LMFDB - Embedded newform 6027.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
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 861)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.39091$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{4} +1.39091 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.00000 q^{4} +1.39091 q^{5} +1.00000 q^{9} -1.89266 q^{11} +2.00000 q^{12} +3.67447 q^{13} -1.39091 q^{15} +4.00000 q^{16} -2.00000 q^{17} -1.60909 q^{19} -2.78181 q^{20} +0.609094 q^{23} -3.06538 q^{25} -1.00000 q^{27} -2.78181 q^{29} +1.50175 q^{31} +1.89266 q^{33} -2.00000 q^{36} -1.00000 q^{37} -3.67447 q^{39} -1.00000 q^{41} -3.71643 q^{43} +3.78532 q^{44} +1.39091 q^{45} +11.2416 q^{47} -4.00000 q^{48} +2.00000 q^{51} -7.34895 q^{52} +7.34895 q^{53} -2.63251 q^{55} +1.60909 q^{57} -3.34895 q^{59} +2.78181 q^{60} -10.6325 q^{61} -8.00000 q^{64} +5.11085 q^{65} +5.67447 q^{67} +4.00000 q^{68} -0.609094 q^{69} -9.34895 q^{71} -3.71643 q^{73} +3.06538 q^{75} +3.21819 q^{76} +1.82377 q^{79} +5.56363 q^{80} +1.00000 q^{81} +12.5252 q^{83} -2.78181 q^{85} +2.78181 q^{87} -7.24161 q^{89} -1.21819 q^{92} -1.50175 q^{93} -2.23810 q^{95} +11.5636 q^{97} -1.89266 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 6 q^{4} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 6 * q^4 + 3 * q^9	$ 3 q - 3 q^{3} - 6 q^{4} + 3 q^{9} + 6 q^{12} - 3 q^{13} + 12 q^{16} - 6 q^{17} - 9 q^{19} + 6 q^{23} + 9 q^{25} - 3 q^{27} + 3 q^{31} - 6 q^{36} - 3 q^{37} + 3 q^{39} - 3 q^{41} - 21 q^{43} - 12 q^{48} + 6 q^{51} + 6 q^{52} - 6 q^{53} + 30 q^{55} + 9 q^{57} + 18 q^{59} + 6 q^{61} - 24 q^{64} + 18 q^{65} + 3 q^{67} + 12 q^{68} - 6 q^{69} - 21 q^{73} - 9 q^{75} + 18 q^{76} + 21 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{89} - 12 q^{92} - 3 q^{93} + 24 q^{95} + 18 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 - 6 * q^4 + 3 * q^9 + 6 * q^12 - 3 * q^13 + 12 * q^16 - 6 * q^17 - 9 * q^19 + 6 * q^23 + 9 * q^25 - 3 * q^27 + 3 * q^31 - 6 * q^36 - 3 * q^37 + 3 * q^39 - 3 * q^41 - 21 * q^43 - 12 * q^48 + 6 * q^51 + 6 * q^52 - 6 * q^53 + 30 * q^55 + 9 * q^57 + 18 * q^59 + 6 * q^61 - 24 * q^64 + 18 * q^65 + 3 * q^67 + 12 * q^68 - 6 * q^69 - 21 * q^73 - 9 * q^75 + 18 * q^76 + 21 * q^79 + 3 * q^81 - 6 * q^83 + 12 * q^89 - 12 * q^92 - 3 * q^93 + 24 * q^95 + 18 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	1.39091	0.622032	0.311016	−	0.950405i	$-0.399331\pi$
$5$	1.39091	0.622032	0.311016	+	0.950405i	$0.399331\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.89266	−0.570659	−0.285329	−	0.958430i	$-0.592103\pi$
$11$	−1.89266	−0.570659	−0.285329	+	0.958430i	$0.592103\pi$
$12$	2.00000	0.577350
$13$	3.67447	1.01912	0.509558	−	0.860436i	$-0.329809\pi$
$13$	3.67447	1.01912	0.509558	+	0.860436i	$0.329809\pi$
$14$	0	0
$15$	−1.39091	−0.359130
$16$	4.00000	1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.60909	−0.369151	−0.184576	−	0.982818i	$-0.559091\pi$
$19$	−1.60909	−0.369151	−0.184576	+	0.982818i	$0.559091\pi$
$20$	−2.78181	−0.622032
$21$	0	0
$22$	0	0
$23$	0.609094	0.127005	0.0635024	−	0.997982i	$-0.479773\pi$
$23$	0.609094	0.127005	0.0635024	+	0.997982i	$0.479773\pi$
$24$	0	0
$25$	−3.06538	−0.613076
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.78181	−0.516570	−0.258285	−	0.966069i	$-0.583157\pi$
$29$	−2.78181	−0.516570	−0.258285	+	0.966069i	$0.583157\pi$
$30$	0	0
$31$	1.50175	0.269723	0.134862	−	0.990864i	$-0.456941\pi$
$31$	1.50175	0.269723	0.134862	+	0.990864i	$0.456941\pi$
$32$	0	0
$33$	1.89266	0.329470
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−3.67447	−0.588387
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.71643	−0.566751	−0.283375	−	0.959009i	$-0.591454\pi$
$43$	−3.71643	−0.566751	−0.283375	+	0.959009i	$0.591454\pi$
$44$	3.78532	0.570659
$45$	1.39091	0.207344
$46$	0	0
$47$	11.2416	1.63976	0.819878	−	0.572538i	$-0.194041\pi$
$47$	11.2416	1.63976	0.819878	+	0.572538i	$0.194041\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	−7.34895	−1.01912
$53$	7.34895	1.00946	0.504728	−	0.863279i	$-0.331593\pi$
$53$	7.34895	1.00946	0.504728	+	0.863279i	$0.331593\pi$
$54$	0	0
$55$	−2.63251	−0.354968
$56$	0	0
$57$	1.60909	0.213130
$58$	0	0
$59$	−3.34895	−0.435996	−0.217998	−	0.975949i	$-0.569953\pi$
$59$	−3.34895	−0.435996	−0.217998	+	0.975949i	$0.569953\pi$
$60$	2.78181	0.359130
$61$	−10.6325	−1.36135	−0.680677	−	0.732584i	$-0.738314\pi$
$61$	−10.6325	−1.36135	−0.680677	+	0.732584i	$0.738314\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	5.11085	0.633923
$66$	0	0
$67$	5.67447	0.693247	0.346624	−	0.938004i	$-0.387328\pi$
$67$	5.67447	0.693247	0.346624	+	0.938004i	$0.387328\pi$
$68$	4.00000	0.485071
$69$	−0.609094	−0.0733263
$70$	0	0
$71$	−9.34895	−1.10952	−0.554758	−	0.832012i	$-0.687189\pi$
$71$	−9.34895	−1.10952	−0.554758	+	0.832012i	$0.687189\pi$
