LMFDB - Embedded newform 7360.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} +1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +O(q^{10})$	$q-2.79129 q^{3} +1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} -0.791288 q^{11} -5.79129 q^{13} -2.79129 q^{15} +0.791288 q^{17} +5.79129 q^{19} -5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -7.58258 q^{29} +3.37386 q^{31} +2.20871 q^{33} +1.79129 q^{35} +4.00000 q^{37} +16.1652 q^{39} -6.79129 q^{41} +11.1652 q^{43} +4.79129 q^{45} +4.41742 q^{47} -3.79129 q^{49} -2.20871 q^{51} -6.00000 q^{53} -0.791288 q^{55} -16.1652 q^{57} -13.5826 q^{59} -10.3739 q^{61} +8.58258 q^{63} -5.79129 q^{65} +11.1652 q^{67} +2.79129 q^{69} -8.37386 q^{71} +12.7477 q^{73} -2.79129 q^{75} -1.41742 q^{77} -8.00000 q^{79} -0.417424 q^{81} -6.00000 q^{83} +0.791288 q^{85} +21.1652 q^{87} +15.1652 q^{89} -10.3739 q^{91} -9.41742 q^{93} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - q^7 + 5 * q^9	$ 2 q - q^{3} + 2 q^{5} - q^{7} + 5 q^{9} + 3 q^{11} - 7 q^{13} - q^{15} - 3 q^{17} + 7 q^{19} - 10 q^{21} - 2 q^{23} + 2 q^{25} - 10 q^{27} - 6 q^{29} - 7 q^{31} + 9 q^{33} - q^{35} + 8 q^{37} + 14 q^{39} - 9 q^{41} + 4 q^{43} + 5 q^{45} + 18 q^{47} - 3 q^{49} - 9 q^{51} - 12 q^{53} + 3 q^{55} - 14 q^{57} - 18 q^{59} - 7 q^{61} + 8 q^{63} - 7 q^{65} + 4 q^{67} + q^{69} - 3 q^{71} - 2 q^{73} - q^{75} - 12 q^{77} - 16 q^{79} - 10 q^{81} - 12 q^{83} - 3 q^{85} + 24 q^{87} + 12 q^{89} - 7 q^{91} - 28 q^{93} + 7 q^{95} + 7 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - q^7 + 5 * q^9 + 3 * q^11 - 7 * q^13 - q^15 - 3 * q^17 + 7 * q^19 - 10 * q^21 - 2 * q^23 + 2 * q^25 - 10 * q^27 - 6 * q^29 - 7 * q^31 + 9 * q^33 - q^35 + 8 * q^37 + 14 * q^39 - 9 * q^41 + 4 * q^43 + 5 * q^45 + 18 * q^47 - 3 * q^49 - 9 * q^51 - 12 * q^53 + 3 * q^55 - 14 * q^57 - 18 * q^59 - 7 * q^61 + 8 * q^63 - 7 * q^65 + 4 * q^67 + q^69 - 3 * q^71 - 2 * q^73 - q^75 - 12 * q^77 - 16 * q^79 - 10 * q^81 - 12 * q^83 - 3 * q^85 + 24 * q^87 + 12 * q^89 - 7 * q^91 - 28 * q^93 + 7 * q^95 + 7 * 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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.79129	0.677043	0.338522	−	0.940959i	$-0.390073\pi$
$7$	1.79129	0.677043	0.338522	+	0.940959i	$0.390073\pi$
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	−0.791288	−0.238582	−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288	−0.238582	−0.119291	+	0.992859i	$0.538062\pi$
$12$	0	0
$13$	−5.79129	−1.60621	−0.803107	−	0.595835i	$-0.796821\pi$
$13$	−5.79129	−1.60621	−0.803107	+	0.595835i	$0.796821\pi$
$14$	0	0
$15$	−2.79129	−0.720707
$16$	0	0
$17$	0.791288	0.191915	0.0959577	−	0.995385i	$-0.469409\pi$
$17$	0.791288	0.191915	0.0959577	+	0.995385i	$0.469409\pi$
$18$	0	0
$19$	5.79129	1.32861	0.664306	−	0.747460i	$-0.268727\pi$
$19$	5.79129	1.32861	0.664306	+	0.747460i	$0.268727\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−7.58258	−1.40805	−0.704024	−	0.710176i	$-0.748615\pi$
$29$	−7.58258	−1.40805	−0.704024	+	0.710176i	$0.748615\pi$
$30$	0	0
$31$	3.37386	0.605964	0.302982	−	0.952996i	$-0.402018\pi$
$31$	3.37386	0.605964	0.302982	+	0.952996i	$0.402018\pi$
$32$	0	0
$33$	2.20871	0.384487
$34$	0	0
$35$	1.79129	0.302783
$36$	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
$39$	16.1652	2.58850
$40$	0	0
$41$	−6.79129	−1.06062	−0.530310	−	0.847804i	$-0.677925\pi$
$41$	−6.79129	−1.06062	−0.530310	+	0.847804i	$0.677925\pi$
$42$	0	0
$43$	11.1652	1.70267	0.851335	−	0.524623i	$-0.175794\pi$
$43$	11.1652	1.70267	0.851335	+	0.524623i	$0.175794\pi$
$44$	0	0
$45$	4.79129	0.714243
$46$	0	0
$47$	4.41742	0.644348	0.322174	−	0.946681i	$-0.395586\pi$
$47$	4.41742	0.644348	0.322174	+	0.946681i	$0.395586\pi$
$48$	0	0
$49$	−3.79129	−0.541613
$50$	0	0
$51$	−2.20871	−0.309282
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−0.791288	−0.106697
$56$	0	0
$57$	−16.1652	−2.14113
$58$	0	0
$59$	−13.5826	−1.76830	−0.884150	−	0.467202i	$-0.845262\pi$
$59$	−13.5826	−1.76830	−0.884150	+	0.467202i	$0.845262\pi$
$60$	0	0
$61$	−10.3739	−1.32824	−0.664119	−	0.747627i	$-0.731193\pi$
$61$	−10.3739	−1.32824	−0.664119	+	0.747627i	$0.731193\pi$
$62$	0	0
$63$	8.58258	1.08130
$64$	0	0
$65$	−5.79129	−0.718321
$66$	0	0
$67$	11.1652	1.36404	0.682020	−	0.731333i	$-0.261102\pi$
$67$	11.1652	1.36404	0.682020	+	0.731333i	$0.261102\pi$
$68$	0	0
$69$	2.79129	0.336032
$70$	0	0
$71$	−8.37386	−0.993795	−0.496897	−	0.867809i	$-0.665527\pi$
$71$	−8.37386	−0.993795	−0.496897	+	0.867809i	$0.665527\pi$
