LMFDB - Embedded newform 7360.2.a.bq.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7360,2,Mod(1,7360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,2,0,1,0,5,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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			$-1.79129$ 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)

$f(q)$	$=$	$q-1.79129 q^{3} +1.00000 q^{5} +2.79129 q^{7} +0.208712 q^{9} -3.79129 q^{11} -1.20871 q^{13} -1.79129 q^{15} -3.79129 q^{17} -1.20871 q^{19} -5.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +1.58258 q^{29} +10.3739 q^{31} +6.79129 q^{33} +2.79129 q^{35} +4.00000 q^{37} +2.16515 q^{39} -2.20871 q^{41} +7.16515 q^{43} +0.208712 q^{45} -13.5826 q^{47} +0.791288 q^{49} +6.79129 q^{51} -6.00000 q^{53} -3.79129 q^{55} +2.16515 q^{57} +4.41742 q^{59} +3.37386 q^{61} +0.582576 q^{63} -1.20871 q^{65} +7.16515 q^{67} -1.79129 q^{69} -5.37386 q^{71} -14.7477 q^{73} -1.79129 q^{75} -10.5826 q^{77} +8.00000 q^{79} -9.58258 q^{81} +6.00000 q^{83} -3.79129 q^{85} -2.83485 q^{87} -3.16515 q^{89} -3.37386 q^{91} -18.5826 q^{93} -1.20871 q^{95} +14.9564 q^{97} -0.791288 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.79129	−1.03420	−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129	−1.03420	−0.517100	+	0.855925i	$0.672989\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.79129	1.05501	0.527504	−	0.849553i	$-0.323128\pi$
$7$	2.79129	1.05501	0.527504	+	0.849553i	$0.323128\pi$
$8$	0	0
$9$	0.208712	0.0695707
$10$	0	0
$11$	−3.79129	−1.14312	−0.571558	−	0.820562i	$-0.693661\pi$
$11$	−3.79129	−1.14312	−0.571558	+	0.820562i	$0.693661\pi$
$12$	0	0
$13$	−1.20871	−0.335236	−0.167618	−	0.985852i	$-0.553608\pi$
$13$	−1.20871	−0.335236	−0.167618	+	0.985852i	$0.553608\pi$
$14$	0	0
$15$	−1.79129	−0.462509
$16$	0	0
$17$	−3.79129	−0.919522	−0.459761	−	0.888043i	$-0.652065\pi$
$17$	−3.79129	−0.919522	−0.459761	+	0.888043i	$0.652065\pi$
$18$	0	0
$19$	−1.20871	−0.277298	−0.138649	−	0.990342i	$-0.544276\pi$
$19$	−1.20871	−0.277298	−0.138649	+	0.990342i	$0.544276\pi$
$20$	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
$29$	1.58258	0.293877	0.146938	−	0.989146i	$-0.453058\pi$
$29$	1.58258	0.293877	0.146938	+	0.989146i	$0.453058\pi$
$30$	0	0
$31$	10.3739	1.86320	0.931600	−	0.363484i	$-0.118413\pi$
$31$	10.3739	1.86320	0.931600	+	0.363484i	$0.118413\pi$
$32$	0	0
$33$	6.79129	1.18221
$34$	0	0
$35$	2.79129	0.471814
$36$	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
$39$	2.16515	0.346702
$40$	0	0
$41$	−2.20871	−0.344943	−0.172471	−	0.985015i	$-0.555175\pi$
$41$	−2.20871	−0.344943	−0.172471	+	0.985015i	$0.555175\pi$
$42$	0	0
$43$	7.16515	1.09268	0.546338	−	0.837565i	$-0.316022\pi$
$43$	7.16515	1.09268	0.546338	+	0.837565i	$0.316022\pi$
$44$	0	0
$45$	0.208712	0.0311130
$46$	0	0
$47$	−13.5826	−1.98122	−0.990611	−	0.136710i	$-0.956347\pi$
$47$	−13.5826	−1.98122	−0.990611	+	0.136710i	$0.956347\pi$
$48$	0	0
$49$	0.791288	0.113041
$50$	0	0
$51$	6.79129	0.950971
$52$	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
$55$	−3.79129	−0.511217
$56$	0	0
$57$	2.16515	0.286781
$58$	0	0
$59$	4.41742	0.575100	0.287550	−	0.957766i	$-0.407159\pi$
$59$	4.41742	0.575100	0.287550	+	0.957766i	$0.407159\pi$
$60$	0	0
$61$	3.37386	0.431979	0.215989	−	0.976396i	$-0.430702\pi$
$61$	3.37386	0.431979	0.215989	+	0.976396i	$0.430702\pi$
$62$	0	0
$63$	0.582576	0.0733976
$64$	0	0
$65$	−1.20871	−0.149922
$66$	0	0
$67$	7.16515	0.875363	0.437681	−	0.899130i	$-0.355800\pi$
$67$	7.16515	0.875363	0.437681	+	0.899130i	$0.355800\pi$
$68$	0	0
$69$	−1.79129	−0.215646
$70$	0	0
$71$	−5.37386	−0.637760	−0.318880	−	0.947795i	$-0.603307\pi$
$71$	−5.37386	−0.637760	−0.318880	+	0.947795i	$0.603307\pi$
$72$	0	0
$73$	−14.7477	−1.72609	−0.863045	−	0.505126i	$-0.831446\pi$
$73$	−14.7477	−1.72609	−0.863045	+	0.505126i	$0.831446\pi$
$74$	0	0
$75$	−1.79129	−0.206840
$76$	0	0
$77$	−10.5826	−1.20600
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−3.79129	−0.411223
$86$	0	0
$87$	−2.83485	−0.303928
$88$	0	0
$89$	−3.16515	−0.335505	−0.167753	−	0.985829i	$-0.553651\pi$
$89$	−3.16515	−0.335505	−0.167753	+	0.985829i	$0.553651\pi$
$90$	0	0
$91$	−3.37386	−0.353677
$92$	0	0
$93$	−18.5826	−1.92692
$94$	0	0
$95$	−1.20871	−0.124011
$96$	0	0