$72$	0	0
$73$	−3.71643	−0.434976	−0.217488	−	0.976063i	$-0.569786\pi$
$73$	−3.71643	−0.434976	−0.217488	+	0.976063i	$0.569786\pi$
$74$	0	0
$75$	3.06538	0.353960
$76$	3.21819	0.369151
$77$	0	0
$78$	0	0
$79$	1.82377	0.205190	0.102595	−	0.994723i	$-0.467285\pi$
$79$	1.82377	0.205190	0.102595	+	0.994723i	$0.467285\pi$
$80$	5.56363	0.622032
$81$	1.00000	0.111111
$82$	0	0
$83$	12.5252	1.37482	0.687408	−	0.726271i	$-0.258748\pi$
$83$	12.5252	1.37482	0.687408	+	0.726271i	$0.258748\pi$
$84$	0	0
$85$	−2.78181	−0.301730
$86$	0	0
$87$	2.78181	0.298242
$88$	0	0
$89$	−7.24161	−0.767609	−0.383804	−	0.923414i	$-0.625386\pi$
$89$	−7.24161	−0.767609	−0.383804	+	0.923414i	$0.625386\pi$
$90$	0	0
$91$	0	0
$92$	−1.21819	−0.127005
$93$	−1.50175	−0.155725
$94$	0	0
$95$	−2.23810	−0.229624
$96$	0	0
$97$	11.5636	1.17411	0.587054	−	0.809548i	$-0.300288\pi$
$97$	11.5636	1.17411	0.587054	+	0.809548i	$0.300288\pi$
$98$	0	0
$99$	−1.89266	−0.190220
$100$	6.13076	0.613076
$101$	−7.56363	−0.752609	−0.376304	−	0.926496i	$-0.622805\pi$
$101$	−7.56363	−0.752609	−0.376304	+	0.926496i	$0.622805\pi$
$102$	0	0
$103$	−5.78181	−0.569699	−0.284849	−	0.958572i	$-0.591944\pi$
$103$	−5.78181	−0.569699	−0.284849	+	0.958572i	$0.591944\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.39091	0.134464	0.0672320	−	0.997737i	$-0.478583\pi$
$107$	1.39091	0.134464	0.0672320	+	0.997737i	$0.478583\pi$
$108$	2.00000	0.192450
$109$	−5.73985	−0.549778	−0.274889	−	0.961476i	$-0.588641\pi$
$109$	−5.73985	−0.549778	−0.274889	+	0.961476i	$0.588641\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	13.7434	1.29287	0.646433	−	0.762970i	$-0.276260\pi$
$113$	13.7434	1.29287	0.646433	+	0.762970i	$0.276260\pi$
$114$	0	0
$115$	0.847192	0.0790011
$116$	5.56363	0.516570
$117$	3.67447	0.339705
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.41784	−0.674349
$122$	0	0
$123$	1.00000	0.0901670
$124$	−3.00351	−0.269723
$125$	−11.2182	−1.00339
$126$	0	0
$127$	11.5671	1.02642	0.513209	−	0.858264i	$-0.328457\pi$
$127$	11.5671	1.02642	0.513209	+	0.858264i	$0.328457\pi$
$128$	0	0
$129$	3.71643	0.327214
$130$	0	0
$131$	−15.3909	−1.34471	−0.672355	−	0.740229i	$-0.734717\pi$
$131$	−15.3909	−1.34471	−0.672355	+	0.740229i	$0.734717\pi$
$132$	−3.78532	−0.329470
$133$	0	0
$134$	0	0
$135$	−1.39091	−0.119710
$136$	0	0
$137$	11.3489	0.969606	0.484803	−	0.874623i	$-0.338891\pi$
$137$	11.3489	0.969606	0.484803	+	0.874623i	$0.338891\pi$
$138$	0	0
$139$	16.9161	1.43480	0.717402	−	0.696660i	$-0.245331\pi$
$139$	16.9161	1.43480	0.717402	+	0.696660i	$0.245331\pi$
$140$	0	0
$141$	−11.2416	−0.946714
$142$	0	0
$143$	−6.95453	−0.581567
$144$	4.00000	0.333333
$145$	−3.86924	−0.321323
$146$	0	0
$147$	0	0
$148$	2.00000	0.164399
$149$	5.89266	0.482746	0.241373	−	0.970432i	$-0.422402\pi$
$149$	5.89266	0.482746	0.241373	+	0.970432i	$0.422402\pi$
$150$	0	0
$151$	18.4178	1.49882	0.749411	−	0.662105i	$-0.230337\pi$
$151$	18.4178	1.49882	0.749411	+	0.662105i	$0.230337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	2.08880	0.167776
$156$	7.34895	0.588387
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−7.34895	−0.582809
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.84719	0.692966	0.346483	−	0.938056i	$-0.387376\pi$
$163$	8.84719	0.692966	0.346483	+	0.938056i	$0.387376\pi$
$164$	2.00000	0.156174
$165$	2.63251	0.204941
$166$	0	0
$167$	−15.3489	−1.18774	−0.593869	−	0.804562i	$-0.702400\pi$
$167$	−15.3489	−1.18774	−0.593869	+	0.804562i	$0.702400\pi$
$168$	0	0
$169$	0.501754	0.0385965
$170$	0	0
$171$	−1.60909	−0.123050
$172$	7.43287	0.566751
$173$	−17.3489	−1.31902	−0.659508	−	0.751698i	$-0.729235\pi$
$173$	−17.3489	−1.31902	−0.659508	+	0.751698i	$0.729235\pi$
$174$	0	0
$175$	0	0
$176$	−7.57064	−0.570659
$177$	3.34895	0.251722
$178$	0	0
$179$	9.00351	0.672954	0.336477	−	0.941692i	$-0.390765\pi$
$179$	9.00351	0.672954	0.336477	+	0.941692i	$0.390765\pi$
$180$	−2.78181	−0.207344
$181$	−9.39442	−0.698281	−0.349141	−	0.937070i	$-0.613526\pi$
$181$	−9.39442	−0.698281	−0.349141	+	0.937070i	$0.613526\pi$
$182$	0	0
$183$	10.6325	0.785978
$184$	0	0
$185$	−1.39091	−0.102261
$186$	0	0
$187$	3.78532	0.276810
$188$	−22.4832	−1.63976
$189$	0	0
$190$	0	0
$191$	−12.0234	−0.869984	−0.434992	−	0.900434i	$-0.643249\pi$
$191$	−12.0234	−0.869984	−0.434992	+	0.900434i	$0.643249\pi$
$192$	8.00000	0.577350
$193$	−6.24161	−0.449281	−0.224640	−	0.974442i	$-0.572121\pi$
$193$	−6.24161	−0.449281	−0.224640	+	0.974442i	$0.572121\pi$
$194$	0	0
$195$	−5.11085	−0.365995
$196$	0	0
$197$	−17.2650	−1.23008	−0.615041	−	0.788495i	$-0.710861\pi$
$197$	−17.2650	−1.23008	−0.615041	+	0.788495i	$0.710861\pi$
$198$	0	0
$199$	−9.34895	−0.662729	−0.331365	−	0.943503i	$-0.607509\pi$
$199$	−9.34895	−0.662729	−0.331365	+	0.943503i	$0.607509\pi$
$200$	0	0
$201$	−5.67447	−0.400246
$202$	0	0
$203$	0	0
$204$	−4.00000	−0.280056
$205$	−1.39091	−0.0971451