$72$	0	0
$73$	12.7477	1.49201	0.746004	−	0.665941i	$-0.231970\pi$
$73$	12.7477	1.49201	0.746004	+	0.665941i	$0.231970\pi$
$74$	0	0
$75$	−2.79129	−0.322310
$76$	0	0
$77$	−1.41742	−0.161530
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0.791288	0.0858272
$86$	0	0
$87$	21.1652	2.26914
$88$	0	0
$89$	15.1652	1.60750	0.803751	−	0.594965i	$-0.202834\pi$
$89$	15.1652	1.60750	0.803751	+	0.594965i	$0.202834\pi$
$90$	0	0
$91$	−10.3739	−1.08748
$92$	0	0
$93$	−9.41742	−0.976541
$94$	0	0
$95$	5.79129	0.594174
$96$	0	0
$97$	−7.95644	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$97$	−7.95644	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$98$	0	0
$99$	−3.79129	−0.381039
$100$	0	0
$101$	−4.41742	−0.439550	−0.219775	−	0.975551i	$-0.570532\pi$
$101$	−4.41742	−0.439550	−0.219775	+	0.975551i	$0.570532\pi$
$102$	0	0
$103$	6.37386	0.628035	0.314018	−	0.949417i	$-0.398325\pi$
$103$	6.37386	0.628035	0.314018	+	0.949417i	$0.398325\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	4.41742	0.427049	0.213524	−	0.976938i	$-0.431506\pi$
$107$	4.41742	0.427049	0.213524	+	0.976938i	$0.431506\pi$
$108$	0	0
$109$	3.37386	0.323158	0.161579	−	0.986860i	$-0.448341\pi$
$109$	3.37386	0.323158	0.161579	+	0.986860i	$0.448341\pi$
$110$	0	0
$111$	−11.1652	−1.05975
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−27.7477	−2.56528
$118$	0	0
$119$	1.41742	0.129935
$120$	0	0
$121$	−10.3739	−0.943079
$122$	0	0
$123$	18.9564	1.70924
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.7477	−1.13118	−0.565589	−	0.824687i	$-0.691351\pi$
$127$	−12.7477	−1.13118	−0.565589	+	0.824687i	$0.691351\pi$
$128$	0	0
$129$	−31.1652	−2.74394
$130$	0	0
$131$	−9.16515	−0.800763	−0.400381	−	0.916349i	$-0.631122\pi$
$131$	−9.16515	−0.800763	−0.400381	+	0.916349i	$0.631122\pi$
$132$	0	0
$133$	10.3739	0.899528
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	3.79129	0.323912	0.161956	−	0.986798i	$-0.448220\pi$
$137$	3.79129	0.323912	0.161956	+	0.986798i	$0.448220\pi$
$138$	0	0
$139$	12.7477	1.08125	0.540624	−	0.841264i	$-0.318188\pi$
$139$	12.7477	1.08125	0.540624	+	0.841264i	$0.318188\pi$
$140$	0	0
$141$	−12.3303	−1.03840
$142$	0	0
$143$	4.58258	0.383214
$144$	0	0
$145$	−7.58258	−0.629699
$146$	0	0
$147$	10.5826	0.872836
$148$	0	0
$149$	−8.20871	−0.672484	−0.336242	−	0.941776i	$-0.609156\pi$
$149$	−8.20871	−0.672484	−0.336242	+	0.941776i	$0.609156\pi$
$150$	0	0
$151$	10.7913	0.878183	0.439091	−	0.898442i	$-0.355300\pi$
$151$	10.7913	0.878183	0.439091	+	0.898442i	$0.355300\pi$
$152$	0	0
$153$	3.79129	0.306507
$154$	0	0
$155$	3.37386	0.270995
$156$	0	0
$157$	14.7477	1.17700	0.588498	−	0.808498i	$-0.299719\pi$
$157$	14.7477	1.17700	0.588498	+	0.808498i	$0.299719\pi$
$158$	0	0
$159$	16.7477	1.32818
$160$	0	0
$161$	−1.79129	−0.141173
$162$	0	0
$163$	8.62614	0.675651	0.337826	−	0.941209i	$-0.390309\pi$
$163$	8.62614	0.675651	0.337826	+	0.941209i	$0.390309\pi$
$164$	0	0
$165$	2.20871	0.171948
$166$	0	0
$167$	−18.3303	−1.41844	−0.709221	−	0.704987i	$-0.750953\pi$
$167$	−18.3303	−1.41844	−0.709221	+	0.704987i	$0.750953\pi$
$168$	0	0
$169$	20.5390	1.57992
$170$	0	0
$171$	27.7477	2.12192
$172$	0	0
$173$	18.7913	1.42868	0.714338	−	0.699801i	$-0.246728\pi$
$173$	18.7913	1.42868	0.714338	+	0.699801i	$0.246728\pi$
$174$	0	0
$175$	1.79129	0.135409
$176$	0	0
$177$	37.9129	2.84971
$178$	0	0
$179$	10.7477	0.803323	0.401661	−	0.915788i	$-0.368433\pi$
$179$	10.7477	0.803323	0.401661	+	0.915788i	$0.368433\pi$
$180$	0	0
$181$	18.5390	1.37799	0.688997	−	0.724764i	$-0.258051\pi$
$181$	18.5390	1.37799	0.688997	+	0.724764i	$0.258051\pi$
$182$	0	0
$183$	28.9564	2.14052
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−0.626136	−0.0457876
$188$	0	0
$189$	−8.95644	−0.651485
$190$	0	0
$191$	−25.5826	−1.85109	−0.925545	−	0.378637i	$-0.876393\pi$
$191$	−25.5826	−1.85109	−0.925545	+	0.378637i	$0.876393\pi$
$192$	0	0
$193$	−20.7477	−1.49345	−0.746727	−	0.665131i	$-0.768376\pi$
$193$	−20.7477	−1.49345	−0.746727	+	0.665131i	$0.768376\pi$
$194$	0	0
$195$	16.1652	1.15761
$196$	0	0
$197$	11.5390	0.822121	0.411060	−	0.911608i	$-0.365159\pi$
$197$	11.5390	0.822121	0.411060	+	0.911608i	$0.365159\pi$
$198$	0	0
$199$	16.3303	1.15762	0.578812	−	0.815461i	$-0.303516\pi$
$199$	16.3303	1.15762	0.578812	+	0.815461i	$0.303516\pi$
$200$	0	0
$201$	−31.1652	−2.19822
$202$	0	0
$203$	−13.5826	−0.953310
$204$	0	0
$205$	−6.79129	−0.474324
$206$	0	0
$207$	−4.79129	−0.333018
$208$	0	0
$209$	−4.58258	−0.316983
