LMFDB - Embedded newform 9576.2.a.cd.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:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
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			$3.67370$ 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)

$f(q)$	$=$	$q+0.911179 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+0.911179 q^{5} +1.00000 q^{7} -2.00000 q^{11} +4.25857 q^{17} -1.00000 q^{19} -9.34740 q^{23} -4.16975 q^{25} +8.43622 q^{29} +0.177642 q^{31} +0.911179 q^{35} -11.1698 q^{37} +6.00000 q^{41} +9.34740 q^{43} -6.91118 q^{47} +1.00000 q^{49} -14.2586 q^{53} -1.82236 q^{55} +3.82236 q^{61} +4.00000 q^{67} -6.08093 q^{71} +14.5171 q^{73} -2.00000 q^{77} -13.5250 q^{79} -17.9033 q^{83} +3.88032 q^{85} -10.0000 q^{89} -0.911179 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{7}+O(q^{10}) $ 3 * q + 3 * q^7	$ 3 q + 3 q^{7} - 6 q^{11} - 10 q^{17} - 3 q^{19} - 8 q^{23} + 13 q^{25} + 8 q^{29} + 6 q^{31} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 18 q^{47} + 3 q^{49} - 20 q^{53} + 6 q^{61} + 12 q^{67} + 10 q^{71} - 2 q^{73} - 6 q^{77} - 26 q^{79} - 20 q^{83} + 8 q^{85} - 30 q^{89} + 18 q^{97}+O(q^{100}) $ 3 * q + 3 * q^7 - 6 * q^11 - 10 * q^17 - 3 * q^19 - 8 * q^23 + 13 * q^25 + 8 * q^29 + 6 * q^31 - 8 * q^37 + 18 * q^41 + 8 * q^43 - 18 * q^47 + 3 * q^49 - 20 * q^53 + 6 * q^61 + 12 * q^67 + 10 * q^71 - 2 * q^73 - 6 * q^77 - 26 * q^79 - 20 * q^83 + 8 * q^85 - 30 * q^89 + 18 * 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
$5$	0.911179	0.407492	0.203746	−	0.979024i	$-0.434688\pi$
$5$	0.911179	0.407492	0.203746	+	0.979024i	$0.434688\pi$
$6$	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
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.25857	1.03286	0.516428	−	0.856331i	$-0.327261\pi$
$17$	4.25857	1.03286	0.516428	+	0.856331i	$0.327261\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.34740	−1.94907	−0.974533	−	0.224243i	$-0.928009\pi$
$23$	−9.34740	−1.94907	−0.974533	+	0.224243i	$0.928009\pi$
$24$	0	0
$25$	−4.16975	−0.833951
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.43622	1.56657	0.783283	−	0.621665i	$-0.213544\pi$
$29$	8.43622	1.56657	0.783283	+	0.621665i	$0.213544\pi$
$30$	0	0
$31$	0.177642	0.0319055	0.0159528	−	0.999873i	$-0.494922\pi$
$31$	0.177642	0.0319055	0.0159528	+	0.999873i	$0.494922\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.911179	0.154017
$36$	0	0
$37$	−11.1698	−1.83630	−0.918148	−	0.396237i	$-0.870316\pi$
$37$	−11.1698	−1.83630	−0.918148	+	0.396237i	$0.870316\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	9.34740	1.42546	0.712732	−	0.701436i	$-0.247457\pi$
$43$	9.34740	1.42546	0.712732	+	0.701436i	$0.247457\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.91118	−1.00810	−0.504050	−	0.863675i	$-0.668157\pi$
$47$	−6.91118	−1.00810	−0.504050	+	0.863675i	$0.668157\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−14.2586	−1.95857	−0.979283	−	0.202497i	$-0.935094\pi$
$53$	−14.2586	−1.95857	−0.979283	+	0.202497i	$0.935094\pi$
$54$	0	0
$55$	−1.82236	−0.245727
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	3.82236	0.489403	0.244701	−	0.969598i	$-0.421310\pi$
$61$	3.82236	0.489403	0.244701	+	0.969598i	$0.421310\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.08093	−0.721674	−0.360837	−	0.932629i	$-0.617509\pi$
$71$	−6.08093	−0.721674	−0.360837	+	0.932629i	$0.617509\pi$
$72$	0	0
$73$	14.5171	1.69910	0.849552	−	0.527505i	$-0.176872\pi$
$73$	14.5171	1.69910	0.849552	+	0.527505i	$0.176872\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−13.5250	−1.52169	−0.760843	−	0.648936i	$-0.775214\pi$
$79$	−13.5250	−1.52169	−0.760843	+	0.648936i	$0.775214\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.9033	−1.96514	−0.982571	−	0.185889i	$-0.940483\pi$
$83$	−17.9033	−1.96514	−0.982571	+	0.185889i	$0.940483\pi$
$84$	0	0
$85$	3.88032	0.420880
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.911179	−0.0934850
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.73354	0.271997	0.135999	−	0.990709i	$-0.456576\pi$
$101$	2.73354	0.271997	0.135999	+	0.990709i	$0.456576\pi$
$102$	0	0
$103$	−4.65260	−0.458435	−0.229217	−	0.973375i	$-0.573617\pi$
