LMFDB - Embedded newform 9576.2.a.bi.1.2

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.23607 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.47214 q^{13} -1.23607 q^{17} -1.00000 q^{19} +2.00000 q^{23} -3.47214 q^{25} +5.70820 q^{29} -1.52786 q^{31} +1.23607 q^{35} +8.47214 q^{37} -8.47214 q^{41} +8.00000 q^{43} +1.70820 q^{47} +1.00000 q^{49} +11.2361 q^{53} -2.47214 q^{55} -8.94427 q^{59} -13.4164 q^{61} -5.52786 q^{65} -1.52786 q^{67} +7.70820 q^{71} +12.4721 q^{73} -2.00000 q^{77} +4.00000 q^{79} -13.7082 q^{83} -1.52786 q^{85} +4.47214 q^{89} -4.47214 q^{91} -1.23607 q^{95} -17.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 16 q^{43} - 10 q^{47} + 2 q^{49} + 18 q^{53} + 4 q^{55} - 20 q^{65}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^29 - 12 * q^31 - 2 * q^35 + 8 * q^37 - 8 * q^41 + 16 * q^43 - 10 * q^47 + 2 * q^49 + 18 * q^53 + 4 * q^55 - 20 * q^65 - 12 * q^67 + 2 * q^71 + 16 * q^73 - 4 * q^77 + 8 * q^79 - 14 * q^83 - 12 * q^85 + 2 * q^95 - 8 * 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
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.23607	−0.299791	−0.149895	−	0.988702i	$-0.547894\pi$
$17$	−1.23607	−0.299791	−0.149895	+	0.988702i	$0.547894\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.70820	1.05999	0.529993	−	0.848002i	$-0.322194\pi$
$29$	5.70820	1.05999	0.529993	+	0.848002i	$0.322194\pi$
$30$	0	0
$31$	−1.52786	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	8.47214	1.39281	0.696405	−	0.717649i	$-0.254782\pi$
$37$	8.47214	1.39281	0.696405	+	0.717649i	$0.254782\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.47214	−1.32313	−0.661563	−	0.749890i	$-0.730106\pi$
$41$	−8.47214	−1.32313	−0.661563	+	0.749890i	$0.730106\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.70820	0.249167	0.124584	−	0.992209i	$-0.460241\pi$
$47$	1.70820	0.249167	0.124584	+	0.992209i	$0.460241\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−13.4164	−1.71780	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	−13.4164	−1.71780	−0.858898	+	0.512148i	$0.828850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.52786	−0.685647
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.70820	0.914796	0.457398	−	0.889262i	$-0.348782\pi$
$71$	7.70820	0.914796	0.457398	+	0.889262i	$0.348782\pi$
$72$	0	0
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.7082	−1.50467	−0.752335	−	0.658780i	$-0.771073\pi$
$83$	−13.7082	−1.50467	−0.752335	+	0.658780i	$0.771073\pi$
$84$	0	0
$85$	−1.52786	−0.165720
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.23607	−0.126818
$96$	0	0
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.7639	−1.07105	−0.535526	−	0.844519i	$-0.679886\pi$
$101$	−10.7639	−1.07105	−0.535526	+	0.844519i	$0.679886\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.1803	−1.75756	−0.878780	−	0.477227i	$-0.841642\pi$
$107$	−18.1803	−1.75756	−0.878780	+	0.477227i	$0.841642\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.23607	−0.680712	−0.340356	−	0.940297i	$-0.610548\pi$
$113$	−7.23607	−0.680712	−0.340356	+	0.940297i	$0.610548\pi$
$114$	0	0
$115$	2.47214	0.230528
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−15.4164	−1.36798	−0.683992	−	0.729489i	$-0.739758\pi$
$127$	−15.4164	−1.36798	−0.683992	+	0.729489i	$0.739758\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.2361	−1.33118	−0.665591	−	0.746317i	$-0.731820\pi$
$131$	−15.2361	−1.33118	−0.665591	+	0.746317i	$0.731820\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	9.52786	0.808143	0.404071	−	0.914727i	$-0.367595\pi$
$139$	9.52786	0.808143	0.404071	+	0.914727i	$0.367595\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	7.05573	0.585946