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	23.3739	1.60155
$214$	0	0
$215$	11.1652	0.761457
$216$	0	0
$217$	6.04356	0.410264
$218$	0	0
$219$	−35.5826	−2.40445
$220$	0	0
$221$	−4.58258	−0.308257
$222$	0	0
$223$	7.16515	0.479814	0.239907	−	0.970796i	$-0.422883\pi$
$223$	7.16515	0.479814	0.239907	+	0.970796i	$0.422883\pi$
$224$	0	0
$225$	4.79129	0.319419
$226$	0	0
$227$	−22.7477	−1.50982	−0.754910	−	0.655829i	$-0.772319\pi$
$227$	−22.7477	−1.50982	−0.754910	+	0.655829i	$0.772319\pi$
$228$	0	0
$229$	−20.3303	−1.34346	−0.671732	−	0.740794i	$-0.734449\pi$
$229$	−20.3303	−1.34346	−0.671732	+	0.740794i	$0.734449\pi$
$230$	0	0
$231$	3.95644	0.260315
$232$	0	0
$233$	−1.58258	−0.103678	−0.0518390	−	0.998655i	$-0.516508\pi$
$233$	−1.58258	−0.103678	−0.0518390	+	0.998655i	$0.516508\pi$
$234$	0	0
$235$	4.41742	0.288161
$236$	0	0
$237$	22.3303	1.45051
$238$	0	0
$239$	15.1652	0.980952	0.490476	−	0.871455i	$-0.336823\pi$
$239$	15.1652	0.980952	0.490476	+	0.871455i	$0.336823\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	−3.79129	−0.242216
$246$	0	0
$247$	−33.5390	−2.13404
$248$	0	0
$249$	16.7477	1.06134
$250$	0	0
$251$	26.2087	1.65428	0.827140	−	0.561996i	$-0.189967\pi$
$251$	26.2087	1.65428	0.827140	+	0.561996i	$0.189967\pi$
$252$	0	0
$253$	0.791288	0.0497478
$254$	0	0
$255$	−2.20871	−0.138315
$256$	0	0
$257$	−4.74773	−0.296155	−0.148078	−	0.988976i	$-0.547309\pi$
$257$	−4.74773	−0.296155	−0.148078	+	0.988976i	$0.547309\pi$
$258$	0	0
$259$	7.16515	0.445221
$260$	0	0
$261$	−36.3303	−2.24879
$262$	0	0
$263$	−11.2087	−0.691159	−0.345579	−	0.938390i	$-0.612318\pi$
$263$	−11.2087	−0.691159	−0.345579	+	0.938390i	$0.612318\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−42.3303	−2.59057
$268$	0	0
$269$	10.7477	0.655300	0.327650	−	0.944799i	$-0.393743\pi$
$269$	10.7477	0.655300	0.327650	+	0.944799i	$0.393743\pi$
$270$	0	0
$271$	−18.1216	−1.10081	−0.550404	−	0.834898i	$-0.685526\pi$
$271$	−18.1216	−1.10081	−0.550404	+	0.834898i	$0.685526\pi$
$272$	0	0
$273$	28.9564	1.75252
$274$	0	0
$275$	−0.791288	−0.0477165
$276$	0	0
$277$	−17.1652	−1.03135	−0.515677	−	0.856783i	$-0.672460\pi$
$277$	−17.1652	−1.03135	−0.515677	+	0.856783i	$0.672460\pi$
$278$	0	0
$279$	16.1652	0.967782
$280$	0	0
$281$	10.7477	0.641156	0.320578	−	0.947222i	$-0.396123\pi$
$281$	10.7477	0.641156	0.320578	+	0.947222i	$0.396123\pi$
$282$	0	0
$283$	8.33030	0.495185	0.247593	−	0.968864i	$-0.420361\pi$
$283$	8.33030	0.495185	0.247593	+	0.968864i	$0.420361\pi$
$284$	0	0
$285$	−16.1652	−0.957541
$286$	0	0
$287$	−12.1652	−0.718086
$288$	0	0
$289$	−16.3739	−0.963168
$290$	0	0
$291$	22.2087	1.30190
$292$	0	0
$293$	−27.4955	−1.60630	−0.803151	−	0.595776i	$-0.796845\pi$
$293$	−27.4955	−1.60630	−0.803151	+	0.595776i	$0.796845\pi$
$294$	0	0
$295$	−13.5826	−0.790808
$296$	0	0
$297$	3.95644	0.229576
$298$	0	0
$299$	5.79129	0.334919
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	12.3303	0.708357
$304$	0	0
$305$	−10.3739	−0.594006
$306$	0	0
$307$	−15.5390	−0.886858	−0.443429	−	0.896309i	$-0.646238\pi$
$307$	−15.5390	−0.886858	−0.443429	+	0.896309i	$0.646238\pi$
$308$	0	0
$309$	−17.7913	−1.01211
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−4.62614	−0.261485	−0.130742	−	0.991416i	$-0.541736\pi$
$313$	−4.62614	−0.261485	−0.130742	+	0.991416i	$0.541736\pi$
$314$	0	0
$315$	8.58258	0.483573
$316$	0	0
$317$	−9.79129	−0.549934	−0.274967	−	0.961454i	$-0.588667\pi$
$317$	−9.79129	−0.549934	−0.274967	+	0.961454i	$0.588667\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−12.3303	−0.688210
$322$	0	0
$323$	4.58258	0.254981
$324$	0	0
$325$	−5.79129	−0.321243
$326$	0	0
$327$	−9.41742	−0.520785
$328$	0	0
$329$	7.91288	0.436251
$330$	0	0
$331$	−20.7477	−1.14040	−0.570199	−	0.821507i	$-0.693134\pi$
$331$	−20.7477	−1.14040	−0.570199	+	0.821507i	$0.693134\pi$
$332$	0	0
$333$	19.1652	1.05024
$334$	0	0
$335$	11.1652	0.610017
$336$	0	0
$337$	−12.2087	−0.665051	−0.332525	−	0.943094i	$-0.607901\pi$
$337$	−12.2087	−0.665051	−0.332525	+	0.943094i	$0.607901\pi$
$338$	0	0
$339$	−16.7477	−0.909612
$340$	0	0
$341$	−2.66970	−0.144572
$342$	0	0
$343$	−19.3303	−1.04374
$344$	0	0
$345$	2.79129	0.150278
$346$	0	0
$347$	5.20871	0.279618	0.139809	−	0.990178i	$-0.455351\pi$
$347$	5.20871	0.279618	0.139809	+	0.990178i	$0.455351\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	28.9564	1.54558
$352$	0	0
$353$	3.16515	0.168464	0.0842320	−	0.996446i	$-0.473156\pi$
$353$	3.16515	0.168464	0.0842320	+	0.996446i	$0.473156\pi$
$354$	0	0