$206$	0	0
$207$	0.609094	0.0423349
$208$	14.6979	1.01912
$209$	3.04547	0.210659
$210$	0	0
$211$	−15.5706	−1.07193	−0.535964	−	0.844241i	$-0.680052\pi$
$211$	−15.5706	−1.07193	−0.535964	+	0.844241i	$0.680052\pi$
$212$	−14.6979	−1.00946
$213$	9.34895	0.640579
$214$	0	0
$215$	−5.16921	−0.352537
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.71643	0.251133
$220$	5.26503	0.354968
$221$	−7.34895	−0.494344
$222$	0	0
$223$	−7.06889	−0.473368	−0.236684	−	0.971587i	$-0.576061\pi$
$223$	−7.06889	−0.473368	−0.236684	+	0.971587i	$0.576061\pi$
$224$	0	0
$225$	−3.06538	−0.204359
$226$	0	0
$227$	3.34895	0.222277	0.111139	−	0.993805i	$-0.464550\pi$
$227$	3.34895	0.222277	0.111139	+	0.993805i	$0.464550\pi$
$228$	−3.21819	−0.213130
$229$	−24.3070	−1.60625	−0.803125	−	0.595810i	$-0.796831\pi$
$229$	−24.3070	−1.60625	−0.803125	+	0.595810i	$0.796831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.10734	−0.531129	−0.265565	−	0.964093i	$-0.585558\pi$
$233$	−8.10734	−0.531129	−0.265565	+	0.964093i	$0.585558\pi$
$234$	0	0
$235$	15.6360	1.01998
$236$	6.69789	0.435996
$237$	−1.82377	−0.118467
$238$	0	0
$239$	6.99649	0.452565	0.226283	−	0.974062i	$-0.427343\pi$
$239$	6.99649	0.452565	0.226283	+	0.974062i	$0.427343\pi$
$240$	−5.56363	−0.359130
$241$	−6.71643	−0.432643	−0.216322	−	0.976322i	$-0.569406\pi$
$241$	−6.71643	−0.432643	−0.216322	+	0.976322i	$0.569406\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	21.2650	1.36135
$245$	0	0
$246$	0	0
$247$	−5.91257	−0.376208
$248$	0	0
$249$	−12.5252	−0.793751
$250$	0	0
$251$	−8.73985	−0.551655	−0.275827	−	0.961207i	$-0.588952\pi$
$251$	−8.73985	−0.551655	−0.275827	+	0.961207i	$0.588952\pi$
$252$	0	0
$253$	−1.15281	−0.0724764
$254$	0	0
$255$	2.78181	0.174204
$256$	16.0000	1.00000
$257$	−22.5906	−1.40916	−0.704580	−	0.709625i	$-0.748865\pi$
$257$	−22.5906	−1.40916	−0.704580	+	0.709625i	$0.748865\pi$
$258$	0	0
$259$	0	0
$260$	−10.2217	−0.633923
$261$	−2.78181	−0.172190
$262$	0	0
$263$	−14.4598	−0.891629	−0.445815	−	0.895125i	$-0.647086\pi$
$263$	−14.4598	−0.891629	−0.445815	+	0.895125i	$0.647086\pi$
$264$	0	0
$265$	10.2217	0.627914
$266$	0	0
$267$	7.24161	0.443179
$268$	−11.3489	−0.693247
$269$	3.91608	0.238768	0.119384	−	0.992848i	$-0.461908\pi$
$269$	3.91608	0.238768	0.119384	+	0.992848i	$0.461908\pi$
$270$	0	0
$271$	−28.0468	−1.70372	−0.851862	−	0.523766i	$-0.824527\pi$
$271$	−28.0468	−1.70372	−0.851862	+	0.523766i	$0.824527\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	0	0
$275$	5.80172	0.349857
$276$	1.21819	0.0733263
$277$	5.71643	0.343467	0.171734	−	0.985143i	$-0.445063\pi$
$277$	5.71643	0.343467	0.171734	+	0.985143i	$0.445063\pi$
$278$	0	0
$279$	1.50175	0.0899077
$280$	0	0
$281$	10.5906	0.631779	0.315890	−	0.948796i	$-0.397697\pi$
$281$	10.5906	0.631779	0.315890	+	0.948796i	$0.397697\pi$
$282$	0	0
$283$	−14.7853	−0.878896	−0.439448	−	0.898268i	$-0.644826\pi$
$283$	−14.7853	−0.878896	−0.439448	+	0.898268i	$0.644826\pi$
$284$	18.6979	1.10952
$285$	2.23810	0.132574
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−11.5636	−0.677872
$292$	7.43287	0.434976
$293$	−11.6710	−0.681825	−0.340913	−	0.940095i	$-0.610736\pi$
$293$	−11.6710	−0.681825	−0.340913	+	0.940095i	$0.610736\pi$
$294$	0	0
$295$	−4.65807	−0.271203
$296$	0	0
$297$	1.89266	0.109823
$298$	0	0
$299$	2.23810	0.129433
$300$	−6.13076	−0.353960
$301$	0	0
$302$	0	0
$303$	7.56363	0.434519
$304$	−6.43637	−0.369151
$305$	−14.7888	−0.846806
$306$	0	0
$307$	−14.5636	−0.831190	−0.415595	−	0.909550i	$-0.636427\pi$
$307$	−14.5636	−0.831190	−0.415595	+	0.909550i	$0.636427\pi$
$308$	0	0
$309$	5.78181	0.328916
$310$	0	0
$311$	20.8123	1.18015	0.590077	−	0.807347i	$-0.299097\pi$
$311$	20.8123	1.18015	0.590077	+	0.807347i	$0.299097\pi$
$312$	0	0
$313$	12.3724	0.699328	0.349664	−	0.936875i	$-0.386296\pi$
$313$	12.3724	0.699328	0.349664	+	0.936875i	$0.386296\pi$
$314$	0	0
$315$	0	0
$316$	−3.64754	−0.205190
$317$	9.80874	0.550914	0.275457	−	0.961313i	$-0.411171\pi$
$317$	9.80874	0.550914	0.275457	+	0.961313i	$0.411171\pi$
$318$	0	0
$319$	5.26503	0.294785
$320$	−11.1273	−0.622032
$321$	−1.39091	−0.0776328
$322$	0	0
$323$	3.21819	0.179065
$324$	−2.00000	−0.111111
$325$	−11.2637	−0.624795
$326$	0	0
$327$	5.73985	0.317415
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.52517	0.0838312	0.0419156	−	0.999121i	$-0.486654\pi$
$331$	1.52517	0.0838312	0.0419156	+	0.999121i	$0.486654\pi$
$332$	−25.0503	−1.37482
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	7.89266	0.431222
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	0	0
$339$	−13.7434	−0.746437
$340$	5.56363	0.301730
$341$	−2.84231	−0.153920
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.847192	−0.0456113
$346$	0	0
$347$	0.0839192	0.00450502	0.00225251	−	0.999997i	$-0.499283\pi$
$347$	0.0839192	0.00450502	0.00225251	+	0.999997i	$0.499283\pi$
