LMFDB - Embedded newform 6027.2.a.q.1.1

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:	$0$
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 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.55247$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} -2.55247 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} -2.55247 q^{5} +1.00000 q^{9} +6.17255 q^{11} -2.00000 q^{12} +2.06760 q^{13} -2.55247 q^{15} +4.00000 q^{16} +2.00000 q^{17} +0.447525 q^{19} +5.10495 q^{20} -0.552475 q^{23} +1.51513 q^{25} +1.00000 q^{27} -5.10495 q^{29} +7.72503 q^{31} +6.17255 q^{33} -2.00000 q^{36} -1.00000 q^{37} +2.06760 q^{39} +1.00000 q^{41} -10.6201 q^{43} -12.3451 q^{44} -2.55247 q^{45} +8.30775 q^{47} +4.00000 q^{48} +2.00000 q^{51} -4.13520 q^{52} -4.13520 q^{53} -15.7553 q^{55} +0.447525 q^{57} -8.13520 q^{59} +5.10495 q^{60} -7.75528 q^{61} -8.00000 q^{64} -5.27750 q^{65} -0.0676012 q^{67} -4.00000 q^{68} -0.552475 q^{69} +2.13520 q^{71} +10.6201 q^{73} +1.51513 q^{75} -0.895051 q^{76} +16.7926 q^{79} -10.2099 q^{80} +1.00000 q^{81} +13.9278 q^{83} -5.10495 q^{85} -5.10495 q^{87} -12.3078 q^{89} +1.10495 q^{92} +7.72503 q^{93} -1.14230 q^{95} -16.2099 q^{97} +6.17255 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−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
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.17255	1.86109	0.930547	−	0.366172i	$-0.119332\pi$
$11$	6.17255	1.86109	0.930547	+	0.366172i	$0.119332\pi$
$12$	−2.00000	−0.577350
$13$	2.06760	0.573449	0.286725	−	0.958013i	$-0.407433\pi$
$13$	2.06760	0.573449	0.286725	+	0.958013i	$0.407433\pi$
$14$	0	0
$15$	−2.55247	−0.659046
$16$	4.00000	1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	0.447525	0.102669	0.0513347	−	0.998682i	$-0.483652\pi$
$19$	0.447525	0.102669	0.0513347	+	0.998682i	$0.483652\pi$
$20$	5.10495	1.14150
$21$	0	0
$22$	0	0
$23$	−0.552475	−0.115199	−0.0575995	−	0.998340i	$-0.518345\pi$
$23$	−0.552475	−0.115199	−0.0575995	+	0.998340i	$0.518345\pi$
$24$	0	0
$25$	1.51513	0.303025
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.10495	−0.947965	−0.473983	−	0.880534i	$-0.657184\pi$
$29$	−5.10495	−0.947965	−0.473983	+	0.880534i	$0.657184\pi$
$30$	0	0
$31$	7.72503	1.38746	0.693728	−	0.720237i	$-0.255967\pi$
$31$	7.72503	1.38746	0.693728	+	0.720237i	$0.255967\pi$
$32$	0	0
$33$	6.17255	1.07450
$34$	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
$39$	2.06760	0.331081
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−10.6201	−1.61955	−0.809773	−	0.586743i	$-0.800410\pi$
$43$	−10.6201	−1.61955	−0.809773	+	0.586743i	$0.800410\pi$
$44$	−12.3451	−1.86109
$45$	−2.55247	−0.380500
$46$	0	0
$47$	8.30775	1.21181	0.605905	−	0.795537i	$-0.292811\pi$
$47$	8.30775	1.21181	0.605905	+	0.795537i	$0.292811\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	−4.13520	−0.573449
$53$	−4.13520	−0.568014	−0.284007	−	0.958822i	$-0.591664\pi$
$53$	−4.13520	−0.568014	−0.284007	+	0.958822i	$0.591664\pi$
$54$	0	0
$55$	−15.7553	−2.12444
$56$	0	0
$57$	0.447525	0.0592762
$58$	0	0
$59$	−8.13520	−1.05911	−0.529557	−	0.848275i	$-0.677642\pi$
$59$	−8.13520	−1.05911	−0.529557	+	0.848275i	$0.677642\pi$
$60$	5.10495	0.659046
$61$	−7.75528	−0.992962	−0.496481	−	0.868048i	$-0.665375\pi$
$61$	−7.75528	−0.992962	−0.496481	+	0.868048i	$0.665375\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−5.27750	−0.654593
$66$	0	0
$67$	−0.0676012	−0.00825881	−0.00412940	−	0.999991i	$-0.501314\pi$
$67$	−0.0676012	−0.00825881	−0.00412940	+	0.999991i	$0.501314\pi$
$68$	−4.00000	−0.485071
$69$	−0.552475	−0.0665101
$70$	0	0
$71$	2.13520	0.253402	0.126701	−	0.991941i	$-0.459561\pi$
$71$	2.13520	0.253402	0.126701	+	0.991941i	$0.459561\pi$
$72$	0	0
$73$	10.6201	1.24299	0.621493	−	0.783420i	$-0.286526\pi$
$73$	10.6201	1.24299	0.621493	+	0.783420i	$0.286526\pi$
$74$	0	0
$75$	1.51513	0.174952
$76$	−0.895051	−0.102669
$77$	0	0
$78$	0	0
$79$	16.7926	1.88932	0.944659	−	0.328054i	$-0.106393\pi$
$79$	16.7926	1.88932	0.944659	+	0.328054i	$0.106393\pi$
$80$	−10.2099	−1.14150
$81$	1.00000	0.111111
$82$	0	0
$83$	13.9278	1.52878	0.764389	−	0.644755i	$-0.223041\pi$
$83$	13.9278	1.52878	0.764389	+	0.644755i	$0.223041\pi$
$84$	0	0
$85$	−5.10495	−0.553709
$86$	0	0
$87$	−5.10495	−0.547308
$88$	0	0
$89$	−12.3078	−1.30462	−0.652310	−	0.757953i	$-0.726200\pi$
$89$	−12.3078	−1.30462	−0.652310	+	0.757953i	$0.726200\pi$
$90$	0	0
$91$	0	0
$92$	1.10495	0.115199
$93$	7.72503	0.801048
$94$	0	0
$95$	−1.14230	−0.117197
$96$	0	0
$97$	−16.2099	−1.64587	−0.822933	−	0.568139i	$-0.807664\pi$
