LMFDB - Embedded newform 9576.2.a.be.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:	$0$
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-3.23607 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-3.23607 q^{5} -1.00000 q^{7} +4.47214 q^{11} +4.47214 q^{13} +3.23607 q^{17} +1.00000 q^{19} +8.47214 q^{23} +5.47214 q^{25} +2.76393 q^{29} -2.47214 q^{31} +3.23607 q^{35} +4.47214 q^{37} -2.00000 q^{41} +8.00000 q^{43} +1.23607 q^{47} +1.00000 q^{49} +1.23607 q^{53} -14.4721 q^{55} -4.00000 q^{59} -12.4721 q^{61} -14.4721 q^{65} -10.4721 q^{67} +0.763932 q^{71} -4.47214 q^{73} -4.47214 q^{77} -12.0000 q^{79} +7.70820 q^{83} -10.4721 q^{85} -6.94427 q^{89} -4.47214 q^{91} -3.23607 q^{95} +12.4721 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} + 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 2 q^{49} - 2 q^{53} - 20 q^{55} - 8 q^{59} - 16 q^{61} - 20 q^{65} - 12 q^{67} + 6 q^{71} - 24 q^{79} + 2 q^{83} - 12 q^{85} + 4 q^{89} - 2 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^17 + 2 * q^19 + 8 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 + 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 2 * q^49 - 2 * q^53 - 20 * q^55 - 8 * q^59 - 16 * q^61 - 20 * q^65 - 12 * q^67 + 6 * q^71 - 24 * q^79 + 2 * q^83 - 12 * q^85 + 4 * q^89 - 2 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.23607	0.784862	0.392431	−	0.919781i	$-0.371634\pi$
$17$	3.23607	0.784862	0.392431	+	0.919781i	$0.371634\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.47214	1.76656	0.883281	−	0.468844i	$-0.155329\pi$
$23$	8.47214	1.76656	0.883281	+	0.468844i	$0.155329\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	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
$47$	1.23607	0.180299	0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607	0.180299	0.0901495	+	0.995928i	$0.471266\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.23607	0.169787	0.0848935	−	0.996390i	$-0.472945\pi$
$53$	1.23607	0.169787	0.0848935	+	0.996390i	$0.472945\pi$
$54$	0	0
$55$	−14.4721	−1.95142
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.47214	−0.509647
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.70820	0.846085	0.423043	−	0.906110i	$-0.360962\pi$
$83$	7.70820	0.846085	0.423043	+	0.906110i	$0.360962\pi$
$84$	0	0
$85$	−10.4721	−1.13586
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.94427	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$89$	−6.94427	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.23607	−0.332014
$96$	0	0
$97$	12.4721	1.26635	0.633177	−	0.774007i	$-0.281751\pi$
$97$	12.4721	1.26635	0.633177	+	0.774007i	$0.281751\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	0	0
$103$	8.94427	0.881305	0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427	0.881305	0.440653	+	0.897678i	$0.354747\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.70820	0.551833	0.275916	−	0.961182i	$-0.411019\pi$
$107$	5.70820	0.551833	0.275916	+	0.961182i	$0.411019\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.7639	1.01259	0.506293	−	0.862362i	$-0.331016\pi$
$113$	10.7639	1.01259	0.506293	+	0.862362i	$0.331016\pi$
$114$	0	0
$115$	−27.4164	−2.55659
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.23607	−0.296650
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−1.52786	−0.135576	−0.0677880	−	0.997700i	$-0.521594\pi$
$127$	−1.52786	−0.135576	−0.0677880	+	0.997700i	$0.521594\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.2361	1.15644	0.578220	−	0.815881i	$-0.303747\pi$
$131$	13.2361	1.15644	0.578220	+	0.815881i	$0.303747\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.9443	1.44765	0.723823	−	0.689985i	$-0.242383\pi$
$137$	16.9443	1.44765	0.723823	+	0.689985i	$0.242383\pi$