$103$	−4.65260	−0.458435	−0.229217	+	0.973375i	$0.573617\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.258574	−0.0249973	−0.0124987	−	0.999922i	$-0.503979\pi$
$107$	−0.258574	−0.0249973	−0.0124987	+	0.999922i	$0.503979\pi$
$108$	0	0
$109$	−8.99211	−0.861288	−0.430644	−	0.902522i	$-0.641714\pi$
$109$	−8.99211	−0.861288	−0.430644	+	0.902522i	$0.641714\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.0809	1.13648	0.568239	−	0.822863i	$-0.307625\pi$
$113$	12.0809	1.13648	0.568239	+	0.822863i	$0.307625\pi$
$114$	0	0
$115$	−8.51715	−0.794228
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.25857	0.390383
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.35528	−0.747319
$126$	0	0
$127$	−0.652604	−0.0579093	−0.0289546	−	0.999581i	$-0.509218\pi$
$127$	−0.652604	−0.0579093	−0.0289546	+	0.999581i	$0.509218\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.61386	0.228374	0.114187	−	0.993459i	$-0.463574\pi$
$131$	2.61386	0.228374	0.114187	+	0.993459i	$0.463574\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.64472	0.311389	0.155695	−	0.987805i	$-0.450238\pi$
$137$	3.64472	0.311389	0.155695	+	0.987805i	$0.450238\pi$
$138$	0	0
$139$	−9.82236	−0.833121	−0.416561	−	0.909108i	$-0.636765\pi$
$139$	−9.82236	−0.833121	−0.416561	+	0.909108i	$0.636765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.68690	0.638362
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.1698	−1.40660	−0.703300	−	0.710893i	$-0.748291\pi$
$149$	−17.1698	−1.40660	−0.703300	+	0.710893i	$0.748291\pi$
$150$	0	0
$151$	−13.5250	−1.10065	−0.550326	−	0.834950i	$-0.685497\pi$
$151$	−13.5250	−1.10065	−0.550326	+	0.834950i	$0.685497\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.161864	0.0130012
$156$	0	0
$157$	15.8224	1.26276	0.631381	−	0.775473i	$-0.282489\pi$
$157$	15.8224	1.26276	0.631381	+	0.775473i	$0.282489\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.34740	−0.736678
$162$	0	0
$163$	−4.99211	−0.391012	−0.195506	−	0.980702i	$-0.562635\pi$
$163$	−4.99211	−0.391012	−0.195506	+	0.980702i	$0.562635\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.5171	1.27814	0.639068	−	0.769151i	$-0.279320\pi$
$167$	16.5171	1.27814	0.639068	+	0.769151i	$0.279320\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.69479	−0.661053	−0.330526	−	0.943797i	$-0.607226\pi$
$173$	−8.69479	−0.661053	−0.330526	+	0.943797i	$0.607226\pi$
$174$	0	0
$175$	−4.16975	−0.315204
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.9534	1.11767	0.558834	−	0.829280i	$-0.311249\pi$
$179$	14.9534	1.11767	0.558834	+	0.829280i	$0.311249\pi$
$180$	0	0
$181$	23.6869	1.76063	0.880317	−	0.474386i	$-0.157330\pi$
$181$	23.6869	1.76063	0.880317	+	0.474386i	$0.157330\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.1776	−0.748275
$186$	0	0
$187$	−8.51715	−0.622836
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.99211	−0.650646	−0.325323	−	0.945603i	$-0.605473\pi$
$191$	−8.99211	−0.650646	−0.325323	+	0.945603i	$0.605473\pi$
$192$	0	0
$193$	−10.5171	−0.757041	−0.378520	−	0.925593i	$-0.623567\pi$
$193$	−10.5171	−0.757041	−0.378520	+	0.925593i	$0.623567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.6869	−0.975151	−0.487576	−	0.873081i	$-0.662119\pi$
$197$	−13.6869	−0.975151	−0.487576	+	0.873081i	$0.662119\pi$
$198$	0	0
$199$	18.3395	1.30005	0.650027	−	0.759911i	$-0.274758\pi$
$199$	18.3395	1.30005	0.650027	+	0.759911i	$0.274758\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.43622	0.592106
$204$	0	0
$205$	5.46707	0.381837
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	10.6948	0.736260	0.368130	−	0.929774i	$-0.379998\pi$
$211$	10.6948	0.736260	0.368130	+	0.929774i	$0.379998\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.51715	0.580865
$216$	0	0
$217$	0.177642	0.0120592
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	18.5171	1.24000	0.620000	−	0.784602i	$-0.287133\pi$
$223$	18.5171	1.24000	0.620000	+	0.784602i	$0.287133\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.8224	−0.917422	−0.458711	−	0.888585i	$-0.651689\pi$
$227$	−13.8224	−0.917422	−0.458711	+	0.888585i	$0.651689\pi$
$228$	0	0
