LMFDB - Embedded newform 6864.2.a.bv.1.2

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			1.2
Root			$-1.39091$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.39091 q^{5} +3.28357 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.39091 q^{5} +3.28357 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.39091 q^{15} -2.67447 q^{17} -3.28357 q^{19} +3.28357 q^{21} +3.28357 q^{23} -3.06538 q^{25} +1.00000 q^{27} +1.89266 q^{29} -3.39091 q^{31} +1.00000 q^{33} -4.56713 q^{35} +4.00000 q^{37} -1.00000 q^{39} +5.28357 q^{41} +1.89266 q^{43} -1.39091 q^{45} +7.34895 q^{47} +3.78181 q^{49} -2.67447 q^{51} +11.3489 q^{53} -1.39091 q^{55} -3.28357 q^{57} -1.21819 q^{59} +13.3489 q^{61} +3.28357 q^{63} +1.39091 q^{65} +3.17623 q^{67} +3.28357 q^{69} -11.1343 q^{71} +7.49825 q^{73} -3.06538 q^{75} +3.28357 q^{77} -3.32553 q^{79} +1.00000 q^{81} +7.78532 q^{83} +3.71994 q^{85} +1.89266 q^{87} +6.17272 q^{89} -3.28357 q^{91} -3.39091 q^{93} +4.56713 q^{95} +8.00000 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.39091	−0.622032	−0.311016	−	0.950405i	$-0.600669\pi$
$5$	−1.39091	−0.622032	−0.311016	+	0.950405i	$0.600669\pi$
$6$	0	0
$7$	3.28357	1.24107	0.620536	−	0.784178i	$-0.286915\pi$
$7$	3.28357	1.24107	0.620536	+	0.784178i	$0.286915\pi$
$8$	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
$15$	−1.39091	−0.359130
$16$	0	0
$17$	−2.67447	−0.648655	−0.324328	−	0.945945i	$-0.605138\pi$
$17$	−2.67447	−0.648655	−0.324328	+	0.945945i	$0.605138\pi$
$18$	0	0
$19$	−3.28357	−0.753302	−0.376651	−	0.926355i	$-0.622924\pi$
$19$	−3.28357	−0.753302	−0.376651	+	0.926355i	$0.622924\pi$
$20$	0	0
$21$	3.28357	0.716533
$22$	0	0
$23$	3.28357	0.684671	0.342336	−	0.939578i	$-0.388782\pi$
$23$	3.28357	0.684671	0.342336	+	0.939578i	$0.388782\pi$
$24$	0	0
$25$	−3.06538	−0.613076
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.89266	0.351458	0.175729	−	0.984439i	$-0.443772\pi$
$29$	1.89266	0.351458	0.175729	+	0.984439i	$0.443772\pi$
$30$	0	0
$31$	−3.39091	−0.609025	−0.304512	−	0.952508i	$-0.598493\pi$
$31$	−3.39091	−0.609025	−0.304512	+	0.952508i	$0.598493\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−4.56713	−0.771987
$36$	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
$41$	5.28357	0.825155	0.412577	−	0.910923i	$-0.364629\pi$
$41$	5.28357	0.825155	0.412577	+	0.910923i	$0.364629\pi$
$42$	0	0
$43$	1.89266	0.288628	0.144314	−	0.989532i	$-0.453902\pi$
$43$	1.89266	0.288628	0.144314	+	0.989532i	$0.453902\pi$
$44$	0	0
$45$	−1.39091	−0.207344
$46$	0	0
$47$	7.34895	1.07195	0.535977	−	0.844233i	$-0.319943\pi$
$47$	7.34895	1.07195	0.535977	+	0.844233i	$0.319943\pi$
$48$	0	0
$49$	3.78181	0.540259
$50$	0	0
$51$	−2.67447	−0.374501
$52$	0	0
$53$	11.3489	1.55890	0.779449	−	0.626466i	$-0.215499\pi$
$53$	11.3489	1.55890	0.779449	+	0.626466i	$0.215499\pi$
$54$	0	0
$55$	−1.39091	−0.187550
$56$	0	0
$57$	−3.28357	−0.434919
$58$	0	0
$59$	−1.21819	−0.158594	−0.0792972	−	0.996851i	$-0.525268\pi$
$59$	−1.21819	−0.158594	−0.0792972	+	0.996851i	$0.525268\pi$
$60$	0	0
$61$	13.3489	1.70916	0.854579	−	0.519322i	$-0.173815\pi$
$61$	13.3489	1.70916	0.854579	+	0.519322i	$0.173815\pi$
$62$	0	0
$63$	3.28357	0.413691
$64$	0	0
$65$	1.39091	0.172521
$66$	0	0
$67$	3.17623	0.388038	0.194019	−	0.980998i	$-0.437848\pi$
$67$	3.17623	0.388038	0.194019	+	0.980998i	$0.437848\pi$
$68$	0	0
$69$	3.28357	0.395295
$70$	0	0
$71$	−11.1343	−1.32139	−0.660697	−	0.750652i	$-0.729739\pi$
$71$	−11.1343	−1.32139	−0.660697	+	0.750652i	$0.729739\pi$
$72$	0	0
$73$	7.49825	0.877603	0.438802	−	0.898584i	$-0.355403\pi$
$73$	7.49825	0.877603	0.438802	+	0.898584i	$0.355403\pi$
$74$	0	0
$75$	−3.06538	−0.353960
$76$	0	0
$77$	3.28357	0.374197
$78$	0	0
$79$	−3.32553	−0.374151	−0.187075	−	0.982346i	$-0.559901\pi$
$79$	−3.32553	−0.374151	−0.187075	+	0.982346i	$0.559901\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.78532	0.854550	0.427275	−	0.904122i	$-0.359474\pi$
$83$	7.78532	0.854550	0.427275	+	0.904122i	$0.359474\pi$
$84$	0	0
$85$	3.71994	0.403484
$86$	0	0
$87$	1.89266	0.202915
$88$	0	0
$89$	6.17272	0.654307	0.327153	−	0.944971i	$-0.393911\pi$
$89$	6.17272	0.654307	0.327153	+	0.944971i	$0.393911\pi$
$90$	0	0
$91$	−3.28357	−0.344211
$92$	0	0
$93$	−3.39091	−0.351621
$94$	0	0
$95$	4.56713	0.468578
$96$	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