$97$	14.9564	1.51860	0.759298	−	0.650743i	$-0.225542\pi$
$97$	14.9564	1.51860	0.759298	+	0.650743i	$0.225542\pi$
$98$	0	0
$99$	−0.791288	−0.0795274
$100$	0	0
$101$	−13.5826	−1.35152	−0.675758	−	0.737123i	$-0.736184\pi$
$101$	−13.5826	−1.35152	−0.675758	+	0.737123i	$0.736184\pi$
$102$	0	0
$103$	7.37386	0.726568	0.363284	−	0.931678i	$-0.381655\pi$
$103$	7.37386	0.726568	0.363284	+	0.931678i	$0.381655\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	−13.5826	−1.31308	−0.656539	−	0.754292i	$-0.727980\pi$
$107$	−13.5826	−1.31308	−0.656539	+	0.754292i	$0.727980\pi$
$108$	0	0
$109$	−10.3739	−0.993636	−0.496818	−	0.867855i	$-0.665498\pi$
$109$	−10.3739	−0.993636	−0.496818	+	0.867855i	$0.665498\pi$
$110$	0	0
$111$	−7.16515	−0.680086
$112$	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
$117$	−0.252273	−0.0233226
$118$	0	0
$119$	−10.5826	−0.970103
$120$	0	0
$121$	3.37386	0.306715
$122$	0	0
$123$	3.95644	0.356740
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.7477	−1.30865	−0.654325	−	0.756214i	$-0.727047\pi$
$127$	−14.7477	−1.30865	−0.654325	+	0.756214i	$0.727047\pi$
$128$	0	0
$129$	−12.8348	−1.13005
$130$	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
$133$	−3.37386	−0.292551
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−0.791288	−0.0676043	−0.0338021	−	0.999429i	$-0.510762\pi$
$137$	−0.791288	−0.0676043	−0.0338021	+	0.999429i	$0.510762\pi$
$138$	0	0
$139$	14.7477	1.25089	0.625443	−	0.780270i	$-0.284918\pi$
$139$	14.7477	1.25089	0.625443	+	0.780270i	$0.284918\pi$
$140$	0	0
$141$	24.3303	2.04898
$142$	0	0
$143$	4.58258	0.383214
$144$	0	0
$145$	1.58258	0.131426
$146$	0	0
$147$	−1.41742	−0.116907
$148$	0	0
$149$	−12.7913	−1.04790	−0.523952	−	0.851748i	$-0.675543\pi$
$149$	−12.7913	−1.04790	−0.523952	+	0.851748i	$0.675543\pi$
$150$	0	0
$151$	−6.20871	−0.505258	−0.252629	−	0.967563i	$-0.581295\pi$
$151$	−6.20871	−0.505258	−0.252629	+	0.967563i	$0.581295\pi$
$152$	0	0
$153$	−0.791288	−0.0639718
$154$	0	0
$155$	10.3739	0.833249
$156$	0	0
$157$	−12.7477	−1.01738	−0.508690	−	0.860950i	$-0.669870\pi$
$157$	−12.7477	−1.01738	−0.508690	+	0.860950i	$0.669870\pi$
$158$	0	0
$159$	10.7477	0.852350
$160$	0	0
$161$	2.79129	0.219984
$162$	0	0
$163$	−22.3739	−1.75246	−0.876228	−	0.481897i	$-0.839948\pi$
$163$	−22.3739	−1.75246	−0.876228	+	0.481897i	$0.839948\pi$
$164$	0	0
$165$	6.79129	0.528701
$166$	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
$169$	−11.5390	−0.887617
$170$	0	0
$171$	−0.252273	−0.0192918
$172$	0	0
$173$	14.2087	1.08027	0.540134	−	0.841579i	$-0.318373\pi$
$173$	14.2087	1.08027	0.540134	+	0.841579i	$0.318373\pi$
$174$	0	0
$175$	2.79129	0.211002
$176$	0	0
$177$	−7.91288	−0.594768
$178$	0	0
$179$	16.7477	1.25178	0.625892	−	0.779910i	$-0.284735\pi$
$179$	16.7477	1.25178	0.625892	+	0.779910i	$0.284735\pi$
$180$	0	0
$181$	−13.5390	−1.00635	−0.503174	−	0.864185i	$-0.667834\pi$
$181$	−13.5390	−1.00635	−0.503174	+	0.864185i	$0.667834\pi$
$182$	0	0
$183$	−6.04356	−0.446753
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	14.3739	1.05112
$188$	0	0
$189$	13.9564	1.01518
$190$	0	0
$191$	16.4174	1.18792	0.593962	−	0.804493i	$-0.297563\pi$
$191$	16.4174	1.18792	0.593962	+	0.804493i	$0.297563\pi$
$192$	0	0
$193$	6.74773	0.485712	0.242856	−	0.970062i	$-0.421916\pi$
$193$	6.74773	0.485712	0.242856	+	0.970062i	$0.421916\pi$
$194$	0	0
$195$	2.16515	0.155050
$196$	0	0
$197$	−20.5390	−1.46334	−0.731672	−	0.681657i	$-0.761260\pi$
$197$	−20.5390	−1.46334	−0.731672	+	0.681657i	$0.761260\pi$
$198$	0	0
$199$	20.3303	1.44118	0.720588	−	0.693363i	$-0.243872\pi$
$199$	20.3303	1.44118	0.720588	+	0.693363i	$0.243872\pi$
$200$	0	0
$201$	−12.8348	−0.905300
$202$	0	0
$203$	4.41742	0.310042
$204$	0	0
$205$	−2.20871	−0.154263
$206$	0	0
$207$	0.208712	0.0145065
$208$	0	0
$209$	4.58258	0.316983
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	9.62614	0.659572
$214$	0	0
$215$	7.16515	0.488659
$216$	0	0
$217$	28.9564	1.96569
$218$	0	0
$219$	26.4174	1.78512
$220$	0	0
$221$	4.58258	0.308257
$222$	0	0
$223$	11.1652	0.747674	0.373837	−	0.927494i	$-0.378042\pi$
$223$	11.1652	0.747674	0.373837	+	0.927494i	$0.378042\pi$
$224$	0	0
$225$	0.208712	0.0139141
$226$	0	0
$227$	−4.74773	−0.315118	−0.157559	−	0.987510i	$-0.550362\pi$
$227$	−4.74773	−0.315118	−0.157559	+	0.987510i	$0.550362\pi$