$138$	0	0
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	−8.94427	−0.742781
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.52786	0.125167	0.0625837	−	0.998040i	$-0.480066\pi$
$149$	1.52786	0.125167	0.0625837	+	0.998040i	$0.480066\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−22.3607	−1.78458	−0.892288	−	0.451466i	$-0.850901\pi$
$157$	−22.3607	−1.78458	−0.892288	+	0.451466i	$0.850901\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.47214	−0.667698
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.47214	0.500829	0.250414	−	0.968139i	$-0.419433\pi$
$167$	6.47214	0.500829	0.250414	+	0.968139i	$0.419433\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.9443	−0.832078	−0.416039	−	0.909347i	$-0.636582\pi$
$173$	−10.9443	−0.832078	−0.416039	+	0.909347i	$0.636582\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.70820	−0.426651	−0.213326	−	0.976981i	$-0.568430\pi$
$179$	−5.70820	−0.426651	−0.213326	+	0.976981i	$0.568430\pi$
$180$	0	0
$181$	−4.47214	−0.332411	−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214	−0.332411	−0.166206	+	0.986091i	$0.553152\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−14.4721	−1.06401
$186$	0	0
$187$	14.4721	1.05831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.5279	1.12356	0.561778	−	0.827288i	$-0.310117\pi$
$191$	15.5279	1.12356	0.561778	+	0.827288i	$0.310117\pi$
$192$	0	0
$193$	8.47214	0.609838	0.304919	−	0.952378i	$-0.401371\pi$
$193$	8.47214	0.609838	0.304919	+	0.952378i	$0.401371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−12.9443	−0.917595	−0.458798	−	0.888541i	$-0.651720\pi$
$199$	−12.9443	−0.917595	−0.458798	+	0.888541i	$0.651720\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.76393	−0.193990
$204$	0	0
$205$	6.47214	0.452034
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.47214	0.309344
$210$	0	0
$211$	2.47214	0.170189	0.0850944	−	0.996373i	$-0.472881\pi$
$211$	2.47214	0.170189	0.0850944	+	0.996373i	$0.472881\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−25.8885	−1.76558
$216$	0	0
$217$	2.47214	0.167820
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.4721	0.973501
$222$	0	0
$223$	3.41641	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$223$	3.41641	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.4164	0.757734	0.378867	−	0.925451i	$-0.376314\pi$
$227$	11.4164	0.757734	0.378867	+	0.925451i	$0.376314\pi$
$228$	0	0
$229$	5.05573	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$229$	5.05573	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.47214	0.424004	0.212002	−	0.977269i	$-0.432002\pi$
$233$	6.47214	0.424004	0.212002	+	0.977269i	$0.432002\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.9443	−1.48414	−0.742071	−	0.670322i	$-0.766156\pi$
$239$	−22.9443	−1.48414	−0.742071	+	0.670322i	$0.766156\pi$
$240$	0	0
$241$	−8.47214	−0.545738	−0.272869	−	0.962051i	$-0.587973\pi$
$241$	−8.47214	−0.545738	−0.272869	+	0.962051i	$0.587973\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	4.47214	0.284555
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.1803	1.40001	0.700005	−	0.714138i	$-0.253181\pi$
$251$	22.1803	1.40001	0.700005	+	0.714138i	$0.253181\pi$
$252$	0	0
$253$	37.8885	2.38203
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.52786	0.220062	0.110031	−	0.993928i	$-0.464905\pi$
$257$	3.52786	0.220062	0.110031	+	0.993928i	$0.464905\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.9443	−0.921503	−0.460752	−	0.887529i	$-0.652420\pi$
$263$	−14.9443	−0.921503	−0.460752	+	0.887529i	$0.652420\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.4721	1.49209	0.746046	−	0.665894i	$-0.231950\pi$
$269$	24.4721	1.49209	0.746046	+	0.665894i	$0.231950\pi$
$270$	0	0