$97$	−16.2099	−1.64587	−0.822933	+	0.568139i	$0.807664\pi$
$98$	0	0
$99$	6.17255	0.620365
$100$	−3.03025	−0.303025
$101$	12.2099	1.21493	0.607465	−	0.794346i	$-0.292186\pi$
$101$	12.2099	1.21493	0.607465	+	0.794346i	$0.292186\pi$
$102$	0	0
$103$	8.10495	0.798604	0.399302	−	0.916819i	$-0.369252\pi$
$103$	8.10495	0.798604	0.399302	+	0.916819i	$0.369252\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.55247	0.246757	0.123379	−	0.992360i	$-0.460627\pi$
$107$	2.55247	0.246757	0.123379	+	0.992360i	$0.460627\pi$
$108$	−2.00000	−0.192450
$109$	4.58273	0.438946	0.219473	−	0.975619i	$-0.429566\pi$
$109$	4.58273	0.438946	0.219473	+	0.975619i	$0.429566\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−15.0328	−1.41416	−0.707082	−	0.707131i	$-0.749989\pi$
$113$	−15.0328	−1.41416	−0.707082	+	0.707131i	$0.749989\pi$
$114$	0	0
$115$	1.41018	0.131500
$116$	10.2099	0.947965
$117$	2.06760	0.191150
$118$	0	0
$119$	0	0
$120$	0	0
$121$	27.1004	2.46367
$122$	0	0
$123$	1.00000	0.0901670
$124$	−15.4501	−1.38746
$125$	8.89505	0.795598
$126$	0	0
$127$	−2.24015	−0.198781	−0.0993907	−	0.995048i	$-0.531689\pi$
$127$	−2.24015	−0.198781	−0.0993907	+	0.995048i	$0.531689\pi$
$128$	0	0
$129$	−10.6201	−0.935046
$130$	0	0
$131$	16.5525	1.44620	0.723098	−	0.690745i	$-0.242717\pi$
$131$	16.5525	1.44620	0.723098	+	0.690745i	$0.242717\pi$
$132$	−12.3451	−1.07450
$133$	0	0
$134$	0	0
$135$	−2.55247	−0.219682
$136$	0	0
$137$	−0.135202	−0.0115511	−0.00577556	−	0.999983i	$-0.501838\pi$
$137$	−0.135202	−0.0115511	−0.00577556	+	0.999983i	$0.501838\pi$
$138$	0	0
$139$	8.37535	0.710388	0.355194	−	0.934793i	$-0.384415\pi$
$139$	8.37535	0.710388	0.355194	+	0.934793i	$0.384415\pi$
$140$	0	0
$141$	8.30775	0.699639
$142$	0	0
$143$	12.7624	1.06724
$144$	4.00000	0.333333
$145$	13.0303	1.08210
$146$	0	0
$147$	0	0
$148$	2.00000	0.164399
$149$	−2.17255	−0.177982	−0.0889911	−	0.996032i	$-0.528364\pi$
$149$	−2.17255	−0.177982	−0.0889911	+	0.996032i	$0.528364\pi$
$150$	0	0
$151$	−16.1004	−1.31023	−0.655115	−	0.755529i	$-0.727380\pi$
$151$	−16.1004	−1.31023	−0.655115	+	0.755529i	$0.727380\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−19.7179	−1.58378
$156$	−4.13520	−0.331081
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−4.13520	−0.327943
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.58982	0.516155	0.258077	−	0.966124i	$-0.416911\pi$
$163$	6.58982	0.516155	0.258077	+	0.966124i	$0.416911\pi$
$164$	−2.00000	−0.156174
$165$	−15.7553	−1.22655
$166$	0	0
$167$	3.86480	0.299067	0.149534	−	0.988757i	$-0.452223\pi$
$167$	3.86480	0.299067	0.149534	+	0.988757i	$0.452223\pi$
$168$	0	0
$169$	−8.72503	−0.671156
$170$	0	0
$171$	0.447525	0.0342231
$172$	21.2402	1.61955
$173$	5.86480	0.445892	0.222946	−	0.974831i	$-0.428433\pi$
$173$	5.86480	0.445892	0.222946	+	0.974831i	$0.428433\pi$
$174$	0	0
$175$	0	0
$176$	24.6902	1.86109
$177$	−8.13520	−0.611479
$178$	0	0
$179$	−9.45005	−0.706330	−0.353165	−	0.935561i	$-0.614895\pi$
$179$	−9.45005	−0.706330	−0.353165	+	0.935561i	$0.614895\pi$
$180$	5.10495	0.380500
$181$	−7.89758	−0.587022	−0.293511	−	0.955956i	$-0.594824\pi$
$181$	−7.89758	−0.587022	−0.293511	+	0.955956i	$0.594824\pi$
$182$	0	0
$183$	−7.75528	−0.573287
$184$	0	0
$185$	2.55247	0.187662
$186$	0	0
$187$	12.3451	0.902763
$188$	−16.6155	−1.21181
$189$	0	0
$190$	0	0
$191$	5.20280	0.376462	0.188231	−	0.982125i	$-0.439725\pi$
$191$	5.20280	0.376462	0.188231	+	0.982125i	$0.439725\pi$
$192$	−8.00000	−0.577350
$193$	13.3078	0.957913	0.478957	−	0.877839i	$-0.341015\pi$
$193$	13.3078	0.957913	0.478957	+	0.877839i	$0.341015\pi$
$194$	0	0
$195$	−5.27750	−0.377930
$196$	0	0
$197$	19.5106	1.39007	0.695035	−	0.718976i	$-0.255389\pi$
$197$	19.5106	1.39007	0.695035	+	0.718976i	$0.255389\pi$
$198$	0	0
$199$	−2.13520	−0.151360	−0.0756802	−	0.997132i	$-0.524113\pi$
$199$	−2.13520	−0.151360	−0.0756802	+	0.997132i	$0.524113\pi$
$200$	0	0
$201$	−0.0676012	−0.00476822
$202$	0	0
$203$	0	0
$204$	−4.00000	−0.280056
$205$	−2.55247	−0.178273
$206$	0	0
$207$	−0.552475	−0.0383996
$208$	8.27040	0.573449
$209$	2.76237	0.191077
$210$	0	0
$211$	16.6902	1.14900	0.574500	−	0.818504i	$-0.305196\pi$
$211$	16.6902	1.14900	0.574500	+	0.818504i	$0.305196\pi$
$212$	8.27040	0.568014
$213$	2.13520	0.146302
$214$	0	0
$215$	27.1075	1.84871
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.6201	0.717638
$220$	31.5106	2.12444
$221$	4.13520	0.278164
$222$	0	0
$223$	−15.9652	−1.06911	−0.534554	−	0.845135i	$-0.679520\pi$
$223$	−15.9652	−1.06911	−0.534554	+	0.845135i	$0.679520\pi$
$224$	0	0
$225$	1.51513	0.101008
$226$	0	0
$227$	8.13520	0.539952	0.269976	−	0.962867i	$-0.412984\pi$