$101$	−3.89266	−0.387334	−0.193667	−	0.981067i	$-0.562038\pi$
$101$	−3.89266	−0.387334	−0.193667	+	0.981067i	$0.562038\pi$
$102$	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
$105$	−4.56713	−0.445707
$106$	0	0
$107$	10.7818	1.04232	0.521159	−	0.853460i	$-0.325500\pi$
$107$	10.7818	1.04232	0.521159	+	0.853460i	$0.325500\pi$
$108$	0	0
$109$	−0.632514	−0.0605838	−0.0302919	−	0.999541i	$-0.509644\pi$
$109$	−0.632514	−0.0605838	−0.0302919	+	0.999541i	$0.509644\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	3.21819	0.302742	0.151371	−	0.988477i	$-0.451631\pi$
$113$	3.21819	0.302742	0.151371	+	0.988477i	$0.451631\pi$
$114$	0	0
$115$	−4.56713	−0.425887
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−8.78181	−0.805027
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.28357	0.476403
$124$	0	0
$125$	11.2182	1.00339
$126$	0	0
$127$	−9.45629	−0.839110	−0.419555	−	0.907730i	$-0.637814\pi$
$127$	−9.45629	−0.839110	−0.419555	+	0.907730i	$0.637814\pi$
$128$	0	0
$129$	1.89266	0.166639
$130$	0	0
$131$	21.2650	1.85793	0.928967	−	0.370162i	$-0.120698\pi$
$131$	21.2650	1.85793	0.928967	+	0.370162i	$0.120698\pi$
$132$	0	0
$133$	−10.7818	−0.934902
$134$	0	0
$135$	−1.39091	−0.119710
$136$	0	0
$137$	0.609094	0.0520384	0.0260192	−	0.999661i	$-0.491717\pi$
$137$	0.609094	0.0520384	0.0260192	+	0.999661i	$0.491717\pi$
$138$	0	0
$139$	−20.0234	−1.69836	−0.849182	−	0.528100i	$-0.822905\pi$
$139$	−20.0234	−1.69836	−0.849182	+	0.528100i	$0.822905\pi$
$140$	0	0
$141$	7.34895	0.618893
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.63251	−0.218618
$146$	0	0
$147$	3.78181	0.311919
$148$	0	0
$149$	−6.28708	−0.515057	−0.257529	−	0.966271i	$-0.582908\pi$
$149$	−6.28708	−0.515057	−0.257529	+	0.966271i	$0.582908\pi$
$150$	0	0
$151$	−16.4178	−1.33606	−0.668032	−	0.744132i	$-0.732863\pi$
$151$	−16.4178	−1.33606	−0.668032	+	0.744132i	$0.732863\pi$
$152$	0	0
$153$	−2.67447	−0.216218
$154$	0	0
$155$	4.71643	0.378833
$156$	0	0
$157$	19.6360	1.56713	0.783563	−	0.621313i	$-0.213400\pi$
$157$	19.6360	1.56713	0.783563	+	0.621313i	$0.213400\pi$
$158$	0	0
$159$	11.3489	0.900030
$160$	0	0
$161$	10.7818	0.849726
$162$	0	0
$163$	0.387397	0.0303433	0.0151717	−	0.999885i	$-0.495171\pi$
$163$	0.387397	0.0303433	0.0151717	+	0.999885i	$0.495171\pi$
$164$	0	0
$165$	−1.39091	−0.108282
$166$	0	0
$167$	9.00351	0.696712	0.348356	−	0.937362i	$-0.386740\pi$
$167$	9.00351	0.696712	0.348356	+	0.937362i	$0.386740\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.28357	−0.251101
$172$	0	0
$173$	−14.4598	−1.09936	−0.549679	−	0.835376i	$-0.685250\pi$
$173$	−14.4598	−1.09936	−0.549679	+	0.835376i	$0.685250\pi$
$174$	0	0
$175$	−10.0654	−0.760871
$176$	0	0
$177$	−1.21819	−0.0915646
$178$	0	0
$179$	16.6325	1.24317	0.621586	−	0.783346i	$-0.286489\pi$
$179$	16.6325	1.24317	0.621586	+	0.783346i	$0.286489\pi$
$180$	0	0
$181$	−21.1996	−1.57576	−0.787879	−	0.615830i	$-0.788821\pi$
$181$	−21.1996	−1.57576	−0.787879	+	0.615830i	$0.788821\pi$
$182$	0	0
$183$	13.3489	0.986783
$184$	0	0
$185$	−5.56363	−0.409046
$186$	0	0
$187$	−2.67447	−0.195577
$188$	0	0
$189$	3.28357	0.238844
$190$	0	0
$191$	−7.78532	−0.563326	−0.281663	−	0.959513i	$-0.590886\pi$
$191$	−7.78532	−0.563326	−0.281663	+	0.959513i	$0.590886\pi$
$192$	0	0
$193$	2.06538	0.148669	0.0743346	−	0.997233i	$-0.476317\pi$
$193$	2.06538	0.148669	0.0743346	+	0.997233i	$0.476317\pi$
$194$	0	0
$195$	1.39091	0.0996049
$196$	0	0
$197$	18.6325	1.32751	0.663756	−	0.747949i	$-0.268961\pi$
$197$	18.6325	1.32751	0.663756	+	0.747949i	$0.268961\pi$
$198$	0	0
$199$	5.56363	0.394395	0.197197	−	0.980364i	$-0.436816\pi$
$199$	5.56363	0.394395	0.197197	+	0.980364i	$0.436816\pi$
$200$	0	0
$201$	3.17623	0.224034
$202$	0	0
$203$	6.21468	0.436185
$204$	0	0
$205$	−7.34895	−0.513273
$206$	0	0
$207$	3.28357	0.228224
$208$	0	0
$209$	−3.28357	−0.227129
$210$	0	0
$211$	19.2416	1.32465	0.662323	−	0.749218i	$-0.269570\pi$
$211$	19.2416	1.32465	0.662323	+	0.749218i	$0.269570\pi$
$212$	0	0
$213$	−11.1343	−0.762907
$214$	0	0
$215$	−2.63251	−0.179536
$216$	0	0
$217$	−11.1343	−0.755843
$218$	0	0
$219$	7.49825	0.506684
$220$	0	0
$221$	2.67447	0.179905
$222$	0	0
$223$	−23.3070	−1.56075	−0.780376	−	0.625311i	$-0.784972\pi$
$223$	−23.3070	−1.56075	−0.780376	+	0.625311i	$0.784972\pi$
$224$	0	0
$225$	−3.06538	−0.204359
$226$	0	0
$227$	12.2801	0.815056	0.407528	−	0.913193i	$-0.366391\pi$