$271$	−14.4721	−0.879120	−0.439560	−	0.898213i	$-0.644866\pi$
$271$	−14.4721	−0.879120	−0.439560	+	0.898213i	$0.644866\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.4721	1.47573
$276$	0	0
$277$	4.47214	0.268705	0.134352	−	0.990934i	$-0.457105\pi$
$277$	4.47214	0.268705	0.134352	+	0.990934i	$0.457105\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.1803	−0.845928	−0.422964	−	0.906146i	$-0.639010\pi$
$281$	−14.1803	−0.845928	−0.422964	+	0.906146i	$0.639010\pi$
$282$	0	0
$283$	−24.9443	−1.48278	−0.741392	−	0.671073i	$-0.765834\pi$
$283$	−24.9443	−1.48278	−0.741392	+	0.671073i	$0.765834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−6.52786	−0.383992
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	12.9443	0.753645
$296$	0	0
$297$	0	0
$298$	0	0
$299$	37.8885	2.19115
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	40.3607	2.31105
$306$	0	0
$307$	26.4721	1.51084	0.755422	−	0.655238i	$-0.227432\pi$
$307$	26.4721	1.51084	0.755422	+	0.655238i	$0.227432\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.18034	0.350455	0.175227	−	0.984528i	$-0.443934\pi$
$311$	6.18034	0.350455	0.175227	+	0.984528i	$0.443934\pi$
$312$	0	0
$313$	12.4721	0.704967	0.352483	−	0.935818i	$-0.385337\pi$
$313$	12.4721	0.704967	0.352483	+	0.935818i	$0.385337\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.1246	−1.52347	−0.761735	−	0.647889i	$-0.775652\pi$
$317$	−27.1246	−1.52347	−0.761735	+	0.647889i	$0.775652\pi$
$318$	0	0
$319$	12.3607	0.692065
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.23607	0.180060
$324$	0	0
$325$	24.4721	1.35747
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.23607	−0.0681466
$330$	0	0
$331$	25.8885	1.42296	0.711482	−	0.702705i	$-0.248025\pi$
$331$	25.8885	1.42296	0.711482	+	0.702705i	$0.248025\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	33.8885	1.85153
$336$	0	0
$337$	−14.9443	−0.814066	−0.407033	−	0.913413i	$-0.633437\pi$
$337$	−14.9443	−0.814066	−0.407033	+	0.913413i	$0.633437\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.0557	−0.598701
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	28.8328	1.54339	0.771693	−	0.635996i	$-0.219410\pi$
$349$	28.8328	1.54339	0.771693	+	0.635996i	$0.219410\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.2918	0.760676	0.380338	−	0.924848i	$-0.375808\pi$
$353$	14.2918	0.760676	0.380338	+	0.924848i	$0.375808\pi$
$354$	0	0
$355$	−2.47214	−0.131207
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.4164	−1.55254	−0.776269	−	0.630401i	$-0.782890\pi$
$359$	−29.4164	−1.55254	−0.776269	+	0.630401i	$0.782890\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.4721	0.757506
$366$	0	0
$367$	−12.9443	−0.675685	−0.337843	−	0.941203i	$-0.609697\pi$
$367$	−12.9443	−0.675685	−0.337843	+	0.941203i	$0.609697\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.23607	−0.0641735
$372$	0	0
$373$	2.94427	0.152449	0.0762243	−	0.997091i	$-0.475713\pi$
$373$	2.94427	0.152449	0.0762243	+	0.997091i	$0.475713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.3607	0.636607
$378$	0	0
$379$	−3.05573	−0.156962	−0.0784811	−	0.996916i	$-0.525007\pi$
$379$	−3.05573	−0.156962	−0.0784811	+	0.996916i	$0.525007\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.4721	−0.535101	−0.267551	−	0.963544i	$-0.586214\pi$
$383$	−10.4721	−0.535101	−0.267551	+	0.963544i	$0.586214\pi$
$384$	0	0
$385$	14.4721	0.737568
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−31.4164	−1.59288	−0.796438	−	0.604721i	$-0.793285\pi$
$389$	−31.4164	−1.59288	−0.796438	+	0.604721i	$0.793285\pi$
$390$	0	0
$391$	27.4164	1.38651
$392$	0	0
$393$	0	0
$394$	0	0
$395$	38.8328	1.95389
$396$	0	0
$397$	−11.8885	−0.596669	−0.298334	−	0.954461i	$-0.596431\pi$