$228$	0	0
$229$	16.3303	1.07914	0.539568	−	0.841942i	$-0.318587\pi$
$229$	16.3303	1.07914	0.539568	+	0.841942i	$0.318587\pi$
$230$	0	0
$231$	18.9564	1.24724
$232$	0	0
$233$	7.58258	0.496751	0.248376	−	0.968664i	$-0.420103\pi$
$233$	7.58258	0.496751	0.248376	+	0.968664i	$0.420103\pi$
$234$	0	0
$235$	−13.5826	−0.886030
$236$	0	0
$237$	−14.3303	−0.930853
$238$	0	0
$239$	3.16515	0.204737	0.102368	−	0.994747i	$-0.467358\pi$
$239$	3.16515	0.204737	0.102368	+	0.994747i	$0.467358\pi$
$240$	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
$243$	2.16515	0.138895
$244$	0	0
$245$	0.791288	0.0505535
$246$	0	0
$247$	1.46099	0.0929603
$248$	0	0
$249$	−10.7477	−0.681110
$250$	0	0
$251$	−30.7913	−1.94353	−0.971764	−	0.235953i	$-0.924179\pi$
$251$	−30.7913	−1.94353	−0.971764	+	0.235953i	$0.924179\pi$
$252$	0	0
$253$	−3.79129	−0.238356
$254$	0	0
$255$	6.79129	0.425287
$256$	0	0
$257$	22.7477	1.41896	0.709482	−	0.704723i	$-0.248929\pi$
$257$	22.7477	1.41896	0.709482	+	0.704723i	$0.248929\pi$
$258$	0	0
$259$	11.1652	0.693769
$260$	0	0
$261$	0.330303	0.0204452
$262$	0	0
$263$	15.7913	0.973733	0.486866	−	0.873477i	$-0.338140\pi$
$263$	15.7913	0.973733	0.486866	+	0.873477i	$0.338140\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	5.66970	0.346980
$268$	0	0
$269$	−16.7477	−1.02113	−0.510563	−	0.859840i	$-0.670563\pi$
$269$	−16.7477	−1.02113	−0.510563	+	0.859840i	$0.670563\pi$
$270$	0	0
$271$	−23.1216	−1.40454	−0.702268	−	0.711912i	$-0.747829\pi$
$271$	−23.1216	−1.40454	−0.702268	+	0.711912i	$0.747829\pi$
$272$	0	0
$273$	6.04356	0.365773
$274$	0	0
$275$	−3.79129	−0.228623
$276$	0	0
$277$	1.16515	0.0700072	0.0350036	−	0.999387i	$-0.488856\pi$
$277$	1.16515	0.0700072	0.0350036	+	0.999387i	$0.488856\pi$
$278$	0	0
$279$	2.16515	0.129624
$280$	0	0
$281$	−16.7477	−0.999086	−0.499543	−	0.866289i	$-0.666499\pi$
$281$	−16.7477	−0.999086	−0.499543	+	0.866289i	$0.666499\pi$
$282$	0	0
$283$	28.3303	1.68406	0.842031	−	0.539429i	$-0.181360\pi$
$283$	28.3303	1.68406	0.842031	+	0.539429i	$0.181360\pi$
$284$	0	0
$285$	2.16515	0.128252
$286$	0	0
$287$	−6.16515	−0.363917
$288$	0	0
$289$	−2.62614	−0.154479
$290$	0	0
$291$	−26.7913	−1.57053
$292$	0	0
$293$	27.4955	1.60630	0.803151	−	0.595776i	$-0.203155\pi$
$293$	27.4955	1.60630	0.803151	+	0.595776i	$0.203155\pi$
$294$	0	0
$295$	4.41742	0.257192
$296$	0	0
$297$	−18.9564	−1.09996
$298$	0	0
$299$	−1.20871	−0.0699016
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	24.3303	1.39774
$304$	0	0
$305$	3.37386	0.193187
$306$	0	0
$307$	−16.5390	−0.943931	−0.471966	−	0.881617i	$-0.656455\pi$
$307$	−16.5390	−0.943931	−0.471966	+	0.881617i	$0.656455\pi$
$308$	0	0
$309$	−13.2087	−0.751417
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−18.3739	−1.03855	−0.519276	−	0.854607i	$-0.673798\pi$
$313$	−18.3739	−1.03855	−0.519276	+	0.854607i	$0.673798\pi$
$314$	0	0
$315$	0.582576	0.0328244
$316$	0	0
$317$	−5.20871	−0.292550	−0.146275	−	0.989244i	$-0.546729\pi$
$317$	−5.20871	−0.292550	−0.146275	+	0.989244i	$0.546729\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	24.3303	1.35799
$322$	0	0
$323$	4.58258	0.254981
$324$	0	0
$325$	−1.20871	−0.0670473
$326$	0	0
$327$	18.5826	1.02762
$328$	0	0
$329$	−37.9129	−2.09020
$330$	0	0
$331$	−6.74773	−0.370889	−0.185444	−	0.982655i	$-0.559372\pi$
$331$	−6.74773	−0.370889	−0.185444	+	0.982655i	$0.559372\pi$
$332$	0	0
$333$	0.834849	0.0457494
$334$	0	0
$335$	7.16515	0.391474
$336$	0	0
$337$	−16.7913	−0.914680	−0.457340	−	0.889292i	$-0.651198\pi$
$337$	−16.7913	−0.914680	−0.457340	+	0.889292i	$0.651198\pi$
$338$	0	0
$339$	−10.7477	−0.583736
$340$	0	0
$341$	−39.3303	−2.12986
$342$	0	0
$343$	−17.3303	−0.935748
$344$	0	0
$345$	−1.79129	−0.0964397
$346$	0	0
$347$	−9.79129	−0.525624	−0.262812	−	0.964847i	$-0.584650\pi$
$347$	−9.79129	−0.525624	−0.262812	+	0.964847i	$0.584650\pi$
$348$	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
$351$	−6.04356	−0.322581
$352$	0	0
$353$	−15.1652	−0.807160	−0.403580	−	0.914944i	$-0.632234\pi$
$353$	−15.1652	−0.807160	−0.403580	+	0.914944i	$0.632234\pi$
$354$	0	0
$355$	−5.37386	−0.285215
$356$	0	0
$357$	18.9564	1.00328
$358$	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
$361$	−17.5390	−0.923106
$362$	0	0
$363$	−6.04356	−0.317205
$364$	0	0
$365$	−14.7477	−0.771931