$227$	12.2801	0.815056	0.407528	+	0.913193i	$0.366391\pi$
$228$	0	0
$229$	22.9126	1.51411	0.757053	−	0.653354i	$-0.226639\pi$
$229$	22.9126	1.51411	0.757053	+	0.653354i	$0.226639\pi$
$230$	0	0
$231$	3.28357	0.216043
$232$	0	0
$233$	5.80874	0.380543	0.190272	−	0.981731i	$-0.439063\pi$
$233$	5.80874	0.380543	0.190272	+	0.981731i	$0.439063\pi$
$234$	0	0
$235$	−10.2217	−0.666790
$236$	0	0
$237$	−3.32553	−0.216016
$238$	0	0
$239$	−5.41433	−0.350224	−0.175112	−	0.984549i	$-0.556029\pi$
$239$	−5.41433	−0.350224	−0.175112	+	0.984549i	$0.556029\pi$
$240$	0	0
$241$	12.3525	0.795692	0.397846	−	0.917452i	$-0.369758\pi$
$241$	12.3525	0.795692	0.397846	+	0.917452i	$0.369758\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.26015	−0.336058
$246$	0	0
$247$	3.28357	0.208928
$248$	0	0
$249$	7.78532	0.493375
$250$	0	0
$251$	−28.0468	−1.77030	−0.885150	−	0.465305i	$-0.845945\pi$
$251$	−28.0468	−1.77030	−0.885150	+	0.465305i	$0.845945\pi$
$252$	0	0
$253$	3.28357	0.206436
$254$	0	0
$255$	3.71994	0.232952
$256$	0	0
$257$	4.99649	0.311673	0.155836	−	0.987783i	$-0.450193\pi$
$257$	4.99649	0.311673	0.155836	+	0.987783i	$0.450193\pi$
$258$	0	0
$259$	13.1343	0.816124
$260$	0	0
$261$	1.89266	0.117153
$262$	0	0
$263$	18.7818	1.15814	0.579068	−	0.815279i	$-0.303417\pi$
$263$	18.7818	1.15814	0.579068	+	0.815279i	$0.303417\pi$
$264$	0	0
$265$	−15.7853	−0.969685
$266$	0	0
$267$	6.17272	0.377764
$268$	0	0
$269$	−3.34895	−0.204189	−0.102094	−	0.994775i	$-0.532554\pi$
$269$	−3.34895	−0.204189	−0.102094	+	0.994775i	$0.532554\pi$
$270$	0	0
$271$	−27.3304	−1.66020	−0.830102	−	0.557612i	$-0.811718\pi$
$271$	−27.3304	−1.66020	−0.830102	+	0.557612i	$0.811718\pi$
$272$	0	0
$273$	−3.28357	−0.198731
$274$	0	0
$275$	−3.06538	−0.184849
$276$	0	0
$277$	−14.1308	−0.849035	−0.424518	−	0.905420i	$-0.639556\pi$
$277$	−14.1308	−0.849035	−0.424518	+	0.905420i	$0.639556\pi$
$278$	0	0
$279$	−3.39091	−0.203008
$280$	0	0
$281$	−10.6325	−0.634283	−0.317141	−	0.948378i	$-0.602723\pi$
$281$	−10.6325	−0.634283	−0.317141	+	0.948378i	$0.602723\pi$
$282$	0	0
$283$	15.8927	0.944721	0.472360	−	0.881406i	$-0.343402\pi$
$283$	15.8927	0.944721	0.472360	+	0.881406i	$0.343402\pi$
$284$	0	0
$285$	4.56713	0.270534
$286$	0	0
$287$	17.3489	1.02408
$288$	0	0
$289$	−9.84719	−0.579247
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	15.0619	0.879924	0.439962	−	0.898016i	$-0.354992\pi$
$293$	15.0619	0.879924	0.439962	+	0.898016i	$0.354992\pi$
$294$	0	0
$295$	1.69438	0.0986509
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.28357	−0.189894
$300$	0	0
$301$	6.21468	0.358208
$302$	0	0
$303$	−3.89266	−0.223628
$304$	0	0
$305$	−18.5671	−1.06315
$306$	0	0
$307$	12.4178	0.708723	0.354362	−	0.935108i	$-0.384698\pi$
$307$	12.4178	0.708723	0.354362	+	0.935108i	$0.384698\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	13.8507	0.785401	0.392701	−	0.919666i	$-0.371541\pi$
$311$	13.8507	0.785401	0.392701	+	0.919666i	$0.371541\pi$
$312$	0	0
$313$	2.84719	0.160933	0.0804664	−	0.996757i	$-0.474359\pi$
$313$	2.84719	0.160933	0.0804664	+	0.996757i	$0.474359\pi$
$314$	0	0
$315$	−4.56713	−0.257329
$316$	0	0
$317$	−9.39792	−0.527840	−0.263920	−	0.964545i	$-0.585015\pi$
$317$	−9.39792	−0.527840	−0.263920	+	0.964545i	$0.585015\pi$
$318$	0	0
$319$	1.89266	0.105969
$320$	0	0
$321$	10.7818	0.601782
$322$	0	0
$323$	8.78181	0.488633
$324$	0	0
$325$	3.06538	0.170037
$326$	0	0
$327$	−0.632514	−0.0349781
$328$	0	0
$329$	24.1308	1.33037
$330$	0	0
$331$	18.9545	1.04184	0.520918	−	0.853607i	$-0.325590\pi$
$331$	18.9545	1.04184	0.520918	+	0.853607i	$0.325590\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	−4.41784	−0.241372
$336$	0	0
$337$	1.21117	0.0659766	0.0329883	−	0.999456i	$-0.489498\pi$
$337$	1.21117	0.0659766	0.0329883	+	0.999456i	$0.489498\pi$
$338$	0	0
$339$	3.21819	0.174788
$340$	0	0
$341$	−3.39091	−0.183628
$342$	0	0
$343$	−10.5671	−0.570572
$344$	0	0
$345$	−4.56713	−0.245886
$346$	0	0
$347$	4.43637	0.238157	0.119079	−	0.992885i	$-0.462006\pi$
$347$	4.43637	0.238157	0.119079	+	0.992885i	$0.462006\pi$
$348$	0	0
$349$	−27.4797	−1.47095	−0.735477	−	0.677549i	$-0.763042\pi$
$349$	−27.4797	−1.47095	−0.735477	+	0.677549i	$0.763042\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−2.60909	−0.138868	−0.0694340	−	0.997587i	$-0.522119\pi$
$353$	−2.60909	−0.138868	−0.0694340	+	0.997587i	$0.522119\pi$