$397$	−11.8885	−0.596669	−0.298334	+	0.954461i	$0.596431\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.1803	1.50713	0.753567	−	0.657371i	$-0.228332\pi$
$401$	30.1803	1.50713	0.753567	+	0.657371i	$0.228332\pi$
$402$	0	0
$403$	−11.0557	−0.550725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−22.3607	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$409$	−22.3607	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−24.9443	−1.22447
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.1246	−0.934298	−0.467149	−	0.884178i	$-0.654719\pi$
$419$	−19.1246	−0.934298	−0.467149	+	0.884178i	$0.654719\pi$
$420$	0	0
$421$	23.3050	1.13581	0.567907	−	0.823093i	$-0.307753\pi$
$421$	23.3050	1.13581	0.567907	+	0.823093i	$0.307753\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	17.7082	0.858974
$426$	0	0
$427$	12.4721	0.603569
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.0689	−1.06302	−0.531510	−	0.847052i	$-0.678375\pi$
$431$	−22.0689	−1.06302	−0.531510	+	0.847052i	$0.678375\pi$
$432$	0	0
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.47214	0.405277
$438$	0	0
$439$	−12.5836	−0.600582	−0.300291	−	0.953848i	$-0.597084\pi$
$439$	−12.5836	−0.600582	−0.300291	+	0.953848i	$0.597084\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.8885	−0.944933	−0.472467	−	0.881349i	$-0.656636\pi$
$443$	−19.8885	−0.944933	−0.472467	+	0.881349i	$0.656636\pi$
$444$	0	0
$445$	22.4721	1.06528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.7639	−0.507981	−0.253991	−	0.967207i	$-0.581743\pi$
$449$	−10.7639	−0.507981	−0.253991	+	0.967207i	$0.581743\pi$
$450$	0	0
$451$	−8.94427	−0.421169
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.4721	0.678464
$456$	0	0
$457$	−5.41641	−0.253369	−0.126684	−	0.991943i	$-0.540434\pi$
$457$	−5.41641	−0.253369	−0.126684	+	0.991943i	$0.540434\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.763932	0.0355799	0.0177899	−	0.999842i	$-0.494337\pi$
$461$	0.763932	0.0355799	0.0177899	+	0.999842i	$0.494337\pi$
$462$	0	0
$463$	9.88854	0.459560	0.229780	−	0.973243i	$-0.426199\pi$
$463$	9.88854	0.459560	0.229780	+	0.973243i	$0.426199\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.81966	−0.0842038	−0.0421019	−	0.999113i	$-0.513405\pi$
$467$	−1.81966	−0.0842038	−0.0421019	+	0.999113i	$0.513405\pi$
$468$	0	0
$469$	10.4721	0.483558
$470$	0	0
$471$	0	0
$472$	0	0
$473$	35.7771	1.64503
$474$	0	0
$475$	5.47214	0.251079
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.1246	−0.873826	−0.436913	−	0.899504i	$-0.643928\pi$
$479$	−19.1246	−0.873826	−0.436913	+	0.899504i	$0.643928\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−40.3607	−1.83268
$486$	0	0
$487$	−3.41641	−0.154812	−0.0774061	−	0.997000i	$-0.524664\pi$
$487$	−3.41641	−0.154812	−0.0774061	+	0.997000i	$0.524664\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	8.94427	0.402830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.763932	−0.0342670
$498$	0	0
$499$	2.11146	0.0945218	0.0472609	−	0.998883i	$-0.484951\pi$
$499$	2.11146	0.0945218	0.0472609	+	0.998883i	$0.484951\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.81966	−0.259486	−0.129743	−	0.991548i	$-0.541415\pi$
$503$	−5.81966	−0.259486	−0.129743	+	0.991548i	$0.541415\pi$
$504$	0	0
$505$	13.5279	0.601982
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.4721	1.79390	0.896948	−	0.442136i	$-0.145779\pi$
$509$	40.4721	1.79390	0.896948	+	0.442136i	$0.145779\pi$
$510$	0	0
$511$	4.47214	0.197836
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−28.9443	−1.27544
$516$	0	0
$517$	5.52786	0.243115
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.4164	1.11351	0.556757	−	0.830676i	$-0.312046\pi$
$521$	25.4164	1.11351	0.556757	+	0.830676i	$0.312046\pi$
$522$	0	0
$523$	−30.4721	−1.33245	−0.666227	−	0.745749i	$-0.732092\pi$