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.5279	1.10825	0.554123	−	0.832435i	$-0.313054\pi$
$149$	13.5279	1.10825	0.554123	+	0.832435i	$0.313054\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.88854	−0.151691
$156$	0	0
$157$	−3.52786	−0.281554	−0.140777	−	0.990041i	$-0.544960\pi$
$157$	−3.52786	−0.281554	−0.140777	+	0.990041i	$0.544960\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.4164	−1.81202	−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164	−1.81202	−0.906008	+	0.423261i	$0.860886\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.4721	−1.55647	−0.778234	−	0.627975i	$-0.783884\pi$
$173$	−20.4721	−1.55647	−0.778234	+	0.627975i	$0.783884\pi$
$174$	0	0
$175$	−3.47214	−0.262469
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.70820	−0.576138	−0.288069	−	0.957610i	$-0.593013\pi$
$179$	−7.70820	−0.576138	−0.288069	+	0.957610i	$0.593013\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4721	0.769927
$186$	0	0
$187$	2.47214	0.180780
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	−8.47214	−0.609838	−0.304919	−	0.952378i	$-0.598629\pi$
$193$	−8.47214	−0.609838	−0.304919	+	0.952378i	$0.598629\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.70820	0.400637
$204$	0	0
$205$	−10.4721	−0.731406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−6.47214	−0.445560	−0.222780	−	0.974869i	$-0.571513\pi$
$211$	−6.47214	−0.445560	−0.222780	+	0.974869i	$0.571513\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.88854	0.674393
$216$	0	0
$217$	−1.52786	−0.103718
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.52786	0.371844
$222$	0	0
$223$	−13.5279	−0.905893	−0.452946	−	0.891538i	$-0.649627\pi$
$223$	−13.5279	−0.905893	−0.452946	+	0.891538i	$0.649627\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.52786	0.366897	0.183449	−	0.983029i	$-0.441274\pi$
$227$	5.52786	0.366897	0.183449	+	0.983029i	$0.441274\pi$
$228$	0	0
$229$	14.9443	0.987545	0.493773	−	0.869591i	$-0.335618\pi$
$229$	14.9443	0.987545	0.493773	+	0.869591i	$0.335618\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.41641	0.485865	0.242933	−	0.970043i	$-0.421891\pi$
$233$	7.41641	0.485865	0.242933	+	0.970043i	$0.421891\pi$
$234$	0	0
$235$	2.11146	0.137736
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.4164	0.867835	0.433918	−	0.900953i	$-0.357131\pi$
$239$	13.4164	0.867835	0.433918	+	0.900953i	$0.357131\pi$
$240$	0	0
$241$	21.4164	1.37955	0.689776	−	0.724023i	$-0.257709\pi$
$241$	21.4164	1.37955	0.689776	+	0.724023i	$0.257709\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.23607	0.0789695
$246$	0	0
$247$	4.47214	0.284555
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7639	0.805652	0.402826	−	0.915277i	$-0.368028\pi$
$251$	12.7639	0.805652	0.402826	+	0.915277i	$0.368028\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.94427	0.183659	0.0918293	−	0.995775i	$-0.470729\pi$
$257$	2.94427	0.183659	0.0918293	+	0.995775i	$0.470729\pi$
$258$	0	0
$259$	8.47214	0.526433
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.4721	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$263$	−20.4721	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$264$	0	0
$265$	13.8885	0.853166
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	7.41641	0.450515	0.225257	−	0.974299i	$-0.427678\pi$
$271$	7.41641	0.450515	0.225257	+	0.974299i	$0.427678\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.94427	0.418755
$276$	0	0
$277$	21.4164	1.28679	0.643394	−	0.765536i	$-0.277526\pi$
$277$	21.4164	1.28679	0.643394	+	0.765536i	$0.277526\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.2361	−0.670288	−0.335144	−	0.942167i	$-0.608785\pi$
$281$	−11.2361	−0.670288	−0.335144	+	0.942167i	$0.608785\pi$
$282$	0	0