$229$	−27.0343	−1.78648	−0.893238	−	0.449583i	$-0.851572\pi$
$229$	−27.0343	−1.78648	−0.893238	+	0.449583i	$0.851572\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.517149	0.0338795	0.0169398	−	0.999857i	$-0.494608\pi$
$233$	0.517149	0.0338795	0.0169398	+	0.999857i	$0.494608\pi$
$234$	0	0
$235$	−6.29732	−0.410792
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.8645	−1.15556	−0.577781	−	0.816192i	$-0.696081\pi$
$239$	−17.8645	−1.15556	−0.577781	+	0.816192i	$0.696081\pi$
$240$	0	0
$241$	−17.0501	−1.09829	−0.549146	−	0.835726i	$-0.685047\pi$
$241$	−17.0501	−1.09829	−0.549146	+	0.835726i	$0.685047\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.911179	0.0582131
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.9033	−1.63500	−0.817501	−	0.575928i	$-0.804641\pi$
$251$	−25.9033	−1.63500	−0.817501	+	0.575928i	$0.804641\pi$
$252$	0	0
$253$	18.6948	1.17533
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.46707	−0.465783	−0.232892	−	0.972503i	$-0.574819\pi$
$257$	−7.46707	−0.465783	−0.232892	+	0.972503i	$0.574819\pi$
$258$	0	0
$259$	−11.1698	−0.694055
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.830247	−0.0511952	−0.0255976	−	0.999672i	$-0.508149\pi$
$263$	−0.830247	−0.0511952	−0.0255976	+	0.999672i	$0.508149\pi$
$264$	0	0
$265$	−12.9921	−0.798099
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.8567	1.51554	0.757769	−	0.652523i	$-0.226290\pi$
$269$	24.8567	1.51554	0.757769	+	0.652523i	$0.226290\pi$
$270$	0	0
$271$	20.5171	1.24633	0.623164	−	0.782091i	$-0.285847\pi$
$271$	20.5171	1.24633	0.623164	+	0.782091i	$0.285847\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.33951	0.502891
$276$	0	0
$277$	2.83025	0.170053	0.0850265	−	0.996379i	$-0.472902\pi$
$277$	2.83025	0.170053	0.0850265	+	0.996379i	$0.472902\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.43622	−0.503262	−0.251631	−	0.967823i	$-0.580967\pi$
$281$	−8.43622	−0.503262	−0.251631	+	0.967823i	$0.580967\pi$
$282$	0	0
$283$	29.9842	1.78238	0.891188	−	0.453633i	$-0.149872\pi$
$283$	29.9842	1.78238	0.891188	+	0.453633i	$0.149872\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	1.13546	0.0667915
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.3896	−1.60012	−0.800058	−	0.599922i	$-0.795198\pi$
$293$	−27.3896	−1.60012	−0.800058	+	0.599922i	$0.795198\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	9.34740	0.538775
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.48285	0.199427
$306$	0	0
$307$	−22.2198	−1.26815	−0.634076	−	0.773270i	$-0.718620\pi$
$307$	−22.2198	−1.26815	−0.634076	+	0.773270i	$0.718620\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.4283	−0.874860	−0.437430	−	0.899252i	$-0.644111\pi$
$311$	−15.4283	−0.874860	−0.437430	+	0.899252i	$0.644111\pi$
$312$	0	0
$313$	30.8724	1.74501	0.872507	−	0.488602i	$-0.162493\pi$
$313$	30.8724	1.74501	0.872507	+	0.488602i	$0.162493\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.791502	0.0444552	0.0222276	−	0.999753i	$-0.492924\pi$
$317$	0.791502	0.0444552	0.0222276	+	0.999753i	$0.492924\pi$
$318$	0	0
$319$	−16.8724	−0.944675
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.25857	−0.236953
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.91118	−0.381026
$330$	0	0
$331$	16.5171	0.907865	0.453932	−	0.891036i	$-0.350021\pi$
$331$	16.5171	0.907865	0.453932	+	0.891036i	$0.350021\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.64472	0.199132
$336$	0	0
$337$	9.64472	0.525381	0.262691	−	0.964880i	$-0.415390\pi$
$337$	9.64472	0.525381	0.262691	+	0.964880i	$0.415390\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.355285	−0.0192397
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.1619	−1.18971	−0.594856	−	0.803833i	$-0.702791\pi$
$347$	−22.1619	−1.18971	−0.594856	+	0.803833i	$0.702791\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.9033	−1.05935	−0.529673	−	0.848202i	$-0.677685\pi$
$353$	−19.9033	−1.05935	−0.529673	+	0.848202i	$0.677685\pi$
$354$	0	0
$355$	−5.54082	−0.294076
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.70268	−0.512088	−0.256044	−	0.966665i	$-0.582419\pi$
$359$	−9.70268	−0.512088	−0.256044	+	0.966665i	$0.582419\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.2277	0.692370