$227$	8.13520	0.539952	0.269976	+	0.962867i	$0.412984\pi$
$228$	−0.895051	−0.0592762
$229$	0.177120	0.0117044	0.00585222	−	0.999983i	$-0.498137\pi$
$229$	0.177120	0.0117044	0.00585222	+	0.999983i	$0.498137\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.1726	−1.05950	−0.529750	−	0.848154i	$-0.677714\pi$
$233$	−16.1726	−1.05950	−0.529750	+	0.848154i	$0.677714\pi$
$234$	0	0
$235$	−21.2053	−1.38328
$236$	16.2704	1.05911
$237$	16.7926	1.09080
$238$	0	0
$239$	25.4501	1.64623	0.823113	−	0.567877i	$-0.192235\pi$
$239$	25.4501	1.64623	0.823113	+	0.567877i	$0.192235\pi$
$240$	−10.2099	−0.659046
$241$	13.6201	0.877346	0.438673	−	0.898647i	$-0.355449\pi$
$241$	13.6201	0.877346	0.438673	+	0.898647i	$0.355449\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	15.5106	0.992962
$245$	0	0
$246$	0	0
$247$	0.925304	0.0588757
$248$	0	0
$249$	13.9278	0.882640
$250$	0	0
$251$	−1.58273	−0.0999009	−0.0499504	−	0.998752i	$-0.515906\pi$
$251$	−1.58273	−0.0999009	−0.0499504	+	0.998752i	$0.515906\pi$
$252$	0	0
$253$	−3.41018	−0.214396
$254$	0	0
$255$	−5.10495	−0.319684
$256$	16.0000	1.00000
$257$	−8.44296	−0.526657	−0.263329	−	0.964706i	$-0.584820\pi$
$257$	−8.44296	−0.526657	−0.263329	+	0.964706i	$0.584820\pi$
$258$	0	0
$259$	0	0
$260$	10.5550	0.654593
$261$	−5.10495	−0.315988
$262$	0	0
$263$	7.41270	0.457087	0.228543	−	0.973534i	$-0.426604\pi$
$263$	7.41270	0.457087	0.228543	+	0.973534i	$0.426604\pi$
$264$	0	0
$265$	10.5550	0.648388
$266$	0	0
$267$	−12.3078	−0.753222
$268$	0.135202	0.00825881
$269$	21.3754	1.30328	0.651639	−	0.758529i	$-0.274082\pi$
$269$	21.3754	1.30328	0.651639	+	0.758529i	$0.274082\pi$
$270$	0	0
$271$	−6.40561	−0.389113	−0.194556	−	0.980891i	$-0.562327\pi$
$271$	−6.40561	−0.389113	−0.194556	+	0.980891i	$0.562327\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	0	0
$275$	9.35220	0.563959
$276$	1.10495	0.0665101
$277$	12.6201	0.758267	0.379133	−	0.925342i	$-0.376222\pi$
$277$	12.6201	0.758267	0.379133	+	0.925342i	$0.376222\pi$
$278$	0	0
$279$	7.72503	0.462485
$280$	0	0
$281$	−20.4430	−1.21952	−0.609762	−	0.792584i	$-0.708735\pi$
$281$	−20.4430	−1.21952	−0.609762	+	0.792584i	$0.708735\pi$
$282$	0	0
$283$	−1.34510	−0.0799580	−0.0399790	−	0.999201i	$-0.512729\pi$
$283$	−1.34510	−0.0799580	−0.0399790	+	0.999201i	$0.512729\pi$
$284$	−4.27040	−0.253402
$285$	−1.14230	−0.0676638
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−16.2099	−0.950241
$292$	−21.2402	−1.24299
$293$	24.3824	1.42444	0.712219	−	0.701957i	$-0.247690\pi$
$293$	24.3824	1.42444	0.712219	+	0.701957i	$0.247690\pi$
$294$	0	0
$295$	20.7649	1.20898
$296$	0	0
$297$	6.17255	0.358168
$298$	0	0
$299$	−1.14230	−0.0660608
$300$	−3.03025	−0.174952
$301$	0	0
$302$	0	0
$303$	12.2099	0.701440
$304$	1.79010	0.102669
$305$	19.7952	1.13347
$306$	0	0
$307$	19.2099	1.09637	0.548183	−	0.836358i	$-0.315320\pi$
$307$	19.2099	1.09637	0.548183	+	0.836358i	$0.315320\pi$
$308$	0	0
$309$	8.10495	0.461074
$310$	0	0
$311$	30.9980	1.75773	0.878866	−	0.477068i	$-0.158300\pi$
$311$	30.9980	1.75773	0.878866	+	0.477068i	$0.158300\pi$
$312$	0	0
$313$	16.3380	0.923479	0.461739	−	0.887016i	$-0.347226\pi$
$313$	16.3380	0.923479	0.461739	+	0.887016i	$0.347226\pi$
$314$	0	0
$315$	0	0
$316$	−33.5853	−1.88932
$317$	−23.5479	−1.32258	−0.661291	−	0.750129i	$-0.729991\pi$
$317$	−23.5479	−1.32258	−0.661291	+	0.750129i	$0.729991\pi$
$318$	0	0
$319$	−31.5106	−1.76425
$320$	20.4198	1.14150
$321$	2.55247	0.142465
$322$	0	0
$323$	0.895051	0.0498020
$324$	−2.00000	−0.111111
$325$	3.13268	0.173770
$326$	0	0
$327$	4.58273	0.253425
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.9278	−1.37016	−0.685079	−	0.728469i	$-0.740232\pi$
$331$	−24.9278	−1.37016	−0.685079	+	0.728469i	$0.740232\pi$
$332$	−27.8557	−1.52878
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	0.172550	0.00942744
$336$	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
$339$	−15.0328	−0.816468
$340$	10.2099	0.553709
$341$	47.6831	2.58219
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.41018	0.0759214
$346$	0	0
$347$	25.3754	1.36222	0.681110	−	0.732181i	$-0.261497\pi$
$347$	25.3754	1.36222	0.681110	+	0.732181i	$0.261497\pi$
$348$	10.2099	0.547308
$349$	−29.9652	−1.60400	−0.802000	−	0.597325i	$-0.796230\pi$
$349$	−29.9652	−1.60400	−0.802000	+	0.597325i	$0.796230\pi$
$350$	0	0
$351$	2.06760	0.110360
$352$	0	0
$353$	15.1049	0.803955	0.401978	−	0.915649i	$-0.368323\pi$
$353$	15.1049	0.803955	0.401978	+	0.915649i	$0.368323\pi$
$354$	0	0
$355$	−5.45005	−0.289259
$356$	24.6155	1.30462
$357$	0	0
$358$	0	0
