32.6297	−1.24854	−0.624270	−	0.781208i	$-0.714604\pi$
$683$	−32.6297	−1.24854	−0.624270	+	0.781208i	$0.714604\pi$
$684$	0	0
$685$	1.41018	0.0538801
$686$	0	0
$687$	16.0747	0.613288
$688$	0	0
$689$	0.135202	0.00515080
$690$	0	0
$691$	−17.7179	−0.674022	−0.337011	−	0.941501i	$-0.609416\pi$
$691$	−17.7179	−0.674022	−0.337011	+	0.941501i	$0.609416\pi$
$692$	0	0
$693$	−3.62008	−0.137515
$694$	0	0
$695$	7.13977	0.270827
$696$	0	0
$697$	−4.96975	−0.188243
$698$	0	0
$699$	−27.5479	−1.04196
$700$	0	0
$701$	2.50346	0.0945545	0.0472772	−	0.998882i	$-0.484946\pi$
$701$	2.50346	0.0945545	0.0472772	+	0.998882i	$0.484946\pi$
$702$	0	0
$703$	14.4803	0.546135
$704$	0	0
$705$	10.5550	0.397524
$706$	0	0
$707$	−15.1049	−0.568080
$708$	0	0
$709$	8.83454	0.331788	0.165894	−	0.986144i	$-0.446949\pi$
$709$	8.83454	0.331788	0.165894	+	0.986144i	$0.446949\pi$
$710$	0	0
$711$	−9.06760	−0.340062
$712$	0	0
$713$	16.4803	0.617192
$714$	0	0
$715$	2.55247	0.0954571
$716$	0	0
$717$	10.6503	0.397744
$718$	0	0
$719$	−29.9395	−1.11655	−0.558277	−	0.829654i	$-0.688537\pi$
$719$	−29.9395	−1.11655	−0.558277	+	0.829654i	$0.688537\pi$
$720$	0	0
$721$	28.9606	1.07855
$722$	0	0
$723$	−17.5853	−0.654003
$724$	0	0
$725$	−9.35220	−0.347332
$726$	0	0
$727$	−29.0353	−1.07686	−0.538430	−	0.842670i	$-0.680982\pi$
$727$	−29.0353	−1.07686	−0.538430	+	0.842670i	$0.680982\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.9349	−0.700333
$732$	0	0
$733$	1.97432	0.0729230	0.0364615	−	0.999335i	$-0.488391\pi$
$733$	1.97432	0.0729230	0.0364615	+	0.999335i	$0.488391\pi$
$734$	0	0
$735$	−15.5827	−0.574778
$736$	0	0
$737$	−11.7926	−0.434387
$738$	0	0
$739$	5.68058	0.208964	0.104482	−	0.994527i	$-0.466682\pi$
$739$	5.68058	0.208964	0.104482	+	0.994527i	$0.466682\pi$
$740$	0	0
$741$	−3.62008	−0.132987
$742$	0	0
$743$	−54.0656	−1.98347	−0.991736	−	0.128292i	$-0.959051\pi$
$743$	−54.0656	−1.98347	−0.991736	+	0.128292i	$0.959051\pi$
$744$	0	0
$745$	−48.6760	−1.78335
$746$	0	0
$747$	−8.34510	−0.305331
$748$	0	0
$749$	−47.4409	−1.73345
$750$	0	0
$751$	−4.07470	−0.148688	−0.0743439	−	0.997233i	$-0.523686\pi$
$751$	−4.07470	−0.148688	−0.0743439	+	0.997233i	$0.523686\pi$
$752$	0	0
$753$	6.40561	0.233433
$754$	0	0
$755$	−46.2008	−1.68142
$756$	0	0
$757$	1.96518	0.0714256	0.0357128	−	0.999362i	$-0.488630\pi$
$757$	1.96518	0.0714256	0.0357128	+	0.999362i	$0.488630\pi$
$758$	0	0
$759$	−3.62008	−0.131400
$760$	0	0
$761$	28.1751	1.02135	0.510673	−	0.859775i	$-0.329396\pi$
$761$	28.1751	1.02135	0.510673	+	0.859775i	$0.329396\pi$
$762$	0	0
$763$	−64.2755	−2.32693
$764$	0	0
$765$	−7.82997	−0.283093
$766$	0	0
$767$	−1.10495	−0.0398974
$768$	0	0
$769$	7.51970	0.271167	0.135584	−	0.990766i	$-0.456709\pi$
$769$	7.51970	0.271167	0.135584	+	0.990766i	$0.456709\pi$
$770$	0	0
$771$	23.4501	0.844533
$772$	0	0
$773$	−2.61298	−0.0939824	−0.0469912	−	0.998895i	$-0.514963\pi$
$773$	−2.61298	−0.0939824	−0.0469912	+	0.998895i	$0.514963\pi$
$774$	0	0