$523$	−30.4721	−1.33245	−0.666227	+	0.745749i	$0.732092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	48.7771	2.12074
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.94427	−0.387419
$534$	0	0
$535$	−18.4721	−0.798620
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.47214	0.192629
$540$	0	0
$541$	−0.111456	−0.00479188	−0.00239594	−	0.999997i	$-0.500763\pi$
$541$	−0.111456	−0.00479188	−0.00239594	+	0.999997i	$0.500763\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.52786	0.408129
$546$	0	0
$547$	38.8328	1.66037	0.830186	−	0.557487i	$-0.188234\pi$
$547$	38.8328	1.66037	0.830186	+	0.557487i	$0.188234\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.76393	0.117747
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.4164	1.16167	0.580835	−	0.814021i	$-0.302726\pi$
$557$	27.4164	1.16167	0.580835	+	0.814021i	$0.302726\pi$
$558$	0	0
$559$	35.7771	1.51321
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.41641	−0.143984	−0.0719922	−	0.997405i	$-0.522936\pi$
$563$	−3.41641	−0.143984	−0.0719922	+	0.997405i	$0.522936\pi$
$564$	0	0
$565$	−34.8328	−1.46543
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.2918	−0.515299	−0.257649	−	0.966238i	$-0.582948\pi$
$569$	−12.2918	−0.515299	−0.257649	+	0.966238i	$0.582948\pi$
$570$	0	0
$571$	−29.8885	−1.25080	−0.625398	−	0.780306i	$-0.715063\pi$
$571$	−29.8885	−1.25080	−0.625398	+	0.780306i	$0.715063\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	46.3607	1.93337
$576$	0	0
$577$	1.41641	0.0589658	0.0294829	−	0.999565i	$-0.490614\pi$
$577$	1.41641	0.0589658	0.0294829	+	0.999565i	$0.490614\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.70820	−0.319790
$582$	0	0
$583$	5.52786	0.228941
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.7082	−1.47384	−0.736918	−	0.675983i	$-0.763719\pi$
$587$	−35.7082	−1.47384	−0.736918	+	0.675983i	$0.763719\pi$
$588$	0	0
$589$	−2.47214	−0.101863
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.2918	−1.07967	−0.539837	−	0.841769i	$-0.681514\pi$
$593$	−26.2918	−1.07967	−0.539837	+	0.841769i	$0.681514\pi$
$594$	0	0
$595$	10.4721	0.429316
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.5967	−1.12757	−0.563786	−	0.825921i	$-0.690656\pi$
$599$	−27.5967	−1.12757	−0.563786	+	0.825921i	$0.690656\pi$
$600$	0	0
$601$	45.4164	1.85257	0.926287	−	0.376819i	$-0.122982\pi$
$601$	45.4164	1.85257	0.926287	+	0.376819i	$0.122982\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−29.1246	−1.18408
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.52786	0.223633
$612$	0	0
$613$	−34.3607	−1.38781	−0.693907	−	0.720064i	$-0.744112\pi$
$613$	−34.3607	−1.38781	−0.693907	+	0.720064i	$0.744112\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.0557	1.25026	0.625128	−	0.780522i	$-0.285047\pi$
$617$	31.0557	1.25026	0.625128	+	0.780522i	$0.285047\pi$
$618$	0	0
$619$	31.4164	1.26273	0.631366	−	0.775485i	$-0.282495\pi$
$619$	31.4164	1.26273	0.631366	+	0.775485i	$0.282495\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.94427	0.278216
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.4721	0.577042
$630$	0	0
$631$	0.944272	0.0375909	0.0187954	−	0.999823i	$-0.494017\pi$
$631$	0.944272	0.0375909	0.0187954	+	0.999823i	$0.494017\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.94427	0.196207
$636$	0	0
$637$	4.47214	0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.2918	0.801478	0.400739	−	0.916192i	$-0.368753\pi$
$641$	20.2918	0.801478	0.400739	+	0.916192i	$0.368753\pi$
$642$	0	0
$643$	32.9443	1.29920	0.649598	−	0.760278i	$-0.274937\pi$
$643$	32.9443	1.29920	0.649598	+	0.760278i	$0.274937\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.6525	−0.969189	−0.484594	−	0.874739i	$-0.661033\pi$
