LMFDB - Embedded newform 9576.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.00000 q^{7} -2.85410 q^{11} +5.47214 q^{13} +3.61803 q^{17} -1.00000 q^{19} -2.23607 q^{23} -4.00000 q^{25} +6.09017 q^{29} -10.0902 q^{31} -1.00000 q^{35} -9.47214 q^{37} +0.381966 q^{41} -2.00000 q^{43} +0.708204 q^{47} +1.00000 q^{49} +11.0902 q^{53} -2.85410 q^{55} -7.94427 q^{59} -5.00000 q^{61} +5.47214 q^{65} -15.3262 q^{67} -9.00000 q^{71} -5.85410 q^{73} +2.85410 q^{77} +14.9443 q^{79} -5.38197 q^{83} +3.61803 q^{85} +7.23607 q^{89} -5.47214 q^{91} -1.00000 q^{95} -4.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} + q^{11} + 2 q^{13} + 5 q^{17} - 2 q^{19} - 8 q^{25} + q^{29} - 9 q^{31} - 2 q^{35} - 10 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} + 11 q^{53} + q^{55} + 2 q^{59} - 10 q^{61}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + q^11 + 2 * q^13 + 5 * q^17 - 2 * q^19 - 8 * q^25 + q^29 - 9 * q^31 - 2 * q^35 - 10 * q^37 + 3 * q^41 - 4 * q^43 - 12 * q^47 + 2 * q^49 + 11 * q^53 + q^55 + 2 * q^59 - 10 * q^61 + 2 * q^65 - 15 * q^67 - 18 * q^71 - 5 * q^73 - q^77 + 12 * q^79 - 13 * q^83 + 5 * q^85 + 10 * q^89 - 2 * q^91 - 2 * q^95 - 18 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.85410	−0.860544	−0.430272	−	0.902699i	$-0.641582\pi$
$11$	−2.85410	−0.860544	−0.430272	+	0.902699i	$0.641582\pi$
$12$	0	0
$13$	5.47214	1.51770	0.758849	−	0.651267i	$-0.225762\pi$
$13$	5.47214	1.51770	0.758849	+	0.651267i	$0.225762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.61803	0.877502	0.438751	−	0.898609i	$-0.355421\pi$
$17$	3.61803	0.877502	0.438751	+	0.898609i	$0.355421\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.23607	−0.466252	−0.233126	−	0.972446i	$-0.574896\pi$
$23$	−2.23607	−0.466252	−0.233126	+	0.972446i	$0.574896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.09017	1.13092	0.565458	−	0.824777i	$-0.308699\pi$
$29$	6.09017	1.13092	0.565458	+	0.824777i	$0.308699\pi$
$30$	0	0
$31$	−10.0902	−1.81225	−0.906124	−	0.423012i	$-0.860973\pi$
$31$	−10.0902	−1.81225	−0.906124	+	0.423012i	$0.860973\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.47214	−1.55721	−0.778605	−	0.627515i	$-0.784072\pi$
$37$	−9.47214	−1.55721	−0.778605	+	0.627515i	$0.784072\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.381966	0.0596531	0.0298265	−	0.999555i	$-0.490505\pi$
$41$	0.381966	0.0596531	0.0298265	+	0.999555i	$0.490505\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.708204	0.103302	0.0516511	−	0.998665i	$-0.483552\pi$
$47$	0.708204	0.103302	0.0516511	+	0.998665i	$0.483552\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.0902	1.52335	0.761676	−	0.647958i	$-0.224377\pi$
$53$	11.0902	1.52335	0.761676	+	0.647958i	$0.224377\pi$
$54$	0	0
$55$	−2.85410	−0.384847
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.94427	−1.03426	−0.517128	−	0.855908i	$-0.672999\pi$
$59$	−7.94427	−1.03426	−0.517128	+	0.855908i	$0.672999\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.47214	0.678735
$66$	0	0
$67$	−15.3262	−1.87240	−0.936199	−	0.351470i	$-0.885682\pi$
$67$	−15.3262	−1.87240	−0.936199	+	0.351470i	$0.885682\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	0	0
$73$	−5.85410	−0.685171	−0.342585	−	0.939487i	$-0.611303\pi$
$73$	−5.85410	−0.685171	−0.342585	+	0.939487i	$0.611303\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.85410	0.325255
$78$	0	0
$79$	14.9443	1.68136	0.840681	−	0.541531i	$-0.182155\pi$
$79$	14.9443	1.68136	0.840681	+	0.541531i	$0.182155\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.38197	−0.590748	−0.295374	−	0.955382i	$-0.595444\pi$
$83$	−5.38197	−0.590748	−0.295374	+	0.955382i	$0.595444\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.23607	0.767022	0.383511	−	0.923536i	$-0.374715\pi$
$89$	7.23607	0.767022	0.383511	+	0.923536i	$0.374715\pi$
$90$	0	0
$91$	−5.47214	−0.573636
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−4.52786	−0.459735	−0.229867	−	0.973222i	$-0.573829\pi$
$97$	−4.52786	−0.459735	−0.229867	+	0.973222i	$0.573829\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.70820	−0.866499	−0.433249	−	0.901274i	$-0.642633\pi$
$101$	−8.70820	−0.866499	−0.433249	+	0.901274i	$0.642633\pi$
$102$	0	0
$103$	15.4721	1.52451	0.762257	−	0.647274i	$-0.224091\pi$