$366$	0	0
$367$	−0.834849	−0.0435787	−0.0217894	−	0.999763i	$-0.506936\pi$
$367$	−0.834849	−0.0435787	−0.0217894	+	0.999763i	$0.506936\pi$
$368$	0	0
$369$	−0.460985	−0.0239979
$370$	0	0
$371$	−16.7477	−0.869499
$372$	0	0
$373$	14.7477	0.763608	0.381804	−	0.924243i	$-0.375303\pi$
$373$	14.7477	0.763608	0.381804	+	0.924243i	$0.375303\pi$
$374$	0	0
$375$	−1.79129	−0.0925017
$376$	0	0
$377$	−1.91288	−0.0985183
$378$	0	0
$379$	−7.37386	−0.378770	−0.189385	−	0.981903i	$-0.560649\pi$
$379$	−7.37386	−0.378770	−0.189385	+	0.981903i	$0.560649\pi$
$380$	0	0
$381$	26.4174	1.35341
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−10.5826	−0.539338
$386$	0	0
$387$	1.49545	0.0760182
$388$	0	0
$389$	−29.7042	−1.50606	−0.753031	−	0.657986i	$-0.771409\pi$
$389$	−29.7042	−1.50606	−0.753031	+	0.657986i	$0.771409\pi$
$390$	0	0
$391$	−3.79129	−0.191734
$392$	0	0
$393$	16.4174	0.828150
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−16.5390	−0.830069	−0.415035	−	0.909806i	$-0.636231\pi$
$397$	−16.5390	−0.830069	−0.415035	+	0.909806i	$0.636231\pi$
$398$	0	0
$399$	6.04356	0.302556
$400$	0	0
$401$	22.7477	1.13597	0.567984	−	0.823040i	$-0.307724\pi$
$401$	22.7477	1.13597	0.567984	+	0.823040i	$0.307724\pi$
$402$	0	0
$403$	−12.5390	−0.624613
$404$	0	0
$405$	−9.58258	−0.476162
$406$	0	0
$407$	−15.1652	−0.751709
$408$	0	0
$409$	−22.7913	−1.12696	−0.563478	−	0.826131i	$-0.690537\pi$
$409$	−22.7913	−1.12696	−0.563478	+	0.826131i	$0.690537\pi$
$410$	0	0
$411$	1.41742	0.0699164
$412$	0	0
$413$	12.3303	0.606735
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	−26.4174	−1.29367
$418$	0	0
$419$	−39.1652	−1.91334	−0.956671	−	0.291170i	$-0.905956\pi$
$419$	−39.1652	−1.91334	−0.956671	+	0.291170i	$0.905956\pi$
$420$	0	0
$421$	23.1216	1.12688	0.563439	−	0.826158i	$-0.309478\pi$
$421$	23.1216	1.12688	0.563439	+	0.826158i	$0.309478\pi$
$422$	0	0
$423$	−2.83485	−0.137835
$424$	0	0
$425$	−3.79129	−0.183904
$426$	0	0
$427$	9.41742	0.455741
$428$	0	0
$429$	−8.20871	−0.396320
$430$	0	0
$431$	−19.9129	−0.959170	−0.479585	−	0.877496i	$-0.659213\pi$
$431$	−19.9129	−0.959170	−0.479585	+	0.877496i	$0.659213\pi$
$432$	0	0
$433$	1.53901	0.0739603	0.0369802	−	0.999316i	$-0.488226\pi$
$433$	1.53901	0.0739603	0.0369802	+	0.999316i	$0.488226\pi$
$434$	0	0
$435$	−2.83485	−0.135921
$436$	0	0
$437$	−1.20871	−0.0578205
$438$	0	0
$439$	25.5390	1.21891	0.609455	−	0.792820i	$-0.291388\pi$
$439$	25.5390	1.21891	0.609455	+	0.792820i	$0.291388\pi$
$440$	0	0
$441$	0.165151	0.00786435
$442$	0	0
$443$	35.2087	1.67282	0.836408	−	0.548107i	$-0.184651\pi$
$443$	35.2087	1.67282	0.836408	+	0.548107i	$0.184651\pi$
$444$	0	0
$445$	−3.16515	−0.150043
$446$	0	0
$447$	22.9129	1.08374
$448$	0	0
$449$	25.1216	1.18556	0.592781	−	0.805364i	$-0.298030\pi$
$449$	25.1216	1.18556	0.592781	+	0.805364i	$0.298030\pi$
$450$	0	0
$451$	8.37386	0.394310
$452$	0	0
$453$	11.1216	0.522538
$454$	0	0
$455$	−3.37386	−0.158169
$456$	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
$459$	−18.9564	−0.884811
$460$	0	0
$461$	1.25227	0.0583242	0.0291621	−	0.999575i	$-0.490716\pi$
$461$	1.25227	0.0583242	0.0291621	+	0.999575i	$0.490716\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	0	0
$465$	−18.5826	−0.861746
$466$	0	0
$467$	−25.9129	−1.19911	−0.599553	−	0.800335i	$-0.704655\pi$
$467$	−25.9129	−1.19911	−0.599553	+	0.800335i	$0.704655\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	22.8348	1.05217
$472$	0	0
$473$	−27.1652	−1.24905
$474$	0	0
$475$	−1.20871	−0.0554595
$476$	0	0
$477$	−1.25227	−0.0573376
$478$	0	0
$479$	−39.4955	−1.80459	−0.902297	−	0.431116i	$-0.858120\pi$
$479$	−39.4955	−1.80459	−0.902297	+	0.431116i	$0.858120\pi$
$480$	0	0
$481$	−4.83485	−0.220450
$482$	0	0
$483$	−5.00000	−0.227508
$484$	0	0
$485$	14.9564	0.679137
$486$	0	0
$487$	15.5826	0.706114	0.353057	−	0.935602i	$-0.385142\pi$
$487$	15.5826	0.706114	0.353057	+	0.935602i	$0.385142\pi$
$488$	0	0
$489$	40.0780	1.81239
$490$	0	0
$491$	16.7477	0.755814	0.377907	−	0.925843i	$-0.376644\pi$
$491$	16.7477	0.755814	0.377907	+	0.925843i	$0.376644\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	−0.791288	−0.0355657
$496$	0	0
$497$	−15.0000	−0.672842
$498$	0	0
$499$	−23.1652	−1.03701	−0.518507	−	0.855073i	$-0.673512\pi$
$499$	−23.1652	−1.03701	−0.518507	+	0.855073i	$0.673512\pi$