$647$	−24.6525	−0.969189	−0.484594	+	0.874739i	$0.661033\pi$
$648$	0	0
$649$	−17.8885	−0.702187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.8885	1.63923	0.819613	−	0.572918i	$-0.194189\pi$
$653$	41.8885	1.63923	0.819613	+	0.572918i	$0.194189\pi$
$654$	0	0
$655$	−42.8328	−1.67362
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.2918	1.18000	0.590000	−	0.807403i	$-0.299128\pi$
$659$	30.2918	1.18000	0.590000	+	0.807403i	$0.299128\pi$
$660$	0	0
$661$	−1.41641	−0.0550919	−0.0275459	−	0.999621i	$-0.508769\pi$
$661$	−1.41641	−0.0550919	−0.0275459	+	0.999621i	$0.508769\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.23607	0.125489
$666$	0	0
$667$	23.4164	0.906687
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−55.7771	−2.15325
$672$	0	0
$673$	44.2492	1.70568	0.852841	−	0.522171i	$-0.174878\pi$
$673$	44.2492	1.70568	0.852841	+	0.522171i	$0.174878\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.4164	0.669367	0.334683	−	0.942331i	$-0.391371\pi$
$677$	17.4164	0.669367	0.334683	+	0.942331i	$0.391371\pi$
$678$	0	0
$679$	−12.4721	−0.478637
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.2361	0.889103	0.444552	−	0.895753i	$-0.353363\pi$
$683$	23.2361	0.889103	0.444552	+	0.895753i	$0.353363\pi$
$684$	0	0
$685$	−54.8328	−2.09505
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.52786	0.210595
$690$	0	0
$691$	13.5279	0.514624	0.257312	−	0.966328i	$-0.417163\pi$
$691$	13.5279	0.514624	0.257312	+	0.966328i	$0.417163\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.8885	1.28547
$696$	0	0
$697$	−6.47214	−0.245150
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.8328	1.46670	0.733348	−	0.679854i	$-0.237957\pi$
$701$	38.8328	1.46670	0.733348	+	0.679854i	$0.237957\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.18034	0.157218
$708$	0	0
$709$	44.2492	1.66181	0.830907	−	0.556411i	$-0.187822\pi$
$709$	44.2492	1.66181	0.830907	+	0.556411i	$0.187822\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.9443	−0.784369
$714$	0	0
$715$	−64.7214	−2.42044
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.875388	0.0326465	0.0163232	−	0.999867i	$-0.494804\pi$
$719$	0.875388	0.0326465	0.0163232	+	0.999867i	$0.494804\pi$
$720$	0	0
$721$	−8.94427	−0.333102
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.1246	0.561714
$726$	0	0
$727$	−40.3607	−1.49689	−0.748447	−	0.663194i	$-0.769200\pi$
$727$	−40.3607	−1.49689	−0.748447	+	0.663194i	$0.769200\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	25.8885	0.957522
$732$	0	0
$733$	20.4721	0.756156	0.378078	−	0.925774i	$-0.376585\pi$
$733$	20.4721	0.756156	0.378078	+	0.925774i	$0.376585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−46.8328	−1.72511
$738$	0	0
$739$	−17.8885	−0.658041	−0.329020	−	0.944323i	$-0.606718\pi$
$739$	−17.8885	−0.658041	−0.329020	+	0.944323i	$0.606718\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	26.6525	0.977785	0.488892	−	0.872344i	$-0.337401\pi$
$743$	26.6525	0.977785	0.488892	+	0.872344i	$0.337401\pi$
$744$	0	0
$745$	−4.94427	−0.181144
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.70820	−0.208573
$750$	0	0
$751$	−34.8328	−1.27107	−0.635534	−	0.772073i	$-0.719220\pi$
$751$	−34.8328	−1.27107	−0.635534	+	0.772073i	$0.719220\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−64.7214	−2.35545
$756$	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
$761$	−43.2361	−1.56731	−0.783653	−	0.621199i	$-0.786646\pi$
$761$	−43.2361	−1.56731	−0.783653	+	0.621199i	$0.786646\pi$
$762$	0	0
$763$	2.94427	0.106590
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.8885	−0.645918
$768$	0	0
$769$	30.3607	1.09483	0.547417	−	0.836860i	$-0.315611\pi$