$355$	−8.37386	−0.444439
$356$	0	0
$357$	−3.95644	−0.209397
$358$	0	0
$359$	−9.16515	−0.483718	−0.241859	−	0.970311i	$-0.577757\pi$
$359$	−9.16515	−0.483718	−0.241859	+	0.970311i	$0.577757\pi$
$360$	0	0
$361$	14.5390	0.765211
$362$	0	0
$363$	28.9564	1.51982
$364$	0	0
$365$	12.7477	0.667247
$366$	0	0
$367$	19.1652	1.00041	0.500206	−	0.865906i	$-0.333257\pi$
$367$	19.1652	1.00041	0.500206	+	0.865906i	$0.333257\pi$
$368$	0	0
$369$	−32.5390	−1.69391
$370$	0	0
$371$	−10.7477	−0.557994
$372$	0	0
$373$	−12.7477	−0.660052	−0.330026	−	0.943972i	$-0.607058\pi$
$373$	−12.7477	−0.660052	−0.330026	+	0.943972i	$0.607058\pi$
$374$	0	0
$375$	−2.79129	−0.144141
$376$	0	0
$377$	43.9129	2.26163
$378$	0	0
$379$	−6.37386	−0.327403	−0.163702	−	0.986510i	$-0.552343\pi$
$379$	−6.37386	−0.327403	−0.163702	+	0.986510i	$0.552343\pi$
$380$	0	0
$381$	35.5826	1.82295
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−1.41742	−0.0722386
$386$	0	0
$387$	53.4955	2.71933
$388$	0	0
$389$	20.7042	1.04974	0.524871	−	0.851182i	$-0.324113\pi$
$389$	20.7042	1.04974	0.524871	+	0.851182i	$0.324113\pi$
$390$	0	0
$391$	−0.791288	−0.0400171
$392$	0	0
$393$	25.5826	1.29047
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	15.5390	0.779881	0.389940	−	0.920840i	$-0.372496\pi$
$397$	15.5390	0.779881	0.389940	+	0.920840i	$0.372496\pi$
$398$	0	0
$399$	−28.9564	−1.44964
$400$	0	0
$401$	−4.74773	−0.237090	−0.118545	−	0.992949i	$-0.537823\pi$
$401$	−4.74773	−0.237090	−0.118545	+	0.992949i	$0.537823\pi$
$402$	0	0
$403$	−19.5390	−0.973308
$404$	0	0
$405$	−0.417424	−0.0207420
$406$	0	0
$407$	−3.16515	−0.156891
$408$	0	0
$409$	−18.2087	−0.900363	−0.450181	−	0.892937i	$-0.648641\pi$
$409$	−18.2087	−0.900363	−0.450181	+	0.892937i	$0.648641\pi$
$410$	0	0
$411$	−10.5826	−0.522000
$412$	0	0
$413$	−24.3303	−1.19722
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−35.5826	−1.74249
$418$	0	0
$419$	20.8348	1.01785	0.508924	−	0.860811i	$-0.330043\pi$
$419$	20.8348	1.01785	0.508924	+	0.860811i	$0.330043\pi$
$420$	0	0
$421$	−18.1216	−0.883192	−0.441596	−	0.897214i	$-0.645588\pi$
$421$	−18.1216	−0.883192	−0.441596	+	0.897214i	$0.645588\pi$
$422$	0	0
$423$	21.1652	1.02908
$424$	0	0
$425$	0.791288	0.0383831
$426$	0	0
$427$	−18.5826	−0.899274
$428$	0	0
$429$	−12.7913	−0.617569
$430$	0	0
$431$	−25.9129	−1.24818	−0.624090	−	0.781353i	$-0.714530\pi$
$431$	−25.9129	−1.24818	−0.624090	+	0.781353i	$0.714530\pi$
$432$	0	0
$433$	−30.5390	−1.46761	−0.733806	−	0.679359i	$-0.762258\pi$
$433$	−30.5390	−1.46761	−0.733806	+	0.679359i	$0.762258\pi$
$434$	0	0
$435$	21.1652	1.01479
$436$	0	0
$437$	−5.79129	−0.277035
$438$	0	0
$439$	6.53901	0.312090	0.156045	−	0.987750i	$-0.450125\pi$
$439$	6.53901	0.312090	0.156045	+	0.987750i	$0.450125\pi$
$440$	0	0
$441$	−18.1652	−0.865007
$442$	0	0
$443$	−39.7913	−1.89054	−0.945271	−	0.326288i	$-0.894202\pi$
$443$	−39.7913	−1.89054	−0.945271	+	0.326288i	$0.894202\pi$
$444$	0	0
$445$	15.1652	0.718897
$446$	0	0
$447$	22.9129	1.08374
$448$	0	0
$449$	−16.1216	−0.760825	−0.380412	−	0.924817i	$-0.624218\pi$
$449$	−16.1216	−0.760825	−0.380412	+	0.924817i	$0.624218\pi$
$450$	0	0
$451$	5.37386	0.253045
$452$	0	0
$453$	−30.1216	−1.41524
$454$	0	0
$455$	−10.3739	−0.486334
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	−3.95644	−0.184671
$460$	0	0
$461$	28.7477	1.33892	0.669458	−	0.742850i	$-0.266527\pi$
$461$	28.7477	1.33892	0.669458	+	0.742850i	$0.266527\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	0	0
$465$	−9.41742	−0.436723
$466$	0	0
$467$	−19.9129	−0.921458	−0.460729	−	0.887541i	$-0.652412\pi$
$467$	−19.9129	−0.921458	−0.460729	+	0.887541i	$0.652412\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	−41.1652	−1.89679
$472$	0	0
$473$	−8.83485	−0.406227
$474$	0	0
$475$	5.79129	0.265723
$476$	0	0
$477$	−28.7477	−1.31627
$478$	0	0
$479$	−15.4955	−0.708005	−0.354003	−	0.935244i	$-0.615180\pi$
$479$	−15.4955	−0.708005	−0.354003	+	0.935244i	$0.615180\pi$
$480$	0	0
$481$	−23.1652	−1.05624
$482$	0	0
$483$	5.00000	0.227508
$484$	0	0
$485$	−7.95644	−0.361283
$486$	0	0
$487$	−6.41742	−0.290801	−0.145401	−	0.989373i	$-0.546447\pi$
$487$	−6.41742	−0.290801	−0.145401	+	0.989373i	$0.546447\pi$
$488$	0	0
$489$	−24.0780	−1.08885
$490$	0	0
$491$	10.7477	0.485038	0.242519	−	0.970147i	$-0.422026\pi$
$491$	10.7477	0.485038	0.242519	+	0.970147i	$0.422026\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−3.79129	−0.170406
$496$	0	0
$497$	−15.0000	−0.672842
$498$	0	0