$366$	0	0
$367$	−33.3896	−1.74292	−0.871461	−	0.490465i	$-0.836827\pi$
$367$	−33.3896	−1.74292	−0.871461	+	0.490465i	$0.836827\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.2586	−0.740268
$372$	0	0
$373$	32.9921	1.70827	0.854133	−	0.520054i	$-0.174088\pi$
$373$	32.9921	1.70827	0.854133	+	0.520054i	$0.174088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−19.2119	−0.986851	−0.493426	−	0.869788i	$-0.664255\pi$
$379$	−19.2119	−0.986851	−0.493426	+	0.869788i	$0.664255\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.2277	−0.880295	−0.440148	−	0.897925i	$-0.645074\pi$
$383$	−17.2277	−0.880295	−0.440148	+	0.897925i	$0.645074\pi$
$384$	0	0
$385$	−1.82236	−0.0928759
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.1540	1.57957	0.789785	−	0.613384i	$-0.210192\pi$
$389$	31.1540	1.57957	0.789785	+	0.613384i	$0.210192\pi$
$390$	0	0
$391$	−39.8066	−2.01311
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.3237	−0.620074
$396$	0	0
$397$	35.9842	1.80600	0.902998	−	0.429644i	$-0.141361\pi$
$397$	35.9842	1.80600	0.902998	+	0.429644i	$0.141361\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.79150	0.239276	0.119638	−	0.992818i	$-0.461827\pi$
$401$	4.79150	0.239276	0.119638	+	0.992818i	$0.461827\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.3395	1.10733
$408$	0	0
$409$	−22.3975	−1.10748	−0.553742	−	0.832688i	$-0.686801\pi$
$409$	−22.3975	−1.10748	−0.553742	+	0.832688i	$0.686801\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.3131	−0.800778
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.9534	0.632813	0.316407	−	0.948624i	$-0.397524\pi$
$419$	12.9534	0.632813	0.316407	+	0.948624i	$0.397524\pi$
$420$	0	0
$421$	−30.2198	−1.47282	−0.736412	−	0.676533i	$-0.763482\pi$
$421$	−30.2198	−1.47282	−0.736412	+	0.676533i	$0.763482\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17.7572	−0.861351
$426$	0	0
$427$	3.82236	0.184977
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.6139	−0.607588	−0.303794	−	0.952738i	$-0.598253\pi$
$431$	−12.6139	−0.607588	−0.303794	+	0.952738i	$0.598253\pi$
$432$	0	0
$433$	9.05008	0.434919	0.217459	−	0.976069i	$-0.430223\pi$
$433$	9.05008	0.434919	0.217459	+	0.976069i	$0.430223\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.34740	0.447147
$438$	0	0
$439$	14.1619	0.675909	0.337954	−	0.941162i	$-0.390265\pi$
$439$	14.1619	0.675909	0.337954	+	0.941162i	$0.390265\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.8224	−1.13183	−0.565917	−	0.824462i	$-0.691478\pi$
$443$	−23.8224	−1.13183	−0.565917	+	0.824462i	$0.691478\pi$
$444$	0	0
$445$	−9.11179	−0.431940
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.20850	−0.340190	−0.170095	−	0.985428i	$-0.554407\pi$
$449$	−7.20850	−0.340190	−0.170095	+	0.985428i	$0.554407\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.88032	−0.0879578	−0.0439789	−	0.999032i	$-0.514003\pi$
$457$	−1.88032	−0.0879578	−0.0439789	+	0.999032i	$0.514003\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.2164	−0.475825	−0.237912	−	0.971287i	$-0.576463\pi$
$461$	−10.2164	−0.475825	−0.237912	+	0.971287i	$0.576463\pi$
$462$	0	0
$463$	−21.9842	−1.02169	−0.510847	−	0.859672i	$-0.670668\pi$
$463$	−21.9842	−1.02169	−0.510847	+	0.859672i	$0.670668\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.9033	0.643368	0.321684	−	0.946847i	$-0.395751\pi$
$467$	13.9033	0.643368	0.321684	+	0.946847i	$0.395751\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18.6948	−0.859587
$474$	0	0
$475$	4.16975	0.191321
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.1231	−0.462537	−0.231269	−	0.972890i	$-0.574288\pi$
$479$	−10.1231	−0.462537	−0.231269	+	0.972890i	$0.574288\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.46707	0.248247
$486$	0	0
$487$	7.70268	0.349042	0.174521	−	0.984653i	$-0.444162\pi$
$487$	7.70268	0.349042	0.174521	+	0.984653i	$0.444162\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.1776	−0.730087	−0.365043	−	0.930991i	$-0.618946\pi$
$491$	−16.1776	−0.730087	−0.365043	+	0.930991i	$0.618946\pi$
$492$	0	0
$493$	35.9263	1.61804