$354$	0	0
$355$	15.4867	0.821950
$356$	0	0
$357$	−8.78181	−0.464783
$358$	0	0
$359$	11.8507	0.625456	0.312728	−	0.949843i	$-0.398757\pi$
$359$	11.8507	0.625456	0.312728	+	0.949843i	$0.398757\pi$
$360$	0	0
$361$	−8.21819	−0.432536
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−10.4294	−0.545897
$366$	0	0
$367$	7.78532	0.406390	0.203195	−	0.979138i	$-0.434867\pi$
$367$	7.78532	0.406390	0.203195	+	0.979138i	$0.434867\pi$
$368$	0	0
$369$	5.28357	0.275052
$370$	0	0
$371$	37.2650	1.93470
$372$	0	0
$373$	10.3454	0.535666	0.267833	−	0.963465i	$-0.413692\pi$
$373$	10.3454	0.535666	0.267833	+	0.963465i	$0.413692\pi$
$374$	0	0
$375$	11.2182	0.579305
$376$	0	0
$377$	−1.89266	−0.0974770
$378$	0	0
$379$	7.87412	0.404466	0.202233	−	0.979337i	$-0.435180\pi$
$379$	7.87412	0.404466	0.202233	+	0.979337i	$0.435180\pi$
$380$	0	0
$381$	−9.45629	−0.484460
$382$	0	0
$383$	16.9196	0.864551	0.432275	−	0.901742i	$-0.357711\pi$
$383$	16.9196	0.864551	0.432275	+	0.901742i	$0.357711\pi$
$384$	0	0
$385$	−4.56713	−0.232763
$386$	0	0
$387$	1.89266	0.0962094
$388$	0	0
$389$	19.4797	0.987660	0.493830	−	0.869558i	$-0.335596\pi$
$389$	19.4797	0.987660	0.493830	+	0.869558i	$0.335596\pi$
$390$	0	0
$391$	−8.78181	−0.444115
$392$	0	0
$393$	21.2650	1.07268
$394$	0	0
$395$	4.62550	0.232734
$396$	0	0
$397$	11.8322	0.593839	0.296920	−	0.954902i	$-0.404041\pi$
$397$	11.8322	0.593839	0.296920	+	0.954902i	$0.404041\pi$
$398$	0	0
$399$	−10.7818	−0.539766
$400$	0	0
$401$	−1.60558	−0.0801791	−0.0400895	−	0.999196i	$-0.512764\pi$
$401$	−1.60558	−0.0801791	−0.0400895	+	0.999196i	$0.512764\pi$
$402$	0	0
$403$	3.39091	0.168913
$404$	0	0
$405$	−1.39091	−0.0691147
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	4.63251	0.229063	0.114532	−	0.993420i	$-0.463463\pi$
$409$	4.63251	0.229063	0.114532	+	0.993420i	$0.463463\pi$
$410$	0	0
$411$	0.609094	0.0300444
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−10.8287	−0.531558
$416$	0	0
$417$	−20.0234	−0.980551
$418$	0	0
$419$	20.6325	1.00796	0.503982	−	0.863714i	$-0.331868\pi$
$419$	20.6325	1.00796	0.503982	+	0.863714i	$0.331868\pi$
$420$	0	0
$421$	21.7014	1.05766	0.528831	−	0.848727i	$-0.322631\pi$
$421$	21.7014	1.05766	0.528831	+	0.848727i	$0.322631\pi$
$422$	0	0
$423$	7.34895	0.357318
$424$	0	0
$425$	8.19828	0.397675
$426$	0	0
$427$	43.8322	2.12119
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−12.1308	−0.584318	−0.292159	−	0.956370i	$-0.594374\pi$
$431$	−12.1308	−0.584318	−0.292159	+	0.956370i	$0.594374\pi$
$432$	0	0
$433$	8.00702	0.384793	0.192396	−	0.981317i	$-0.438374\pi$
$433$	8.00702	0.384793	0.192396	+	0.981317i	$0.438374\pi$
$434$	0	0
$435$	−2.63251	−0.126219
$436$	0	0
$437$	−10.7818	−0.515764
$438$	0	0
$439$	−7.24161	−0.345623	−0.172812	−	0.984955i	$-0.555285\pi$
$439$	−7.24161	−0.345623	−0.172812	+	0.984955i	$0.555285\pi$
$440$	0	0
$441$	3.78181	0.180086
$442$	0	0
$443$	1.30211	0.0618650	0.0309325	−	0.999521i	$-0.490152\pi$
$443$	1.30211	0.0618650	0.0309325	+	0.999521i	$0.490152\pi$
$444$	0	0
$445$	−8.58567	−0.407000
$446$	0	0
$447$	−6.28708	−0.297368
$448$	0	0
$449$	24.0420	1.13461	0.567305	−	0.823508i	$-0.307986\pi$
$449$	24.0420	1.13461	0.567305	+	0.823508i	$0.307986\pi$
$450$	0	0
$451$	5.28357	0.248793
$452$	0	0
$453$	−16.4178	−0.771377
$454$	0	0
$455$	4.56713	0.214111
$456$	0	0
$457$	−2.13076	−0.0996727	−0.0498364	−	0.998757i	$-0.515870\pi$
$457$	−2.13076	−0.0996727	−0.0498364	+	0.998757i	$0.515870\pi$
$458$	0	0
$459$	−2.67447	−0.124834
$460$	0	0
$461$	25.5451	1.18975	0.594877	−	0.803817i	$-0.297201\pi$
$461$	25.5451	1.18975	0.594877	+	0.803817i	$0.297201\pi$
$462$	0	0
$463$	−23.3140	−1.08349	−0.541747	−	0.840542i	$-0.682237\pi$
$463$	−23.3140	−1.08349	−0.541747	+	0.840542i	$0.682237\pi$
$464$	0	0
$465$	4.71643	0.218719
$466$	0	0
$467$	−0.287076	−0.0132843	−0.00664215	−	0.999978i	$-0.502114\pi$
$467$	−0.287076	−0.0132843	−0.00664215	+	0.999978i	$0.502114\pi$
$468$	0	0
$469$	10.4294	0.481583
$470$	0	0
$471$	19.6360	0.904780
$472$	0	0
$473$	1.89266	0.0870246
$474$	0	0
$475$	10.0654	0.461831
$476$	0	0
$477$	11.3489	0.519633
$478$	0	0
$479$	13.8437	0.632534	0.316267	−	0.948670i	$-0.397570\pi$
$479$	13.8437	0.632534	0.316267	+	0.948670i	$0.397570\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	10.7818	0.490589
$484$	0	0
$485$	−11.1273	−0.505262
$486$	0	0