$103$	15.4721	1.52451	0.762257	+	0.647274i	$0.224091\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.9443	1.34804	0.674022	−	0.738711i	$-0.264565\pi$
$107$	13.9443	1.34804	0.674022	+	0.738711i	$0.264565\pi$
$108$	0	0
$109$	−5.76393	−0.552085	−0.276042	−	0.961145i	$-0.589023\pi$
$109$	−5.76393	−0.552085	−0.276042	+	0.961145i	$0.589023\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.618034	−0.0581397	−0.0290699	−	0.999577i	$-0.509255\pi$
$113$	−0.618034	−0.0581397	−0.0290699	+	0.999577i	$0.509255\pi$
$114$	0	0
$115$	−2.23607	−0.208514
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.61803	−0.331665
$120$	0	0
$121$	−2.85410	−0.259464
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−12.1803	−1.08083	−0.540415	−	0.841398i	$-0.681733\pi$
$127$	−12.1803	−1.08083	−0.540415	+	0.841398i	$0.681733\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.0344	0.964084	0.482042	−	0.876148i	$-0.339895\pi$
$131$	11.0344	0.964084	0.482042	+	0.876148i	$0.339895\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.05573	0.431940	0.215970	−	0.976400i	$-0.430709\pi$
$137$	5.05573	0.431940	0.215970	+	0.976400i	$0.430709\pi$
$138$	0	0
$139$	−11.1803	−0.948304	−0.474152	−	0.880443i	$-0.657245\pi$
$139$	−11.1803	−0.948304	−0.474152	+	0.880443i	$0.657245\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.6180	−1.30605
$144$	0	0
$145$	6.09017	0.505761
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.9443	1.79774	0.898872	−	0.438210i	$-0.144388\pi$
$149$	21.9443	1.79774	0.898872	+	0.438210i	$0.144388\pi$
$150$	0	0
$151$	1.79837	0.146350	0.0731748	−	0.997319i	$-0.476687\pi$
$151$	1.79837	0.146350	0.0731748	+	0.997319i	$0.476687\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.0902	−0.810462
$156$	0	0
$157$	21.0344	1.67873	0.839366	−	0.543567i	$-0.182927\pi$
$157$	21.0344	1.67873	0.839366	+	0.543567i	$0.182927\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.23607	0.176227
$162$	0	0
$163$	4.32624	0.338857	0.169429	−	0.985542i	$-0.445808\pi$
$163$	4.32624	0.338857	0.169429	+	0.985542i	$0.445808\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.1803	−1.48422	−0.742110	−	0.670279i	$-0.766175\pi$
$167$	−19.1803	−1.48422	−0.742110	+	0.670279i	$0.766175\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.18034	−0.317825	−0.158913	−	0.987293i	$-0.550799\pi$
$173$	−4.18034	−0.317825	−0.158913	+	0.987293i	$0.550799\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.85410	−0.213326	−0.106663	−	0.994295i	$-0.534017\pi$
$179$	−2.85410	−0.213326	−0.106663	+	0.994295i	$0.534017\pi$
$180$	0	0
$181$	−10.5623	−0.785090	−0.392545	−	0.919733i	$-0.628405\pi$
$181$	−10.5623	−0.785090	−0.392545	+	0.919733i	$0.628405\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.47214	−0.696405
$186$	0	0
$187$	−10.3262	−0.755129
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.43769	−0.176385	−0.0881927	−	0.996103i	$-0.528109\pi$
$191$	−2.43769	−0.176385	−0.0881927	+	0.996103i	$0.528109\pi$
$192$	0	0
$193$	5.14590	0.370410	0.185205	−	0.982700i	$-0.440705\pi$
$193$	5.14590	0.370410	0.185205	+	0.982700i	$0.440705\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−27.2705	−1.94294	−0.971472	−	0.237156i	$-0.923785\pi$
$197$	−27.2705	−1.94294	−0.971472	+	0.237156i	$0.923785\pi$
$198$	0	0
$199$	21.1803	1.50143	0.750717	−	0.660624i	$-0.229708\pi$
$199$	21.1803	1.50143	0.750717	+	0.660624i	$0.229708\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.09017	−0.427446
$204$	0	0
$205$	0.381966	0.0266777
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.85410	0.197422
$210$	0	0
$211$	−12.9098	−0.888749	−0.444375	−	0.895841i	$-0.646574\pi$
$211$	−12.9098	−0.888749	−0.444375	+	0.895841i	$0.646574\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	10.0902	0.684965
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.7984	1.33178
$222$	0	0
$223$	18.7082	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$223$	18.7082	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.7984	−1.31406	−0.657032	−	0.753863i	$-0.728188\pi$
$227$	−19.7984	−1.31406	−0.657032	+	0.753863i	$0.728188\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.32624	0.217909	0.108955	−	0.994047i	$-0.465250\pi$