$499$	4.83485	0.216438	0.108219	−	0.994127i	$-0.465485\pi$
$499$	4.83485	0.216438	0.108219	+	0.994127i	$0.465485\pi$
$500$	0	0
$501$	51.1652	2.28589
$502$	0	0
$503$	−14.2087	−0.633535	−0.316768	−	0.948503i	$-0.602598\pi$
$503$	−14.2087	−0.633535	−0.316768	+	0.948503i	$0.602598\pi$
$504$	0	0
$505$	−4.41742	−0.196573
$506$	0	0
$507$	−57.3303	−2.54613
$508$	0	0
$509$	−34.7477	−1.54017	−0.770083	−	0.637944i	$-0.779785\pi$
$509$	−34.7477	−1.54017	−0.770083	+	0.637944i	$0.779785\pi$
$510$	0	0
$511$	22.8348	1.01015
$512$	0	0
$513$	−28.9564	−1.27846
$514$	0	0
$515$	6.37386	0.280866
$516$	0	0
$517$	−3.49545	−0.153730
$518$	0	0
$519$	−52.4519	−2.30238
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	17.1652	0.750580	0.375290	−	0.926908i	$-0.377543\pi$
$523$	17.1652	0.750580	0.375290	+	0.926908i	$0.377543\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	2.66970	0.116294
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−65.0780	−2.82415
$532$	0	0
$533$	39.3303	1.70358
$534$	0	0
$535$	4.41742	0.190982
$536$	0	0
$537$	−30.0000	−1.29460
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−1.66970	−0.0717859	−0.0358929	−	0.999356i	$-0.511428\pi$
$541$	−1.66970	−0.0717859	−0.0358929	+	0.999356i	$0.511428\pi$
$542$	0	0
$543$	−51.7477	−2.22071
$544$	0	0
$545$	3.37386	0.144520
$546$	0	0
$547$	−26.1216	−1.11688	−0.558439	−	0.829545i	$-0.688600\pi$
$547$	−26.1216	−1.11688	−0.558439	+	0.829545i	$0.688600\pi$
$548$	0	0
$549$	−49.7042	−2.12132
$550$	0	0
$551$	−43.9129	−1.87075
$552$	0	0
$553$	−14.3303	−0.609386
$554$	0	0
$555$	−11.1652	−0.473934
$556$	0	0
$557$	6.33030	0.268224	0.134112	−	0.990966i	$-0.457182\pi$
$557$	6.33030	0.268224	0.134112	+	0.990966i	$0.457182\pi$
$558$	0	0
$559$	−64.6606	−2.73485
$560$	0	0
$561$	1.74773	0.0737891
$562$	0	0
$563$	−15.1652	−0.639135	−0.319567	−	0.947564i	$-0.603538\pi$
$563$	−15.1652	−0.639135	−0.319567	+	0.947564i	$0.603538\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−0.747727	−0.0314016
$568$	0	0
$569$	−39.4955	−1.65574	−0.827868	−	0.560923i	$-0.810446\pi$
$569$	−39.4955	−1.65574	−0.827868	+	0.560923i	$0.810446\pi$
$570$	0	0
$571$	−11.1216	−0.465424	−0.232712	−	0.972546i	$-0.574760\pi$
$571$	−11.1216	−0.465424	−0.232712	+	0.972546i	$0.574760\pi$
$572$	0	0
$573$	71.4083	2.98313
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	41.1652	1.71373	0.856864	−	0.515543i	$-0.172410\pi$
$577$	41.1652	1.71373	0.856864	+	0.515543i	$0.172410\pi$
$578$	0	0
$579$	57.9129	2.40678
$580$	0	0
$581$	−10.7477	−0.445891
$582$	0	0
$583$	4.74773	0.196631
$584$	0	0
$585$	−27.7477	−1.14723
$586$	0	0
$587$	−30.7913	−1.27089	−0.635446	−	0.772145i	$-0.719184\pi$
$587$	−30.7913	−1.27089	−0.635446	+	0.772145i	$0.719184\pi$
$588$	0	0
$589$	19.5390	0.805091
$590$	0	0
$591$	−32.2087	−1.32489
$592$	0	0
$593$	−31.9129	−1.31050	−0.655252	−	0.755410i	$-0.727438\pi$
$593$	−31.9129	−1.31050	−0.655252	+	0.755410i	$0.727438\pi$
$594$	0	0
$595$	1.41742	0.0581087
$596$	0	0
$597$	−45.5826	−1.86557
$598$	0	0
$599$	−1.12159	−0.0458270	−0.0229135	−	0.999737i	$-0.507294\pi$
$599$	−1.12159	−0.0458270	−0.0229135	+	0.999737i	$0.507294\pi$
$600$	0	0
$601$	−18.2087	−0.742749	−0.371374	−	0.928483i	$-0.621113\pi$
$601$	−18.2087	−0.742749	−0.371374	+	0.928483i	$0.621113\pi$
$602$	0	0
$603$	53.4955	2.17850
$604$	0	0
$605$	−10.3739	−0.421758
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	37.9129	1.53631
$610$	0	0
$611$	−25.5826	−1.03496
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	18.9564	0.764397
$616$	0	0
$617$	−23.8693	−0.960943	−0.480471	−	0.877010i	$-0.659534\pi$
$617$	−23.8693	−0.960943	−0.480471	+	0.877010i	$0.659534\pi$
$618$	0	0
$619$	−1.79129	−0.0719979	−0.0359990	−	0.999352i	$-0.511461\pi$
$619$	−1.79129	−0.0719979	−0.0359990	+	0.999352i	$0.511461\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	27.1652	1.08835
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.7913	0.510835
$628$	0	0
$629$	3.16515	0.126203
$630$	0	0
$631$	−27.9129	−1.11119	−0.555597	−	0.831452i	$-0.687510\pi$
$631$	−27.9129	−1.11119	−0.555597	+	0.831452i	$0.687510\pi$
$632$	0	0
$633$	27.9129	1.10944
$634$	0	0
$635$	−12.7477	−0.505878
$636$	0	0
$637$	21.9564	0.869946
$638$	0	0
$639$	−40.1216	−1.58719
$640$	0	0
$641$	15.1652	0.598987	0.299494	−	0.954098i	$-0.403182\pi$
$641$	15.1652	0.598987	0.299494	+	0.954098i	$0.403182\pi$
$642$	0	0
$643$	6.74773	0.266104	0.133052	−	0.991109i	$-0.457522\pi$
$643$	6.74773	0.266104	0.133052	+	0.991109i	$0.457522\pi$
$644$	0	0
$645$	−31.1652	−1.22713