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.08093	−0.272767
$498$	0	0
$499$	3.28943	0.147255	0.0736276	−	0.997286i	$-0.476542\pi$
$499$	3.28943	0.147255	0.0736276	+	0.997286i	$0.476542\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.9613	0.800853	0.400426	−	0.916329i	$-0.368862\pi$
$503$	17.9613	0.800853	0.400426	+	0.916329i	$0.368862\pi$
$504$	0	0
$505$	2.49074	0.110836
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.5171	−1.17535	−0.587676	−	0.809096i	$-0.699957\pi$
$509$	−26.5171	−1.17535	−0.587676	+	0.809096i	$0.699957\pi$
$510$	0	0
$511$	14.5171	0.642201
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.23935	−0.186808
$516$	0	0
$517$	13.8224	0.607907
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.8724	−1.00206	−0.501030	−	0.865430i	$-0.667045\pi$
$521$	−22.8724	−1.00206	−0.501030	+	0.865430i	$0.667045\pi$
$522$	0	0
$523$	7.52504	0.329047	0.164523	−	0.986373i	$-0.447391\pi$
$523$	7.52504	0.329047	0.164523	+	0.986373i	$0.447391\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.756503	0.0329538
$528$	0	0
$529$	64.3738	2.79886
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−0.235607	−0.0101862
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	22.6790	0.975047	0.487523	−	0.873110i	$-0.337900\pi$
$541$	22.6790	0.975047	0.487523	+	0.873110i	$0.337900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.19342	−0.350968
$546$	0	0
$547$	−21.8224	−0.933057	−0.466528	−	0.884506i	$-0.654495\pi$
$547$	−21.8224	−0.933057	−0.466528	+	0.884506i	$0.654495\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.43622	−0.359395
$552$	0	0
$553$	−13.5250	−0.575143
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.5593	1.80330	0.901648	−	0.432470i	$-0.142358\pi$
$557$	42.5593	1.80330	0.901648	+	0.432470i	$0.142358\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.46707	0.230410	0.115205	−	0.993342i	$-0.463248\pi$
$563$	5.46707	0.230410	0.115205	+	0.993342i	$0.463248\pi$
$564$	0	0
$565$	11.0079	0.463105
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.0809	−0.506459	−0.253230	−	0.967406i	$-0.581493\pi$
$569$	−12.0809	−0.506459	−0.253230	+	0.967406i	$0.581493\pi$
$570$	0	0
$571$	37.0343	1.54984	0.774919	−	0.632061i	$-0.217791\pi$
$571$	37.0343	1.54984	0.774919	+	0.632061i	$0.217791\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38.9763	1.62543
$576$	0	0
$577$	8.17764	0.340440	0.170220	−	0.985406i	$-0.445552\pi$
$577$	8.17764	0.340440	0.170220	+	0.985406i	$0.445552\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.9033	−0.742754
$582$	0	0
$583$	28.5171	1.18106
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.5638	−0.642386	−0.321193	−	0.947014i	$-0.604084\pi$
$587$	−15.5638	−0.642386	−0.321193	+	0.947014i	$0.604084\pi$
$588$	0	0
$589$	−0.177642	−0.00731963
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.0034	1.56061	0.780307	−	0.625396i	$-0.215063\pi$
$593$	38.0034	1.56061	0.780307	+	0.625396i	$0.215063\pi$
$594$	0	0
$595$	3.88032	0.159078
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−47.8258	−1.95411	−0.977055	−	0.212989i	$-0.931680\pi$
$599$	−47.8258	−1.95411	−0.977055	+	0.212989i	$0.931680\pi$
$600$	0	0
$601$	36.3395	1.48232	0.741160	−	0.671329i	$-0.234276\pi$
$601$	36.3395	1.48232	0.741160	+	0.671329i	$0.234276\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.37825	−0.259313
$606$	0	0
$607$	−14.0422	−0.569955	−0.284977	−	0.958534i	$-0.591986\pi$
$607$	−14.0422	−0.569955	−0.284977	+	0.958534i	$0.591986\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−15.5093	−0.626413	−0.313207	−	0.949685i	$-0.601403\pi$
$613$	−15.5093	−0.626413	−0.313207	+	0.949685i	$0.601403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.6948	−1.23573	−0.617863	−	0.786286i	$-0.712001\pi$
$617$	−30.6948	−1.23573	−0.617863	+	0.786286i	$0.712001\pi$
$618$	0	0
$619$	13.4671	0.541287	0.270644	−	0.962680i	$-0.412763\pi$
$619$	13.4671	0.541287	0.270644	+	0.962680i	$0.412763\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.0000	−0.400642
$624$	0	0
$625$	13.2356	0.529424
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−47.5672	−1.89663
$630$	0	0
$631$	−41.3474	−1.64601	−0.823007	−	0.568031i	$-0.807705\pi$