$348$	−5.56363	−0.298242
$349$	6.93111	0.371014	0.185507	−	0.982643i	$-0.440607\pi$
$349$	6.93111	0.371014	0.185507	+	0.982643i	$0.440607\pi$
$350$	0	0
$351$	−3.67447	−0.196129
$352$	0	0
$353$	−12.7818	−0.680307	−0.340154	−	0.940370i	$-0.610479\pi$
$353$	−12.7818	−0.680307	−0.340154	+	0.940370i	$0.610479\pi$
$354$	0	0
$355$	−13.0035	−0.690155
$356$	14.4832	0.767609
$357$	0	0
$358$	0	0
$359$	36.7399	1.93906	0.969528	−	0.244982i	$-0.0787820\pi$
$359$	36.7399	1.93906	0.969528	+	0.244982i	$0.0787820\pi$
$360$	0	0
$361$	−16.4108	−0.863727
$362$	0	0
$363$	7.41784	0.389335
$364$	0	0
$365$	−5.16921	−0.270569
$366$	0	0
$367$	−9.84719	−0.514019	−0.257010	−	0.966409i	$-0.582737\pi$
$367$	−9.84719	−0.514019	−0.257010	+	0.966409i	$0.582737\pi$
$368$	2.43637	0.127005
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	0	0
$372$	3.00351	0.155725
$373$	12.5636	0.650520	0.325260	−	0.945625i	$-0.394548\pi$
$373$	12.5636	0.650520	0.325260	+	0.945625i	$0.394548\pi$
$374$	0	0
$375$	11.2182	0.579305
$376$	0	0
$377$	−10.2217	−0.526444
$378$	0	0
$379$	−32.1157	−1.64967	−0.824837	−	0.565370i	$-0.808733\pi$
$379$	−32.1157	−1.64967	−0.824837	+	0.565370i	$0.808733\pi$
$380$	4.47620	0.229624
$381$	−11.5671	−0.592602
$382$	0	0
$383$	−31.4797	−1.60854	−0.804269	−	0.594266i	$-0.797443\pi$
$383$	−31.4797	−1.60854	−0.804269	+	0.594266i	$0.797443\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.71643	−0.188917
$388$	−23.1273	−1.17411
$389$	−28.3105	−1.43540	−0.717700	−	0.696353i	$-0.754805\pi$
$389$	−28.3105	−1.43540	−0.717700	+	0.696353i	$0.754805\pi$
$390$	0	0
$391$	−1.21819	−0.0616064
$392$	0	0
$393$	15.3909	0.776369
$394$	0	0
$395$	2.53670	0.127635
$396$	3.78532	0.190220
$397$	22.4633	1.12740	0.563700	−	0.825979i	$-0.309377\pi$
$397$	22.4633	1.12740	0.563700	+	0.825979i	$0.309377\pi$
$398$	0	0
$399$	0	0
$400$	−12.2615	−0.613076
$401$	−25.7014	−1.28347	−0.641733	−	0.766928i	$-0.721784\pi$
$401$	−25.7014	−1.28347	−0.641733	+	0.766928i	$0.721784\pi$
$402$	0	0
$403$	5.51816	0.274879
$404$	15.1273	0.752609
$405$	1.39091	0.0691147
$406$	0	0
$407$	1.89266	0.0938157
$408$	0	0
$409$	−11.4178	−0.564576	−0.282288	−	0.959330i	$-0.591093\pi$
$409$	−11.4178	−0.564576	−0.282288	+	0.959330i	$0.591093\pi$
$410$	0	0
$411$	−11.3489	−0.559802
$412$	11.5636	0.569699
$413$	0	0
$414$	0	0
$415$	17.4213	0.855180
$416$	0	0
$417$	−16.9161	−0.828384
$418$	0	0
$419$	6.21468	0.303607	0.151803	−	0.988411i	$-0.451492\pi$
$419$	6.21468	0.303607	0.151803	+	0.988411i	$0.451492\pi$
$420$	0	0
$421$	−3.66746	−0.178741	−0.0893704	−	0.995998i	$-0.528485\pi$
$421$	−3.66746	−0.178741	−0.0893704	+	0.995998i	$0.528485\pi$
$422$	0	0
$423$	11.2416	0.546586
$424$	0	0
$425$	6.13076	0.297386
$426$	0	0
$427$	0	0
$428$	−2.78181	−0.134464
$429$	6.95453	0.335768
$430$	0	0
$431$	33.6524	1.62098	0.810490	−	0.585752i	$-0.199201\pi$
$431$	33.6524	1.62098	0.810490	+	0.585752i	$0.199201\pi$
$432$	−4.00000	−0.192450
$433$	−6.06187	−0.291315	−0.145657	−	0.989335i	$-0.546530\pi$
$433$	−6.06187	−0.291315	−0.145657	+	0.989335i	$0.546530\pi$
$434$	0	0
$435$	3.86924	0.185516
$436$	11.4797	0.549778
$437$	−0.980089	−0.0468840
$438$	0	0
$439$	3.98146	0.190025	0.0950124	−	0.995476i	$-0.469711\pi$
$439$	3.98146	0.190025	0.0950124	+	0.995476i	$0.469711\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.64754	0.0782772	0.0391386	−	0.999234i	$-0.487539\pi$
$443$	1.64754	0.0782772	0.0391386	+	0.999234i	$0.487539\pi$
$444$	−2.00000	−0.0949158
$445$	−10.0724	−0.477477
$446$	0	0
$447$	−5.89266	−0.278713
$448$	0	0
$449$	27.8741	1.31546	0.657731	−	0.753253i	$-0.271517\pi$
$449$	27.8741	1.31546	0.657731	+	0.753253i	$0.271517\pi$
$450$	0	0
$451$	1.89266	0.0891219
$452$	−27.4867	−1.29287
$453$	−18.4178	−0.865345
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.9615	0.653094	0.326547	−	0.945181i	$-0.394115\pi$
$457$	13.9615	0.653094	0.326547	+	0.945181i	$0.394115\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	−1.69438	−0.0790011
$461$	−10.3944	−0.484116	−0.242058	−	0.970262i	$-0.577822\pi$
$461$	−10.3944	−0.484116	−0.242058	+	0.970262i	$0.577822\pi$
$462$	0	0
$463$	−17.1073	−0.795045	−0.397523	−	0.917592i	$-0.630130\pi$
$463$	−17.1073	−0.795045	−0.397523	+	0.917592i	$0.630130\pi$
$464$	−11.1273	−0.516570
$465$	−2.08880	−0.0968658
$466$	0	0
$467$	−0.739853	−0.0342363	−0.0171182	−	0.999853i	$-0.505449\pi$
$467$	−0.739853	−0.0342363	−0.0171182	+	0.999853i	$0.505449\pi$
$468$	−7.34895	−0.339705
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	7.03395	0.323421
$474$	0	0
$475$	4.93248	0.226318
$476$	0	0
$477$	7.34895	0.336485
$478$	0	0
$479$	−8.54371	−0.390372	−0.195186	−	0.980766i	$-0.562531\pi$
$479$	−8.54371	−0.390372	−0.195186	+	0.980766i	$0.562531\pi$
$480$	0	0
$481$	−3.67447	−0.167542
$482$	0	0
$483$	0	0
$484$	14.8357	0.674349
$485$	16.0839	0.730333
$486$	0	0
