LMFDB - Embedded newform 9408.2.a.eh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.74657 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.74657 q^{5} +1.00000 q^{9} -1.54364 q^{11} -6.03677 q^{13} +2.74657 q^{15} +7.49314 q^{17} +6.03677 q^{19} -7.49314 q^{23} +2.54364 q^{25} -1.00000 q^{27} -1.25343 q^{29} +5.29021 q^{31} +1.54364 q^{33} -4.94950 q^{37} +6.03677 q^{39} -5.08727 q^{41} -3.45636 q^{43} -2.74657 q^{45} +9.49314 q^{47} -7.49314 q^{51} +3.83384 q^{53} +4.23970 q^{55} -6.03677 q^{57} -5.54364 q^{59} +14.5804 q^{61} +16.5804 q^{65} +4.03677 q^{67} +7.49314 q^{69} -5.49314 q^{71} +12.5436 q^{73} -2.54364 q^{75} -7.79707 q^{79} +1.00000 q^{81} +6.52991 q^{83} -20.5804 q^{85} +1.25343 q^{87} -9.49314 q^{89} -5.29021 q^{93} -16.5804 q^{95} +1.54364 q^{97} -1.54364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} - 3 q^{37} - 3 q^{39} - 6 q^{41} - 15 q^{43} + 12 q^{47} - 6 q^{51} - 6 q^{53} - 12 q^{55} + 3 q^{57} - 12 q^{59} + 18 q^{61} + 24 q^{65} - 9 q^{67} + 6 q^{69} + 33 q^{73} - 3 q^{75} - 27 q^{79} + 3 q^{81} - 18 q^{83} - 36 q^{85} + 12 q^{87} - 12 q^{89} - 3 q^{93} - 24 q^{95}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 - 3 * q^37 - 3 * q^39 - 6 * q^41 - 15 * q^43 + 12 * q^47 - 6 * q^51 - 6 * q^53 - 12 * q^55 + 3 * q^57 - 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 + 6 * q^69 + 33 * q^73 - 3 * q^75 - 27 * q^79 + 3 * q^81 - 18 * q^83 - 36 * q^85 + 12 * q^87 - 12 * q^89 - 3 * q^93 - 24 * q^95

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.74657	−1.22830	−0.614151	−	0.789188i	$-0.710502\pi$
$5$	−2.74657	−1.22830	−0.614151	+	0.789188i	$0.710502\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.54364	−0.465424	−0.232712	−	0.972546i	$-0.574760\pi$
$11$	−1.54364	−0.465424	−0.232712	+	0.972546i	$0.574760\pi$
$12$	0	0
$13$	−6.03677	−1.67430	−0.837150	−	0.546974i	$-0.815780\pi$
$13$	−6.03677	−1.67430	−0.837150	+	0.546974i	$0.815780\pi$
$14$	0	0
$15$	2.74657	0.709161
$16$	0	0
$17$	7.49314	1.81735	0.908676	−	0.417501i	$-0.137094\pi$
$17$	7.49314	1.81735	0.908676	+	0.417501i	$0.137094\pi$
$18$	0	0
$19$	6.03677	1.38493	0.692465	−	0.721451i	$-0.256525\pi$
$19$	6.03677	1.38493	0.692465	+	0.721451i	$0.256525\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.49314	−1.56243	−0.781213	−	0.624264i	$-0.785399\pi$
$23$	−7.49314	−1.56243	−0.781213	+	0.624264i	$0.785399\pi$
$24$	0	0
$25$	2.54364	0.508727
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.25343	−0.232756	−0.116378	−	0.993205i	$-0.537128\pi$
$29$	−1.25343	−0.232756	−0.116378	+	0.993205i	$0.537128\pi$
$30$	0	0
$31$	5.29021	0.950149	0.475074	−	0.879946i	$-0.342421\pi$
$31$	5.29021	0.950149	0.475074	+	0.879946i	$0.342421\pi$
$32$	0	0
$33$	1.54364	0.268713
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.94950	−0.813693	−0.406846	−	0.913497i	$-0.633372\pi$