$233$	3.32624	0.217909	0.108955	+	0.994047i	$0.465250\pi$
$234$	0	0
$235$	0.708204	0.0461981
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.5967	1.46166	0.730831	−	0.682558i	$-0.239133\pi$
$239$	22.5967	1.46166	0.730831	+	0.682558i	$0.239133\pi$
$240$	0	0
$241$	−27.1246	−1.74725	−0.873625	−	0.486600i	$-0.838237\pi$
$241$	−27.1246	−1.74725	−0.873625	+	0.486600i	$0.838237\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.47214	−0.348184
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.85410	0.558866	0.279433	−	0.960165i	$-0.409854\pi$
$251$	8.85410	0.558866	0.279433	+	0.960165i	$0.409854\pi$
$252$	0	0
$253$	6.38197	0.401231
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.9787	−1.68289	−0.841443	−	0.540346i	$-0.818293\pi$
$257$	−26.9787	−1.68289	−0.841443	+	0.540346i	$0.818293\pi$
$258$	0	0
$259$	9.47214	0.588570
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.61803	−0.531411	−0.265705	−	0.964054i	$-0.585605\pi$
$263$	−8.61803	−0.531411	−0.265705	+	0.964054i	$0.585605\pi$
$264$	0	0
$265$	11.0902	0.681264
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.1459	−1.10637	−0.553187	−	0.833057i	$-0.686589\pi$
$269$	−18.1459	−1.10637	−0.553187	+	0.833057i	$0.686589\pi$
$270$	0	0
$271$	−20.5623	−1.24907	−0.624536	−	0.780996i	$-0.714712\pi$
$271$	−20.5623	−1.24907	−0.624536	+	0.780996i	$0.714712\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.4164	0.688435
$276$	0	0
$277$	−31.1246	−1.87010	−0.935048	−	0.354520i	$-0.884644\pi$
$277$	−31.1246	−1.87010	−0.935048	+	0.354520i	$0.884644\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.76393	−0.463157	−0.231579	−	0.972816i	$-0.574389\pi$
$281$	−7.76393	−0.463157	−0.231579	+	0.972816i	$0.574389\pi$
$282$	0	0
$283$	−13.3820	−0.795475	−0.397738	−	0.917499i	$-0.630205\pi$
$283$	−13.3820	−0.795475	−0.397738	+	0.917499i	$0.630205\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.381966	−0.0225467
$288$	0	0
$289$	−3.90983	−0.229990
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−31.6525	−1.84916	−0.924579	−	0.380991i	$-0.875583\pi$
$293$	−31.6525	−1.84916	−0.924579	+	0.380991i	$0.875583\pi$
$294$	0	0
$295$	−7.94427	−0.462533
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.2361	−0.707630
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.00000	−0.286299
$306$	0	0
$307$	10.6738	0.609184	0.304592	−	0.952483i	$-0.401480\pi$
$307$	10.6738	0.609184	0.304592	+	0.952483i	$0.401480\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.56231	0.372114	0.186057	−	0.982539i	$-0.440429\pi$
$311$	6.56231	0.372114	0.186057	+	0.982539i	$0.440429\pi$
$312$	0	0
$313$	−25.5967	−1.44681	−0.723407	−	0.690422i	$-0.757425\pi$
$313$	−25.5967	−1.44681	−0.723407	+	0.690422i	$0.757425\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.4164	−1.42753	−0.713764	−	0.700386i	$-0.753011\pi$
$317$	−25.4164	−1.42753	−0.713764	+	0.700386i	$0.753011\pi$
$318$	0	0
$319$	−17.3820	−0.973203
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.61803	−0.201313
$324$	0	0
$325$	−21.8885	−1.21416
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.708204	−0.0390445
$330$	0	0
$331$	−2.14590	−0.117949	−0.0589746	−	0.998259i	$-0.518783\pi$
$331$	−2.14590	−0.117949	−0.0589746	+	0.998259i	$0.518783\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.3262	−0.837362
$336$	0	0
$337$	17.0344	0.927925	0.463963	−	0.885855i	$-0.346427\pi$
$337$	17.0344	0.927925	0.463963	+	0.885855i	$0.346427\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	28.7984	1.55952
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.9098	0.800402	0.400201	−	0.916427i	$-0.368940\pi$
$347$	14.9098	0.800402	0.400201	+	0.916427i	$0.368940\pi$
$348$	0	0
$349$	5.67376	0.303710	0.151855	−	0.988403i	$-0.451475\pi$
$349$	5.67376	0.303710	0.151855	+	0.988403i	$0.451475\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.43769	−0.502318	−0.251159	−	0.967946i	$-0.580812\pi$
$353$	−9.43769	−0.502318	−0.251159	+	0.967946i	$0.580812\pi$
$354$	0	0
$355$	−9.00000	−0.477670
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.8541	−0.783970	−0.391985	−	0.919972i	$-0.628211\pi$
$359$	−14.8541	−0.783970	−0.391985	+	0.919972i	$0.628211\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.85410	−0.306418
$366$	0	0