$631$	−41.3474	−1.64601	−0.823007	+	0.568031i	$0.807705\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.594639	−0.0235975
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.1152	−0.991992	−0.495996	−	0.868325i	$-0.665197\pi$
$641$	−25.1152	−0.991992	−0.495996	+	0.868325i	$0.665197\pi$
$642$	0	0
$643$	−1.30521	−0.0514724	−0.0257362	−	0.999669i	$-0.508193\pi$
$643$	−1.30521	−0.0514724	−0.0257362	+	0.999669i	$0.508193\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.13890	0.162717	0.0813584	−	0.996685i	$-0.474074\pi$
$647$	4.13890	0.162717	0.0813584	+	0.996685i	$0.474074\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.0264	1.09676	0.548379	−	0.836230i	$-0.315245\pi$
$653$	28.0264	1.09676	0.548379	+	0.836230i	$0.315245\pi$
$654$	0	0
$655$	2.38169	0.0930604
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.7414	0.924835	0.462417	−	0.886662i	$-0.346982\pi$
$659$	23.7414	0.924835	0.462417	+	0.886662i	$0.346982\pi$
$660$	0	0
$661$	−5.66049	−0.220168	−0.110084	−	0.993922i	$-0.535112\pi$
$661$	−5.66049	−0.220168	−0.110084	+	0.993922i	$0.535112\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.911179	−0.0353340
$666$	0	0
$667$	−78.8567	−3.05334
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.64472	−0.295121
$672$	0	0
$673$	2.87243	0.110724	0.0553621	−	0.998466i	$-0.482369\pi$
$673$	2.87243	0.110724	0.0553621	+	0.998466i	$0.482369\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.21194	−0.354044	−0.177022	−	0.984207i	$-0.556646\pi$
$677$	−9.21194	−0.354044	−0.177022	+	0.984207i	$0.556646\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.0809	−0.691848	−0.345924	−	0.938263i	$-0.612434\pi$
$683$	−18.0809	−0.691848	−0.345924	+	0.938263i	$0.612434\pi$
$684$	0	0
$685$	3.32099	0.126888
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	47.2119	1.79603	0.898013	−	0.439968i	$-0.145010\pi$
$691$	47.2119	1.79603	0.898013	+	0.439968i	$0.145010\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.94992	−0.339490
$696$	0	0
$697$	25.5514	0.967830
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.3474	0.730741	0.365371	−	0.930862i	$-0.380942\pi$
$701$	19.3474	0.730741	0.365371	+	0.930862i	$0.380942\pi$
$702$	0	0
$703$	11.1698	0.421275
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.73354	0.102805
$708$	0	0
$709$	24.2198	0.909595	0.454797	−	0.890595i	$-0.349712\pi$
$709$	24.2198	0.909595	0.454797	+	0.890595i	$0.349712\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.66049	−0.0621860
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−52.3008	−1.95049	−0.975245	−	0.221129i	$-0.929026\pi$
$719$	−52.3008	−1.95049	−0.975245	+	0.221129i	$0.929026\pi$
$720$	0	0
$721$	−4.65260	−0.173272
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−35.1769	−1.30644
$726$	0	0
$727$	−28.8724	−1.07082	−0.535410	−	0.844593i	$-0.679843\pi$
$727$	−28.8724	−1.07082	−0.535410	+	0.844593i	$0.679843\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	39.8066	1.47230
$732$	0	0
$733$	−37.8066	−1.39642	−0.698209	−	0.715894i	$-0.746019\pi$
$733$	−37.8066	−1.39642	−0.698209	+	0.715894i	$0.746019\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	7.36317	0.270859	0.135429	−	0.990787i	$-0.456759\pi$
$739$	7.36317	0.270859	0.135429	+	0.990787i	$0.456759\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.4204	0.455662	0.227831	−	0.973701i	$-0.426837\pi$
$743$	12.4204	0.455662	0.227831	+	0.973701i	$0.426837\pi$
$744$	0	0
$745$	−15.6447	−0.573178
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.258574	−0.00944810
$750$	0	0
$751$	−39.2699	−1.43298	−0.716490	−	0.697598i	$-0.754252\pi$
$751$	−39.2699	−1.43298	−0.716490	+	0.697598i	$0.754252\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.3237	−0.448506
$756$	0	0
$757$	−15.9842	−0.580956	−0.290478	−	0.956882i	$-0.593814\pi$
$757$	−15.9842	−0.580956	−0.290478	+	0.956882i	$0.593814\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.43622	−0.233313	−0.116656	−	0.993172i	$-0.537218\pi$
$761$	−6.43622	−0.233313	−0.116656	+	0.993172i	$0.537218\pi$
$762$	0	0
$763$	−8.99211	−0.325536