$359$	26.4173	1.39425	0.697125	−	0.716949i	$-0.254462\pi$
$359$	26.4173	1.39425	0.697125	+	0.716949i	$0.254462\pi$
$360$	0	0
$361$	−18.7997	−0.989459
$362$	0	0
$363$	27.1004	1.42240
$364$	0	0
$365$	−27.1075	−1.41887
$366$	0	0
$367$	7.58982	0.396186	0.198093	−	0.980183i	$-0.436525\pi$
$367$	7.58982	0.396186	0.198093	+	0.980183i	$0.436525\pi$
$368$	−2.20990	−0.115199
$369$	1.00000	0.0520579
$370$	0	0
$371$	0	0
$372$	−15.4501	−0.801048
$373$	17.2099	0.891095	0.445547	−	0.895258i	$-0.353009\pi$
$373$	17.2099	0.891095	0.445547	+	0.895258i	$0.353009\pi$
$374$	0	0
$375$	8.89505	0.459338
$376$	0	0
$377$	−10.5550	−0.543610
$378$	0	0
$379$	25.3708	1.30321	0.651605	−	0.758559i	$-0.274096\pi$
$379$	25.3708	1.30321	0.651605	+	0.758559i	$0.274096\pi$
$380$	2.28459	0.117197
$381$	−2.24015	−0.114766
$382$	0	0
$383$	10.8345	0.553619	0.276810	−	0.960925i	$-0.410723\pi$
$383$	10.8345	0.553619	0.276810	+	0.960925i	$0.410723\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.6201	−0.539849
$388$	32.4198	1.64587
$389$	14.2729	0.723666	0.361833	−	0.932243i	$-0.382151\pi$
$389$	14.2729	0.723666	0.361833	+	0.932243i	$0.382151\pi$
$390$	0	0
$391$	−1.10495	−0.0558797
$392$	0	0
$393$	16.5525	0.834962
$394$	0	0
$395$	−42.8628	−2.15666
$396$	−12.3451	−0.620365
$397$	17.8628	0.896506	0.448253	−	0.893907i	$-0.352046\pi$
$397$	17.8628	0.896506	0.448253	+	0.893907i	$0.352046\pi$
$398$	0	0
$399$	0	0
$400$	6.06051	0.303025
$401$	15.7205	0.785042	0.392521	−	0.919743i	$-0.371603\pi$
$401$	15.7205	0.785042	0.392521	+	0.919743i	$0.371603\pi$
$402$	0	0
$403$	15.9723	0.795636
$404$	−24.4198	−1.21493
$405$	−2.55247	−0.126833
$406$	0	0
$407$	−6.17255	−0.305962
$408$	0	0
$409$	−23.1004	−1.14224	−0.571120	−	0.820866i	$-0.693491\pi$
$409$	−23.1004	−1.14224	−0.571120	+	0.820866i	$0.693491\pi$
$410$	0	0
$411$	−0.135202	−0.00666905
$412$	−16.2099	−0.798604
$413$	0	0
$414$	0	0
$415$	−35.5504	−1.74510
$416$	0	0
$417$	8.37535	0.410143
$418$	0	0
$419$	−22.3451	−1.09163	−0.545815	−	0.837906i	$-0.683780\pi$
$419$	−22.3451	−1.09163	−0.545815	+	0.837906i	$0.683780\pi$
$420$	0	0
$421$	−34.8325	−1.69763	−0.848816	−	0.528688i	$-0.822684\pi$
$421$	−34.8325	−1.69763	−0.848816	+	0.528688i	$0.822684\pi$
$422$	0	0
$423$	8.30775	0.403937
$424$	0	0
$425$	3.03025	0.146989
$426$	0	0
$427$	0	0
$428$	−5.10495	−0.246757
$429$	12.7624	0.616173
$430$	0	0
$431$	16.4920	0.794390	0.397195	−	0.917734i	$-0.369984\pi$
$431$	16.4920	0.794390	0.397195	+	0.917734i	$0.369984\pi$
$432$	4.00000	0.192450
$433$	19.9349	0.958011	0.479006	−	0.877812i	$-0.340997\pi$
$433$	19.9349	0.958011	0.479006	+	0.877812i	$0.340997\pi$
$434$	0	0
$435$	13.0303	0.624753
$436$	−9.16546	−0.438946
$437$	−0.247246	−0.0118274
$438$	0	0
$439$	25.8905	1.23569	0.617843	−	0.786302i	$-0.288007\pi$
$439$	25.8905	1.23569	0.617843	+	0.786302i	$0.288007\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.5853	1.50066	0.750330	−	0.661063i	$-0.229894\pi$
$443$	31.5853	1.50066	0.750330	+	0.661063i	$0.229894\pi$
$444$	2.00000	0.0949158
$445$	31.4152	1.48922
$446$	0	0
$447$	−2.17255	−0.102758
$448$	0	0
$449$	−10.0630	−0.474904	−0.237452	−	0.971399i	$-0.576312\pi$
$449$	−10.0630	−0.474904	−0.237452	+	0.971399i	$0.576312\pi$
$450$	0	0
$451$	6.17255	0.290654
$452$	30.0656	1.41416
$453$	−16.1004	−0.756462
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.1377	−0.801669	−0.400835	−	0.916150i	$-0.631280\pi$
$457$	−17.1377	−0.801669	−0.400835	+	0.916150i	$0.631280\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	−2.82035	−0.131500
$461$	−6.89758	−0.321252	−0.160626	−	0.987015i	$-0.551351\pi$
$461$	−6.89758	−0.321252	−0.160626	+	0.987015i	$0.551351\pi$
$462$	0	0
$463$	−25.1726	−1.16987	−0.584934	−	0.811081i	$-0.698880\pi$
$463$	−25.1726	−1.16987	−0.584934	+	0.811081i	$0.698880\pi$
$464$	−20.4198	−0.947965
$465$	−19.7179	−0.914397
$466$	0	0
$467$	−9.58273	−0.443436	−0.221718	−	0.975111i	$-0.571166\pi$
$467$	−9.58273	−0.443436	−0.221718	+	0.975111i	$0.571166\pi$
$468$	−4.13520	−0.191150
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−65.5530	−3.01413
$474$	0	0
$475$	0.678058	0.0311114
$476$	0	0
$477$	−4.13520	−0.189338
$478$	0	0
$479$	11.9627	0.546588	0.273294	−	0.961931i	$-0.411887\pi$
$479$	11.9627	0.546588	0.273294	+	0.961931i	$0.411887\pi$
$480$	0	0
$481$	−2.06760	−0.0942745
$482$	0	0
$483$	0	0
$484$	−54.2008	−2.46367
$485$	41.3754	1.87876
$486$	0	0
$487$	−19.7507	−0.894990	−0.447495	−	0.894286i	$-0.647684\pi$
$487$	−19.7507	−0.894990	−0.447495	+	0.894286i	$0.647684\pi$
$488$	0	0
$489$	6.58982	0.298002
$490$	0	0
$491$	4.75985	0.214809	0.107404	−	0.994215i	$-0.465746\pi$