$646$	0	0
$647$	−21.1652	−0.832088	−0.416044	−	0.909344i	$-0.636584\pi$
$647$	−21.1652	−0.832088	−0.416044	+	0.909344i	$0.636584\pi$
$648$	0	0
$649$	10.7477	0.421885
$650$	0	0
$651$	−16.8693	−0.661161
$652$	0	0
$653$	−3.46099	−0.135439	−0.0677194	−	0.997704i	$-0.521572\pi$
$653$	−3.46099	−0.135439	−0.0677194	+	0.997704i	$0.521572\pi$
$654$	0	0
$655$	−9.16515	−0.358112
$656$	0	0
$657$	61.0780	2.38288
$658$	0	0
$659$	−8.83485	−0.344157	−0.172078	−	0.985083i	$-0.555048\pi$
$659$	−8.83485	−0.344157	−0.172078	+	0.985083i	$0.555048\pi$
$660$	0	0
$661$	25.6261	0.996741	0.498371	−	0.866964i	$-0.333932\pi$
$661$	25.6261	0.996741	0.498371	+	0.866964i	$0.333932\pi$
$662$	0	0
$663$	12.7913	0.496772
$664$	0	0
$665$	10.3739	0.402281
$666$	0	0
$667$	7.58258	0.293599
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	8.20871	0.316894
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−42.6606	−1.63958	−0.819790	−	0.572664i	$-0.805910\pi$
$677$	−42.6606	−1.63958	−0.819790	+	0.572664i	$0.805910\pi$
$678$	0	0
$679$	−14.2523	−0.546952
$680$	0	0
$681$	63.4955	2.43315
$682$	0	0
$683$	−11.3739	−0.435209	−0.217604	−	0.976037i	$-0.569824\pi$
$683$	−11.3739	−0.435209	−0.217604	+	0.976037i	$0.569824\pi$
$684$	0	0
$685$	3.79129	0.144858
$686$	0	0
$687$	56.7477	2.16506
$688$	0	0
$689$	34.7477	1.32378
$690$	0	0
$691$	42.7477	1.62620	0.813100	−	0.582124i	$-0.197778\pi$
$691$	42.7477	1.62620	0.813100	+	0.582124i	$0.197778\pi$
$692$	0	0
$693$	−6.79129	−0.257980
$694$	0	0
$695$	12.7477	0.483549
$696$	0	0
$697$	−5.37386	−0.203550
$698$	0	0
$699$	4.41742	0.167082
$700$	0	0
$701$	−23.3739	−0.882819	−0.441409	−	0.897306i	$-0.645521\pi$
$701$	−23.3739	−0.882819	−0.441409	+	0.897306i	$0.645521\pi$
$702$	0	0
$703$	23.1652	0.873690
$704$	0	0
$705$	−12.3303	−0.464386
$706$	0	0
$707$	−7.91288	−0.297594
$708$	0	0
$709$	−2.46099	−0.0924242	−0.0462121	−	0.998932i	$-0.514715\pi$
$709$	−2.46099	−0.0924242	−0.0462121	+	0.998932i	$0.514715\pi$
$710$	0	0
$711$	−38.3303	−1.43750
$712$	0	0
$713$	−3.37386	−0.126352
$714$	0	0
$715$	4.58258	0.171379
$716$	0	0
$717$	−42.3303	−1.58085
$718$	0	0
$719$	−2.53901	−0.0946893	−0.0473446	−	0.998879i	$-0.515076\pi$
$719$	−2.53901	−0.0946893	−0.0473446	+	0.998879i	$0.515076\pi$
$720$	0	0
$721$	11.4174	0.425207
$722$	0	0
$723$	78.1561	2.90666
$724$	0	0
$725$	−7.58258	−0.281610
$726$	0	0
$727$	−39.1216	−1.45094	−0.725470	−	0.688254i	$-0.758377\pi$
$727$	−39.1216	−1.45094	−0.725470	+	0.688254i	$0.758377\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	8.83485	0.326769
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	0	0
$735$	10.5826	0.390344
$736$	0	0
$737$	−8.83485	−0.325436
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	93.6170	3.43911
$742$	0	0
$743$	12.9564	0.475326	0.237663	−	0.971348i	$-0.423619\pi$
$743$	12.9564	0.475326	0.237663	+	0.971348i	$0.423619\pi$
$744$	0	0
$745$	−8.20871	−0.300744
$746$	0	0
$747$	−28.7477	−1.05182
$748$	0	0
$749$	7.91288	0.289130
$750$	0	0
$751$	8.74773	0.319209	0.159605	−	0.987181i	$-0.448978\pi$
$751$	8.74773	0.319209	0.159605	+	0.987181i	$0.448978\pi$
$752$	0	0
$753$	−73.1561	−2.66596
$754$	0	0
$755$	10.7913	0.392735
$756$	0	0
$757$	10.3303	0.375461	0.187731	−	0.982221i	$-0.439887\pi$
$757$	10.3303	0.375461	0.187731	+	0.982221i	$0.439887\pi$
$758$	0	0
$759$	−2.20871	−0.0801712
$760$	0	0
$761$	−11.0436	−0.400329	−0.200164	−	0.979762i	$-0.564148\pi$
$761$	−11.0436	−0.400329	−0.200164	+	0.979762i	$0.564148\pi$
$762$	0	0
$763$	6.04356	0.218792
$764$	0	0
$765$	3.79129	0.137074
$766$	0	0
$767$	78.6606	2.84027
$768$	0	0
$769$	−40.3303	−1.45435	−0.727174	−	0.686453i	$-0.759167\pi$
$769$	−40.3303	−1.45435	−0.727174	+	0.686453i	$0.759167\pi$
$770$	0	0
$771$	13.2523	0.477269
$772$	0	0
$773$	−33.4955	−1.20475	−0.602374	−	0.798214i	$-0.705778\pi$
$773$	−33.4955	−1.20475	−0.602374	+	0.798214i	$0.705778\pi$
$774$	0	0
$775$	3.37386	0.121193
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	−39.3303	−1.40915
$780$	0	0
$781$	6.62614	0.237102
$782$	0	0
$783$	37.9129	1.35490
$784$	0	0
$785$	14.7477	0.526369
$786$	0	0
$787$	−17.5826	−0.626751	−0.313376	−	0.949629i	$-0.601460\pi$
$787$	−17.5826	−0.626751	−0.313376	+	0.949629i	$0.601460\pi$
$788$	0	0
$789$	31.2867	1.11384
$790$	0	0
$791$	10.7477	0.382145
$792$	0	0
$793$	60.0780	2.13343
$794$	0	0
$795$	16.7477	0.593981
$796$	0	0
$797$	4.08712	0.144773	0.0723866	−	0.997377i	$-0.476938\pi$
$797$	4.08712	0.144773	0.0723866	+	0.997377i	$0.476938\pi$
$798$	0	0
$799$	3.49545	0.123660