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−43.1962	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$769$	−43.1962	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.0343	1.69171	0.845853	−	0.533416i	$-0.179092\pi$
$773$	47.0343	1.69171	0.845853	+	0.533416i	$0.179092\pi$
$774$	0	0
$775$	−0.740725	−0.0266076
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	12.1619	0.435186
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.4170	0.514565
$786$	0	0
$787$	−16.1197	−0.574604	−0.287302	−	0.957840i	$-0.592758\pi$
$787$	−16.1197	−0.574604	−0.287302	+	0.957840i	$0.592758\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.0809	0.429548
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.6447	−1.05007	−0.525035	−	0.851081i	$-0.675948\pi$
$797$	−29.6447	−1.05007	−0.525035	+	0.851081i	$0.675948\pi$
$798$	0	0
$799$	−29.4318	−1.04122
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−29.0343	−1.02460
$804$	0	0
$805$	−8.51715	−0.300190
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.6790	0.586403	0.293201	−	0.956051i	$-0.405279\pi$
$809$	16.6790	0.586403	0.293201	+	0.956051i	$0.405279\pi$
$810$	0	0
$811$	0.594639	0.0208806	0.0104403	−	0.999945i	$-0.496677\pi$
$811$	0.594639	0.0208806	0.0104403	+	0.999945i	$0.496677\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.54871	−0.159334
$816$	0	0
$817$	−9.34740	−0.327024
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.5250	−0.611628	−0.305814	−	0.952091i	$-0.598929\pi$
$821$	−17.5250	−0.611628	−0.305814	+	0.952091i	$0.598929\pi$
$822$	0	0
$823$	−20.2815	−0.706970	−0.353485	−	0.935440i	$-0.615003\pi$
$823$	−20.2815	−0.706970	−0.353485	+	0.935440i	$0.615003\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.42044	−0.292807	−0.146404	−	0.989225i	$-0.546770\pi$
$827$	−8.42044	−0.292807	−0.146404	+	0.989225i	$0.546770\pi$
$828$	0	0
$829$	9.34740	0.324648	0.162324	−	0.986737i	$-0.448101\pi$
$829$	9.34740	0.324648	0.162324	+	0.986737i	$0.448101\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.25857	0.147551
$834$	0	0
$835$	15.0501	0.520829
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.3553	−0.426552	−0.213276	−	0.976992i	$-0.568413\pi$
$839$	−12.3553	−0.426552	−0.213276	+	0.976992i	$0.568413\pi$
$840$	0	0
$841$	42.1698	1.45413
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.8453	−0.407492
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	104.408	3.57906
$852$	0	0
$853$	−9.05008	−0.309869	−0.154934	−	0.987925i	$-0.549517\pi$
$853$	−9.05008	−0.309869	−0.154934	+	0.987925i	$0.549517\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.0343	−0.376924	−0.188462	−	0.982080i	$-0.560350\pi$
$857$	−11.0343	−0.376924	−0.188462	+	0.982080i	$0.560350\pi$
$858$	0	0
$859$	−11.0501	−0.377024	−0.188512	−	0.982071i	$-0.560366\pi$
$859$	−11.0501	−0.377024	−0.188512	+	0.982071i	$0.560366\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.2929	1.13330	0.566651	−	0.823958i	$-0.308239\pi$
$863$	33.2929	1.13330	0.566651	+	0.823958i	$0.308239\pi$
$864$	0	0
$865$	−7.92251	−0.269373
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.0501	0.917611
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.35528	−0.282460
$876$	0	0
$877$	34.0264	1.14899	0.574495	−	0.818508i	$-0.305198\pi$
$877$	34.0264	1.14899	0.574495	+	0.818508i	$0.305198\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.09671	0.138022	0.0690108	−	0.997616i	$-0.478016\pi$
$881$	4.09671	0.138022	0.0690108	+	0.997616i	$0.478016\pi$
$882$	0	0
$883$	−27.2894	−0.918362	−0.459181	−	0.888343i	$-0.651857\pi$
$883$	−27.2894	−0.918362	−0.459181	+	0.888343i	$0.651857\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.9225	−1.74339	−0.871694	−	0.490051i	$-0.836978\pi$
$887$	−51.9225	−1.74339	−0.871694	+	0.490051i	$0.836978\pi$
$888$	0	0
$889$	−0.652604	−0.0218877
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.91118	0.231274
$894$	0	0
$895$	13.6252	0.455440
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.49863	0.0499821
$900$	0	0
$901$	−60.7212	−2.02292
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.5830	0.717443
$906$	0	0
$907$	35.0501	1.16382	0.581909	−	0.813254i	$-0.302306\pi$