$775$	−6.89758	−0.247768
$776$	0	0
$777$	−14.4803	−0.519478
$778$	0	0
$779$	−5.86480	−0.210128
$780$	0	0
$781$	16.4803	0.589712
$782$	0	0
$783$	−6.17255	−0.220589
$784$	0	0
$785$	43.9162	1.56744
$786$	0	0
$787$	3.07927	0.109764	0.0548820	−	0.998493i	$-0.482522\pi$
$787$	3.07927	0.109764	0.0548820	+	0.998493i	$0.482522\pi$
$788$	0	0
$789$	21.1049	0.751356
$790$	0	0
$791$	−3.24015	−0.115207
$792$	0	0
$793$	−1.86480	−0.0662209
$794$	0	0
$795$	0.345101	0.0122395
$796$	0	0
$797$	22.1494	0.784572	0.392286	−	0.919843i	$-0.371684\pi$
$797$	22.1494	0.784572	0.392286	+	0.919843i	$0.371684\pi$
$798$	0	0
$799$	−12.6852	−0.448768
$800$	0	0
$801$	9.65742	0.341228
$802$	0	0
$803$	16.7250	0.590213
$804$	0	0
$805$	−33.4501	−1.17896
$806$	0	0
$807$	8.13520	0.286373
$808$	0	0
$809$	−17.5479	−0.616951	−0.308476	−	0.951232i	$-0.599819\pi$
$809$	−17.5479	−0.616951	−0.308476	+	0.951232i	$0.599819\pi$
$810$	0	0
$811$	18.5898	0.652777	0.326388	−	0.945236i	$-0.394168\pi$
$811$	18.5898	0.652777	0.326388	+	0.945236i	$0.394168\pi$
$812$	0	0
$813$	14.0257	0.491902
$814$	0	0
$815$	−51.0559	−1.78841
$816$	0	0
$817$	−22.3451	−0.781756
$818$	0	0
$819$	3.62008	0.126496
$820$	0	0
$821$	−8.93492	−0.311831	−0.155915	−	0.987770i	$-0.549833\pi$
$821$	−8.93492	−0.311831	−0.155915	+	0.987770i	$0.549833\pi$
$822$	0	0
$823$	8.61551	0.300318	0.150159	−	0.988662i	$-0.452021\pi$
$823$	8.61551	0.300318	0.150159	+	0.988662i	$0.452021\pi$
$824$	0	0
$825$	1.51513	0.0527499
$826$	0	0
$827$	11.1398	0.387368	0.193684	−	0.981064i	$-0.437956\pi$
$827$	11.1398	0.387368	0.193684	+	0.981064i	$0.437956\pi$
$828$	0	0
$829$	7.79010	0.270561	0.135281	−	0.990807i	$-0.456806\pi$
$829$	7.79010	0.270561	0.135281	+	0.990807i	$0.456806\pi$
$830$	0	0
$831$	−4.96975	−0.172399
$832$	0	0
$833$	18.7275	0.648871
$834$	0	0
$835$	24.1210	0.834742
$836$	0	0
$837$	−4.55247	−0.157357
$838$	0	0
$839$	−27.3895	−0.945592	−0.472796	−	0.881172i	$-0.656755\pi$
$839$	−27.3895	−0.945592	−0.472796	+	0.881172i	$0.656755\pi$
$840$	0	0
$841$	9.10038	0.313806
$842$	0	0
$843$	7.75528	0.267106
$844$	0	0
$845$	−2.55247	−0.0878078
$846$	0	0
$847$	−3.62008	−0.124387
$848$	0	0
$849$	7.82745	0.268637
$850$	0	0
$851$	−14.4803	−0.496378
$852$	0	0
$853$	−11.7205	−0.401301	−0.200650	−	0.979663i	$-0.564306\pi$
$853$	−11.7205	−0.401301	−0.200650	+	0.979663i	$0.564306\pi$
$854$	0	0
$855$	−9.24015	−0.316007
$856$	0	0
$857$	−13.9021	−0.474888	−0.237444	−	0.971401i	$-0.576310\pi$
$857$	−13.9021	−0.474888	−0.237444	+	0.971401i	$0.576310\pi$
$858$	0	0
$859$	53.5761	1.82799	0.913997	−	0.405722i	$-0.132980\pi$
$859$	53.5761	1.82799	0.913997	+	0.405722i	$0.132980\pi$
$860$	0	0
$861$	5.86480	0.199872
$862$	0	0
$863$	−15.6549	−0.532899	−0.266449	−	0.963849i	$-0.585851\pi$
$863$	−15.6549	−0.532899	−0.266449	+	0.963849i	$0.585851\pi$
$864$	0	0
$865$	−18.9207	−0.643324
$866$	0	0
$867$	−7.58982	−0.257764
$868$	0	0
$869$	−9.06760	−0.307597
$870$	0	0
$871$	11.7926	0.399578