$367$	−13.4721	−0.703240	−0.351620	−	0.936143i	$-0.614369\pi$
$367$	−13.4721	−0.703240	−0.351620	+	0.936143i	$0.614369\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.0902	−0.575773
$372$	0	0
$373$	33.9230	1.75647	0.878233	−	0.478233i	$-0.158723\pi$
$373$	33.9230	1.75647	0.878233	+	0.478233i	$0.158723\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.3262	1.71639
$378$	0	0
$379$	−10.8197	−0.555769	−0.277884	−	0.960615i	$-0.589633\pi$
$379$	−10.8197	−0.555769	−0.277884	+	0.960615i	$0.589633\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.4164	0.634449	0.317224	−	0.948351i	$-0.397249\pi$
$383$	12.4164	0.634449	0.317224	+	0.948351i	$0.397249\pi$
$384$	0	0
$385$	2.85410	0.145459
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.50658	−0.380599	−0.190299	−	0.981726i	$-0.560946\pi$
$389$	−7.50658	−0.380599	−0.190299	+	0.981726i	$0.560946\pi$
$390$	0	0
$391$	−8.09017	−0.409137
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.9443	0.751928
$396$	0	0
$397$	13.1246	0.658705	0.329353	−	0.944207i	$-0.393169\pi$
$397$	13.1246	0.658705	0.329353	+	0.944207i	$0.393169\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.7984	1.78769	0.893843	−	0.448381i	$-0.147999\pi$
$401$	35.7984	1.78769	0.893843	+	0.448381i	$0.147999\pi$
$402$	0	0
$403$	−55.2148	−2.75044
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.0344	1.34005
$408$	0	0
$409$	16.7984	0.830626	0.415313	−	0.909679i	$-0.363672\pi$
$409$	16.7984	0.830626	0.415313	+	0.909679i	$0.363672\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.94427	0.390912
$414$	0	0
$415$	−5.38197	−0.264190
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.29180	−0.0631084	−0.0315542	−	0.999502i	$-0.510046\pi$
$419$	−1.29180	−0.0631084	−0.0315542	+	0.999502i	$0.510046\pi$
$420$	0	0
$421$	−9.47214	−0.461644	−0.230822	−	0.972996i	$-0.574141\pi$
$421$	−9.47214	−0.461644	−0.230822	+	0.972996i	$0.574141\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−14.4721	−0.702002
$426$	0	0
$427$	5.00000	0.241967
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.88854	−0.379978	−0.189989	−	0.981786i	$-0.560845\pi$
$431$	−7.88854	−0.379978	−0.189989	+	0.981786i	$0.560845\pi$
$432$	0	0
$433$	−26.1246	−1.25547	−0.627734	−	0.778428i	$-0.716018\pi$
$433$	−26.1246	−1.25547	−0.627734	+	0.778428i	$0.716018\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.23607	0.106966
$438$	0	0
$439$	7.41641	0.353966	0.176983	−	0.984214i	$-0.443366\pi$
$439$	7.41641	0.353966	0.176983	+	0.984214i	$0.443366\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.38197	0.350728	0.175364	−	0.984504i	$-0.443890\pi$
$443$	7.38197	0.350728	0.175364	+	0.984504i	$0.443890\pi$
$444$	0	0
$445$	7.23607	0.343023
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.03444	0.379169	0.189584	−	0.981864i	$-0.439286\pi$
$449$	8.03444	0.379169	0.189584	+	0.981864i	$0.439286\pi$
$450$	0	0
$451$	−1.09017	−0.0513341
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.47214	−0.256538
$456$	0	0
$457$	−20.5066	−0.959257	−0.479629	−	0.877472i	$-0.659229\pi$
$457$	−20.5066	−0.959257	−0.479629	+	0.877472i	$0.659229\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.09017	0.330222	0.165111	−	0.986275i	$-0.447202\pi$
$461$	7.09017	0.330222	0.165111	+	0.986275i	$0.447202\pi$
$462$	0	0
$463$	16.4164	0.762935	0.381468	−	0.924382i	$-0.375419\pi$
$463$	16.4164	0.762935	0.381468	+	0.924382i	$0.375419\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.5623	−0.905236	−0.452618	−	0.891705i	$-0.649510\pi$
$467$	−19.5623	−0.905236	−0.452618	+	0.891705i	$0.649510\pi$
$468$	0	0
$469$	15.3262	0.707700
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.70820	0.262463
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.965558	0.0441175	0.0220587	−	0.999757i	$-0.492978\pi$
$479$	0.965558	0.0441175	0.0220587	+	0.999757i	$0.492978\pi$
$480$	0	0
$481$	−51.8328	−2.36337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.52786	−0.205600
$486$	0	0
$487$	−21.4721	−0.972995	−0.486498	−	0.873682i	$-0.661726\pi$
$487$	−21.4721	−0.972995	−0.486498	+	0.873682i	$0.661726\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.41641	−0.154180	−0.0770902	−	0.997024i	$-0.524563\pi$
$491$	−3.41641	−0.154180	−0.0770902	+	0.997024i	$0.524563\pi$