$487$	30.8322	1.39714	0.698569	−	0.715542i	$-0.253820\pi$
$487$	30.8322	1.39714	0.698569	+	0.715542i	$0.253820\pi$
$488$	0	0
$489$	−8.84719	−0.400084
$490$	0	0
$491$	18.5671	0.837923	0.418962	−	0.908004i	$-0.362394\pi$
$491$	18.5671	0.837923	0.418962	+	0.908004i	$0.362394\pi$
$492$	−2.00000	−0.0901670
$493$	5.56363	0.250573
$494$	0	0
$495$	−2.63251	−0.118323
$496$	6.00702	0.269723
$497$	0	0
$498$	0	0
$499$	2.52167	0.112885	0.0564426	−	0.998406i	$-0.482024\pi$
$499$	2.52167	0.112885	0.0564426	+	0.998406i	$0.482024\pi$
$500$	22.4364	1.00339
$501$	15.3489	0.685740
$502$	0	0
$503$	15.5870	0.694992	0.347496	−	0.937681i	$-0.387032\pi$
$503$	15.5870	0.694992	0.347496	+	0.937681i	$0.387032\pi$
$504$	0	0
$505$	−10.5203	−0.468147
$506$	0	0
$507$	−0.501754	−0.0222837
$508$	−23.1343	−1.02642
$509$	−9.70140	−0.430007	−0.215004	−	0.976613i	$-0.568976\pi$
$509$	−9.70140	−0.430007	−0.215004	+	0.976613i	$0.568976\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.60909	0.0710432
$514$	0	0
$515$	−8.04196	−0.354371
$516$	−7.43287	−0.327214
$517$	−21.2765	−0.935742
$518$	0	0
$519$	17.3489	0.761534
$520$	0	0
$521$	−34.3689	−1.50573	−0.752864	−	0.658177i	$-0.771328\pi$
$521$	−34.3689	−1.50573	−0.752864	+	0.658177i	$0.771328\pi$
$522$	0	0
$523$	−7.49474	−0.327722	−0.163861	−	0.986483i	$-0.552395\pi$
$523$	−7.49474	−0.327722	−0.163861	+	0.986483i	$0.552395\pi$
$524$	30.7818	1.34471
$525$	0	0
$526$	0	0
$527$	−3.00351	−0.130835
$528$	7.57064	0.329470
$529$	−22.6290	−0.983870
$530$	0	0
$531$	−3.34895	−0.145332
$532$	0	0
$533$	−3.67447	−0.159159
$534$	0	0
$535$	1.93462	0.0836409
$536$	0	0
$537$	−9.00351	−0.388530
$538$	0	0
$539$	0	0
$540$	2.78181	0.119710
$541$	9.71643	0.417742	0.208871	−	0.977943i	$-0.433021\pi$
$541$	9.71643	0.417742	0.208871	+	0.977943i	$0.433021\pi$
$542$	0	0
$543$	9.39442	0.403153
$544$	0	0
$545$	−7.98360	−0.341980
$546$	0	0
$547$	−38.7447	−1.65661	−0.828303	−	0.560281i	$-0.810693\pi$
$547$	−38.7447	−1.65661	−0.828303	+	0.560281i	$0.810693\pi$
$548$	−22.6979	−0.969606
$549$	−10.6325	−0.453785
$550$	0	0
$551$	4.47620	0.190692
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.39091	0.0590407
$556$	−33.8322	−1.43480
$557$	15.1577	0.642252	0.321126	−	0.947037i	$-0.395939\pi$
$557$	15.1577	0.642252	0.321126	+	0.947037i	$0.395939\pi$
$558$	0	0
$559$	−13.6559	−0.577584
$560$	0	0
$561$	−3.78532	−0.159816
$562$	0	0
$563$	9.11085	0.383976	0.191988	−	0.981397i	$-0.438506\pi$
$563$	9.11085	0.383976	0.191988	+	0.981397i	$0.438506\pi$
$564$	22.4832	0.946714
$565$	19.1157	0.804205
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.6559	−1.03363	−0.516815	−	0.856097i	$-0.672883\pi$
$569$	−24.6559	−1.03363	−0.516815	+	0.856097i	$0.672883\pi$
$570$	0	0
$571$	−43.7283	−1.82997	−0.914987	−	0.403484i	$-0.867799\pi$
$571$	−43.7283	−1.82997	−0.914987	+	0.403484i	$0.867799\pi$
$572$	13.9091	0.581567
$573$	12.0234	0.502286
$574$	0	0
$575$	−1.86710	−0.0778636
$576$	−8.00000	−0.333333
$577$	−22.9651	−0.956048	−0.478024	−	0.878347i	$-0.658647\pi$
$577$	−22.9651	−0.956048	−0.478024	+	0.878347i	$0.658647\pi$
$578$	0	0
$579$	6.24161	0.259392
$580$	7.73848	0.321323
$581$	0	0
$582$	0	0
$583$	−13.9091	−0.576055
$584$	0	0
$585$	5.11085	0.211308
$586$	0	0
$587$	−31.0199	−1.28033	−0.640164	−	0.768238i	$-0.721134\pi$
$587$	−31.0199	−1.28033	−0.640164	+	0.768238i	$0.721134\pi$
$588$	0	0
$589$	−2.41646	−0.0995686
$590$	0	0
$591$	17.2650	0.710188
$592$	−4.00000	−0.164399
$593$	−18.3759	−0.754607	−0.377303	−	0.926090i	$-0.623149\pi$
$593$	−18.3759	−0.754607	−0.377303	+	0.926090i	$0.623149\pi$
$594$	0	0
$595$	0	0
$596$	−11.7853	−0.482746
$597$	9.34895	0.382627
$598$	0	0
$599$	28.5252	1.16551	0.582754	−	0.812649i	$-0.301975\pi$
$599$	28.5252	1.16551	0.582754	+	0.812649i	$0.301975\pi$
$600$	0	0
$601$	5.32904	0.217376	0.108688	−	0.994076i	$-0.465335\pi$
$601$	5.32904	0.217376	0.108688	+	0.994076i	$0.465335\pi$
$602$	0	0
$603$	5.67447	0.231082
$604$	−36.8357	−1.49882
$605$	−10.3175	−0.419467
$606$	0	0
$607$	19.7888	0.803204	0.401602	−	0.915814i	$-0.368454\pi$
$607$	19.7888	0.803204	0.401602	+	0.915814i	$0.368454\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	41.3070	1.67110
$612$	4.00000	0.161690
$613$	−25.4867	−1.02940	−0.514700	−	0.857371i	$-0.672097\pi$
$613$	−25.4867	−1.02940	−0.514700	+	0.857371i	$0.672097\pi$
$614$	0	0
$615$	1.39091	0.0560868
$616$	0	0
$617$	3.51679	0.141580	0.0707902	−	0.997491i	$-0.477448\pi$
$617$	3.51679	0.141580	0.0707902	+	0.997491i	$0.477448\pi$
$618$	0	0
$619$	28.7668	1.15623	0.578117	−	0.815954i	$-0.303788\pi$
$619$	28.7668	1.15623	0.578117	+	0.815954i	$0.303788\pi$
$620$	−4.17760	−0.167776
$621$	−0.609094	−0.0244421
$622$	0	0
$623$	0	0
$624$	−14.6979	−0.588387
$625$	−0.276549	−0.0110620
$626$	0	0
$627$	−3.04547	−0.121624
$628$	20.0000	0.798087
$629$	2.00000	0.0797452
$630$	0	0
$631$	−10.6511	−0.424012	−0.212006	−	0.977268i	$-0.568000\pi$