$283$	0.944272	0.0561311	0.0280656	−	0.999606i	$-0.491065\pi$
$283$	0.944272	0.0561311	0.0280656	+	0.999606i	$0.491065\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.47214	−0.500094
$288$	0	0
$289$	−15.4721	−0.910126
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.52786	−0.439783	−0.219891	−	0.975524i	$-0.570570\pi$
$293$	−7.52786	−0.439783	−0.219891	+	0.975524i	$0.570570\pi$
$294$	0	0
$295$	−11.0557	−0.643689
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.94427	−0.517261
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.5836	−0.949574
$306$	0	0
$307$	−17.5279	−1.00037	−0.500184	−	0.865919i	$-0.666734\pi$
$307$	−17.5279	−1.00037	−0.500184	+	0.865919i	$0.666734\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.34752	−0.0764111	−0.0382055	−	0.999270i	$-0.512164\pi$
$311$	−1.34752	−0.0764111	−0.0382055	+	0.999270i	$0.512164\pi$
$312$	0	0
$313$	5.41641	0.306153	0.153077	−	0.988214i	$-0.451082\pi$
$313$	5.41641	0.306153	0.153077	+	0.988214i	$0.451082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.6525	0.822965	0.411483	−	0.911418i	$-0.365011\pi$
$317$	14.6525	0.822965	0.411483	+	0.911418i	$0.365011\pi$
$318$	0	0
$319$	−11.4164	−0.639196
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.23607	0.0687767
$324$	0	0
$325$	15.5279	0.861331
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.70820	0.0941763
$330$	0	0
$331$	−11.0557	−0.607678	−0.303839	−	0.952723i	$-0.598268\pi$
$331$	−11.0557	−0.607678	−0.303839	+	0.952723i	$0.598268\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.88854	−0.103182
$336$	0	0
$337$	−6.94427	−0.378279	−0.189139	−	0.981950i	$-0.560570\pi$
$337$	−6.94427	−0.378279	−0.189139	+	0.981950i	$0.560570\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.05573	0.165477
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.3607	−1.62985	−0.814923	−	0.579569i	$-0.803221\pi$
$347$	−30.3607	−1.62985	−0.814923	+	0.579569i	$0.803221\pi$
$348$	0	0
$349$	34.9443	1.87052	0.935262	−	0.353956i	$-0.115164\pi$
$349$	34.9443	1.87052	0.935262	+	0.353956i	$0.115164\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.1803	−1.60634	−0.803169	−	0.595752i	$-0.796854\pi$
$353$	−30.1803	−1.60634	−0.803169	+	0.595752i	$0.796854\pi$
$354$	0	0
$355$	9.52786	0.505687
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.9443	−0.788729	−0.394364	−	0.918954i	$-0.629035\pi$
$359$	−14.9443	−0.788729	−0.394364	+	0.918954i	$0.629035\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.4164	0.806932
$366$	0	0
$367$	20.9443	1.09328	0.546641	−	0.837367i	$-0.315906\pi$
$367$	20.9443	1.09328	0.546641	+	0.837367i	$0.315906\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.2361	0.583348
$372$	0	0
$373$	31.8885	1.65113	0.825563	−	0.564310i	$-0.190858\pi$
$373$	31.8885	1.65113	0.825563	+	0.564310i	$0.190858\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−25.5279	−1.31475
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.52786	−0.0780702	−0.0390351	−	0.999238i	$-0.512428\pi$
$383$	−1.52786	−0.0780702	−0.0390351	+	0.999238i	$0.512428\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.3607	−1.23514	−0.617568	−	0.786518i	$-0.711882\pi$
$389$	−24.3607	−1.23514	−0.617568	+	0.786518i	$0.711882\pi$
$390$	0	0
$391$	−2.47214	−0.125021
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.94427	0.248773
$396$	0	0
$397$	−19.8885	−0.998177	−0.499089	−	0.866551i	$-0.666332\pi$
$397$	−19.8885	−0.998177	−0.499089	+	0.866551i	$0.666332\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.76393	−0.237899	−0.118950	−	0.992900i	$-0.537953\pi$
$401$	−4.76393	−0.237899	−0.118950	+	0.992900i	$0.537953\pi$
$402$	0	0
$403$	6.83282	0.340367
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9443	−0.839896