$800$	0	0
$801$	72.6606	2.56734
$802$	0	0
$803$	−10.0871	−0.355967
$804$	0	0
$805$	−1.79129	−0.0631346
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	33.9564	1.19384	0.596922	−	0.802299i	$-0.296390\pi$
$809$	33.9564	1.19384	0.596922	+	0.802299i	$0.296390\pi$
$810$	0	0
$811$	−2.08712	−0.0732887	−0.0366444	−	0.999328i	$-0.511667\pi$
$811$	−2.08712	−0.0732887	−0.0366444	+	0.999328i	$0.511667\pi$
$812$	0	0
$813$	50.5826	1.77401
$814$	0	0
$815$	8.62614	0.302160
$816$	0	0
$817$	64.6606	2.26219
$818$	0	0
$819$	−49.7042	−1.73680
$820$	0	0
$821$	−21.1652	−0.738669	−0.369334	−	0.929297i	$-0.620414\pi$
$821$	−21.1652	−0.738669	−0.369334	+	0.929297i	$0.620414\pi$
$822$	0	0
$823$	−22.8348	−0.795973	−0.397986	−	0.917391i	$-0.630291\pi$
$823$	−22.8348	−0.795973	−0.397986	+	0.917391i	$0.630291\pi$
$824$	0	0
$825$	2.20871	0.0768975
$826$	0	0
$827$	−23.0780	−0.802502	−0.401251	−	0.915968i	$-0.631424\pi$
$827$	−23.0780	−0.802502	−0.401251	+	0.915968i	$0.631424\pi$
$828$	0	0
$829$	−23.4955	−0.816031	−0.408015	−	0.912975i	$-0.633779\pi$
$829$	−23.4955	−0.816031	−0.408015	+	0.912975i	$0.633779\pi$
$830$	0	0
$831$	47.9129	1.66208
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−18.3303	−0.634346
$836$	0	0
$837$	−16.8693	−0.583089
$838$	0	0
$839$	31.5826	1.09035	0.545176	−	0.838322i	$-0.316463\pi$
$839$	31.5826	1.09035	0.545176	+	0.838322i	$0.316463\pi$
$840$	0	0
$841$	28.4955	0.982602
$842$	0	0
$843$	−30.0000	−1.03325
$844$	0	0
$845$	20.5390	0.706564
$846$	0	0
$847$	−18.5826	−0.638505
$848$	0	0
$849$	−23.2523	−0.798016
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−40.5390	−1.38803	−0.694015	−	0.719961i	$-0.744160\pi$
$853$	−40.5390	−1.38803	−0.694015	+	0.719961i	$0.744160\pi$
$854$	0	0
$855$	27.7477	0.948952
$856$	0	0
$857$	−9.16515	−0.313076	−0.156538	−	0.987672i	$-0.550033\pi$
$857$	−9.16515	−0.313076	−0.156538	+	0.987672i	$0.550033\pi$
$858$	0	0
$859$	−26.7477	−0.912621	−0.456310	−	0.889821i	$-0.650829\pi$
$859$	−26.7477	−0.912621	−0.456310	+	0.889821i	$0.650829\pi$
$860$	0	0
$861$	33.9564	1.15723
$862$	0	0
$863$	22.4174	0.763098	0.381549	−	0.924349i	$-0.375391\pi$
$863$	22.4174	0.763098	0.381549	+	0.924349i	$0.375391\pi$
$864$	0	0
$865$	18.7913	0.638923
$866$	0	0
$867$	45.7042	1.55219
$868$	0	0
$869$	6.33030	0.214741
$870$	0	0
$871$	−64.6606	−2.19094
$872$	0	0
$873$	−38.1216	−1.29022
$874$	0	0
$875$	1.79129	0.0605566
$876$	0	0
$877$	42.7042	1.44202	0.721009	−	0.692926i	$-0.243679\pi$
$877$	42.7042	1.44202	0.721009	+	0.692926i	$0.243679\pi$
$878$	0	0
$879$	76.7477	2.58864
$880$	0	0
$881$	30.3303	1.02185	0.510927	−	0.859624i	$-0.329302\pi$
$881$	30.3303	1.02185	0.510927	+	0.859624i	$0.329302\pi$
$882$	0	0
$883$	−34.9564	−1.17638	−0.588189	−	0.808724i	$-0.700159\pi$
$883$	−34.9564	−1.17638	−0.588189	+	0.808724i	$0.700159\pi$
$884$	0	0
$885$	37.9129	1.27443
$886$	0	0
$887$	−15.1652	−0.509196	−0.254598	−	0.967047i	$-0.581943\pi$
$887$	−15.1652	−0.509196	−0.254598	+	0.967047i	$0.581943\pi$
$888$	0	0
$889$	−22.8348	−0.765856
$890$	0	0
$891$	0.330303	0.0110656
$892$	0	0
$893$	25.5826	0.856088
$894$	0	0
$895$	10.7477	0.359257
$896$	0	0
$897$	−16.1652	−0.539739
$898$	0	0
$899$	−25.5826	−0.853227
$900$	0	0
$901$	−4.74773	−0.158170
$902$	0	0
$903$	−55.8258	−1.85776
$904$	0	0
$905$	18.5390	0.616258
$906$	0	0
$907$	6.74773	0.224055	0.112027	−	0.993705i	$-0.464266\pi$
$907$	6.74773	0.224055	0.112027	+	0.993705i	$0.464266\pi$
$908$	0	0
$909$	−21.1652	−0.702004
$910$	0	0
$911$	−4.41742	−0.146356	−0.0731779	−	0.997319i	$-0.523314\pi$
$911$	−4.41742	−0.146356	−0.0731779	+	0.997319i	$0.523314\pi$
$912$	0	0
$913$	4.74773	0.157127
$914$	0	0
$915$	28.9564	0.957270
$916$	0	0
$917$	−16.4174	−0.542151
$918$	0	0
$919$	55.1652	1.81973	0.909865	−	0.414904i	$-0.136185\pi$
$919$	55.1652	1.81973	0.909865	+	0.414904i	$0.136185\pi$
$920$	0	0
$921$	43.3739	1.42922
$922$	0	0
$923$	48.4955	1.59625
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	30.5390	1.00303
$928$	0	0
$929$	15.4955	0.508389	0.254195	−	0.967153i	$-0.418190\pi$
$929$	15.4955	0.508389	0.254195	+	0.967153i	$0.418190\pi$
$930$	0	0
$931$	−21.9564	−0.719593
$932$	0	0
$933$	33.4955	1.09659
$934$	0	0
$935$	−0.626136	−0.0204769
$936$	0	0
$937$	44.6261	1.45787	0.728936	−	0.684582i	$-0.240015\pi$
$937$	44.6261	1.45787	0.728936	+	0.684582i	$0.240015\pi$
$938$	0	0
$939$	12.9129	0.421396
$940$	0	0
$941$	−32.0436	−1.04459	−0.522295	−	0.852765i	$-0.674924\pi$
$941$	−32.0436	−1.04459	−0.522295	+	0.852765i	$0.674924\pi$
$942$	0	0