$631$	−10.6511	−0.424012	−0.212006	+	0.977268i	$0.568000\pi$
$632$	0	0
$633$	15.5706	0.618877
$634$	0	0
$635$	16.0888	0.638465
$636$	14.6979	0.582809
$637$	0	0
$638$	0	0
$639$	−9.34895	−0.369839
$640$	0	0
$641$	−32.0234	−1.26485	−0.632425	−	0.774622i	$-0.717940\pi$
$641$	−32.0234	−1.26485	−0.632425	+	0.774622i	$0.717940\pi$
$642$	0	0
$643$	−38.6409	−1.52385	−0.761924	−	0.647666i	$-0.775745\pi$
$643$	−38.6409	−1.52385	−0.761924	+	0.647666i	$0.775745\pi$
$644$	0	0
$645$	5.16921	0.203537
$646$	0	0
$647$	19.7014	0.774542	0.387271	−	0.921966i	$-0.373418\pi$
$647$	19.7014	0.774542	0.387271	+	0.921966i	$0.373418\pi$
$648$	0	0
$649$	6.33842	0.248805
$650$	0	0
$651$	0	0
$652$	−17.6944	−0.692966
$653$	9.72482	0.380562	0.190281	−	0.981730i	$-0.439060\pi$
$653$	9.72482	0.380562	0.190281	+	0.981730i	$0.439060\pi$
$654$	0	0
$655$	−21.4073	−0.836453
$656$	−4.00000	−0.156174
$657$	−3.71643	−0.144992
$658$	0	0
$659$	−27.1273	−1.05673	−0.528364	−	0.849018i	$-0.677194\pi$
$659$	−27.1273	−1.05673	−0.528364	+	0.849018i	$0.677194\pi$
$660$	−5.26503	−0.204941
$661$	−45.5486	−1.77164	−0.885818	−	0.464034i	$-0.846402\pi$
$661$	−45.5486	−1.77164	−0.885818	+	0.464034i	$0.846402\pi$
$662$	0	0
$663$	7.34895	0.285409
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.69438	−0.0656068
$668$	30.6979	1.18774
$669$	7.06889	0.273299
$670$	0	0
$671$	20.1237	0.776868
$672$	0	0
$673$	−13.6675	−0.526842	−0.263421	−	0.964681i	$-0.584851\pi$
$673$	−13.6675	−0.526842	−0.263421	+	0.964681i	$0.584851\pi$
$674$	0	0
$675$	3.06538	0.117987
$676$	−1.00351	−0.0385965
$677$	28.9615	1.11308	0.556541	−	0.830820i	$-0.312128\pi$
$677$	28.9615	1.11308	0.556541	+	0.830820i	$0.312128\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.34895	−0.128332
$682$	0	0
$683$	−0.590554	−0.0225969	−0.0112985	−	0.999936i	$-0.503596\pi$
$683$	−0.590554	−0.0225969	−0.0112985	+	0.999936i	$0.503596\pi$
$684$	3.21819	0.123050
$685$	15.7853	0.603126
$686$	0	0
$687$	24.3070	0.927369
$688$	−14.8657	−0.566751
$689$	27.0035	1.02875
$690$	0	0
$691$	26.1507	0.994818	0.497409	−	0.867516i	$-0.334285\pi$
$691$	26.1507	0.994818	0.497409	+	0.867516i	$0.334285\pi$
$692$	34.6979	1.31902
$693$	0	0
$694$	0	0
$695$	23.5287	0.892494
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	8.10734	0.306648
$700$	0	0
$701$	10.1797	0.384483	0.192242	−	0.981348i	$-0.438424\pi$
$701$	10.1797	0.384483	0.192242	+	0.981348i	$0.438424\pi$
$702$	0	0
$703$	1.60909	0.0606881
$704$	15.1413	0.570659
$705$	−15.6360	−0.588887
$706$	0	0
$707$	0	0
$708$	−6.69789	−0.251722
$709$	−2.42936	−0.0912364	−0.0456182	−	0.998959i	$-0.514526\pi$
$709$	−2.42936	−0.0912364	−0.0456182	+	0.998959i	$0.514526\pi$
$710$	0	0
$711$	1.82377	0.0683968
$712$	0	0
$713$	0.914709	0.0342561
$714$	0	0
$715$	−9.67310	−0.361753
$716$	−18.0070	−0.672954
$717$	−6.99649	−0.261289
$718$	0	0
$719$	−49.6409	−1.85129	−0.925647	−	0.378389i	$-0.876478\pi$
$719$	−49.6409	−1.85129	−0.925647	+	0.378389i	$0.876478\pi$
$720$	5.56363	0.207344
$721$	0	0
$722$	0	0
$723$	6.71643	0.249787
$724$	18.7888	0.698281
$725$	8.52731	0.316696
$726$	0	0
$727$	−13.8962	−0.515380	−0.257690	−	0.966228i	$-0.582961\pi$
$727$	−13.8962	−0.515380	−0.257690	+	0.966228i	$0.582961\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.43287	0.274914
$732$	−21.2650	−0.785978
$733$	−13.6140	−0.502844	−0.251422	−	0.967878i	$-0.580898\pi$
$733$	−13.6140	−0.502844	−0.251422	+	0.967878i	$0.580898\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.7399	−0.395608
$738$	0	0
$739$	−25.2801	−0.929942	−0.464971	−	0.885326i	$-0.653935\pi$
$739$	−25.2801	−0.929942	−0.464971	+	0.885326i	$0.653935\pi$
$740$	2.78181	0.102261
$741$	5.91257	0.217204
$742$	0	0
$743$	12.2147	0.448113	0.224057	−	0.974576i	$-0.428070\pi$
$743$	12.2147	0.448113	0.224057	+	0.974576i	$0.428070\pi$
$744$	0	0
$745$	8.19614	0.300283
$746$	0	0
$747$	12.5252	0.458272
$748$	−7.57064	−0.276810
$749$	0	0
$750$	0	0
$751$	−33.5906	−1.22574	−0.612868	−	0.790185i	$-0.709984\pi$
$751$	−33.5906	−1.22574	−0.612868	+	0.790185i	$0.709984\pi$
$752$	44.9664	1.63976
$753$	8.73985	0.318498
$754$	0	0
$755$	25.6175	0.932316
$756$	0	0
$757$	−33.3374	−1.21167	−0.605835	−	0.795591i	$-0.707161\pi$
$757$	−33.3374	−1.21167	−0.605835	+	0.795591i	$0.707161\pi$
$758$	0	0
$759$	1.15281	0.0418443
$760$	0	0
$761$	−20.1308	−0.729739	−0.364870	−	0.931059i	$-0.618886\pi$
$761$	−20.1308	−0.729739	−0.364870	+	0.931059i	$0.618886\pi$
$762$	0	0
$763$	0	0
$764$	24.0468	0.869984
$765$	−2.78181	−0.100577
$766$	0	0
$767$	−12.3056	−0.444330
$768$	−16.0000	−0.577350
$769$	1.49474	0.0539016	0.0269508	−	0.999637i	$-0.491420\pi$
$769$	1.49474	0.0539016	0.0269508	+	0.999637i	$0.491420\pi$
$770$	0	0
$771$	22.5906	0.813579
$772$	12.4832	0.449281
$773$	27.0738	0.973776	0.486888	−	0.873464i	$-0.338132\pi$
$773$	27.0738	0.973776	0.486888	+	0.873464i	$0.338132\pi$