$408$	0	0
$409$	35.3050	1.74572	0.872859	−	0.487973i	$-0.162264\pi$
$409$	35.3050	1.74572	0.872859	+	0.487973i	$0.162264\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.94427	−0.440119
$414$	0	0
$415$	−16.9443	−0.831762
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.5967	−1.34819	−0.674095	−	0.738645i	$-0.735466\pi$
$419$	−27.5967	−1.34819	−0.674095	+	0.738645i	$0.735466\pi$
$420$	0	0
$421$	15.5279	0.756782	0.378391	−	0.925646i	$-0.376478\pi$
$421$	15.5279	0.756782	0.378391	+	0.925646i	$0.376478\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.29180	0.208183
$426$	0	0
$427$	−13.4164	−0.649265
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.6525	1.38014	0.690071	−	0.723742i	$-0.257579\pi$
$431$	28.6525	1.38014	0.690071	+	0.723742i	$0.257579\pi$
$432$	0	0
$433$	10.5836	0.508615	0.254307	−	0.967123i	$-0.418152\pi$
$433$	10.5836	0.508615	0.254307	+	0.967123i	$0.418152\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	12.3607	0.589943	0.294972	−	0.955506i	$-0.404690\pi$
$439$	12.3607	0.589943	0.294972	+	0.955506i	$0.404690\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.4164	0.637433	0.318716	−	0.947850i	$-0.396748\pi$
$443$	13.4164	0.637433	0.318716	+	0.947850i	$0.396748\pi$
$444$	0	0
$445$	5.52786	0.262046
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.2361	1.09658	0.548289	−	0.836289i	$-0.315279\pi$
$449$	23.2361	1.09658	0.548289	+	0.836289i	$0.315279\pi$
$450$	0	0
$451$	16.9443	0.797875
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.52786	−0.259150
$456$	0	0
$457$	21.4164	1.00182	0.500909	−	0.865500i	$-0.332999\pi$
$457$	21.4164	1.00182	0.500909	+	0.865500i	$0.332999\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	28.0689	1.30730	0.653649	−	0.756798i	$-0.273237\pi$
$461$	28.0689	1.30730	0.653649	+	0.756798i	$0.273237\pi$
$462$	0	0
$463$	−41.8885	−1.94673	−0.973363	−	0.229270i	$-0.926366\pi$
$463$	−41.8885	−1.94673	−0.973363	+	0.229270i	$0.926366\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.7639	0.960840	0.480420	−	0.877039i	$-0.340484\pi$
$467$	20.7639	0.960840	0.480420	+	0.877039i	$0.340484\pi$
$468$	0	0
$469$	−1.52786	−0.0705502
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	3.47214	0.159313
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.7082	−1.17464	−0.587319	−	0.809356i	$-0.699817\pi$
$479$	−25.7082	−1.17464	−0.587319	+	0.809356i	$0.699817\pi$
$480$	0	0
$481$	−37.8885	−1.72757
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.5279	−0.977530
$486$	0	0
$487$	−39.4164	−1.78613	−0.893064	−	0.449930i	$-0.851449\pi$
$487$	−39.4164	−1.78613	−0.893064	+	0.449930i	$0.851449\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.3050	1.41277	0.706386	−	0.707826i	$-0.250324\pi$
$491$	31.3050	1.41277	0.706386	+	0.707826i	$0.250324\pi$
$492$	0	0
$493$	−7.05573	−0.317774
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.70820	0.345760
$498$	0	0
$499$	5.88854	0.263607	0.131804	−	0.991276i	$-0.457923\pi$
$499$	5.88854	0.263607	0.131804	+	0.991276i	$0.457923\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.29180	−0.102186	−0.0510931	−	0.998694i	$-0.516271\pi$
$503$	−2.29180	−0.102186	−0.0510931	+	0.998694i	$0.516271\pi$
$504$	0	0
$505$	−13.3050	−0.592063
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.8885	−1.23614	−0.618069	−	0.786124i	$-0.712085\pi$
$509$	−27.8885	−1.23614	−0.618069	+	0.786124i	$0.712085\pi$
$510$	0	0
$511$	12.4721	0.551735
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.94427	0.217871
$516$	0	0
$517$	−3.41641	−0.150253
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.11146	0.355369	0.177685	−	0.984087i	$-0.443139\pi$
$521$	8.11146	0.355369	0.177685	+	0.984087i	$0.443139\pi$
$522$	0	0