$500$	0	0
$501$	32.8348	1.46695
$502$	0	0
$503$	18.7913	0.837862	0.418931	−	0.908018i	$-0.362405\pi$
$503$	18.7913	0.837862	0.418931	+	0.908018i	$0.362405\pi$
$504$	0	0
$505$	−13.5826	−0.604417
$506$	0	0
$507$	20.6697	0.917973
$508$	0	0
$509$	−7.25227	−0.321451	−0.160726	−	0.986999i	$-0.551383\pi$
$509$	−7.25227	−0.321451	−0.160726	+	0.986999i	$0.551383\pi$
$510$	0	0
$511$	−41.1652	−1.82104
$512$	0	0
$513$	−6.04356	−0.266830
$514$	0	0
$515$	7.37386	0.324931
$516$	0	0
$517$	51.4955	2.26477
$518$	0	0
$519$	−25.4519	−1.11721
$520$	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
$523$	1.16515	0.0509485	0.0254743	−	0.999675i	$-0.491890\pi$
$523$	1.16515	0.0509485	0.0254743	+	0.999675i	$0.491890\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	−39.3303	−1.71325
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.921970	0.0400101
$532$	0	0
$533$	2.66970	0.115637
$534$	0	0
$535$	−13.5826	−0.587226
$536$	0	0
$537$	−30.0000	−1.29460
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−38.3303	−1.64795	−0.823974	−	0.566627i	$-0.808248\pi$
$541$	−38.3303	−1.64795	−0.823974	+	0.566627i	$0.808248\pi$
$542$	0	0
$543$	24.2523	1.04076
$544$	0	0
$545$	−10.3739	−0.444367
$546$	0	0
$547$	−15.1216	−0.646553	−0.323276	−	0.946305i	$-0.604784\pi$
$547$	−15.1216	−0.646553	−0.323276	+	0.946305i	$0.604784\pi$
$548$	0	0
$549$	0.704166	0.0300531
$550$	0	0
$551$	−1.91288	−0.0814914
$552$	0	0
$553$	22.3303	0.949581
$554$	0	0
$555$	−7.16515	−0.304144
$556$	0	0
$557$	−30.3303	−1.28514	−0.642568	−	0.766229i	$-0.722131\pi$
$557$	−30.3303	−1.28514	−0.642568	+	0.766229i	$0.722131\pi$
$558$	0	0
$559$	−8.66061	−0.366305
$560$	0	0
$561$	−25.7477	−1.08707
$562$	0	0
$563$	−3.16515	−0.133395	−0.0666976	−	0.997773i	$-0.521246\pi$
$563$	−3.16515	−0.133395	−0.0666976	+	0.997773i	$0.521246\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	−26.7477	−1.12330
$568$	0	0
$569$	15.4955	0.649603	0.324802	−	0.945782i	$-0.394702\pi$
$569$	15.4955	0.649603	0.324802	+	0.945782i	$0.394702\pi$
$570$	0	0
$571$	−30.1216	−1.26055	−0.630275	−	0.776372i	$-0.717058\pi$
$571$	−30.1216	−1.26055	−0.630275	+	0.776372i	$0.717058\pi$
$572$	0	0
$573$	−29.4083	−1.22855
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	22.8348	0.950627	0.475314	−	0.879816i	$-0.342335\pi$
$577$	22.8348	0.950627	0.475314	+	0.879816i	$0.342335\pi$
$578$	0	0
$579$	−12.0871	−0.502324
$580$	0	0
$581$	16.7477	0.694813
$582$	0	0
$583$	22.7477	0.942115
$584$	0	0
$585$	−0.252273	−0.0104302
$586$	0	0
$587$	26.2087	1.08175	0.540875	−	0.841103i	$-0.318093\pi$
$587$	26.2087	1.08175	0.540875	+	0.841103i	$0.318093\pi$
$588$	0	0
$589$	−12.5390	−0.516661
$590$	0	0
$591$	36.7913	1.51339
$592$	0	0
$593$	13.9129	0.571333	0.285667	−	0.958329i	$-0.407785\pi$
$593$	13.9129	0.571333	0.285667	+	0.958329i	$0.407785\pi$
$594$	0	0
$595$	−10.5826	−0.433843
$596$	0	0
$597$	−36.4174	−1.49047
$598$	0	0
$599$	−40.1216	−1.63932	−0.819662	−	0.572848i	$-0.805839\pi$
$599$	−40.1216	−1.63932	−0.819662	+	0.572848i	$0.805839\pi$
$600$	0	0
$601$	−22.7913	−0.929676	−0.464838	−	0.885396i	$-0.653887\pi$
$601$	−22.7913	−0.929676	−0.464838	+	0.885396i	$0.653887\pi$
$602$	0	0
$603$	1.49545	0.0608996
$604$	0	0
$605$	3.37386	0.137167
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−7.91288	−0.320646
$610$	0	0
$611$	16.4174	0.664178
$612$	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
$615$	3.95644	0.159539
$616$	0	0
$617$	44.8693	1.80637	0.903185	−	0.429251i	$-0.141222\pi$
$617$	44.8693	1.80637	0.903185	+	0.429251i	$0.141222\pi$
$618$	0	0
$619$	−2.79129	−0.112191	−0.0560957	−	0.998425i	$-0.517865\pi$
$619$	−2.79129	−0.112191	−0.0560957	+	0.998425i	$0.517865\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	−8.83485	−0.353961
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.20871	−0.327824
$628$	0	0
$629$	−15.1652	−0.604674
$630$	0	0
$631$	−17.9129	−0.713100	−0.356550	−	0.934276i	$-0.616047\pi$
$631$	−17.9129	−0.713100	−0.356550	+	0.934276i	$0.616047\pi$
$632$	0	0
$633$	−17.9129	−0.711973
$634$	0	0
$635$	−14.7477	−0.585246
$636$	0	0
$637$	−0.956439	−0.0378955
$638$	0	0
$639$	−1.12159	−0.0443694
$640$	0	0
$641$	−3.16515	−0.125016	−0.0625080	−	0.998044i	$-0.519910\pi$
$641$	−3.16515	−0.125016	−0.0625080	+	0.998044i	$0.519910\pi$
$642$	0	0