$492$	0	0
$493$	22.0344	0.992381
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.00000	0.403705
$498$	0	0
$499$	30.3951	1.36067	0.680336	−	0.732900i	$-0.261834\pi$
$499$	30.3951	1.36067	0.680336	+	0.732900i	$0.261834\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.8328	0.661363	0.330681	−	0.943742i	$-0.392721\pi$
$503$	14.8328	0.661363	0.330681	+	0.943742i	$0.392721\pi$
$504$	0	0
$505$	−8.70820	−0.387510
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	5.85410	0.258970
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.4721	0.681784
$516$	0	0
$517$	−2.02129	−0.0888961
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.1803	−1.54128	−0.770639	−	0.637272i	$-0.780063\pi$
$521$	−35.1803	−1.54128	−0.770639	+	0.637272i	$0.780063\pi$
$522$	0	0
$523$	−6.23607	−0.272684	−0.136342	−	0.990662i	$-0.543535\pi$
$523$	−6.23607	−0.272684	−0.136342	+	0.990662i	$0.543535\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.5066	−1.59025
$528$	0	0
$529$	−18.0000	−0.782609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.09017	0.0905353
$534$	0	0
$535$	13.9443	0.602863
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.85410	−0.122935
$540$	0	0
$541$	11.8885	0.511128	0.255564	−	0.966792i	$-0.417739\pi$
$541$	11.8885	0.511128	0.255564	+	0.966792i	$0.417739\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.76393	−0.246900
$546$	0	0
$547$	−16.9098	−0.723012	−0.361506	−	0.932370i	$-0.617737\pi$
$547$	−16.9098	−0.723012	−0.361506	+	0.932370i	$0.617737\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.09017	−0.259450
$552$	0	0
$553$	−14.9443	−0.635495
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.5623	1.88817	0.944083	−	0.329709i	$-0.106950\pi$
$557$	44.5623	1.88817	0.944083	+	0.329709i	$0.106950\pi$
$558$	0	0
$559$	−10.9443	−0.462893
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.8885	1.21751	0.608753	−	0.793359i	$-0.291670\pi$
$563$	28.8885	1.21751	0.608753	+	0.793359i	$0.291670\pi$
$564$	0	0
$565$	−0.618034	−0.0260009
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.7771	−1.62562	−0.812810	−	0.582529i	$-0.802063\pi$
$569$	−38.7771	−1.62562	−0.812810	+	0.582529i	$0.802063\pi$
$570$	0	0
$571$	−26.4164	−1.10549	−0.552746	−	0.833350i	$-0.686420\pi$
$571$	−26.4164	−1.10549	−0.552746	+	0.833350i	$0.686420\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.94427	0.373002
$576$	0	0
$577$	24.5066	1.02022	0.510111	−	0.860109i	$-0.329604\pi$
$577$	24.5066	1.02022	0.510111	+	0.860109i	$0.329604\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.38197	0.223282
$582$	0	0
$583$	−31.6525	−1.31091
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.4164	0.718852	0.359426	−	0.933174i	$-0.382972\pi$
$587$	17.4164	0.718852	0.359426	+	0.933174i	$0.382972\pi$
$588$	0	0
$589$	10.0902	0.415758
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.1803	−0.541252	−0.270626	−	0.962685i	$-0.587231\pi$
$593$	−13.1803	−0.541252	−0.270626	+	0.962685i	$0.587231\pi$
$594$	0	0
$595$	−3.61803	−0.148325
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.2705	0.869089	0.434545	−	0.900650i	$-0.356909\pi$
$599$	21.2705	0.869089	0.434545	+	0.900650i	$0.356909\pi$
$600$	0	0
$601$	19.3820	0.790607	0.395303	−	0.918551i	$-0.370639\pi$
$601$	19.3820	0.790607	0.395303	+	0.918551i	$0.370639\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.85410	−0.116036
$606$	0	0
$607$	−2.88854	−0.117242	−0.0586212	−	0.998280i	$-0.518670\pi$
$607$	−2.88854	−0.117242	−0.0586212	+	0.998280i	$0.518670\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.87539	0.156781
$612$	0	0
$613$	11.5623	0.466997	0.233499	−	0.972357i	$-0.424983\pi$
$613$	11.5623	0.466997	0.233499	+	0.972357i	$0.424983\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.8673	−1.68551	−0.842756	−	0.538296i	$-0.819068\pi$
$617$	−41.8673	−1.68551	−0.842756	+	0.538296i	$0.819068\pi$
$618$	0	0
$619$	−19.6738	−0.790755	−0.395378	−	0.918519i	$-0.629386\pi$
$619$	−19.6738	−0.790755	−0.395378	+	0.918519i	$0.629386\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.23607	−0.289907
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.2705	−1.36645
$630$	0	0
$631$	−43.9443	−1.74939	−0.874697	−	0.484670i	$-0.838940\pi$