$491$	4.75985	0.214809	0.107404	+	0.994215i	$0.465746\pi$
$492$	−2.00000	−0.0901670
$493$	−10.2099	−0.459831
$494$	0	0
$495$	−15.7553	−0.708147
$496$	30.9001	1.38746
$497$	0	0
$498$	0	0
$499$	−5.47778	−0.245219	−0.122610	−	0.992455i	$-0.539126\pi$
$499$	−5.47778	−0.245219	−0.122610	+	0.992455i	$0.539126\pi$
$500$	−17.7901	−0.795598
$501$	3.86480	0.172666
$502$	0	0
$503$	−3.00709	−0.134080	−0.0670399	−	0.997750i	$-0.521355\pi$
$503$	−3.00709	−0.134080	−0.0670399	+	0.997750i	$0.521355\pi$
$504$	0	0
$505$	−31.1655	−1.38684
$506$	0	0
$507$	−8.72503	−0.387492
$508$	4.48030	0.198781
$509$	−31.7205	−1.40598	−0.702992	−	0.711198i	$-0.748153\pi$
$509$	−31.7205	−1.40598	−0.702992	+	0.711198i	$0.748153\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.447525	0.0197587
$514$	0	0
$515$	−20.6877	−0.911608
$516$	21.2402	0.935046
$517$	51.2800	2.25529
$518$	0	0
$519$	5.86480	0.257436
$520$	0	0
$521$	24.1120	1.05637	0.528184	−	0.849130i	$-0.322873\pi$
$521$	24.1120	1.05637	0.528184	+	0.849130i	$0.322873\pi$
$522$	0	0
$523$	35.1751	1.53810	0.769049	−	0.639189i	$-0.220730\pi$
$523$	35.1751	1.53810	0.769049	+	0.639189i	$0.220730\pi$
$524$	−33.1049	−1.44620
$525$	0	0
$526$	0	0
$527$	15.4501	0.673015
$528$	24.6902	1.07450
$529$	−22.6948	−0.986729
$530$	0	0
$531$	−8.13520	−0.353038
$532$	0	0
$533$	2.06760	0.0895578
$534$	0	0
$535$	−6.51513	−0.281673
$536$	0	0
$537$	−9.45005	−0.407800
$538$	0	0
$539$	0	0
$540$	5.10495	0.219682
$541$	16.6201	0.714553	0.357276	−	0.933999i	$-0.383705\pi$
$541$	16.6201	0.714553	0.357276	+	0.933999i	$0.383705\pi$
$542$	0	0
$543$	−7.89758	−0.338917
$544$	0	0
$545$	−11.6973	−0.501057
$546$	0	0
$547$	18.6760	0.798529	0.399264	−	0.916836i	$-0.369266\pi$
$547$	18.6760	0.798529	0.399264	+	0.916836i	$0.369266\pi$
$548$	0.270405	0.0115511
$549$	−7.75528	−0.330987
$550$	0	0
$551$	−2.28459	−0.0973270
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.55247	0.108347
$556$	−16.7507	−0.710388
$557$	−29.6831	−1.25771	−0.628857	−	0.777521i	$-0.716477\pi$
$557$	−29.6831	−1.25771	−0.628857	+	0.777521i	$0.716477\pi$
$558$	0	0
$559$	−21.9581	−0.928728
$560$	0	0
$561$	12.3451	0.521211
$562$	0	0
$563$	1.27750	0.0538402	0.0269201	−	0.999638i	$-0.491430\pi$
$563$	1.27750	0.0538402	0.0269201	+	0.999638i	$0.491430\pi$
$564$	−16.6155	−0.699639
$565$	38.3708	1.61427
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.9581	0.459387	0.229693	−	0.973263i	$-0.426228\pi$
$569$	10.9581	0.459387	0.229693	+	0.973263i	$0.426228\pi$
$570$	0	0
$571$	33.3733	1.39663	0.698315	−	0.715791i	$-0.253934\pi$
$571$	33.3733	1.39663	0.698315	+	0.715791i	$0.253934\pi$
$572$	−25.5247	−1.06724
$573$	5.20280	0.217350
$574$	0	0
$575$	−0.837069	−0.0349082
$576$	−8.00000	−0.333333
$577$	−26.5878	−1.10686	−0.553432	−	0.832894i	$-0.686682\pi$
$577$	−26.5878	−1.10686	−0.553432	+	0.832894i	$0.686682\pi$
$578$	0	0
$579$	13.3078	0.553051
$580$	−26.0605	−1.08210
$581$	0	0
$582$	0	0
$583$	−25.5247	−1.05713
$584$	0	0
$585$	−5.27750	−0.218198
$586$	0	0
$587$	32.2472	1.33099	0.665493	−	0.746404i	$-0.268221\pi$
$587$	32.2472	1.33099	0.665493	+	0.746404i	$0.268221\pi$
$588$	0	0
$589$	3.45714	0.142449
$590$	0	0
$591$	19.5106	0.802557
$592$	−4.00000	−0.164399
$593$	−28.7881	−1.18218	−0.591092	−	0.806604i	$-0.701303\pi$
$593$	−28.7881	−1.18218	−0.591092	+	0.806604i	$0.701303\pi$
$594$	0	0
$595$	0	0
$596$	4.34510	0.177982
$597$	−2.13520	−0.0873880
$598$	0	0
$599$	2.07217	0.0846666	0.0423333	−	0.999104i	$-0.486521\pi$
$599$	2.07217	0.0846666	0.0423333	+	0.999104i	$0.486521\pi$
$600$	0	0
$601$	7.38245	0.301136	0.150568	−	0.988600i	$-0.451890\pi$
$601$	7.38245	0.301136	0.150568	+	0.988600i	$0.451890\pi$
$602$	0	0
$603$	−0.0676012	−0.00275294
$604$	32.2008	1.31023
$605$	−69.1730	−2.81228
$606$	0	0
$607$	14.7952	0.600517	0.300258	−	0.953858i	$-0.402927\pi$
$607$	14.7952	0.600517	0.300258	+	0.953858i	$0.402927\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.1771	0.694912
$612$	−4.00000	−0.161690
$613$	32.0656	1.29512	0.647558	−	0.762016i	$-0.275790\pi$
$613$	32.0656	1.29512	0.647558	+	0.762016i	$0.275790\pi$
$614$	0	0
$615$	−2.55247	−0.102926
$616$	0	0
$617$	42.6155	1.71564	0.857818	−	0.513954i	$-0.171820\pi$
$617$	42.6155	1.71564	0.857818	+	0.513954i	$0.171820\pi$
$618$	0	0
$619$	17.2356	0.692757	0.346378	−	0.938095i	$-0.387411\pi$
$619$	17.2356	0.692757	0.346378	+	0.938095i	$0.387411\pi$
$620$	39.4359	1.58378
$621$	−0.552475	−0.0221700
$622$	0	0
$623$	0	0
$624$	8.27040	0.331081
$625$	−30.2800	−1.21120
$626$	0	0
$627$	2.76237	0.110319
$628$	−20.0000	−0.798087