$643$	20.7477	0.818210	0.409105	−	0.912487i	$-0.365841\pi$
$643$	20.7477	0.818210	0.409105	+	0.912487i	$0.365841\pi$
$644$	0	0
$645$	−12.8348	−0.505372
$646$	0	0
$647$	2.83485	0.111449	0.0557247	−	0.998446i	$-0.482253\pi$
$647$	2.83485	0.111449	0.0557247	+	0.998446i	$0.482253\pi$
$648$	0	0
$649$	−16.7477	−0.657406
$650$	0	0
$651$	−51.8693	−2.03292
$652$	0	0
$653$	−35.5390	−1.39075	−0.695375	−	0.718647i	$-0.744762\pi$
$653$	−35.5390	−1.39075	−0.695375	+	0.718647i	$0.744762\pi$
$654$	0	0
$655$	−9.16515	−0.358112
$656$	0	0
$657$	−3.07803	−0.120085
$658$	0	0
$659$	27.1652	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$659$	27.1652	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$660$	0	0
$661$	39.3739	1.53147	0.765733	−	0.643159i	$-0.222376\pi$
$661$	39.3739	1.53147	0.765733	+	0.643159i	$0.222376\pi$
$662$	0	0
$663$	−8.20871	−0.318800
$664$	0	0
$665$	−3.37386	−0.130833
$666$	0	0
$667$	1.58258	0.0612776
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	−12.7913	−0.493802
$672$	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
$677$	30.6606	1.17838	0.589191	−	0.807993i	$-0.299446\pi$
$677$	30.6606	1.17838	0.589191	+	0.807993i	$0.299446\pi$
$678$	0	0
$679$	41.7477	1.60213
$680$	0	0
$681$	8.50455	0.325895
$682$	0	0
$683$	−2.37386	−0.0908334	−0.0454167	−	0.998968i	$-0.514462\pi$
$683$	−2.37386	−0.0908334	−0.0454167	+	0.998968i	$0.514462\pi$
$684$	0	0
$685$	−0.791288	−0.0302336
$686$	0	0
$687$	−29.2523	−1.11604
$688$	0	0
$689$	7.25227	0.276290
$690$	0	0
$691$	−15.2523	−0.580224	−0.290112	−	0.956993i	$-0.593693\pi$
$691$	−15.2523	−0.580224	−0.290112	+	0.956993i	$0.593693\pi$
$692$	0	0
$693$	−2.20871	−0.0839020
$694$	0	0
$695$	14.7477	0.559413
$696$	0	0
$697$	8.37386	0.317183
$698$	0	0
$699$	−13.5826	−0.513740
$700$	0	0
$701$	−9.62614	−0.363574	−0.181787	−	0.983338i	$-0.558188\pi$
$701$	−9.62614	−0.363574	−0.181787	+	0.983338i	$0.558188\pi$
$702$	0	0
$703$	−4.83485	−0.182350
$704$	0	0
$705$	24.3303	0.916332
$706$	0	0
$707$	−37.9129	−1.42586
$708$	0	0
$709$	−34.5390	−1.29714	−0.648570	−	0.761155i	$-0.724633\pi$
$709$	−34.5390	−1.29714	−0.648570	+	0.761155i	$0.724633\pi$
$710$	0	0
$711$	1.66970	0.0626185
$712$	0	0
$713$	10.3739	0.388504
$714$	0	0
$715$	4.58258	0.171379
$716$	0	0
$717$	−5.66970	−0.211739
$718$	0	0
$719$	−29.5390	−1.10162	−0.550810	−	0.834631i	$-0.685681\pi$
$719$	−29.5390	−1.10162	−0.550810	+	0.834631i	$0.685681\pi$
$720$	0	0
$721$	20.5826	0.766535
$722$	0	0
$723$	50.1561	1.86532
$724$	0	0
$725$	1.58258	0.0587754
$726$	0	0
$727$	−2.12159	−0.0786854	−0.0393427	−	0.999226i	$-0.512526\pi$
$727$	−2.12159	−0.0786854	−0.0393427	+	0.999226i	$0.512526\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	−27.1652	−1.00474
$732$	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
$735$	−1.41742	−0.0522825
$736$	0	0
$737$	−27.1652	−1.00064
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	−2.61704	−0.0961395
$742$	0	0
$743$	9.95644	0.365266	0.182633	−	0.983181i	$-0.441538\pi$
$743$	9.95644	0.365266	0.182633	+	0.983181i	$0.441538\pi$
$744$	0	0
$745$	−12.7913	−0.468637
$746$	0	0
$747$	1.25227	0.0458183
$748$	0	0
$749$	−37.9129	−1.38531
$750$	0	0
$751$	18.7477	0.684114	0.342057	−	0.939679i	$-0.388876\pi$
$751$	18.7477	0.684114	0.342057	+	0.939679i	$0.388876\pi$
$752$	0	0
$753$	55.1561	2.01000
$754$	0	0
$755$	−6.20871	−0.225958
$756$	0	0
$757$	−26.3303	−0.956991	−0.478496	−	0.878090i	$-0.658818\pi$
$757$	−26.3303	−0.956991	−0.478496	+	0.878090i	$0.658818\pi$
$758$	0	0
$759$	6.79129	0.246508
$760$	0	0
$761$	−33.9564	−1.23092	−0.615460	−	0.788168i	$-0.711030\pi$
$761$	−33.9564	−1.23092	−0.615460	+	0.788168i	$0.711030\pi$
$762$	0	0
$763$	−28.9564	−1.04829
$764$	0	0
$765$	−0.791288	−0.0286091
$766$	0	0
$767$	−5.33939	−0.192794
$768$	0	0
$769$	−3.66970	−0.132333	−0.0661663	−	0.997809i	$-0.521077\pi$
$769$	−3.66970	−0.132333	−0.0661663	+	0.997809i	$0.521077\pi$
$770$	0	0
$771$	−40.7477	−1.46749
$772$	0	0
$773$	21.4955	0.773138	0.386569	−	0.922261i	$-0.373660\pi$
$773$	21.4955	0.773138	0.386569	+	0.922261i	$0.373660\pi$
$774$	0	0
$775$	10.3739	0.372640
$776$	0	0
$777$	−20.0000	−0.717496
$778$	0	0
$779$	2.66970	0.0956518
$780$	0	0
$781$	20.3739	0.729034
$782$	0	0
$783$	7.91288	0.282783
$784$	0	0
$785$	−12.7477	−0.454986
$786$	0	0