$523$	41.3050	1.80614	0.903070	−	0.429494i	$-0.141308\pi$
$523$	41.3050	1.80614	0.903070	+	0.429494i	$0.141308\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.88854	0.0822663
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	37.8885	1.64114
$534$	0	0
$535$	−22.4721	−0.971555
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.3607	0.529473
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.70820	−0.243178
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.3050	1.41118	0.705588	−	0.708622i	$-0.250683\pi$
$557$	33.3050	1.41118	0.705588	+	0.708622i	$0.250683\pi$
$558$	0	0
$559$	−35.7771	−1.51321
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.4721	0.778508	0.389254	−	0.921131i	$-0.372733\pi$
$563$	18.4721	0.778508	0.389254	+	0.921131i	$0.372733\pi$
$564$	0	0
$565$	−8.94427	−0.376288
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.1246	−1.89172	−0.945861	−	0.324572i	$-0.894780\pi$
$569$	−45.1246	−1.89172	−0.945861	+	0.324572i	$0.894780\pi$
$570$	0	0
$571$	−13.8885	−0.581217	−0.290609	−	0.956842i	$-0.593858\pi$
$571$	−13.8885	−0.581217	−0.290609	+	0.956842i	$0.593858\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.94427	−0.289596
$576$	0	0
$577$	−23.5279	−0.979478	−0.489739	−	0.871869i	$-0.662908\pi$
$577$	−23.5279	−0.979478	−0.489739	+	0.871869i	$0.662908\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.7082	−0.568712
$582$	0	0
$583$	−22.4721	−0.930701
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.1246	−1.86249	−0.931246	−	0.364391i	$-0.881277\pi$
$587$	−45.1246	−1.86249	−0.931246	+	0.364391i	$0.881277\pi$
$588$	0	0
$589$	1.52786	0.0629545
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−46.5410	−1.91121	−0.955605	−	0.294650i	$-0.904797\pi$
$593$	−46.5410	−1.91121	−0.955605	+	0.294650i	$0.904797\pi$
$594$	0	0
$595$	−1.52786	−0.0626363
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.5967	−1.53616	−0.768081	−	0.640353i	$-0.778788\pi$
$599$	−37.5967	−1.53616	−0.768081	+	0.640353i	$0.778788\pi$
$600$	0	0
$601$	27.3050	1.11379	0.556896	−	0.830582i	$-0.311992\pi$
$601$	27.3050	1.11379	0.556896	+	0.830582i	$0.311992\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.65248	−0.351773
$606$	0	0
$607$	−11.0557	−0.448738	−0.224369	−	0.974504i	$-0.572032\pi$
$607$	−11.0557	−0.448738	−0.224369	+	0.974504i	$0.572032\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.63932	−0.309054
$612$	0	0
$613$	42.3607	1.71093	0.855466	−	0.517859i	$-0.173271\pi$
$613$	42.3607	1.71093	0.855466	+	0.517859i	$0.173271\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.88854	−0.237064	−0.118532	−	0.992950i	$-0.537819\pi$
$617$	−5.88854	−0.237064	−0.118532	+	0.992950i	$0.537819\pi$
$618$	0	0
$619$	35.4164	1.42351	0.711753	−	0.702430i	$-0.247902\pi$
$619$	35.4164	1.42351	0.711753	+	0.702430i	$0.247902\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.47214	0.179172
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.4721	−0.417551
$630$	0	0
$631$	8.94427	0.356066	0.178033	−	0.984025i	$-0.443027\pi$
$631$	8.94427	0.356066	0.178033	+	0.984025i	$0.443027\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19.0557	−0.756204
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.2361	1.07576	0.537880	−	0.843021i	$-0.319225\pi$
$641$	27.2361	1.07576	0.537880	+	0.843021i	$0.319225\pi$
$642$	0	0
$643$	−15.0557	−0.593740	−0.296870	−	0.954918i	$-0.595943\pi$
$643$	−15.0557	−0.593740	−0.296870	+	0.954918i	$0.595943\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.7082	0.696181	0.348091	−	0.937461i	$-0.386830\pi$
$647$	17.7082	0.696181	0.348091	+	0.937461i	$0.386830\pi$
$648$	0	0
$649$	17.8885	0.702187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−48.7214	−1.90661	−0.953307	−	0.302003i	$-0.902345\pi$