$487$	7.87412	0.356810	0.178405	−	0.983957i	$-0.442906\pi$
$487$	7.87412	0.356810	0.178405	+	0.983957i	$0.442906\pi$
$488$	0	0
$489$	0.387397	0.0175187
$490$	0	0
$491$	−12.1378	−0.547770	−0.273885	−	0.961762i	$-0.588309\pi$
$491$	−12.1378	−0.547770	−0.273885	+	0.961762i	$0.588309\pi$
$492$	0	0
$493$	−5.06187	−0.227975
$494$	0	0
$495$	−1.39091	−0.0625166
$496$	0	0
$497$	−36.5601	−1.63995
$498$	0	0
$499$	−19.3909	−0.868056	−0.434028	−	0.900899i	$-0.642908\pi$
$499$	−19.3909	−0.868056	−0.434028	+	0.900899i	$0.642908\pi$
$500$	0	0
$501$	9.00351	0.402247
$502$	0	0
$503$	1.47971	0.0659768	0.0329884	−	0.999456i	$-0.489498\pi$
$503$	1.47971	0.0659768	0.0329884	+	0.999456i	$0.489498\pi$
$504$	0	0
$505$	5.41433	0.240934
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−22.0049	−0.975349	−0.487675	−	0.873025i	$-0.662155\pi$
$509$	−22.0049	−0.975349	−0.487675	+	0.873025i	$0.662155\pi$
$510$	0	0
$511$	24.6210	1.08917
$512$	0	0
$513$	−3.28357	−0.144973
$514$	0	0
$515$	11.1273	0.490325
$516$	0	0
$517$	7.34895	0.323206
$518$	0	0
$519$	−14.4598	−0.634715
$520$	0	0
$521$	32.6979	1.43252	0.716260	−	0.697833i	$-0.245852\pi$
$521$	32.6979	1.43252	0.716260	+	0.697833i	$0.245852\pi$
$522$	0	0
$523$	−3.45629	−0.151133	−0.0755664	−	0.997141i	$-0.524076\pi$
$523$	−3.45629	−0.151133	−0.0755664	+	0.997141i	$0.524076\pi$
$524$	0	0
$525$	−10.0654	−0.439289
$526$	0	0
$527$	9.06889	0.395047
$528$	0	0
$529$	−12.2182	−0.531226
$530$	0	0
$531$	−1.21819	−0.0528648
$532$	0	0
$533$	−5.28357	−0.228857
$534$	0	0
$535$	−14.9965	−0.648355
$536$	0	0
$537$	16.6325	0.717746
$538$	0	0
$539$	3.78181	0.162894
$540$	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
$543$	−21.1996	−0.909764
$544$	0	0
$545$	0.879767	0.0376851
$546$	0	0
$547$	−6.67447	−0.285380	−0.142690	−	0.989767i	$-0.545575\pi$
$547$	−6.67447	−0.285380	−0.142690	+	0.989767i	$0.545575\pi$
$548$	0	0
$549$	13.3489	0.569719
$550$	0	0
$551$	−6.21468	−0.264754
$552$	0	0
$553$	−10.9196	−0.464348
$554$	0	0
$555$	−5.56363	−0.236163
$556$	0	0
$557$	−3.85070	−0.163159	−0.0815797	−	0.996667i	$-0.525997\pi$
$557$	−3.85070	−0.163159	−0.0815797	+	0.996667i	$0.525997\pi$
$558$	0	0
$559$	−1.89266	−0.0800510
$560$	0	0
$561$	−2.67447	−0.112916
$562$	0	0
$563$	−22.8287	−0.962113	−0.481057	−	0.876690i	$-0.659747\pi$
$563$	−22.8287	−0.962113	−0.481057	+	0.876690i	$0.659747\pi$
$564$	0	0
$565$	−4.47620	−0.188315
$566$	0	0
$567$	3.28357	0.137897
$568$	0	0
$569$	−30.0304	−1.25894	−0.629471	−	0.777024i	$-0.716728\pi$
$569$	−30.0304	−1.25894	−0.629471	+	0.777024i	$0.716728\pi$
$570$	0	0
$571$	24.6745	1.03259	0.516297	−	0.856409i	$-0.327310\pi$
$571$	24.6745	1.03259	0.516297	+	0.856409i	$0.327310\pi$
$572$	0	0
$573$	−7.78532	−0.325236
$574$	0	0
$575$	−10.0654	−0.419755
$576$	0	0
$577$	43.0972	1.79416	0.897080	−	0.441869i	$-0.145684\pi$
$577$	43.0972	1.79416	0.897080	+	0.441869i	$0.145684\pi$
$578$	0	0
$579$	2.06538	0.0858342
$580$	0	0
$581$	25.5636	1.06056
$582$	0	0
$583$	11.3489	0.470025
$584$	0	0
$585$	1.39091	0.0575069
$586$	0	0
$587$	−44.1776	−1.82340	−0.911702	−	0.410851i	$-0.865232\pi$
$587$	−44.1776	−1.82340	−0.911702	+	0.410851i	$0.865232\pi$
$588$	0	0
$589$	11.1343	0.458780
$590$	0	0
$591$	18.6325	0.766439
$592$	0	0
$593$	33.0689	1.35798	0.678988	−	0.734149i	$-0.262419\pi$
$593$	33.0689	1.35798	0.678988	+	0.734149i	$0.262419\pi$
$594$	0	0
$595$	12.2147	0.500753
$596$	0	0
$597$	5.56363	0.227704
$598$	0	0
$599$	−2.65105	−0.108319	−0.0541596	−	0.998532i	$-0.517248\pi$
$599$	−2.65105	−0.108319	−0.0541596	+	0.998532i	$0.517248\pi$
$600$	0	0
$601$	7.13427	0.291013	0.145506	−	0.989357i	$-0.453519\pi$
$601$	7.13427	0.291013	0.145506	+	0.989357i	$0.453519\pi$
$602$	0	0
$603$	3.17623	0.129346
$604$	0	0
$605$	−1.39091	−0.0565484
$606$	0	0
$607$	−30.5906	−1.24163	−0.620816	−	0.783956i	$-0.713199\pi$
$607$	−30.5906	−1.24163	−0.620816	+	0.783956i	$0.713199\pi$
$608$	0	0
$609$	6.21468	0.251831
$610$	0	0
$611$	−7.34895	−0.297307
$612$	0	0
$613$	−10.0654	−0.406537	−0.203268	−	0.979123i	$-0.565156\pi$
$613$	−10.0654	−0.406537	−0.203268	+	0.979123i	$0.565156\pi$
$614$	0	0
$615$	−7.34895	−0.296338
$616$	0	0
$617$	−22.3944	−0.901565	−0.450783	−	0.892634i	$-0.648855\pi$
$617$	−22.3944	−0.901565	−0.450783	+	0.892634i	$0.648855\pi$
$618$	0	0
$619$	−32.5182	−1.30702	−0.653508	−	0.756920i	$-0.726703\pi$
$619$	−32.5182	−1.30702	−0.653508	+	0.756920i	$0.726703\pi$