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−22.1352	−0.881188	−0.440594	−	0.897706i	$-0.645232\pi$
$631$	−22.1352	−0.881188	−0.440594	+	0.897706i	$0.645232\pi$
$632$	0	0
$633$	16.6902	0.663376
$634$	0	0
$635$	5.71793	0.226909
$636$	8.27040	0.327943
$637$	0	0
$638$	0	0
$639$	2.13520	0.0844673
$640$	0	0
$641$	−14.7972	−0.584454	−0.292227	−	0.956349i	$-0.594396\pi$
$641$	−14.7972	−0.584454	−0.292227	+	0.956349i	$0.594396\pi$
$642$	0	0
$643$	−45.2986	−1.78640	−0.893201	−	0.449657i	$-0.851546\pi$
$643$	−45.2986	−1.78640	−0.893201	+	0.449657i	$0.851546\pi$
$644$	0	0
$645$	27.1075	1.06736
$646$	0	0
$647$	21.7205	0.853919	0.426960	−	0.904271i	$-0.359585\pi$
$647$	21.7205	0.853919	0.426960	+	0.904271i	$0.359585\pi$
$648$	0	0
$649$	−50.2149	−1.97111
$650$	0	0
$651$	0	0
$652$	−13.1796	−0.516155
$653$	−48.9233	−1.91452	−0.957258	−	0.289237i	$-0.906598\pi$
$653$	−48.9233	−1.91452	−0.957258	+	0.289237i	$0.906598\pi$
$654$	0	0
$655$	−42.2498	−1.65084
$656$	4.00000	0.156174
$657$	10.6201	0.414329
$658$	0	0
$659$	−36.4198	−1.41871	−0.709357	−	0.704849i	$-0.751015\pi$
$659$	−36.4198	−1.41871	−0.709357	+	0.704849i	$0.751015\pi$
$660$	31.5106	1.22655
$661$	1.86937	0.0727100	0.0363550	−	0.999339i	$-0.488425\pi$
$661$	1.86937	0.0727100	0.0363550	+	0.999339i	$0.488425\pi$
$662$	0	0
$663$	4.13520	0.160598
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.82035	0.109205
$668$	−7.72960	−0.299067
$669$	−15.9652	−0.617249
$670$	0	0
$671$	−47.8698	−1.84799
$672$	0	0
$673$	−44.8325	−1.72817	−0.864083	−	0.503349i	$-0.832101\pi$
$673$	−44.8325	−1.72817	−0.864083	+	0.503349i	$0.832101\pi$
$674$	0	0
$675$	1.51513	0.0583173
$676$	17.4501	0.671156
$677$	2.13773	0.0821595	0.0410798	−	0.999156i	$-0.486920\pi$
$677$	2.13773	0.0821595	0.0410798	+	0.999156i	$0.486920\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.13520	0.311742
$682$	0	0
$683$	30.4430	1.16487	0.582434	−	0.812878i	$-0.302101\pi$
$683$	30.4430	1.16487	0.582434	+	0.812878i	$0.302101\pi$
$684$	−0.895051	−0.0342231
$685$	0.345101	0.0131856
$686$	0	0
$687$	0.177120	0.00675756
$688$	−42.4803	−1.61955
$689$	−8.54995	−0.325727
$690$	0	0
$691$	−18.2170	−0.693007	−0.346504	−	0.938049i	$-0.612631\pi$
$691$	−18.2170	−0.693007	−0.346504	+	0.938049i	$0.612631\pi$
$692$	−11.7296	−0.445892
$693$	0	0
$694$	0	0
$695$	−21.3779	−0.810909
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	−16.1726	−0.611702
$700$	0	0
$701$	−23.2427	−0.877864	−0.438932	−	0.898520i	$-0.644643\pi$
$701$	−23.2427	−0.877864	−0.438932	+	0.898520i	$0.644643\pi$
$702$	0	0
$703$	−0.447525	−0.0168787
$704$	−49.3804	−1.86109
$705$	−21.2053	−0.798639
$706$	0	0
$707$	0	0
$708$	16.2704	0.611479
$709$	−34.6902	−1.30282	−0.651409	−	0.758727i	$-0.725822\pi$
$709$	−34.6902	−1.30282	−0.651409	+	0.758727i	$0.725822\pi$
$710$	0	0
$711$	16.7926	0.629773
$712$	0	0
$713$	−4.26788	−0.159833
$714$	0	0
$715$	−32.5756	−1.21826
$716$	18.9001	0.706330
$717$	25.4501	0.950450
$718$	0	0
$719$	−34.2986	−1.27912	−0.639561	−	0.768740i	$-0.720884\pi$
$719$	−34.2986	−1.27912	−0.639561	+	0.768740i	$0.720884\pi$
$720$	−10.2099	−0.380500
$721$	0	0
$722$	0	0
$723$	13.6201	0.506536
$724$	15.7952	0.587022
$725$	−7.73464	−0.287257
$726$	0	0
$727$	−12.6226	−0.468146	−0.234073	−	0.972219i	$-0.575206\pi$
$727$	−12.6226	−0.468146	−0.234073	+	0.972219i	$0.575206\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.2402	−0.785595
$732$	15.5106	0.573287
$733$	−34.6458	−1.27967	−0.639835	−	0.768512i	$-0.720997\pi$
$733$	−34.6458	−1.27967	−0.639835	+	0.768512i	$0.720997\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.417272	−0.0153704
$738$	0	0
$739$	−36.8300	−1.35481	−0.677406	−	0.735609i	$-0.736896\pi$
$739$	−36.8300	−1.35481	−0.677406	+	0.735609i	$0.736896\pi$
$740$	−5.10495	−0.187662
$741$	0.925304	0.0339919
$742$	0	0
$743$	28.3451	1.03988	0.519940	−	0.854203i	$-0.325954\pi$
$743$	28.3451	1.03988	0.519940	+	0.854203i	$0.325954\pi$
$744$	0	0
$745$	5.54538	0.203167
$746$	0	0
$747$	13.9278	0.509593
$748$	−24.6902	−0.902763
$749$	0	0
$750$	0	0
$751$	−2.55704	−0.0933079	−0.0466539	−	0.998911i	$-0.514856\pi$
$751$	−2.55704	−0.0933079	−0.0466539	+	0.998911i	$0.514856\pi$
$752$	33.2310	1.21181
$753$	−1.58273	−0.0576778
$754$	0	0
$755$	41.0958	1.49563
$756$	0	0
$757$	44.9258	1.63286	0.816428	−	0.577448i	$-0.195951\pi$
$757$	44.9258	1.63286	0.816428	+	0.577448i	$0.195951\pi$
$758$	0	0
$759$	−3.41018	−0.123782
$760$	0	0
$761$	10.9697	0.397653	0.198827	−	0.980035i	$-0.436287\pi$
$761$	10.9697	0.397653	0.198827	+	0.980035i	$0.436287\pi$
$762$	0	0
$763$	0	0
$764$	−10.4056	−0.376462