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−32.2008	−1.08858
$876$	0	0
$877$	−12.8951	−0.435435	−0.217717	−	0.976012i	$-0.569861\pi$
$877$	−12.8951	−0.435435	−0.217717	+	0.976012i	$0.569861\pi$
$878$	0	0
$879$	28.9349	0.975951
$880$	0	0
$881$	3.24929	0.109471	0.0547357	−	0.998501i	$-0.482568\pi$
$881$	3.24929	0.109471	0.0547357	+	0.998501i	$0.482568\pi$
$882$	0	0
$883$	−1.24929	−0.0420420	−0.0210210	−	0.999779i	$-0.506692\pi$
$883$	−1.24929	−0.0420420	−0.0210210	+	0.999779i	$0.506692\pi$
$884$	0	0
$885$	−2.82035	−0.0948052
$886$	0	0
$887$	53.5156	1.79688	0.898439	−	0.439098i	$-0.144702\pi$
$887$	53.5156	1.79688	0.898439	+	0.439098i	$0.144702\pi$
$888$	0	0
$889$	21.8557	0.733015
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−14.9697	−0.500943
$894$	0	0
$895$	4.48030	0.149760
$896$	0	0
$897$	3.62008	0.120871
$898$	0	0
$899$	28.1004	0.937200
$900$	0	0
$901$	−0.414747	−0.0138172
$902$	0	0
$903$	22.3451	0.743598
$904$	0	0
$905$	−28.0656	−0.932931
$906$	0	0
$907$	29.8648	0.991644	0.495822	−	0.868424i	$-0.334867\pi$
$907$	29.8648	0.991644	0.495822	+	0.868424i	$0.334867\pi$
$908$	0	0
$909$	4.17255	0.138395
$910$	0	0
$911$	−19.6549	−0.651196	−0.325598	−	0.945508i	$-0.605566\pi$
$911$	−19.6549	−0.651196	−0.325598	+	0.945508i	$0.605566\pi$
$912$	0	0
$913$	−8.34510	−0.276183
$914$	0	0
$915$	−4.75985	−0.157356
$916$	0	0
$917$	56.1494	1.85422
$918$	0	0
$919$	−13.3471	−0.440282	−0.220141	−	0.975468i	$-0.570652\pi$
$919$	−13.3471	−0.440282	−0.220141	+	0.975468i	$0.570652\pi$
$920$	0	0
$921$	−22.1004	−0.728232
$922$	0	0
$923$	−16.4803	−0.542456
$924$	0	0
$925$	6.06051	0.199268
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	20.2074	0.662982	0.331491	−	0.943458i	$-0.392448\pi$
$929$	20.2074	0.662982	0.331491	+	0.943458i	$0.392448\pi$
$930$	0	0
$931$	22.1004	0.724311
$932$	0	0
$933$	−6.86023	−0.224594
$934$	0	0
$935$	−7.82997	−0.256068
$936$	0	0
$937$	6.96975	0.227692	0.113846	−	0.993498i	$-0.463683\pi$
$937$	6.96975	0.227692	0.113846	+	0.993498i	$0.463683\pi$
$938$	0	0
$939$	0.589823	0.0192481
$940$	0	0
$941$	−32.7906	−1.06894	−0.534471	−	0.845187i	$-0.679489\pi$
$941$	−32.7906	−1.06894	−0.534471	+	0.845187i	$0.679489\pi$
$942$	0	0
$943$	5.86480	0.190984
$944$	0	0
$945$	9.24015	0.300582
$946$	0	0
$947$	41.8557	1.36013	0.680063	−	0.733154i	$-0.261952\pi$
$947$	41.8557	1.36013	0.680063	+	0.733154i	$0.261952\pi$
$948$	0	0
$949$	−16.7250	−0.542917
$950$	0	0
$951$	26.3476	0.854380
$952$	0	0
$953$	35.5479	1.15151	0.575755	−	0.817622i	$-0.304708\pi$
$953$	35.5479	1.15151	0.575755	+	0.817622i	$0.304708\pi$
$954$	0	0
$955$	−21.3007	−0.689273
$956$	0	0
$957$	−6.17255	−0.199530
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−10.2750	−0.331451
$962$	0	0
$963$	13.1049	0.422301
$964$	0	0
$965$	6.41980	0.206661
$966$	0	0
$967$	6.66452	0.214316	0.107158	−	0.994242i	$-0.465825\pi$
$967$	6.66452	0.214316	0.107158	+	0.994242i	$0.465825\pi$
$968$	0	0
$969$	11.1049	0.356742
$970$	0	0