$620$	0	0
$621$	3.28357	0.131765
$622$	0	0
$623$	20.2685	0.812042
$624$	0	0
$625$	−0.276549	−0.0110620
$626$	0	0
$627$	−3.28357	−0.131133
$628$	0	0
$629$	−10.6979	−0.426553
$630$	0	0
$631$	2.52517	0.100526	0.0502628	−	0.998736i	$-0.483994\pi$
$631$	2.52517	0.100526	0.0502628	+	0.998736i	$0.483994\pi$
$632$	0	0
$633$	19.2416	0.764785
$634$	0	0
$635$	13.1528	0.521953
$636$	0	0
$637$	−3.78181	−0.149841
$638$	0	0
$639$	−11.1343	−0.440465
$640$	0	0
$641$	12.1378	0.479413	0.239707	−	0.970845i	$-0.422949\pi$
$641$	12.1378	0.479413	0.239707	+	0.970845i	$0.422949\pi$
$642$	0	0
$643$	−41.0014	−1.61694	−0.808468	−	0.588540i	$-0.799703\pi$
$643$	−41.0014	−1.61694	−0.808468	+	0.588540i	$0.799703\pi$
$644$	0	0
$645$	−2.63251	−0.103655
$646$	0	0
$647$	−12.9196	−0.507921	−0.253961	−	0.967215i	$-0.581733\pi$
$647$	−12.9196	−0.507921	−0.253961	+	0.967215i	$0.581733\pi$
$648$	0	0
$649$	−1.21819	−0.0478180
$650$	0	0
$651$	−11.1343	−0.436386
$652$	0	0
$653$	−12.5671	−0.491790	−0.245895	−	0.969296i	$-0.579082\pi$
$653$	−12.5671	−0.491790	−0.245895	+	0.969296i	$0.579082\pi$
$654$	0	0
$655$	−29.5777	−1.15569
$656$	0	0
$657$	7.49825	0.292534
$658$	0	0
$659$	−36.4832	−1.42118	−0.710592	−	0.703604i	$-0.751573\pi$
$659$	−36.4832	−1.42118	−0.710592	+	0.703604i	$0.751573\pi$
$660$	0	0
$661$	−44.0539	−1.71350	−0.856748	−	0.515735i	$-0.827519\pi$
$661$	−44.0539	−1.71350	−0.856748	+	0.515735i	$0.827519\pi$
$662$	0	0
$663$	2.67447	0.103868
$664$	0	0
$665$	14.9965	0.581539
$666$	0	0
$667$	6.21468	0.240633
$668$	0	0
$669$	−23.3070	−0.901100
$670$	0	0
$671$	13.3489	0.515330
$672$	0	0
$673$	−24.6909	−0.951763	−0.475882	−	0.879509i	$-0.657871\pi$
$673$	−24.6909	−0.951763	−0.475882	+	0.879509i	$0.657871\pi$
$674$	0	0
$675$	−3.06538	−0.117987
$676$	0	0
$677$	−20.3290	−0.781308	−0.390654	−	0.920538i	$-0.627751\pi$
$677$	−20.3290	−0.781308	−0.390654	+	0.920538i	$0.627751\pi$
$678$	0	0
$679$	26.2685	1.00809
$680$	0	0
$681$	12.2801	0.470573
$682$	0	0
$683$	−18.6909	−0.715186	−0.357593	−	0.933877i	$-0.616403\pi$
$683$	−18.6909	−0.715186	−0.357593	+	0.933877i	$0.616403\pi$
$684$	0	0
$685$	−0.847192	−0.0323696
$686$	0	0
$687$	22.9126	0.874169
$688$	0	0
$689$	−11.3489	−0.432360
$690$	0	0
$691$	4.08880	0.155545	0.0777726	−	0.996971i	$-0.475219\pi$
$691$	4.08880	0.155545	0.0777726	+	0.996971i	$0.475219\pi$
$692$	0	0
$693$	3.28357	0.124732
$694$	0	0
$695$	27.8507	1.05644
$696$	0	0
$697$	−14.1308	−0.535241
$698$	0	0
$699$	5.80874	0.219707
$700$	0	0
$701$	−46.8521	−1.76958	−0.884789	−	0.465992i	$-0.845698\pi$
$701$	−46.8521	−1.76958	−0.884789	+	0.465992i	$0.845698\pi$
$702$	0	0
$703$	−13.1343	−0.495368
$704$	0	0
$705$	−10.2217	−0.384971
$706$	0	0
$707$	−12.7818	−0.480710
$708$	0	0
$709$	29.4797	1.10713	0.553567	−	0.832805i	$-0.313266\pi$
$709$	29.4797	1.10713	0.553567	+	0.832805i	$0.313266\pi$
$710$	0	0
$711$	−3.32553	−0.124717
$712$	0	0
$713$	−11.1343	−0.416982
$714$	0	0
$715$	1.39091	0.0520169
$716$	0	0
$717$	−5.41433	−0.202202
$718$	0	0
$719$	−48.2615	−1.79985	−0.899925	−	0.436044i	$-0.856379\pi$
$719$	−48.2615	−1.79985	−0.899925	+	0.436044i	$0.856379\pi$
$720$	0	0
$721$	−26.2685	−0.978291
$722$	0	0
$723$	12.3525	0.459393
$724$	0	0
$725$	−5.80172	−0.215471
$726$	0	0
$727$	19.3560	0.717873	0.358936	−	0.933362i	$-0.383139\pi$
$727$	19.3560	0.717873	0.358936	+	0.933362i	$0.383139\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.06187	−0.187220
$732$	0	0
$733$	43.3304	1.60045	0.800223	−	0.599703i	$-0.204715\pi$
$733$	43.3304	1.60045	0.800223	+	0.599703i	$0.204715\pi$
$734$	0	0
$735$	−5.26015	−0.194023
$736$	0	0
$737$	3.17623	0.116998
$738$	0	0
$739$	−19.5451	−0.718978	−0.359489	−	0.933149i	$-0.617049\pi$
$739$	−19.5451	−0.718978	−0.359489	+	0.933149i	$0.617049\pi$
$740$	0	0
$741$	3.28357	0.120625
$742$	0	0
$743$	3.48672	0.127915	0.0639577	−	0.997953i	$-0.479628\pi$
$743$	3.48672	0.127915	0.0639577	+	0.997953i	$0.479628\pi$
$744$	0	0
$745$	8.74473	0.320382
$746$	0	0
$747$	7.78532	0.284850
$748$	0	0
$749$	35.4028	1.29359
$750$	0	0
$751$	−10.9126	−0.398205	−0.199103	−	0.979979i	$-0.563803\pi$
$751$	−10.9126	−0.398205	−0.199103	+	0.979979i	$0.563803\pi$
$752$	0	0
$753$	−28.0468	−1.02208
$754$	0	0
$755$	22.8357	0.831075
$756$	0	0
$757$	−21.0689	−0.765762	−0.382881	−	0.923798i	$-0.625068\pi$
$757$	−21.0689	−0.765762	−0.382881	+	0.923798i	$0.625068\pi$
$758$	0	0