$765$	−5.10495	−0.184570
$766$	0	0
$767$	−16.8204	−0.607348
$768$	16.0000	0.577350
$769$	−29.1751	−1.05208	−0.526040	−	0.850460i	$-0.676324\pi$
$769$	−29.1751	−1.05208	−0.526040	+	0.850460i	$0.676324\pi$
$770$	0	0
$771$	−8.44296	−0.304066
$772$	−26.6155	−0.957913
$773$	43.0585	1.54871	0.774353	−	0.632754i	$-0.218076\pi$
$773$	43.0585	1.54871	0.774353	+	0.632754i	$0.218076\pi$
$774$	0	0
$775$	11.7044	0.420434
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.447525	0.0160343
$780$	10.5550	0.377930
$781$	13.1796	0.471605
$782$	0	0
$783$	−5.10495	−0.182436
$784$	0	0
$785$	−25.5247	−0.911017
$786$	0	0
$787$	24.9258	0.888508	0.444254	−	0.895901i	$-0.353469\pi$
$787$	24.9258	0.888508	0.444254	+	0.895901i	$0.353469\pi$
$788$	−39.0211	−1.39007
$789$	7.41270	0.263899
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.0348	−0.569413
$794$	0	0
$795$	10.5550	0.374347
$796$	4.27040	0.151360
$797$	0.137727	0.00487855	0.00243928	−	0.999997i	$-0.499224\pi$
$797$	0.137727	0.00487855	0.00243928	+	0.999997i	$0.499224\pi$
$798$	0	0
$799$	16.6155	0.587814
$800$	0	0
$801$	−12.3078	−0.434873
$802$	0	0
$803$	65.5530	2.31331
$804$	0.135202	0.00476822
$805$	0	0
$806$	0	0
$807$	21.3754	0.752448
$808$	0	0
$809$	1.48235	0.0521166	0.0260583	−	0.999660i	$-0.491704\pi$
$809$	1.48235	0.0521166	0.0260583	+	0.999660i	$0.491704\pi$
$810$	0	0
$811$	−26.1004	−0.916508	−0.458254	−	0.888821i	$-0.651525\pi$
$811$	−26.1004	−0.916508	−0.458254	+	0.888821i	$0.651525\pi$
$812$	0	0
$813$	−6.40561	−0.224654
$814$	0	0
$815$	−16.8204	−0.589191
$816$	8.00000	0.280056
$817$	−4.75275	−0.166278
$818$	0	0
$819$	0	0
$820$	5.10495	0.178273
$821$	−6.69020	−0.233490	−0.116745	−	0.993162i	$-0.537246\pi$
$821$	−6.69020	−0.233490	−0.116745	+	0.993162i	$0.537246\pi$
$822$	0	0
$823$	11.0444	0.384985	0.192493	−	0.981298i	$-0.438343\pi$
$823$	11.0444	0.384985	0.192493	+	0.981298i	$0.438343\pi$
$824$	0	0
$825$	9.35220	0.325602
$826$	0	0
$827$	24.6902	0.858562	0.429281	−	0.903171i	$-0.358767\pi$
$827$	24.6902	0.858562	0.429281	+	0.903171i	$0.358767\pi$
$828$	1.10495	0.0383996
$829$	22.6599	0.787013	0.393506	−	0.919322i	$-0.371262\pi$
$829$	22.6599	0.787013	0.393506	+	0.919322i	$0.371262\pi$
$830$	0	0
$831$	12.6201	0.437786
$832$	−16.5408	−0.573449
$833$	0	0
$834$	0	0
$835$	−9.86480	−0.341385
$836$	−5.52475	−0.191077
$837$	7.72503	0.267016
$838$	0	0
$839$	42.1776	1.45613	0.728066	−	0.685507i	$-0.240419\pi$
$839$	42.1776	1.45613	0.728066	+	0.685507i	$0.240419\pi$
$840$	0	0
$841$	−2.93949	−0.101362
$842$	0	0
$843$	−20.4430	−0.704093
$844$	−33.3804	−1.14900
$845$	22.2704	0.766125
$846$	0	0
$847$	0	0
$848$	−16.5408	−0.568014
$849$	−1.34510	−0.0461637
$850$	0	0
$851$	0.552475	0.0189386
$852$	−4.27040	−0.146302
$853$	28.9652	0.991749	0.495874	−	0.868394i	$-0.334848\pi$
$853$	28.9652	0.991749	0.495874	+	0.868394i	$0.334848\pi$
$854$	0	0
$855$	−1.14230	−0.0390657
$856$	0	0
$857$	−54.6811	−1.86787	−0.933935	−	0.357444i	$-0.883648\pi$
$857$	−54.6811	−1.86787	−0.933935	+	0.357444i	$0.883648\pi$
$858$	0	0
$859$	−30.5898	−1.04371	−0.521856	−	0.853034i	$-0.674760\pi$
$859$	−30.5898	−1.04371	−0.521856	+	0.853034i	$0.674760\pi$
$860$	−54.2149	−1.84871
$861$	0	0
$862$	0	0
$863$	24.7019	0.840861	0.420431	−	0.907325i	$-0.361879\pi$
$863$	24.7019	0.840861	0.420431	+	0.907325i	$0.361879\pi$
$864$	0	0
$865$	−14.9697	−0.508987
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	103.653	3.51620
$870$	0	0
$871$	−0.139772	−0.00473601
$872$	0	0
$873$	−16.2099	−0.548622
$874$	0	0
$875$	0	0
$876$	−21.2402	−0.717638
$877$	−6.23558	−0.210561	−0.105280	−	0.994443i	$-0.533574\pi$
$877$	−6.23558	−0.210561	−0.105280	+	0.994443i	$0.533574\pi$
$878$	0	0
$879$	24.3824	0.822400
$880$	−63.0211	−2.12444
$881$	42.2755	1.42430	0.712148	−	0.702029i	$-0.247722\pi$
$881$	42.2755	1.42430	0.712148	+	0.702029i	$0.247722\pi$
$882$	0	0
$883$	13.9723	0.470204	0.235102	−	0.971971i	$-0.424458\pi$
$883$	13.9723	0.470204	0.235102	+	0.971971i	$0.424458\pi$
$884$	−8.27040	−0.278164
$885$	20.7649	0.698004
$886$	0	0
$887$	30.0656	1.00950	0.504751	−	0.863265i	$-0.331584\pi$
$887$	30.0656	1.00950	0.504751	+	0.863265i	$0.331584\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.17255	0.206788
$892$	31.9304	1.06911
$893$	3.71793	0.124416
$894$	0	0
$895$	24.1210	0.806277
$896$	0	0
$897$	−1.14230	−0.0381402
$898$	0	0
$899$	−39.4359	−1.31526
$900$	−3.03025	−0.101008
$901$	−8.27040	−0.275527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.1584	0.670087
$906$	0	0
$907$	30.0096	0.996453	0.498227	−	0.867047i	$-0.333985\pi$
$907$	30.0096	0.996453	0.498227	+	0.867047i	$0.333985\pi$