$943$	6.79129	0.221155
$944$	0	0
$945$	−8.95644	−0.291353
$946$	0	0
$947$	2.53901	0.0825069	0.0412534	−	0.999149i	$-0.486865\pi$
$947$	2.53901	0.0825069	0.0412534	+	0.999149i	$0.486865\pi$
$948$	0	0
$949$	−73.8258	−2.39649
$950$	0	0
$951$	27.3303	0.886246
$952$	0	0
$953$	5.53901	0.179426	0.0897131	−	0.995968i	$-0.471405\pi$
$953$	5.53901	0.179426	0.0897131	+	0.995968i	$0.471405\pi$
$954$	0	0
$955$	−25.5826	−0.827833
$956$	0	0
$957$	−16.7477	−0.541377
$958$	0	0
$959$	6.79129	0.219302
$960$	0	0
$961$	−19.6170	−0.632808
$962$	0	0
$963$	21.1652	0.682037
$964$	0	0
$965$	−20.7477	−0.667893
$966$	0	0
$967$	32.7477	1.05310	0.526548	−	0.850145i	$-0.323486\pi$
$967$	32.7477	1.05310	0.526548	+	0.850145i	$0.323486\pi$
$968$	0	0
$969$	−12.7913	−0.410915
$970$	0	0
$971$	15.9564	0.512067	0.256033	−	0.966668i	$-0.417584\pi$
$971$	15.9564	0.512067	0.256033	+	0.966668i	$0.417584\pi$
$972$	0	0
$973$	22.8348	0.732052
$974$	0	0
$975$	16.1652	0.517699
$976$	0	0
$977$	−34.1216	−1.09165	−0.545823	−	0.837900i	$-0.683783\pi$
$977$	−34.1216	−1.09165	−0.545823	+	0.837900i	$0.683783\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	16.1652	0.516114
$982$	0	0
$983$	14.3739	0.458455	0.229228	−	0.973373i	$-0.426380\pi$
$983$	14.3739	0.458455	0.229228	+	0.973373i	$0.426380\pi$
$984$	0	0
$985$	11.5390	0.367664
$986$	0	0
$987$	−22.0871	−0.703041
$988$	0	0
$989$	−11.1652	−0.355031
$990$	0	0
$991$	33.2087	1.05491	0.527455	−	0.849583i	$-0.323146\pi$
$991$	33.2087	1.05491	0.527455	+	0.849583i	$0.323146\pi$
$992$	0	0
$993$	57.9129	1.83781
$994$	0	0
$995$	16.3303	0.517705
$996$	0	0
$997$	43.4955	1.37751	0.688757	−	0.724992i	$-0.258157\pi$
$997$	43.4955	1.37751	0.688757	+	0.724992i	$0.258157\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.bk.1.1		2
4.3	odd	2		7360.2.a.bq.1.2		2
8.3	odd	2		230.2.a.a.1.1	✓	2
8.5	even	2		1840.2.a.n.1.2		2
24.11	even	2		2070.2.a.x.1.1		2
40.3	even	4		1150.2.b.g.599.3		4
40.19	odd	2		1150.2.a.o.1.2		2
40.27	even	4		1150.2.b.g.599.2		4
40.29	even	2		9200.2.a.bs.1.1		2
184.91	even	2		5290.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.a.1.1	✓	2	8.3	odd	2
1150.2.a.o.1.2		2	40.19	odd	2
1150.2.b.g.599.2		4	40.27	even	4
1150.2.b.g.599.3		4	40.3	even	4
1840.2.a.n.1.2		2	8.5	even	2
2070.2.a.x.1.1		2	24.11	even	2
5290.2.a.e.1.1		2	184.91	even	2
7360.2.a.bk.1.1		2	1.1	even	1	trivial
7360.2.a.bq.1.2		2	4.3	odd	2
9200.2.a.bs.1.1		2	40.29	even	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} +1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q-2.79129 q^{3} +1.00000 q^{5} +1.79129 q^{7} +4.79129 q^{9} -0.791288 q^{11} -5.79129 q^{13} -2.79129 q^{15} +0.791288 q^{17} +5.79129 q^{19} -5.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -7.58258 q^{29} +3.37386 q^{31} +2.20871 q^{33} +1.79129 q^{35} +4.00000 q^{37} +16.1652 q^{39} -6.79129 q^{41} +11.1652 q^{43} +4.79129 q^{45} +4.41742 q^{47} -3.79129 q^{49} -2.20871 q^{51} -6.00000 q^{53} -0.791288 q^{55} -16.1652 q^{57} -13.5826 q^{59} -10.3739 q^{61} +8.58258 q^{63} -5.79129 q^{65} +11.1652 q^{67} +2.79129 q^{69} -8.37386 q^{71} +12.7477 q^{73} -2.79129 q^{75} -1.41742 q^{77} -8.00000 q^{79} -0.417424 q^{81} -6.00000 q^{83} +0.791288 q^{85} +21.1652 q^{87} +15.1652 q^{89} -10.3739 q^{91} -9.41742 q^{93} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - q^7 + 5 * q^9	\( 2 q - q^{3} + 2 q^{5} - q^{7} + 5 q^{9} + 3 q^{11} - 7 q^{13} - q^{15} - 3 q^{17} + 7 q^{19} - 10 q^{21} - 2 q^{23} + 2 q^{25} - 10 q^{27} - 6 q^{29} - 7 q^{31} + 9 q^{33} - q^{35} + 8 q^{37} + 14 q^{39} - 9 q^{41} + 4 q^{43} + 5 q^{45} + 18 q^{47} - 3 q^{49} - 9 q^{51} - 12 q^{53} + 3 q^{55} - 14 q^{57} - 18 q^{59} - 7 q^{61} + 8 q^{63} - 7 q^{65} + 4 q^{67} + q^{69} - 3 q^{71} - 2 q^{73} - q^{75} - 12 q^{77} - 16 q^{79} - 10 q^{81} - 12 q^{83} - 3 q^{85} + 24 q^{87} + 12 q^{89} - 7 q^{91} - 28 q^{93} + 7 q^{95} + 7 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - q^7 + 5 * q^9 + 3 * q^11 - 7 * q^13 - q^15 - 3 * q^17 + 7 * q^19 - 10 * q^21 - 2 * q^23 + 2 * q^25 - 10 * q^27 - 6 * q^29 - 7 * q^31 + 9 * q^33 - q^35 + 8 * q^37 + 14 * q^39 - 9 * q^41 + 4 * q^43 + 5 * q^45 + 18 * q^47 - 3 * q^49 - 9 * q^51 - 12 * q^53 + 3 * q^55 - 14 * q^57 - 18 * q^59 - 7 * q^61 + 8 * q^63 - 7 * q^65 + 4 * q^67 + q^69 - 3 * q^71 - 2 * q^73 - q^75 - 12 * q^77 - 16 * q^79 - 10 * q^81 - 12 * q^83 - 3 * q^85 + 24 * q^87 + 12 * q^89 - 7 * q^91 - 28 * q^93 + 7 * q^95 + 7 * q^97 - 3 * q^99

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