$653$	−48.7214	−1.90661	−0.953307	+	0.302003i	$0.902345\pi$
$654$	0	0
$655$	−18.8328	−0.735859
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.7639	−1.19839	−0.599196	−	0.800602i	$-0.704513\pi$
$659$	−30.7639	−1.19839	−0.599196	+	0.800602i	$0.704513\pi$
$660$	0	0
$661$	−36.2492	−1.40993	−0.704966	−	0.709241i	$-0.749038\pi$
$661$	−36.2492	−1.40993	−0.704966	+	0.709241i	$0.749038\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.23607	−0.0479327
$666$	0	0
$667$	11.4164	0.442045
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.8328	1.03587
$672$	0	0
$673$	−18.3607	−0.707752	−0.353876	−	0.935292i	$-0.615137\pi$
$673$	−18.3607	−0.707752	−0.353876	+	0.935292i	$0.615137\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−17.4164	−0.668380
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.7639	−0.717982	−0.358991	−	0.933341i	$-0.616879\pi$
$683$	−18.7639	−0.717982	−0.358991	+	0.933341i	$0.616879\pi$
$684$	0	0
$685$	−4.94427	−0.188911
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−50.2492	−1.91434
$690$	0	0
$691$	−24.3607	−0.926724	−0.463362	−	0.886169i	$-0.653357\pi$
$691$	−24.3607	−0.926724	−0.463362	+	0.886169i	$0.653357\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.7771	0.446730
$696$	0	0
$697$	10.4721	0.396660
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.7771	−1.04913	−0.524563	−	0.851372i	$-0.675771\pi$
$701$	−27.7771	−1.04913	−0.524563	+	0.851372i	$0.675771\pi$
$702$	0	0
$703$	−8.47214	−0.319533
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.7639	−0.404819
$708$	0	0
$709$	35.3050	1.32591	0.662953	−	0.748661i	$-0.269303\pi$
$709$	35.3050	1.32591	0.662953	+	0.748661i	$0.269303\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.05573	−0.114438
$714$	0	0
$715$	11.0557	0.413461
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.7639	−1.22189	−0.610944	−	0.791674i	$-0.709210\pi$
$719$	−32.7639	−1.22189	−0.610944	+	0.791674i	$0.709210\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.8197	−0.736084
$726$	0	0
$727$	23.4164	0.868466	0.434233	−	0.900800i	$-0.357019\pi$
$727$	23.4164	0.868466	0.434233	+	0.900800i	$0.357019\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.88854	−0.365741
$732$	0	0
$733$	37.4164	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$733$	37.4164	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.05573	0.112559
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.5967	1.81953	0.909764	−	0.415126i	$-0.136262\pi$
$743$	49.5967	1.81953	0.909764	+	0.415126i	$0.136262\pi$
$744$	0	0
$745$	16.7214	0.612623
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.1803	−0.664295
$750$	0	0
$751$	−32.9443	−1.20215	−0.601077	−	0.799191i	$-0.705261\pi$
$751$	−32.9443	−1.20215	−0.601077	+	0.799191i	$0.705261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.94427	0.179940
$756$	0	0
$757$	−15.8885	−0.577479	−0.288739	−	0.957408i	$-0.593236\pi$
$757$	−15.8885	−0.577479	−0.288739	+	0.957408i	$0.593236\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.2361	0.914807	0.457403	−	0.889259i	$-0.348780\pi$
$761$	25.2361	0.914807	0.457403	+	0.889259i	$0.348780\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.0000	1.44432
$768$	0	0
$769$	19.5279	0.704193	0.352096	−	0.935964i	$-0.385469\pi$
$769$	19.5279	0.704193	0.352096	+	0.935964i	$0.385469\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.2492	1.59153	0.795767	−	0.605603i	$-0.207068\pi$
$773$	44.2492	1.59153	0.795767	+	0.605603i	$0.207068\pi$
$774$	0	0
$775$	5.30495	0.190559
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.47214	0.303546
$780$	0	0
$781$	−15.4164	−0.551642
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.36068	−0.155639
$786$	0	0