$631$	−43.9443	−1.74939	−0.874697	+	0.484670i	$0.838940\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.1803	−0.483362
$636$	0	0
$637$	5.47214	0.216814
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.61803	0.300894	0.150447	−	0.988618i	$-0.451929\pi$
$641$	7.61803	0.300894	0.150447	+	0.988618i	$0.451929\pi$
$642$	0	0
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.0689	−1.69321	−0.846606	−	0.532220i	$-0.821358\pi$
$647$	−43.0689	−1.69321	−0.846606	+	0.532220i	$0.821358\pi$
$648$	0	0
$649$	22.6738	0.890023
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.6525	0.769061	0.384530	−	0.923112i	$-0.374363\pi$
$653$	19.6525	0.769061	0.384530	+	0.923112i	$0.374363\pi$
$654$	0	0
$655$	11.0344	0.431151
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.4508	1.73156	0.865780	−	0.500425i	$-0.166823\pi$
$659$	44.4508	1.73156	0.865780	+	0.500425i	$0.166823\pi$
$660$	0	0
$661$	2.94427	0.114519	0.0572595	−	0.998359i	$-0.481764\pi$
$661$	2.94427	0.114519	0.0572595	+	0.998359i	$0.481764\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	−13.6180	−0.527292
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.2705	0.550907
$672$	0	0
$673$	1.20163	0.0463193	0.0231596	−	0.999732i	$-0.492627\pi$
$673$	1.20163	0.0463193	0.0231596	+	0.999732i	$0.492627\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.7984	0.876213	0.438106	−	0.898923i	$-0.355649\pi$
$677$	22.7984	0.876213	0.438106	+	0.898923i	$0.355649\pi$
$678$	0	0
$679$	4.52786	0.173763
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.05573	−0.308244	−0.154122	−	0.988052i	$-0.549255\pi$
$683$	−8.05573	−0.308244	−0.154122	+	0.988052i	$0.549255\pi$
$684$	0	0
$685$	5.05573	0.193169
$686$	0	0
$687$	0	0
$688$	0	0
$689$	60.6869	2.31199
$690$	0	0
$691$	−17.2361	−0.655691	−0.327845	−	0.944731i	$-0.606323\pi$
$691$	−17.2361	−0.655691	−0.327845	+	0.944731i	$0.606323\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.1803	−0.424094
$696$	0	0
$697$	1.38197	0.0523457
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.11146	−0.0797486	−0.0398743	−	0.999205i	$-0.512696\pi$
$701$	−2.11146	−0.0797486	−0.0398743	+	0.999205i	$0.512696\pi$
$702$	0	0
$703$	9.47214	0.357248
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.70820	0.327506
$708$	0	0
$709$	−37.1803	−1.39634	−0.698168	−	0.715933i	$-0.746001\pi$
$709$	−37.1803	−1.39634	−0.698168	+	0.715933i	$0.746001\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.5623	0.844965
$714$	0	0
$715$	−15.6180	−0.584081
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.7771	1.18509	0.592543	−	0.805539i	$-0.298124\pi$
$719$	31.7771	1.18509	0.592543	+	0.805539i	$0.298124\pi$
$720$	0	0
$721$	−15.4721	−0.576212
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.3607	−0.904733
$726$	0	0
$727$	43.8328	1.62567	0.812835	−	0.582495i	$-0.197923\pi$
$727$	43.8328	1.62567	0.812835	+	0.582495i	$0.197923\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.23607	−0.267636
$732$	0	0
$733$	21.5279	0.795150	0.397575	−	0.917570i	$-0.369852\pi$
$733$	21.5279	0.795150	0.397575	+	0.917570i	$0.369852\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43.7426	1.61128
$738$	0	0
$739$	39.0689	1.43717	0.718586	−	0.695438i	$-0.244790\pi$
$739$	39.0689	1.43717	0.718586	+	0.695438i	$0.244790\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.4164	−1.18924	−0.594621	−	0.804006i	$-0.702698\pi$
$743$	−32.4164	−1.18924	−0.594621	+	0.804006i	$0.702698\pi$
$744$	0	0
$745$	21.9443	0.803976
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.9443	−0.509513
$750$	0	0
$751$	10.7295	0.391525	0.195762	−	0.980651i	$-0.437282\pi$
$751$	10.7295	0.391525	0.195762	+	0.980651i	$0.437282\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.79837	0.0654495
$756$	0	0
$757$	−4.11146	−0.149433	−0.0747167	−	0.997205i	$-0.523805\pi$
$757$	−4.11146	−0.149433	−0.0747167	+	0.997205i	$0.523805\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.888544	−0.0322097	−0.0161048	−	0.999870i	$-0.505127\pi$
$761$	−0.888544	−0.0322097	−0.0161048	+	0.999870i	$0.505127\pi$
$762$	0	0
$763$	5.76393	0.208668
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43.4721	−1.56969
$768$	0	0
$769$	−12.5836	−0.453776	−0.226888	−	0.973921i	$-0.572855\pi$