$787$	8.41742	0.300049	0.150024	−	0.988682i	$-0.452065\pi$
$787$	8.41742	0.300049	0.150024	+	0.988682i	$0.452065\pi$
$788$	0	0
$789$	−28.2867	−1.00703
$790$	0	0
$791$	16.7477	0.595481
$792$	0	0
$793$	−4.07803	−0.144815
$794$	0	0
$795$	10.7477	0.381183
$796$	0	0
$797$	49.9129	1.76800	0.884002	−	0.467482i	$-0.154839\pi$
$797$	49.9129	1.76800	0.884002	+	0.467482i	$0.154839\pi$
$798$	0	0
$799$	51.4955	1.82178
$800$	0	0
$801$	−0.660606	−0.0233413
$802$	0	0
$803$	55.9129	1.97312
$804$	0	0
$805$	2.79129	0.0983800
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	11.0436	0.388271	0.194135	−	0.980975i	$-0.437810\pi$
$809$	11.0436	0.388271	0.194135	+	0.980975i	$0.437810\pi$
$810$	0	0
$811$	47.9129	1.68245	0.841224	−	0.540686i	$-0.181835\pi$
$811$	47.9129	1.68245	0.841224	+	0.540686i	$0.181835\pi$
$812$	0	0
$813$	41.4174	1.45257
$814$	0	0
$815$	−22.3739	−0.783722
$816$	0	0
$817$	−8.66061	−0.302996
$818$	0	0
$819$	−0.704166	−0.0246056
$820$	0	0
$821$	−2.83485	−0.0989369	−0.0494684	−	0.998776i	$-0.515753\pi$
$821$	−2.83485	−0.0989369	−0.0494684	+	0.998776i	$0.515753\pi$
$822$	0	0
$823$	41.1652	1.43493	0.717463	−	0.696596i	$-0.245303\pi$
$823$	41.1652	1.43493	0.717463	+	0.696596i	$0.245303\pi$
$824$	0	0
$825$	6.79129	0.236442
$826$	0	0
$827$	−41.0780	−1.42842	−0.714212	−	0.699930i	$-0.753215\pi$
$827$	−41.0780	−1.42842	−0.714212	+	0.699930i	$0.753215\pi$
$828$	0	0
$829$	31.4955	1.09388	0.546941	−	0.837171i	$-0.315792\pi$
$829$	31.4955	1.09388	0.546941	+	0.837171i	$0.315792\pi$
$830$	0	0
$831$	−2.08712	−0.0724014
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−18.3303	−0.634346
$836$	0	0
$837$	51.8693	1.79287
$838$	0	0
$839$	−22.4174	−0.773935	−0.386968	−	0.922093i	$-0.626478\pi$
$839$	−22.4174	−0.773935	−0.386968	+	0.922093i	$0.626478\pi$
$840$	0	0
$841$	−26.4955	−0.913636
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	−11.5390	−0.396954
$846$	0	0
$847$	9.41742	0.323587
$848$	0	0
$849$	−50.7477	−1.74166
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−8.46099	−0.289699	−0.144849	−	0.989454i	$-0.546270\pi$
$853$	−8.46099	−0.289699	−0.144849	+	0.989454i	$0.546270\pi$
$854$	0	0
$855$	−0.252273	−0.00862755
$856$	0	0
$857$	9.16515	0.313076	0.156538	−	0.987672i	$-0.449967\pi$
$857$	9.16515	0.313076	0.156538	+	0.987672i	$0.449967\pi$
$858$	0	0
$859$	−0.747727	−0.0255121	−0.0127561	−	0.999919i	$-0.504060\pi$
$859$	−0.747727	−0.0255121	−0.0127561	+	0.999919i	$0.504060\pi$
$860$	0	0
$861$	11.0436	0.376364
$862$	0	0
$863$	−31.5826	−1.07508	−0.537542	−	0.843237i	$-0.680647\pi$
$863$	−31.5826	−1.07508	−0.537542	+	0.843237i	$0.680647\pi$
$864$	0	0
$865$	14.2087	0.483111
$866$	0	0
$867$	4.70417	0.159762
$868$	0	0
$869$	−30.3303	−1.02889
$870$	0	0
$871$	−8.66061	−0.293453
$872$	0	0
$873$	3.12159	0.105650
$874$	0	0
$875$	2.79129	0.0943628
$876$	0	0
$877$	−7.70417	−0.260151	−0.130076	−	0.991504i	$-0.541522\pi$
$877$	−7.70417	−0.260151	−0.130076	+	0.991504i	$0.541522\pi$
$878$	0	0
$879$	−49.2523	−1.66124
$880$	0	0
$881$	−6.33030	−0.213273	−0.106637	−	0.994298i	$-0.534008\pi$
$881$	−6.33030	−0.213273	−0.106637	+	0.994298i	$0.534008\pi$
$882$	0	0
$883$	12.0436	0.405298	0.202649	−	0.979251i	$-0.435045\pi$
$883$	12.0436	0.405298	0.202649	+	0.979251i	$0.435045\pi$
$884$	0	0
$885$	−7.91288	−0.265989
$886$	0	0
$887$	−3.16515	−0.106275	−0.0531377	−	0.998587i	$-0.516922\pi$
$887$	−3.16515	−0.106275	−0.0531377	+	0.998587i	$0.516922\pi$
$888$	0	0
$889$	−41.1652	−1.38063
$890$	0	0
$891$	36.3303	1.21711
$892$	0	0
$893$	16.4174	0.549388
$894$	0	0
$895$	16.7477	0.559815
$896$	0	0
$897$	2.16515	0.0722923
$898$	0	0
$899$	16.4174	0.547552
$900$	0	0
$901$	22.7477	0.757837
$902$	0	0
$903$	−35.8258	−1.19221
$904$	0	0
$905$	−13.5390	−0.450052
$906$	0	0
$907$	20.7477	0.688917	0.344458	−	0.938802i	$-0.388063\pi$
$907$	20.7477	0.688917	0.344458	+	0.938802i	$0.388063\pi$
$908$	0	0
$909$	−2.83485	−0.0940260
$910$	0	0
$911$	13.5826	0.450011	0.225005	−	0.974358i	$-0.427760\pi$
$911$	13.5826	0.450011	0.225005	+	0.974358i	$0.427760\pi$
$912$	0	0
$913$	−22.7477	−0.752840
$914$	0	0
$915$	−6.04356	−0.199794
$916$	0	0
$917$	−25.5826	−0.844811
$918$	0	0
$919$	−36.8348	−1.21507	−0.607535	−	0.794293i	$-0.707841\pi$
$919$	−36.8348	−1.21507	−0.607535	+	0.794293i	$0.707841\pi$
$920$	0	0
$921$	29.6261	0.976214
$922$	0	0