$971$	−59.5504	−1.91106	−0.955532	−	0.294887i	$-0.904718\pi$
$971$	−59.5504	−1.91106	−0.955532	+	0.294887i	$0.904718\pi$
$972$	0	0
$973$	10.1261	0.324627
$974$	0	0
$975$	−1.51513	−0.0485229
$976$	0	0
$977$	−36.5575	−1.16958	−0.584789	−	0.811185i	$-0.698823\pi$
$977$	−36.5575	−1.16958	−0.584789	+	0.811185i	$0.698823\pi$
$978$	0	0
$979$	9.65742	0.308653
$980$	0	0
$981$	17.7553	0.566882
$982$	0	0
$983$	39.6599	1.26496	0.632478	−	0.774578i	$-0.282038\pi$
$983$	39.6599	1.26496	0.632478	+	0.774578i	$0.282038\pi$
$984$	0	0
$985$	−0.624646	−0.0199029
$986$	0	0
$987$	14.9697	0.476492
$988$	0	0
$989$	22.3451	0.710533
$990$	0	0
$991$	33.2543	1.05636	0.528179	−	0.849133i	$-0.322875\pi$
$991$	33.2543	1.05636	0.528179	+	0.849133i	$0.322875\pi$
$992$	0	0
$993$	24.7624	0.785810
$994$	0	0
$995$	−26.0605	−0.826174
$996$	0	0
$997$	40.3360	1.27745	0.638726	−	0.769434i	$-0.279462\pi$
$997$	40.3360	1.27745	0.638726	+	0.769434i	$0.279462\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bv.1.1		3
4.3	odd	2		1716.2.a.f.1.1	✓	3
12.11	even	2		5148.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.f.1.1	✓	3	4.3	odd	2
5148.2.a.n.1.3		3	12.11	even	2
6864.2.a.bv.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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.55247 q^{5} -3.62008 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.55247 q^{5} -3.62008 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.55247 q^{15} +3.06760 q^{17} +3.62008 q^{19} -3.62008 q^{21} -3.62008 q^{23} +1.51513 q^{25} +1.00000 q^{27} -6.17255 q^{29} -4.55247 q^{31} +1.00000 q^{33} +9.24015 q^{35} +4.00000 q^{37} -1.00000 q^{39} -1.62008 q^{41} -6.17255 q^{43} -2.55247 q^{45} -4.13520 q^{47} +6.10495 q^{49} +3.06760 q^{51} -0.135202 q^{53} -2.55247 q^{55} +3.62008 q^{57} +1.10495 q^{59} +1.86480 q^{61} -3.62008 q^{63} +2.55247 q^{65} -11.7926 q^{67} -3.62008 q^{69} +16.4803 q^{71} +16.7250 q^{73} +1.51513 q^{75} -3.62008 q^{77} -9.06760 q^{79} +1.00000 q^{81} -8.34510 q^{83} -7.82997 q^{85} -6.17255 q^{87} +9.65742 q^{89} +3.62008 q^{91} -4.55247 q^{93} -9.24015 q^{95} +8.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{9} + 3 q^{11} - 3 q^{13} + 6 q^{17} + 9 q^{25} + 3 q^{27} - 6 q^{31} + 3 q^{33} + 6 q^{35} + 12 q^{37} - 3 q^{39} + 6 q^{41} - 6 q^{47} + 3 q^{49} + 6 q^{51} + 6 q^{53} - 12 q^{59} + 12 q^{61} - 6 q^{67} + 6 q^{71} + 24 q^{73} + 9 q^{75} - 24 q^{79} + 3 q^{81} + 12 q^{83} + 18 q^{85} + 6 q^{89} - 6 q^{93} - 6 q^{95} + 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^11 - 3 * q^13 + 6 * q^17 + 9 * q^25 + 3 * q^27 - 6 * q^31 + 3 * q^33 + 6 * q^35 + 12 * q^37 - 3 * q^39 + 6 * q^41 - 6 * q^47 + 3 * q^49 + 6 * q^51 + 6 * q^53 - 12 * q^59 + 12 * q^61 - 6 * q^67 + 6 * q^71 + 24 * q^73 + 9 * q^75 - 24 * q^79 + 3 * q^81 + 12 * q^83 + 18 * q^85 + 6 * q^89 - 6 * q^93 - 6 * q^95 + 24 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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