$769$	−12.5836	−0.453776	−0.226888	+	0.973921i	$0.572855\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30.1803	−1.08551	−0.542756	−	0.839891i	$-0.682619\pi$
$773$	−30.1803	−1.08551	−0.542756	+	0.839891i	$0.682619\pi$
$774$	0	0
$775$	40.3607	1.44980
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.381966	−0.0136854
$780$	0	0
$781$	25.6869	0.919150
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.0344	0.750751
$786$	0	0
$787$	−34.8328	−1.24166	−0.620828	−	0.783947i	$-0.713203\pi$
$787$	−34.8328	−1.24166	−0.620828	+	0.783947i	$0.713203\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.618034	0.0219748
$792$	0	0
$793$	−27.3607	−0.971606
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.0344	0.957609	0.478805	−	0.877922i	$-0.341070\pi$
$797$	27.0344	0.957609	0.478805	+	0.877922i	$0.341070\pi$
$798$	0	0
$799$	2.56231	0.0906479
$800$	0	0
$801$	0	0
$802$	0	0
$803$	16.7082	0.589620
$804$	0	0
$805$	2.23607	0.0788110
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.2918	−1.17048	−0.585239	−	0.810861i	$-0.698999\pi$
$809$	−33.2918	−1.17048	−0.585239	+	0.810861i	$0.698999\pi$
$810$	0	0
$811$	6.58359	0.231181	0.115591	−	0.993297i	$-0.463124\pi$
$811$	6.58359	0.231181	0.115591	+	0.993297i	$0.463124\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.32624	0.151542
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	0	0
$823$	30.5410	1.06459	0.532297	−	0.846558i	$-0.321329\pi$
$823$	30.5410	1.06459	0.532297	+	0.846558i	$0.321329\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.3050	−1.19290	−0.596450	−	0.802650i	$-0.703423\pi$
$827$	−34.3050	−1.19290	−0.596450	+	0.802650i	$0.703423\pi$
$828$	0	0
$829$	−31.7771	−1.10366	−0.551832	−	0.833955i	$-0.686071\pi$
$829$	−31.7771	−1.10366	−0.551832	+	0.833955i	$0.686071\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.61803	0.125357
$834$	0	0
$835$	−19.1803	−0.663763
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.2492	−1.73480	−0.867398	−	0.497615i	$-0.834209\pi$
$839$	−50.2492	−1.73480	−0.867398	+	0.497615i	$0.834209\pi$
$840$	0	0
$841$	8.09017	0.278971
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.9443	0.582901
$846$	0	0
$847$	2.85410	0.0980681
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.1803	0.726053
$852$	0	0
$853$	−34.1459	−1.16913	−0.584567	−	0.811346i	$-0.698735\pi$
$853$	−34.1459	−1.16913	−0.584567	+	0.811346i	$0.698735\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.9787	1.16069	0.580345	−	0.814370i	$-0.302918\pi$
$857$	33.9787	1.16069	0.580345	+	0.814370i	$0.302918\pi$
$858$	0	0
$859$	26.2705	0.896338	0.448169	−	0.893949i	$-0.352076\pi$
$859$	26.2705	0.896338	0.448169	+	0.893949i	$0.352076\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−55.4508	−1.88757	−0.943784	−	0.330562i	$-0.892762\pi$
$863$	−55.4508	−1.88757	−0.943784	+	0.330562i	$0.892762\pi$
$864$	0	0
$865$	−4.18034	−0.142136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−42.6525	−1.44689
$870$	0	0
$871$	−83.8673	−2.84173
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−52.1803	−1.76200	−0.881002	−	0.473112i	$-0.843131\pi$
$877$	−52.1803	−1.76200	−0.881002	+	0.473112i	$0.843131\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.7984	1.10501	0.552503	−	0.833511i	$-0.313673\pi$
$881$	32.7984	1.10501	0.552503	+	0.833511i	$0.313673\pi$
$882$	0	0
$883$	10.3475	0.348222	0.174111	−	0.984726i	$-0.444295\pi$
$883$	10.3475	0.348222	0.174111	+	0.984726i	$0.444295\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.6525	−1.12994	−0.564970	−	0.825112i	$-0.691112\pi$
$887$	−33.6525	−1.12994	−0.564970	+	0.825112i	$0.691112\pi$
$888$	0	0
$889$	12.1803	0.408515
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.708204	−0.0236991
$894$	0	0
$895$	−2.85410	−0.0954021
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−61.4508	−2.04950
$900$	0	0
$901$	40.1246	1.33674
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5623	−0.351103
$906$	0	0
$907$	32.8197	1.08976	0.544879	−	0.838514i	$-0.316575\pi$
$907$	32.8197	1.08976	0.544879	+	0.838514i	$0.316575\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.52786	−0.116883	−0.0584417	−	0.998291i	$-0.518613\pi$
$911$	−3.52786	−0.116883	−0.0584417	+	0.998291i	$0.518613\pi$
$912$	0	0
$913$	15.3607	0.508364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.0344	−0.364389