$923$	6.49545	0.213800
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	1.53901	0.0505479
$928$	0	0
$929$	−39.4955	−1.29580	−0.647902	−	0.761724i	$-0.724353\pi$
$929$	−39.4955	−1.29580	−0.647902	+	0.761724i	$0.724353\pi$
$930$	0	0
$931$	−0.956439	−0.0313460
$932$	0	0
$933$	−21.4955	−0.703730
$934$	0	0
$935$	14.3739	0.470076
$936$	0	0
$937$	58.3739	1.90699	0.953495	−	0.301407i	$-0.0974564\pi$
$937$	58.3739	1.90699	0.953495	+	0.301407i	$0.0974564\pi$
$938$	0	0
$939$	32.9129	1.07407
$940$	0	0
$941$	−54.9564	−1.79153	−0.895764	−	0.444529i	$-0.853371\pi$
$941$	−54.9564	−1.79153	−0.895764	+	0.444529i	$0.853371\pi$
$942$	0	0
$943$	−2.20871	−0.0719256
$944$	0	0
$945$	13.9564	0.454003
$946$	0	0
$947$	29.5390	0.959889	0.479945	−	0.877299i	$-0.340657\pi$
$947$	29.5390	0.959889	0.479945	+	0.877299i	$0.340657\pi$
$948$	0	0
$949$	17.8258	0.578649
$950$	0	0
$951$	9.33030	0.302556
$952$	0	0
$953$	−26.5390	−0.859683	−0.429842	−	0.902904i	$-0.641431\pi$
$953$	−26.5390	−0.859683	−0.429842	+	0.902904i	$0.641431\pi$
$954$	0	0
$955$	16.4174	0.531255
$956$	0	0
$957$	10.7477	0.347425
$958$	0	0
$959$	−2.20871	−0.0713230
$960$	0	0
$961$	76.6170	2.47152
$962$	0	0
$963$	−2.83485	−0.0913517
$964$	0	0
$965$	6.74773	0.217217
$966$	0	0
$967$	−5.25227	−0.168902	−0.0844509	−	0.996428i	$-0.526914\pi$
$967$	−5.25227	−0.168902	−0.0844509	+	0.996428i	$0.526914\pi$
$968$	0	0
$969$	−8.20871	−0.263702
$970$	0	0
$971$	6.95644	0.223243	0.111621	−	0.993751i	$-0.464396\pi$
$971$	6.95644	0.223243	0.111621	+	0.993751i	$0.464396\pi$
$972$	0	0
$973$	41.1652	1.31969
$974$	0	0
$975$	2.16515	0.0693403
$976$	0	0
$977$	7.12159	0.227840	0.113920	−	0.993490i	$-0.463659\pi$
$977$	7.12159	0.227840	0.113920	+	0.993490i	$0.463659\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−2.16515	−0.0691280
$982$	0	0
$983$	−0.626136	−0.0199707	−0.00998533	−	0.999950i	$-0.503178\pi$
$983$	−0.626136	−0.0199707	−0.00998533	+	0.999950i	$0.503178\pi$
$984$	0	0
$985$	−20.5390	−0.654427
$986$	0	0
$987$	67.9129	2.16169
$988$	0	0
$989$	7.16515	0.227839
$990$	0	0
$991$	−37.7913	−1.20048	−0.600240	−	0.799820i	$-0.704928\pi$
$991$	−37.7913	−1.20048	−0.600240	+	0.799820i	$0.704928\pi$
$992$	0	0
$993$	12.0871	0.383573
$994$	0	0
$995$	20.3303	0.644514
$996$	0	0
$997$	−11.4955	−0.364065	−0.182032	−	0.983293i	$-0.558268\pi$
$997$	−11.4955	−0.364065	−0.182032	+	0.983293i	$0.558268\pi$
$998$	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.bq.1.1		2
4.3	odd	2		7360.2.a.bk.1.2		2
8.3	odd	2		1840.2.a.n.1.1		2
8.5	even	2		230.2.a.a.1.2	✓	2
24.5	odd	2		2070.2.a.x.1.2		2
40.13	odd	4		1150.2.b.g.599.4		4
40.19	odd	2		9200.2.a.bs.1.2		2
40.29	even	2		1150.2.a.o.1.1		2
40.37	odd	4		1150.2.b.g.599.1		4
184.45	odd	2		5290.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.a.1.2	✓	2	8.5	even	2
1150.2.a.o.1.1		2	40.29	even	2
1150.2.b.g.599.1		4	40.37	odd	4
1150.2.b.g.599.4		4	40.13	odd	4
1840.2.a.n.1.1		2	8.3	odd	2
2070.2.a.x.1.2		2	24.5	odd	2
5290.2.a.e.1.2		2	184.45	odd	2
7360.2.a.bk.1.2		2	4.3	odd	2
7360.2.a.bq.1.1		2	1.1	even	1	trivial
9200.2.a.bs.1.2		2	40.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{3} +1.00000 q^{5} +2.79129 q^{7} +0.208712 q^{9} -3.79129 q^{11} -1.20871 q^{13} -1.79129 q^{15} -3.79129 q^{17} -1.20871 q^{19} -5.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +1.58258 q^{29} +10.3739 q^{31} +6.79129 q^{33} +2.79129 q^{35} +4.00000 q^{37} +2.16515 q^{39} -2.20871 q^{41} +7.16515 q^{43} +0.208712 q^{45} -13.5826 q^{47} +0.791288 q^{49} +6.79129 q^{51} -6.00000 q^{53} -3.79129 q^{55} +2.16515 q^{57} +4.41742 q^{59} +3.37386 q^{61} +0.582576 q^{63} -1.20871 q^{65} +7.16515 q^{67} -1.79129 q^{69} -5.37386 q^{71} -14.7477 q^{73} -1.79129 q^{75} -10.5826 q^{77} +8.00000 q^{79} -9.58258 q^{81} +6.00000 q^{83} -3.79129 q^{85} -2.83485 q^{87} -3.16515 q^{89} -3.37386 q^{91} -18.5826 q^{93} -1.20871 q^{95} +14.9564 q^{97} -0.791288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 5 q^{9} - 3 q^{11} - 7 q^{13} + q^{15} - 3 q^{17} - 7 q^{19} - 10 q^{21} + 2 q^{23} + 2 q^{25} + 10 q^{27} - 6 q^{29} + 7 q^{31} + 9 q^{33} + q^{35} + 8 q^{37} - 14 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) 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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