$908$	−16.2704	−0.539952
$909$	12.2099	0.404977
$910$	0	0
$911$	14.4056	0.477279	0.238640	−	0.971108i	$-0.423299\pi$
$911$	14.4056	0.477279	0.238640	+	0.971108i	$0.423299\pi$
$912$	1.79010	0.0592762
$913$	85.9702	2.84520
$914$	0	0
$915$	19.7952	0.654408
$916$	−0.354241	−0.0117044
$917$	0	0
$918$	0	0
$919$	4.37788	0.144413	0.0722065	−	0.997390i	$-0.476996\pi$
$919$	4.37788	0.144413	0.0722065	+	0.997390i	$0.476996\pi$
$920$	0	0
$921$	19.2099	0.632988
$922$	0	0
$923$	4.41475	0.145313
$924$	0	0
$925$	−1.51513	−0.0498171
$926$	0	0
$927$	8.10495	0.266201
$928$	0	0
$929$	−5.59439	−0.183546	−0.0917730	−	0.995780i	$-0.529253\pi$
$929$	−5.59439	−0.183546	−0.0917730	+	0.995780i	$0.529253\pi$
$930$	0	0
$931$	0	0
$932$	32.3451	1.05950
$933$	30.9980	1.01483
$934$	0	0
$935$	−31.5106	−1.03051
$936$	0	0
$937$	34.0328	1.11180	0.555901	−	0.831248i	$-0.312373\pi$
$937$	34.0328	1.11180	0.555901	+	0.831248i	$0.312373\pi$
$938$	0	0
$939$	16.3380	0.533171
$940$	42.4107	1.38328
$941$	2.19571	0.0715781	0.0357890	−	0.999359i	$-0.488606\pi$
$941$	2.19571	0.0715781	0.0357890	+	0.999359i	$0.488606\pi$
$942$	0	0
$943$	−0.552475	−0.0179910
$944$	−32.5408	−1.05911
$945$	0	0
$946$	0	0
$947$	8.14687	0.264738	0.132369	−	0.991201i	$-0.457742\pi$
$947$	8.14687	0.264738	0.132369	+	0.991201i	$0.457742\pi$
$948$	−33.5853	−1.09080
$949$	21.9581	0.712790
$950$	0	0
$951$	−23.5479	−0.763593
$952$	0	0
$953$	−21.5156	−0.696959	−0.348479	−	0.937316i	$-0.613302\pi$
$953$	−21.5156	−0.696959	−0.348479	+	0.937316i	$0.613302\pi$
$954$	0	0
$955$	−13.2800	−0.429731
$956$	−50.9001	−1.64623
$957$	−31.5106	−1.01859
$958$	0	0
$959$	0	0
$960$	20.4198	0.659046
$961$	28.6760	0.925033
$962$	0	0
$963$	2.55247	0.0822523
$964$	−27.2402	−0.877346
$965$	−33.9677	−1.09346
$966$	0	0
$967$	51.8371	1.66697	0.833484	−	0.552544i	$-0.186343\pi$
$967$	51.8371	1.66697	0.833484	+	0.552544i	$0.186343\pi$
$968$	0	0
$969$	0.895051	0.0287532
$970$	0	0
$971$	−5.86480	−0.188210	−0.0941052	−	0.995562i	$-0.529999\pi$
$971$	−5.86480	−0.188210	−0.0941052	+	0.995562i	$0.529999\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	0	0
$975$	3.13268	0.100326
$976$	−31.0211	−0.992962
$977$	61.3804	1.96373	0.981867	−	0.189573i	$-0.0607105\pi$
$977$	61.3804	1.96373	0.981867	+	0.189573i	$0.0607105\pi$
$978$	0	0
$979$	−75.9702	−2.42802
$980$	0	0
$981$	4.58273	0.146315
$982$	0	0
$983$	−4.69682	−0.149805	−0.0749026	−	0.997191i	$-0.523865\pi$
$983$	−4.69682	−0.149805	−0.0749026	+	0.997191i	$0.523865\pi$
$984$	0	0
$985$	−49.8002	−1.58677
$986$	0	0
$987$	0	0
$988$	−1.85061	−0.0588757
$989$	5.86732	0.186570
$990$	0	0
$991$	7.65285	0.243101	0.121550	−	0.992585i	$-0.461213\pi$
$991$	7.65285	0.243101	0.121550	+	0.992585i	$0.461213\pi$
$992$	0	0
$993$	−24.9278	−0.791061
$994$	0	0
$995$	5.45005	0.172778
$996$	−27.8557	−0.882640
$997$	−32.0328	−1.01449	−0.507244	−	0.861802i	$-0.669336\pi$
$997$	−32.0328	−1.01449	−0.507244	+	0.861802i	$0.669336\pi$
$998$	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.q.1.1		3
7.3	odd	6		861.2.i.c.247.1	✓	6
7.5	odd	6		861.2.i.c.739.1	yes	6
7.6	odd	2		6027.2.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.i.c.247.1	✓	6	7.3	odd	6
861.2.i.c.739.1	yes	6	7.5	odd	6
6027.2.a.p.1.3		3	7.6	odd	2
6027.2.a.q.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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} -2.55247 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} -2.55247 q^{5} +1.00000 q^{9} +6.17255 q^{11} -2.00000 q^{12} +2.06760 q^{13} -2.55247 q^{15} +4.00000 q^{16} +2.00000 q^{17} +0.447525 q^{19} +5.10495 q^{20} -0.552475 q^{23} +1.51513 q^{25} +1.00000 q^{27} -5.10495 q^{29} +7.72503 q^{31} +6.17255 q^{33} -2.00000 q^{36} -1.00000 q^{37} +2.06760 q^{39} +1.00000 q^{41} -10.6201 q^{43} -12.3451 q^{44} -2.55247 q^{45} +8.30775 q^{47} +4.00000 q^{48} +2.00000 q^{51} -4.13520 q^{52} -4.13520 q^{53} -15.7553 q^{55} +0.447525 q^{57} -8.13520 q^{59} +5.10495 q^{60} -7.75528 q^{61} -8.00000 q^{64} -5.27750 q^{65} -0.0676012 q^{67} -4.00000 q^{68} -0.552475 q^{69} +2.13520 q^{71} +10.6201 q^{73} +1.51513 q^{75} -0.895051 q^{76} +16.7926 q^{79} -10.2099 q^{80} +1.00000 q^{81} +13.9278 q^{83} -5.10495 q^{85} -5.10495 q^{87} -12.3078 q^{89} +1.10495 q^{92} +7.72503 q^{93} -1.14230 q^{95} -16.2099 q^{97} +6.17255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{4} + 3 q^{9}+O(q^{10}) \) 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

Fields

Representations

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Database

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