$907$	35.0501	1.16382	0.581909	+	0.813254i	$0.302306\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44.4204	1.47171	0.735857	−	0.677137i	$-0.236779\pi$
$911$	44.4204	1.47171	0.735857	+	0.677137i	$0.236779\pi$
$912$	0	0
$913$	35.8066	1.18502
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.61386	0.0863172
$918$	0	0
$919$	−22.6948	−0.748632	−0.374316	−	0.927301i	$-0.622122\pi$
$919$	−22.6948	−0.748632	−0.374316	+	0.927301i	$0.622122\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	46.5751	1.53138
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.7099	−1.43407	−0.717037	−	0.697035i	$-0.754502\pi$
$929$	−43.7099	−1.43407	−0.717037	+	0.697035i	$0.754502\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.76065	−0.253800
$936$	0	0
$937$	−49.3738	−1.61297	−0.806486	−	0.591253i	$-0.798633\pi$
$937$	−49.3738	−1.61297	−0.806486	+	0.591253i	$0.798633\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.1619	0.331267	0.165634	−	0.986187i	$-0.447033\pi$
$941$	10.1619	0.331267	0.165634	+	0.986187i	$0.447033\pi$
$942$	0	0
$943$	−56.0844	−1.82636
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.0501	−0.684036	−0.342018	−	0.939693i	$-0.611110\pi$
$947$	−21.0501	−0.684036	−0.342018	+	0.939693i	$0.611110\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.59808	−0.148946	−0.0744732	−	0.997223i	$-0.523728\pi$
$953$	−4.59808	−0.148946	−0.0744732	+	0.997223i	$0.523728\pi$
$954$	0	0
$955$	−8.19342	−0.265133
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.64472	0.117694
$960$	0	0
$961$	−30.9684	−0.998982
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.58300	−0.308488
$966$	0	0
$967$	−44.6368	−1.43542	−0.717712	−	0.696340i	$-0.754810\pi$
$967$	−44.6368	−1.43542	−0.717712	+	0.696340i	$0.754810\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.4054	−0.879480	−0.439740	−	0.898125i	$-0.644929\pi$
$971$	−27.4054	−0.879480	−0.439740	+	0.898125i	$0.644929\pi$
$972$	0	0
$973$	−9.82236	−0.314890
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.95337	−0.158472	−0.0792361	−	0.996856i	$-0.525248\pi$
$977$	−4.95337	−0.158472	−0.0792361	+	0.996856i	$0.525248\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.0844	1.02333	0.511666	−	0.859184i	$-0.329028\pi$
$983$	32.0844	1.02333	0.511666	+	0.859184i	$0.329028\pi$
$984$	0	0
$985$	−12.4712	−0.397366
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−87.3738	−2.77833
$990$	0	0
$991$	−11.3474	−0.360462	−0.180231	−	0.983624i	$-0.557685\pi$
$991$	−11.3474	−0.360462	−0.180231	+	0.983624i	$0.557685\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.7106	0.529761
$996$	0	0
$997$	−46.6015	−1.47589	−0.737943	−	0.674864i	$-0.764202\pi$
$997$	−46.6015	−1.47589	−0.737943	+	0.674864i	$0.764202\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.cd.1.2	✓	3
3.2	odd	2		9576.2.a.ce.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.cd.1.2	✓	3	1.1	even	1	trivial
9576.2.a.ce.1.2	yes	3	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)

\(f(q)\)	\(=\)	\(q+0.911179 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+0.911179 q^{5} +1.00000 q^{7} -2.00000 q^{11} +4.25857 q^{17} -1.00000 q^{19} -9.34740 q^{23} -4.16975 q^{25} +8.43622 q^{29} +0.177642 q^{31} +0.911179 q^{35} -11.1698 q^{37} +6.00000 q^{41} +9.34740 q^{43} -6.91118 q^{47} +1.00000 q^{49} -14.2586 q^{53} -1.82236 q^{55} +3.82236 q^{61} +4.00000 q^{67} -6.08093 q^{71} +14.5171 q^{73} -2.00000 q^{77} -13.5250 q^{79} -17.9033 q^{83} +3.88032 q^{85} -10.0000 q^{89} -0.911179 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{7}+O(q^{10}) \) 3 * q + 3 * q^7	\( 3 q + 3 q^{7} - 6 q^{11} - 10 q^{17} - 3 q^{19} - 8 q^{23} + 13 q^{25} + 8 q^{29} + 6 q^{31} - 8 q^{37} + 18 q^{41} + 8 q^{43} - 18 q^{47} + 3 q^{49} - 20 q^{53} + 6 q^{61} + 12 q^{67} + 10 q^{71} - 2 q^{73} - 6 q^{77} - 26 q^{79} - 20 q^{83} + 8 q^{85} - 30 q^{89} + 18 q^{97}+O(q^{100}) \) 3 * q + 3 * q^7 - 6 * q^11 - 10 * q^17 - 3 * q^19 - 8 * q^23 + 13 * q^25 + 8 * q^29 + 6 * q^31 - 8 * q^37 + 18 * q^41 + 8 * q^43 - 18 * q^47 + 3 * q^49 - 20 * q^53 + 6 * q^61 + 12 * q^67 + 10 * q^71 - 2 * q^73 - 6 * q^77 - 26 * q^79 - 20 * q^83 + 8 * q^85 - 30 * q^89 + 18 * q^97

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