$787$	−21.5279	−0.767385	−0.383693	−	0.923461i	$-0.625348\pi$
$787$	−21.5279	−0.767385	−0.383693	+	0.923461i	$0.625348\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.23607	−0.257285
$792$	0	0
$793$	60.0000	2.13066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.4164	0.758608	0.379304	−	0.925272i	$-0.376163\pi$
$797$	21.4164	0.758608	0.379304	+	0.925272i	$0.376163\pi$
$798$	0	0
$799$	−2.11146	−0.0746979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−24.9443	−0.880264
$804$	0	0
$805$	2.47214	0.0871313
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.88854	−0.0663977	−0.0331988	−	0.999449i	$-0.510569\pi$
$809$	−1.88854	−0.0663977	−0.0331988	+	0.999449i	$0.510569\pi$
$810$	0	0
$811$	−3.05573	−0.107301	−0.0536506	−	0.998560i	$-0.517086\pi$
$811$	−3.05573	−0.107301	−0.0536506	+	0.998560i	$0.517086\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.7771	−0.692761
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.2492	−1.05571	−0.527853	−	0.849336i	$-0.677003\pi$
$821$	−30.2492	−1.05571	−0.527853	+	0.849336i	$0.677003\pi$
$822$	0	0
$823$	−42.8328	−1.49306	−0.746529	−	0.665353i	$-0.768281\pi$
$823$	−42.8328	−1.49306	−0.746529	+	0.665353i	$0.768281\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.7639	0.930673	0.465337	−	0.885134i	$-0.345933\pi$
$827$	26.7639	0.930673	0.465337	+	0.885134i	$0.345933\pi$
$828$	0	0
$829$	1.41641	0.0491939	0.0245969	−	0.999697i	$-0.492170\pi$
$829$	1.41641	0.0491939	0.0245969	+	0.999697i	$0.492170\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.23607	−0.0428272
$834$	0	0
$835$	−28.9443	−1.00166
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.94427	−0.308791	−0.154395	−	0.988009i	$-0.549343\pi$
$839$	−8.94427	−0.308791	−0.154395	+	0.988009i	$0.549343\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.65248	0.297654
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16.9443	0.580842
$852$	0	0
$853$	−29.7771	−1.01955	−0.509774	−	0.860308i	$-0.670271\pi$
$853$	−29.7771	−1.01955	−0.509774	+	0.860308i	$0.670271\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.2492	1.10161	0.550806	−	0.834633i	$-0.314320\pi$
$857$	32.2492	1.10161	0.550806	+	0.834633i	$0.314320\pi$
$858$	0	0
$859$	−7.05573	−0.240738	−0.120369	−	0.992729i	$-0.538408\pi$
$859$	−7.05573	−0.240738	−0.120369	+	0.992729i	$0.538408\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.1246	1.60414	0.802070	−	0.597230i	$-0.203732\pi$
$863$	47.1246	1.60414	0.802070	+	0.597230i	$0.203732\pi$
$864$	0	0
$865$	−25.3050	−0.860394
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	6.83282	0.231521
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.4721	−0.354023
$876$	0	0
$877$	28.8328	0.973615	0.486808	−	0.873509i	$-0.338161\pi$
$877$	28.8328	0.973615	0.486808	+	0.873509i	$0.338161\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.1803	−1.55586	−0.777928	−	0.628354i	$-0.783729\pi$
$881$	−46.1803	−1.55586	−0.777928	+	0.628354i	$0.783729\pi$
$882$	0	0
$883$	5.88854	0.198165	0.0990826	−	0.995079i	$-0.468409\pi$
$883$	5.88854	0.198165	0.0990826	+	0.995079i	$0.468409\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.5836	−0.825436	−0.412718	−	0.910859i	$-0.635421\pi$
$887$	−24.5836	−0.825436	−0.412718	+	0.910859i	$0.635421\pi$
$888$	0	0
$889$	−15.4164	−0.517050
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.70820	−0.0571629
$894$	0	0
$895$	−9.52786	−0.318481
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.72136	−0.290874
$900$	0	0
$901$	−13.8885	−0.462694
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.36068	−0.144954
$906$	0	0
$907$	−25.5279	−0.847639	−0.423819	−	0.905747i	$-0.639311\pi$