$918$	0	0
$919$	−3.36068	−0.110859	−0.0554293	−	0.998463i	$-0.517653\pi$
$919$	−3.36068	−0.110859	−0.0554293	+	0.998463i	$0.517653\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−49.2492	−1.62106
$924$	0	0
$925$	37.8885	1.24577
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.21478	0.138283	0.0691413	−	0.997607i	$-0.477974\pi$
$929$	4.21478	0.138283	0.0691413	+	0.997607i	$0.477974\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.3262	−0.337704
$936$	0	0
$937$	−1.14590	−0.0374349	−0.0187174	−	0.999825i	$-0.505958\pi$
$937$	−1.14590	−0.0374349	−0.0187174	+	0.999825i	$0.505958\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.36068	−0.207352	−0.103676	−	0.994611i	$-0.533061\pi$
$941$	−6.36068	−0.207352	−0.103676	+	0.994611i	$0.533061\pi$
$942$	0	0
$943$	−0.854102	−0.0278134
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.1033	−1.49816	−0.749078	−	0.662481i	$-0.769503\pi$
$947$	−46.1033	−1.49816	−0.749078	+	0.662481i	$0.769503\pi$
$948$	0	0
$949$	−32.0344	−1.03988
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.4377	−0.370503	−0.185252	−	0.982691i	$-0.559310\pi$
$953$	−11.4377	−0.370503	−0.185252	+	0.982691i	$0.559310\pi$
$954$	0	0
$955$	−2.43769	−0.0788819
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.05573	−0.163258
$960$	0	0
$961$	70.8115	2.28424
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.14590	0.165652
$966$	0	0
$967$	−39.1033	−1.25748	−0.628739	−	0.777616i	$-0.716429\pi$
$967$	−39.1033	−1.25748	−0.628739	+	0.777616i	$0.716429\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.3607	−0.557131	−0.278565	−	0.960417i	$-0.589859\pi$
$971$	−17.3607	−0.557131	−0.278565	+	0.960417i	$0.589859\pi$
$972$	0	0
$973$	11.1803	0.358425
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.5967	−0.690941	−0.345471	−	0.938430i	$-0.612281\pi$
$977$	−21.5967	−0.690941	−0.345471	+	0.938430i	$0.612281\pi$
$978$	0	0
$979$	−20.6525	−0.660056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.4164	−1.54424	−0.772122	−	0.635475i	$-0.780804\pi$
$983$	−48.4164	−1.54424	−0.772122	+	0.635475i	$0.780804\pi$
$984$	0	0
$985$	−27.2705	−0.868911
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.47214	0.142206
$990$	0	0
$991$	35.1246	1.11577	0.557885	−	0.829918i	$-0.311613\pi$
$991$	35.1246	1.11577	0.557885	+	0.829918i	$0.311613\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.1803	0.671462
$996$	0	0
$997$	−24.0344	−0.761178	−0.380589	−	0.924744i	$-0.624279\pi$
$997$	−24.0344	−0.761178	−0.380589	+	0.924744i	$0.624279\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bt.1.1		2
3.2	odd	2		1064.2.a.d.1.1	✓	2
12.11	even	2		2128.2.a.f.1.2		2
21.20	even	2		7448.2.a.z.1.2		2
24.5	odd	2		8512.2.a.q.1.2		2
24.11	even	2		8512.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.d.1.1	✓	2	3.2	odd	2
2128.2.a.f.1.2		2	12.11	even	2
7448.2.a.z.1.2		2	21.20	even	2
8512.2.a.q.1.2		2	24.5	odd	2
8512.2.a.y.1.1		2	24.11	even	2
9576.2.a.bt.1.1		2	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.00000 q^{7} -2.85410 q^{11} +5.47214 q^{13} +3.61803 q^{17} -1.00000 q^{19} -2.23607 q^{23} -4.00000 q^{25} +6.09017 q^{29} -10.0902 q^{31} -1.00000 q^{35} -9.47214 q^{37} +0.381966 q^{41} -2.00000 q^{43} +0.708204 q^{47} +1.00000 q^{49} +11.0902 q^{53} -2.85410 q^{55} -7.94427 q^{59} -5.00000 q^{61} +5.47214 q^{65} -15.3262 q^{67} -9.00000 q^{71} -5.85410 q^{73} +2.85410 q^{77} +14.9443 q^{79} -5.38197 q^{83} +3.61803 q^{85} +7.23607 q^{89} -5.47214 q^{91} -1.00000 q^{95} -4.52786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} + q^{11} + 2 q^{13} + 5 q^{17} - 2 q^{19} - 8 q^{25} + q^{29} - 9 q^{31} - 2 q^{35} - 10 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} + 11 q^{53} + q^{55} + 2 q^{59} - 10 q^{61}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + q^11 + 2 * q^13 + 5 * q^17 - 2 * q^19 - 8 * q^25 + q^29 - 9 * q^31 - 2 * q^35 - 10 * q^37 + 3 * q^41 - 4 * q^43 - 12 * q^47 + 2 * q^49 + 11 * q^53 + q^55 + 2 * q^59 - 10 * q^61 + 2 * q^65 - 15 * q^67 - 18 * q^71 - 5 * q^73 - q^77 + 12 * q^79 - 13 * q^83 + 5 * q^85 + 10 * q^89 - 2 * q^91 - 2 * q^95 - 18 * q^97

Introduction

L-functions

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