$769$	30.3607	1.09483	0.547417	+	0.836860i	$0.315611\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.88854	0.139861	0.0699306	−	0.997552i	$-0.477722\pi$
$773$	3.88854	0.139861	0.0699306	+	0.997552i	$0.477722\pi$
$774$	0	0
$775$	−13.5279	−0.485935
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	3.41641	0.122249
$782$	0	0
$783$	0	0
$784$	0	0
$785$	72.3607	2.58266
$786$	0	0
$787$	−43.4164	−1.54763	−0.773814	−	0.633413i	$-0.781653\pi$
$787$	−43.4164	−1.54763	−0.773814	+	0.633413i	$0.781653\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.7639	−0.382721
$792$	0	0
$793$	−55.7771	−1.98070
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.9443	−1.52116	−0.760582	−	0.649242i	$-0.775086\pi$
$797$	−42.9443	−1.52116	−0.760582	+	0.649242i	$0.775086\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	27.4164	0.966301
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−46.8328	−1.64452	−0.822261	−	0.569110i	$-0.807288\pi$
$811$	−46.8328	−1.64452	−0.822261	+	0.569110i	$0.807288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.3607	−0.570992	−0.285496	−	0.958380i	$-0.592158\pi$
$821$	−16.3607	−0.570992	−0.285496	+	0.958380i	$0.592158\pi$
$822$	0	0
$823$	10.8328	0.377608	0.188804	−	0.982015i	$-0.439539\pi$
$823$	10.8328	0.377608	0.188804	+	0.982015i	$0.439539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.2361	−1.08618	−0.543092	−	0.839673i	$-0.682747\pi$
$827$	−31.2361	−1.08618	−0.543092	+	0.839673i	$0.682747\pi$
$828$	0	0
$829$	−7.52786	−0.261454	−0.130727	−	0.991418i	$-0.541731\pi$
$829$	−7.52786	−0.261454	−0.130727	+	0.991418i	$0.541731\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.23607	0.112123
$834$	0	0
$835$	−20.9443	−0.724806
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.6525	−0.779269
$846$	0	0
$847$	−9.00000	−0.309244
$848$	0	0
$849$	0	0
$850$	0	0
$851$	37.8885	1.29880
$852$	0	0
$853$	−37.7771	−1.29346	−0.646731	−	0.762718i	$-0.723865\pi$
$853$	−37.7771	−1.29346	−0.646731	+	0.762718i	$0.723865\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.8328	−0.848273	−0.424136	−	0.905598i	$-0.639422\pi$
$857$	−24.8328	−0.848273	−0.424136	+	0.905598i	$0.639422\pi$
$858$	0	0
$859$	−16.9443	−0.578131	−0.289066	−	0.957309i	$-0.593345\pi$
$859$	−16.9443	−0.578131	−0.289066	+	0.957309i	$0.593345\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.8197	−0.402346	−0.201173	−	0.979556i	$-0.564475\pi$
$863$	−11.8197	−0.402346	−0.201173	+	0.979556i	$0.564475\pi$
$864$	0	0
$865$	35.4164	1.20419
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−53.6656	−1.82048
$870$	0	0
$871$	−46.8328	−1.58687
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	4.11146	0.138834	0.0694170	−	0.997588i	$-0.477886\pi$
$877$	4.11146	0.138834	0.0694170	+	0.997588i	$0.477886\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.70820	−0.0575509	−0.0287754	−	0.999586i	$-0.509161\pi$
$881$	−1.70820	−0.0575509	−0.0287754	+	0.999586i	$0.509161\pi$
$882$	0	0
$883$	53.8885	1.81349	0.906747	−	0.421675i	$-0.138558\pi$
$883$	53.8885	1.81349	0.906747	+	0.421675i	$0.138558\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.3607	−1.62379	−0.811896	−	0.583802i	$-0.801565\pi$
$887$	−48.3607	−1.62379	−0.811896	+	0.583802i	$0.801565\pi$
$888$	0	0
$889$	1.52786	0.0512429
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.23607	0.0413634
$894$	0	0
$895$	18.4721	0.617455
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.83282	−0.227887
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.4721	0.481070
$906$	0	0
$907$	39.4164	1.30880	0.654400	−	0.756148i	$-0.272921\pi$
$907$	39.4164	1.30880	0.654400	+	0.756148i	$0.272921\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.12461	0.169786	0.0848930	−	0.996390i	$-0.472945\pi$
$911$	5.12461	0.169786	0.0848930	+	0.996390i	$0.472945\pi$
$912$	0	0