$774$	0	0
$775$	−4.60345	−0.165361
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.60909	0.0576518
$780$	10.2217	0.365995
$781$	17.6944	0.633155
$782$	0	0
$783$	2.78181	0.0994139
$784$	0	0
$785$	−13.9091	−0.496436
$786$	0	0
$787$	53.3374	1.90127	0.950637	−	0.310305i	$-0.100431\pi$
$787$	53.3374	1.90127	0.950637	+	0.310305i	$0.100431\pi$
$788$	34.5301	1.23008
$789$	14.4598	0.514782
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−39.0689	−1.38738
$794$	0	0
$795$	−10.2217	−0.362526
$796$	18.6979	0.662729
$797$	30.9615	1.09671	0.548357	−	0.836244i	$-0.315253\pi$
$797$	30.9615	1.09671	0.548357	+	0.836244i	$0.315253\pi$
$798$	0	0
$799$	−22.4832	−0.795399
$800$	0	0
$801$	−7.24161	−0.255870
$802$	0	0
$803$	7.03395	0.248223
$804$	11.3489	0.400246
$805$	0	0
$806$	0	0
$807$	−3.91608	−0.137853
$808$	0	0
$809$	25.6780	0.902790	0.451395	−	0.892324i	$-0.350927\pi$
$809$	25.6780	0.902790	0.451395	+	0.892324i	$0.350927\pi$
$810$	0	0
$811$	−8.41784	−0.295590	−0.147795	−	0.989018i	$-0.547218\pi$
$811$	−8.41784	−0.295590	−0.147795	+	0.989018i	$0.547218\pi$
$812$	0	0
$813$	28.0468	0.983646
$814$	0	0
$815$	12.3056	0.431047
$816$	8.00000	0.280056
$817$	5.98009	0.209217
$818$	0	0
$819$	0	0
$820$	2.78181	0.0971451
$821$	25.5706	0.892422	0.446211	−	0.894928i	$-0.352773\pi$
$821$	25.5706	0.892422	0.446211	+	0.894928i	$0.352773\pi$
$822$	0	0
$823$	27.0433	0.942671	0.471336	−	0.881954i	$-0.343772\pi$
$823$	27.0433	0.942671	0.471336	+	0.881954i	$0.343772\pi$
$824$	0	0
$825$	−5.80172	−0.201990
$826$	0	0
$827$	−7.57064	−0.263257	−0.131629	−	0.991299i	$-0.542021\pi$
$827$	−7.57064	−0.263257	−0.131629	+	0.991299i	$0.542021\pi$
$828$	−1.21819	−0.0423349
$829$	0.439884	0.0152778	0.00763890	−	0.999971i	$-0.497568\pi$
$829$	0.439884	0.0152778	0.00763890	+	0.999971i	$0.497568\pi$
$830$	0	0
$831$	−5.71643	−0.198301
$832$	−29.3958	−1.01912
$833$	0	0
$834$	0	0
$835$	−21.3489	−0.738811
$836$	−6.09094	−0.210659
$837$	−1.50175	−0.0519082
$838$	0	0
$839$	5.11787	0.176688	0.0883442	−	0.996090i	$-0.471842\pi$
$839$	5.11787	0.176688	0.0883442	+	0.996090i	$0.471842\pi$
$840$	0	0
$841$	−21.2615	−0.733156
$842$	0	0
$843$	−10.5906	−0.364758
$844$	31.1413	1.07193
$845$	0.697893	0.0240083
$846$	0	0
$847$	0	0
$848$	29.3958	1.00946
$849$	14.7853	0.507431
$850$	0	0
$851$	−0.609094	−0.0208795
$852$	−18.6979	−0.640579
$853$	−5.93111	−0.203077	−0.101539	−	0.994832i	$-0.532377\pi$
$853$	−5.93111	−0.203077	−0.101539	+	0.994832i	$0.532377\pi$
$854$	0	0
$855$	−2.23810	−0.0765414
$856$	0	0
$857$	−41.9699	−1.43367	−0.716833	−	0.697245i	$-0.754409\pi$
$857$	−41.9699	−1.43367	−0.716833	+	0.697245i	$0.754409\pi$
$858$	0	0
$859$	32.8472	1.12073	0.560366	−	0.828245i	$-0.310661\pi$
$859$	32.8472	1.12073	0.560366	+	0.828245i	$0.310661\pi$
$860$	10.3384	0.352537
$861$	0	0
$862$	0	0
$863$	37.2161	1.26685	0.633425	−	0.773804i	$-0.281649\pi$
$863$	37.2161	1.26685	0.633425	+	0.773804i	$0.281649\pi$
$864$	0	0
$865$	−24.1308	−0.820470
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−3.45178	−0.117094
$870$	0	0
$871$	20.8507	0.706499
$872$	0	0
$873$	11.5636	0.391369
$874$	0	0
$875$	0	0
$876$	−7.43287	−0.251133
$877$	39.7668	1.34283	0.671414	−	0.741082i	$-0.265687\pi$
$877$	39.7668	1.34283	0.671414	+	0.741082i	$0.265687\pi$
$878$	0	0
$879$	11.6710	0.393652
$880$	−10.5301	−0.354968
$881$	19.9231	0.671226	0.335613	−	0.942000i	$-0.391057\pi$
$881$	19.9231	0.671226	0.335613	+	0.942000i	$0.391057\pi$
$882$	0	0
$883$	3.51816	0.118395	0.0591977	−	0.998246i	$-0.481146\pi$
$883$	3.51816	0.118395	0.0591977	+	0.998246i	$0.481146\pi$
$884$	14.6979	0.494344
$885$	4.65807	0.156579
$886$	0	0
$887$	27.4867	0.922914	0.461457	−	0.887163i	$-0.347327\pi$
$887$	27.4867	0.922914	0.461457	+	0.887163i	$0.347327\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.89266	−0.0634065
$892$	14.1378	0.473368
$893$	−18.0888	−0.605319
$894$	0	0
$895$	12.5230	0.418599
$896$	0	0
$897$	−2.23810	−0.0747279
$898$	0	0
$899$	−4.17760	−0.139331
$900$	6.13076	0.204359
$901$	−14.6979	−0.489658
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.0668	−0.434354
$906$	0	0
$907$	22.9744	0.762854	0.381427	−	0.924399i	$-0.375433\pi$
$907$	22.9744	0.762854	0.381427	+	0.924399i	$0.375433\pi$
$908$	−6.69789	−0.222277
$909$	−7.56363	−0.250870
$910$	0	0
$911$	−20.0468	−0.664181	−0.332091	−	0.943247i	$-0.607754\pi$
$911$	−20.0468	−0.664181	−0.332091	+	0.943247i	$0.607754\pi$
$912$	6.43637	0.213130
$913$	−23.7059	−0.784551
$914$	0	0
$915$	14.7888	0.488904
$916$	48.6140	1.60625
$917$	0	0
$918$	0	0
$919$	−40.5287	−1.33692	−0.668459	−	0.743749i	$-0.733046\pi$
$919$	−40.5287	−1.33692	−0.668459	+	0.743749i	$0.733046\pi$
$920$	0	0
$921$	14.5636	0.479888
$922$	0	0
$923$	−34.3525	−1.13072
$924$	0	0
$925$	3.06538	0.100789
$926$	0	0
$927$	−5.78181	−0.189900
$928$	0	0
$929$	40.0468	1.31389	0.656947	−	0.753937i	$-0.271847\pi$