$759$	3.28357	0.119186
$760$	0	0
$761$	0.494737	0.0179342	0.00896709	−	0.999960i	$-0.497146\pi$
$761$	0.494737	0.0179342	0.00896709	+	0.999960i	$0.497146\pi$
$762$	0	0
$763$	−2.07690	−0.0751889
$764$	0	0
$765$	3.71994	0.134495
$766$	0	0
$767$	1.21819	0.0439862
$768$	0	0
$769$	35.1343	1.26697	0.633487	−	0.773753i	$-0.281623\pi$
$769$	35.1343	1.26697	0.633487	+	0.773753i	$0.281623\pi$
$770$	0	0
$771$	4.99649	0.179944
$772$	0	0
$773$	16.8706	0.606794	0.303397	−	0.952864i	$-0.401879\pi$
$773$	16.8706	0.606794	0.303397	+	0.952864i	$0.401879\pi$
$774$	0	0
$775$	10.3944	0.373378
$776$	0	0
$777$	13.1343	0.471189
$778$	0	0
$779$	−17.3489	−0.621591
$780$	0	0
$781$	−11.1343	−0.398415
$782$	0	0
$783$	1.89266	0.0676382
$784$	0	0
$785$	−27.3119	−0.974802
$786$	0	0
$787$	42.1122	1.50114	0.750569	−	0.660792i	$-0.229779\pi$
$787$	42.1122	1.50114	0.750569	+	0.660792i	$0.229779\pi$
$788$	0	0
$789$	18.7818	0.668650
$790$	0	0
$791$	10.5671	0.375724
$792$	0	0
$793$	−13.3489	−0.474035
$794$	0	0
$795$	−15.7853	−0.559848
$796$	0	0
$797$	35.8251	1.26899	0.634496	−	0.772926i	$-0.281208\pi$
$797$	35.8251	1.26899	0.634496	+	0.772926i	$0.281208\pi$
$798$	0	0
$799$	−19.6546	−0.695328
$800$	0	0
$801$	6.17272	0.218102
$802$	0	0
$803$	7.49825	0.264607
$804$	0	0
$805$	−14.9965	−0.528557
$806$	0	0
$807$	−3.34895	−0.117889
$808$	0	0
$809$	15.8087	0.555806	0.277903	−	0.960609i	$-0.410361\pi$
$809$	15.8087	0.555806	0.277903	+	0.960609i	$0.410361\pi$
$810$	0	0
$811$	20.8472	0.732044	0.366022	−	0.930606i	$-0.380720\pi$
$811$	20.8472	0.732044	0.366022	+	0.930606i	$0.380720\pi$
$812$	0	0
$813$	−27.3304	−0.958519
$814$	0	0
$815$	−0.538833	−0.0188745
$816$	0	0
$817$	−6.21468	−0.217424
$818$	0	0
$819$	−3.28357	−0.114737
$820$	0	0
$821$	4.93813	0.172342	0.0861709	−	0.996280i	$-0.472537\pi$
$821$	4.93813	0.172342	0.0861709	+	0.996280i	$0.472537\pi$
$822$	0	0
$823$	−30.4832	−1.06258	−0.531289	−	0.847191i	$-0.678292\pi$
$823$	−30.4832	−1.06258	−0.531289	+	0.847191i	$0.678292\pi$
$824$	0	0
$825$	−3.06538	−0.106723
$826$	0	0
$827$	31.8507	1.10756	0.553779	−	0.832664i	$-0.313185\pi$
$827$	31.8507	1.10756	0.553779	+	0.832664i	$0.313185\pi$
$828$	0	0
$829$	12.4364	0.431933	0.215967	−	0.976401i	$-0.430710\pi$
$829$	12.4364	0.431933	0.215967	+	0.976401i	$0.430710\pi$
$830$	0	0
$831$	−14.1308	−0.490191
$832$	0	0
$833$	−10.1144	−0.350442
$834$	0	0
$835$	−12.5230	−0.433378
$836$	0	0
$837$	−3.39091	−0.117207
$838$	0	0
$839$	−27.2580	−0.941051	−0.470526	−	0.882386i	$-0.655936\pi$
$839$	−27.2580	−0.941051	−0.470526	+	0.882386i	$0.655936\pi$
$840$	0	0
$841$	−25.4178	−0.876477
$842$	0	0
$843$	−10.6325	−0.366203
$844$	0	0
$845$	−1.39091	−0.0478486
$846$	0	0
$847$	3.28357	0.112825
$848$	0	0
$849$	15.8927	0.545435
$850$	0	0
$851$	13.1343	0.450237
$852$	0	0
$853$	29.7014	1.01696	0.508478	−	0.861075i	$-0.330208\pi$
$853$	29.7014	1.01696	0.508478	+	0.861075i	$0.330208\pi$
$854$	0	0
$855$	4.56713	0.156193
$856$	0	0
$857$	−28.8052	−0.983968	−0.491984	−	0.870604i	$-0.663728\pi$
$857$	−28.8052	−0.983968	−0.491984	+	0.870604i	$0.663728\pi$
$858$	0	0
$859$	−40.7518	−1.39043	−0.695216	−	0.718801i	$-0.744691\pi$
$859$	−40.7518	−1.39043	−0.695216	+	0.718801i	$0.744691\pi$
$860$	0	0
$861$	17.3489	0.591251
$862$	0	0
$863$	−31.7853	−1.08198	−0.540992	−	0.841027i	$-0.681951\pi$
$863$	−31.7853	−1.08198	−0.540992	+	0.841027i	$0.681951\pi$
$864$	0	0
$865$	20.1122	0.683836
$866$	0	0
$867$	−9.84719	−0.334428
$868$	0	0
$869$	−3.32553	−0.112811
$870$	0	0
$871$	−3.17623	−0.107622
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	36.8357	1.24527
$876$	0	0
$877$	−15.2182	−0.513882	−0.256941	−	0.966427i	$-0.582715\pi$
$877$	−15.2182	−0.513882	−0.256941	+	0.966427i	$0.582715\pi$
$878$	0	0
$879$	15.0619	0.508024
$880$	0	0
$881$	53.8322	1.81365	0.906826	−	0.421506i	$-0.138498\pi$
$881$	53.8322	1.81365	0.906826	+	0.421506i	$0.138498\pi$
$882$	0	0
$883$	−51.8322	−1.74429	−0.872146	−	0.489246i	$-0.837272\pi$
$883$	−51.8322	−1.74429	−0.872146	+	0.489246i	$0.837272\pi$
$884$	0	0
$885$	1.69438	0.0569561
$886$	0	0
$887$	−22.4902	−0.755148	−0.377574	−	0.925979i	$-0.623242\pi$
$887$	−22.4902	−0.755148	−0.377574	+	0.925979i	$0.623242\pi$
$888$	0	0
$889$	−31.0503	−1.04140
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−24.1308	−0.807505
$894$	0	0
$895$	−23.1343	−0.773293
$896$	0	0
$897$	−3.28357	−0.109635
$898$	0	0