$913$	34.4721	1.14086
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.2361	−0.437093
$918$	0	0
$919$	−9.88854	−0.326193	−0.163096	−	0.986610i	$-0.552148\pi$
$919$	−9.88854	−0.326193	−0.163096	+	0.986610i	$0.552148\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.41641	0.112452
$924$	0	0
$925$	24.4721	0.804639
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.0132	1.41122	0.705608	−	0.708602i	$-0.250674\pi$
$929$	43.0132	1.41122	0.705608	+	0.708602i	$0.250674\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−46.8328	−1.53160
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47.3050	1.54210	0.771049	−	0.636776i	$-0.219732\pi$
$941$	47.3050	1.54210	0.771049	+	0.636776i	$0.219732\pi$
$942$	0	0
$943$	−16.9443	−0.551781
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5279	−0.634570	−0.317285	−	0.948330i	$-0.602771\pi$
$947$	−19.5279	−0.634570	−0.317285	+	0.948330i	$0.602771\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.5410	−0.471030	−0.235515	−	0.971871i	$-0.575678\pi$
$953$	−14.5410	−0.471030	−0.235515	+	0.971871i	$0.575678\pi$
$954$	0	0
$955$	−50.2492	−1.62603
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.9443	−0.547159
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.4164	−0.882565
$966$	0	0
$967$	−18.8328	−0.605623	−0.302811	−	0.953051i	$-0.597925\pi$
$967$	−18.8328	−0.605623	−0.302811	+	0.953051i	$0.597925\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.8328	1.24620	0.623102	−	0.782140i	$-0.285872\pi$
$971$	38.8328	1.24620	0.623102	+	0.782140i	$0.285872\pi$
$972$	0	0
$973$	10.4721	0.335721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.1246	1.12374	0.561868	−	0.827227i	$-0.310083\pi$
$977$	35.1246	1.12374	0.561868	+	0.827227i	$0.310083\pi$
$978$	0	0
$979$	−31.0557	−0.992545
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.9443	1.17834	0.589170	−	0.808009i	$-0.299455\pi$
$983$	36.9443	1.17834	0.589170	+	0.808009i	$0.299455\pi$
$984$	0	0
$985$	−12.9443	−0.412439
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.7771	2.15519
$990$	0	0
$991$	−27.4164	−0.870911	−0.435455	−	0.900210i	$-0.643413\pi$
$991$	−27.4164	−0.870911	−0.435455	+	0.900210i	$0.643413\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	41.8885	1.32796
$996$	0	0
$997$	34.3607	1.08821	0.544107	−	0.839016i	$-0.316869\pi$
$997$	34.3607	1.08821	0.544107	+	0.839016i	$0.316869\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.be.1.1	✓	2	1.1	even	1	trivial
9576.2.a.bq.1.2	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)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-3.23607 q^{5} -1.00000 q^{7} +4.47214 q^{11} +4.47214 q^{13} +3.23607 q^{17} +1.00000 q^{19} +8.47214 q^{23} +5.47214 q^{25} +2.76393 q^{29} -2.47214 q^{31} +3.23607 q^{35} +4.47214 q^{37} -2.00000 q^{41} +8.00000 q^{43} +1.23607 q^{47} +1.00000 q^{49} +1.23607 q^{53} -14.4721 q^{55} -4.00000 q^{59} -12.4721 q^{61} -14.4721 q^{65} -10.4721 q^{67} +0.763932 q^{71} -4.47214 q^{73} -4.47214 q^{77} -12.0000 q^{79} +7.70820 q^{83} -10.4721 q^{85} -6.94427 q^{89} -4.47214 q^{91} -3.23607 q^{95} +12.4721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} + 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 2 q^{49} - 2 q^{53} - 20 q^{55} - 8 q^{59} - 16 q^{61} - 20 q^{65} - 12 q^{67} + 6 q^{71} - 24 q^{79} + 2 q^{83} - 12 q^{85} + 4 q^{89} - 2 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^17 + 2 * q^19 + 8 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 + 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 2 * q^49 - 2 * q^53 - 20 * q^55 - 8 * q^59 - 16 * q^61 - 20 * q^65 - 12 * q^67 + 6 * q^71 - 24 * q^79 + 2 * q^83 - 12 * q^85 + 4 * q^89 - 2 * q^95 + 16 * q^97

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