$929$	40.0468	1.31389	0.656947	+	0.753937i	$0.271847\pi$
$930$	0	0
$931$	0	0
$932$	16.2147	0.531129
$933$	−20.8123	−0.681362
$934$	0	0
$935$	5.26503	0.172185
$936$	0	0
$937$	−5.25664	−0.171727	−0.0858634	−	0.996307i	$-0.527365\pi$
$937$	−5.25664	−0.171727	−0.0858634	+	0.996307i	$0.527365\pi$
$938$	0	0
$939$	−12.3724	−0.403757
$940$	−31.2720	−1.01998
$941$	27.6105	0.900075	0.450038	−	0.893010i	$-0.351411\pi$
$941$	27.6105	0.900075	0.450038	+	0.893010i	$0.351411\pi$
$942$	0	0
$943$	−0.609094	−0.0198348
$944$	−13.3958	−0.435996
$945$	0	0
$946$	0	0
$947$	41.4377	1.34655	0.673273	−	0.739394i	$-0.264888\pi$
$947$	41.4377	1.34655	0.673273	+	0.739394i	$0.264888\pi$
$948$	3.64754	0.118467
$949$	−13.6559	−0.443290
$950$	0	0
$951$	−9.80874	−0.318070
$952$	0	0
$953$	54.4902	1.76511	0.882556	−	0.470208i	$-0.155821\pi$
$953$	54.4902	1.76511	0.882556	+	0.470208i	$0.155821\pi$
$954$	0	0
$955$	−16.7235	−0.541158
$956$	−13.9930	−0.452565
$957$	−5.26503	−0.170194
$958$	0	0
$959$	0	0
$960$	11.1273	0.359130
$961$	−28.7447	−0.927249
$962$	0	0
$963$	1.39091	0.0448213
$964$	13.4329	0.432643
$965$	−8.68149	−0.279467
$966$	0	0
$967$	52.8671	1.70009	0.850046	−	0.526709i	$-0.176574\pi$
$967$	52.8671	1.70009	0.850046	+	0.526709i	$0.176574\pi$
$968$	0	0
$969$	−3.21819	−0.103383
$970$	0	0
$971$	17.3489	0.556754	0.278377	−	0.960472i	$-0.410204\pi$
$971$	17.3489	0.556754	0.278377	+	0.960472i	$0.410204\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	0	0
$975$	11.2637	0.360726
$976$	−42.5301	−1.36135
$977$	−3.14129	−0.100499	−0.0502493	−	0.998737i	$-0.516002\pi$
$977$	−3.14129	−0.100499	−0.0502493	+	0.998737i	$0.516002\pi$
$978$	0	0
$979$	13.7059	0.438043
$980$	0	0
$981$	−5.73985	−0.183259
$982$	0	0
$983$	56.4413	1.80020	0.900098	−	0.435687i	$-0.143495\pi$
$983$	56.4413	1.80020	0.900098	+	0.435687i	$0.143495\pi$
$984$	0	0
$985$	−24.0140	−0.765151
$986$	0	0
$987$	0	0
$988$	11.8251	0.376208
$989$	−2.26366	−0.0719801
$990$	0	0
$991$	−28.0269	−0.890305	−0.445152	−	0.895455i	$-0.646850\pi$
$991$	−28.0269	−0.890305	−0.445152	+	0.895455i	$0.646850\pi$
$992$	0	0
$993$	−1.52517	−0.0484000
$994$	0	0
$995$	−13.0035	−0.412239
$996$	25.0503	0.793751
$997$	3.25664	0.103139	0.0515694	−	0.998669i	$-0.483578\pi$
$997$	3.25664	0.103139	0.0515694	+	0.998669i	$0.483578\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.p.1.2		3
7.2	even	3		861.2.i.c.739.2	yes	6
7.4	even	3		861.2.i.c.247.2	✓	6
7.6	odd	2		6027.2.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.i.c.247.2	✓	6	7.4	even	3
861.2.i.c.739.2	yes	6	7.2	even	3
6027.2.a.p.1.2		3	1.1	even	1	trivial
6027.2.a.q.1.2		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{4} +1.39091 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{4} +1.39091 q^{5} +1.00000 q^{9} -1.89266 q^{11} +2.00000 q^{12} +3.67447 q^{13} -1.39091 q^{15} +4.00000 q^{16} -2.00000 q^{17} -1.60909 q^{19} -2.78181 q^{20} +0.609094 q^{23} -3.06538 q^{25} -1.00000 q^{27} -2.78181 q^{29} +1.50175 q^{31} +1.89266 q^{33} -2.00000 q^{36} -1.00000 q^{37} -3.67447 q^{39} -1.00000 q^{41} -3.71643 q^{43} +3.78532 q^{44} +1.39091 q^{45} +11.2416 q^{47} -4.00000 q^{48} +2.00000 q^{51} -7.34895 q^{52} +7.34895 q^{53} -2.63251 q^{55} +1.60909 q^{57} -3.34895 q^{59} +2.78181 q^{60} -10.6325 q^{61} -8.00000 q^{64} +5.11085 q^{65} +5.67447 q^{67} +4.00000 q^{68} -0.609094 q^{69} -9.34895 q^{71} -3.71643 q^{73} +3.06538 q^{75} +3.21819 q^{76} +1.82377 q^{79} +5.56363 q^{80} +1.00000 q^{81} +12.5252 q^{83} -2.78181 q^{85} +2.78181 q^{87} -7.24161 q^{89} -1.21819 q^{92} -1.50175 q^{93} -2.23810 q^{95} +11.5636 q^{97} -1.89266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 6 q^{4} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 6 * q^4 + 3 * q^9	\( 3 q - 3 q^{3} - 6 q^{4} + 3 q^{9} + 6 q^{12} - 3 q^{13} + 12 q^{16} - 6 q^{17} - 9 q^{19} + 6 q^{23} + 9 q^{25} - 3 q^{27} + 3 q^{31} - 6 q^{36} - 3 q^{37} + 3 q^{39} - 3 q^{41} - 21 q^{43} - 12 q^{48} + 6 q^{51} + 6 q^{52} - 6 q^{53} + 30 q^{55} + 9 q^{57} + 18 q^{59} + 6 q^{61} - 24 q^{64} + 18 q^{65} + 3 q^{67} + 12 q^{68} - 6 q^{69} - 21 q^{73} - 9 q^{75} + 18 q^{76} + 21 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{89} - 12 q^{92} - 3 q^{93} + 24 q^{95} + 18 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 - 6 * q^4 + 3 * q^9 + 6 * q^12 - 3 * q^13 + 12 * q^16 - 6 * q^17 - 9 * q^19 + 6 * q^23 + 9 * q^25 - 3 * q^27 + 3 * q^31 - 6 * q^36 - 3 * q^37 + 3 * q^39 - 3 * q^41 - 21 * q^43 - 12 * q^48 + 6 * q^51 + 6 * q^52 - 6 * q^53 + 30 * q^55 + 9 * q^57 + 18 * q^59 + 6 * q^61 - 24 * q^64 + 18 * q^65 + 3 * q^67 + 12 * q^68 - 6 * q^69 - 21 * q^73 - 9 * q^75 + 18 * q^76 + 21 * q^79 + 3 * q^81 - 6 * q^83 + 12 * q^89 - 12 * q^92 - 3 * q^93 + 24 * q^95 + 18 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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