$899$	−6.41784	−0.214047
$900$	0	0
$901$	−30.3525	−1.01119
$902$	0	0
$903$	6.21468	0.206812
$904$	0	0
$905$	29.4867	0.980172
$906$	0	0
$907$	41.3489	1.37297	0.686485	−	0.727144i	$-0.259153\pi$
$907$	41.3489	1.37297	0.686485	+	0.727144i	$0.259153\pi$
$908$	0	0
$909$	−3.89266	−0.129111
$910$	0	0
$911$	−35.7853	−1.18562	−0.592810	−	0.805342i	$-0.701982\pi$
$911$	−35.7853	−1.18562	−0.592810	+	0.805342i	$0.701982\pi$
$912$	0	0
$913$	7.78532	0.257657
$914$	0	0
$915$	−18.5671	−0.613811
$916$	0	0
$917$	69.8251	2.30583
$918$	0	0
$919$	−49.0269	−1.61725	−0.808625	−	0.588325i	$-0.799788\pi$
$919$	−49.0269	−1.61725	−0.808625	+	0.588325i	$0.799788\pi$
$920$	0	0
$921$	12.4178	0.409182
$922$	0	0
$923$	11.1343	0.366489
$924$	0	0
$925$	−12.2615	−0.403156
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	35.1762	1.15409	0.577047	−	0.816711i	$-0.304205\pi$
$929$	35.1762	1.15409	0.577047	+	0.816711i	$0.304205\pi$
$930$	0	0
$931$	−12.4178	−0.406978
$932$	0	0
$933$	13.8507	0.453452
$934$	0	0
$935$	3.71994	0.121655
$936$	0	0
$937$	16.1308	0.526969	0.263484	−	0.964664i	$-0.415128\pi$
$937$	16.1308	0.526969	0.263484	+	0.964664i	$0.415128\pi$
$938$	0	0
$939$	2.84719	0.0929146
$940$	0	0
$941$	33.9885	1.10799	0.553996	−	0.832519i	$-0.313102\pi$
$941$	33.9885	1.10799	0.553996	+	0.832519i	$0.313102\pi$
$942$	0	0
$943$	17.3489	0.564959
$944$	0	0
$945$	−4.56713	−0.148569
$946$	0	0
$947$	−11.0503	−0.359088	−0.179544	−	0.983750i	$-0.557462\pi$
$947$	−11.0503	−0.359088	−0.179544	+	0.983750i	$0.557462\pi$
$948$	0	0
$949$	−7.49825	−0.243403
$950$	0	0
$951$	−9.39792	−0.304749
$952$	0	0
$953$	2.19126	0.0709818	0.0354909	−	0.999370i	$-0.488701\pi$
$953$	2.19126	0.0709818	0.0354909	+	0.999370i	$0.488701\pi$
$954$	0	0
$955$	10.8287	0.350407
$956$	0	0
$957$	1.89266	0.0611810
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−19.5018	−0.629089
$962$	0	0
$963$	10.7818	0.347439
$964$	0	0
$965$	−2.87275	−0.0924771
$966$	0	0
$967$	15.7598	0.506800	0.253400	−	0.967362i	$-0.418451\pi$
$967$	15.7598	0.506800	0.253400	+	0.967362i	$0.418451\pi$
$968$	0	0
$969$	8.78181	0.282112
$970$	0	0
$971$	−6.57866	−0.211119	−0.105560	−	0.994413i	$-0.533663\pi$
$971$	−6.57866	−0.211119	−0.105560	+	0.994413i	$0.533663\pi$
$972$	0	0
$973$	−65.7482	−2.10779
$974$	0	0
$975$	3.06538	0.0981707
$976$	0	0
$977$	3.83430	0.122670	0.0613350	−	0.998117i	$-0.480464\pi$
$977$	3.83430	0.122670	0.0613350	+	0.998117i	$0.480464\pi$
$978$	0	0
$979$	6.17272	0.197281
$980$	0	0
$981$	−0.632514	−0.0201946
$982$	0	0
$983$	16.5601	0.528186	0.264093	−	0.964497i	$-0.414927\pi$
$983$	16.5601	0.528186	0.264093	+	0.964497i	$0.414927\pi$
$984$	0	0
$985$	−25.9161	−0.825755
$986$	0	0
$987$	24.1308	0.768091
$988$	0	0
$989$	6.21468	0.197615
$990$	0	0
$991$	44.6070	1.41699	0.708493	−	0.705718i	$-0.249375\pi$
$991$	44.6070	1.41699	0.708493	+	0.705718i	$0.249375\pi$
$992$	0	0
$993$	18.9545	0.601504
$994$	0	0
$995$	−7.73848	−0.245326
$996$	0	0
$997$	−40.1846	−1.27266	−0.636330	−	0.771417i	$-0.719548\pi$
$997$	−40.1846	−1.27266	−0.636330	+	0.771417i	$0.719548\pi$
$998$	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.1.2		3
4.3	odd	2		1716.2.a.f.1.2	✓	3
12.11	even	2		5148.2.a.n.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.f.1.2	✓	3	4.3	odd	2
5148.2.a.n.1.2		3	12.11	even	2
6864.2.a.bv.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.39091 q^{5} +3.28357 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.39091 q^{5} +3.28357 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.39091 q^{15} -2.67447 q^{17} -3.28357 q^{19} +3.28357 q^{21} +3.28357 q^{23} -3.06538 q^{25} +1.00000 q^{27} +1.89266 q^{29} -3.39091 q^{31} +1.00000 q^{33} -4.56713 q^{35} +4.00000 q^{37} -1.00000 q^{39} +5.28357 q^{41} +1.89266 q^{43} -1.39091 q^{45} +7.34895 q^{47} +3.78181 q^{49} -2.67447 q^{51} +11.3489 q^{53} -1.39091 q^{55} -3.28357 q^{57} -1.21819 q^{59} +13.3489 q^{61} +3.28357 q^{63} +1.39091 q^{65} +3.17623 q^{67} +3.28357 q^{69} -11.1343 q^{71} +7.49825 q^{73} -3.06538 q^{75} +3.28357 q^{77} -3.32553 q^{79} +1.00000 q^{81} +7.78532 q^{83} +3.71994 q^{85} +1.89266 q^{87} +6.17272 q^{89} -3.28357 q^{91} -3.39091 q^{93} +4.56713 q^{95} +8.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{9}+O(q^{10}) \) 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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