$907$	−25.5279	−0.847639	−0.423819	+	0.905747i	$0.639311\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.5410	−0.879343	−0.439672	−	0.898159i	$-0.644905\pi$
$911$	−26.5410	−0.879343	−0.439672	+	0.898159i	$0.644905\pi$
$912$	0	0
$913$	27.4164	0.907351
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.2361	−0.503139
$918$	0	0
$919$	−35.7771	−1.18018	−0.590089	−	0.807338i	$-0.700907\pi$
$919$	−35.7771	−1.18018	−0.590089	+	0.807338i	$0.700907\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−34.4721	−1.13466
$924$	0	0
$925$	−29.4164	−0.967206
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.40325	−0.210084	−0.105042	−	0.994468i	$-0.533498\pi$
$929$	−6.40325	−0.210084	−0.105042	+	0.994468i	$0.533498\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.05573	0.0999330
$936$	0	0
$937$	16.1115	0.526338	0.263169	−	0.964750i	$-0.415232\pi$
$937$	16.1115	0.526338	0.263169	+	0.964750i	$0.415232\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.8328	0.418338	0.209169	−	0.977880i	$-0.432924\pi$
$941$	12.8328	0.418338	0.209169	+	0.977880i	$0.432924\pi$
$942$	0	0
$943$	−16.9443	−0.551781
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	0	0
$949$	−55.7771	−1.81060
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.3475	0.432369	0.216184	−	0.976353i	$-0.430639\pi$
$953$	13.3475	0.432369	0.216184	+	0.976353i	$0.430639\pi$
$954$	0	0
$955$	−12.3607	−0.399982
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−28.6656	−0.924698
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.4721	−0.337110
$966$	0	0
$967$	18.8328	0.605623	0.302811	−	0.953051i	$-0.402075\pi$
$967$	18.8328	0.605623	0.302811	+	0.953051i	$0.402075\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.8328	−1.50294	−0.751468	−	0.659769i	$-0.770654\pi$
$971$	−46.8328	−1.50294	−0.751468	+	0.659769i	$0.770654\pi$
$972$	0	0
$973$	9.52786	0.305449
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.0689	1.60185	0.800923	−	0.598768i	$-0.204343\pi$
$977$	50.0689	1.60185	0.800923	+	0.598768i	$0.204343\pi$
$978$	0	0
$979$	−8.94427	−0.285860
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.7214	1.55397	0.776985	−	0.629519i	$-0.216748\pi$
$983$	48.7214	1.55397	0.776985	+	0.629519i	$0.216748\pi$
$984$	0	0
$985$	−14.8328	−0.472613
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	52.3607	1.66329	0.831646	−	0.555306i	$-0.187399\pi$
$991$	52.3607	1.66329	0.831646	+	0.555306i	$0.187399\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.8885	−0.820722
$996$	0	0
$997$	−16.4721	−0.521678	−0.260839	−	0.965382i	$-0.583999\pi$
$997$	−16.4721	−0.521678	−0.260839	+	0.965382i	$0.583999\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bi.1.2	✓	2
3.2	odd	2		9576.2.a.bv.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bi.1.2	✓	2	1.1	even	1	trivial
9576.2.a.bv.1.1	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.47214 q^{13} -1.23607 q^{17} -1.00000 q^{19} +2.00000 q^{23} -3.47214 q^{25} +5.70820 q^{29} -1.52786 q^{31} +1.23607 q^{35} +8.47214 q^{37} -8.47214 q^{41} +8.00000 q^{43} +1.70820 q^{47} +1.00000 q^{49} +11.2361 q^{53} -2.47214 q^{55} -8.94427 q^{59} -13.4164 q^{61} -5.52786 q^{65} -1.52786 q^{67} +7.70820 q^{71} +12.4721 q^{73} -2.00000 q^{77} +4.00000 q^{79} -13.7082 q^{83} -1.52786 q^{85} +4.47214 q^{89} -4.47214 q^{91} -1.23607 q^{95} -17.4164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 16 q^{43} - 10 q^{47} + 2 q^{49} + 18 q^{53} + 4 q^{55} - 20 q^{65}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^29 - 12 * q^31 - 2 * q^35 + 8 * q^37 - 8 * q^41 + 16 * q^43 - 10 * q^47 + 2 * q^49 + 18 * q^53 + 4 * q^55 - 20 * q^65 - 12 * q^67 + 2 * q^71 + 16 * q^73 - 4 * q^77 + 8 * q^79 - 14 * q^83 - 12 * q^85 + 2 * q^95 - 8 * q^97

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