$37$	−4.94950	−0.813693	−0.406846	+	0.913497i	$0.633372\pi$
$38$	0	0
$39$	6.03677	0.966657
$40$	0	0
$41$	−5.08727	−0.794499	−0.397249	−	0.917711i	$-0.630035\pi$
$41$	−5.08727	−0.794499	−0.397249	+	0.917711i	$0.630035\pi$
$42$	0	0
$43$	−3.45636	−0.527090	−0.263545	−	0.964647i	$-0.584892\pi$
$43$	−3.45636	−0.527090	−0.263545	+	0.964647i	$0.584892\pi$
$44$	0	0
$45$	−2.74657	−0.409434
$46$	0	0
$47$	9.49314	1.38472	0.692358	−	0.721554i	$-0.256572\pi$
$47$	9.49314	1.38472	0.692358	+	0.721554i	$0.256572\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−7.49314	−1.04925
$52$	0	0
$53$	3.83384	0.526619	0.263309	−	0.964711i	$-0.415186\pi$
$53$	3.83384	0.526619	0.263309	+	0.964711i	$0.415186\pi$
$54$	0	0
$55$	4.23970	0.571682
$56$	0	0
$57$	−6.03677	−0.799590
$58$	0	0
$59$	−5.54364	−0.721720	−0.360860	−	0.932620i	$-0.617517\pi$
$59$	−5.54364	−0.721720	−0.360860	+	0.932620i	$0.617517\pi$
$60$	0	0
$61$	14.5804	1.86683	0.933415	−	0.358798i	$-0.116813\pi$
$61$	14.5804	1.86683	0.933415	+	0.358798i	$0.116813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.5804	2.05655
$66$	0	0
$67$	4.03677	0.493170	0.246585	−	0.969121i	$-0.420691\pi$
$67$	4.03677	0.493170	0.246585	+	0.969121i	$0.420691\pi$
$68$	0	0
$69$	7.49314	0.902068
$70$	0	0
$71$	−5.49314	−0.651915	−0.325958	−	0.945384i	$-0.605687\pi$
$71$	−5.49314	−0.651915	−0.325958	+	0.945384i	$0.605687\pi$
$72$	0	0
$73$	12.5436	1.46812	0.734061	−	0.679084i	$-0.237623\pi$
$73$	12.5436	1.46812	0.734061	+	0.679084i	$0.237623\pi$
$74$	0	0
$75$	−2.54364	−0.293714
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.79707	−0.877239	−0.438619	−	0.898673i	$-0.644532\pi$
$79$	−7.79707	−0.877239	−0.438619	+	0.898673i	$0.644532\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.52991	0.716751	0.358375	−	0.933578i	$-0.383331\pi$
$83$	6.52991	0.716751	0.358375	+	0.933578i	$0.383331\pi$
$84$	0	0
$85$	−20.5804	−2.23226
$86$	0	0
$87$	1.25343	0.134382
$88$	0	0
$89$	−9.49314	−1.00627	−0.503135	−	0.864208i	$-0.667820\pi$
$89$	−9.49314	−1.00627	−0.503135	+	0.864208i	$0.667820\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.29021	−0.548569
$94$	0	0
$95$	−16.5804	−1.70111
$96$	0	0
$97$	1.54364	0.156733	0.0783663	−	0.996925i	$-0.475030\pi$
$97$	1.54364	0.156733	0.0783663	+	0.996925i	$0.475030\pi$
$98$	0	0
$99$	−1.54364	−0.155141
$100$	0	0
$101$	2.58041	0.256760	0.128380	−	0.991725i	$-0.459022\pi$
$101$	2.58041	0.256760	0.128380	+	0.991725i	$0.459022\pi$
$102$	0	0
$103$	6.54364	0.644764	0.322382	−	0.946610i	$-0.395516\pi$
$103$	6.54364	0.644764	0.322382	+	0.946610i	$0.395516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.52991	0.437923	0.218961	−	0.975734i	$-0.429733\pi$
$107$	4.52991	0.437923	0.218961	+	0.975734i	$0.429733\pi$
$108$	0	0
$109$	6.44264	0.617093	0.308546	−	0.951209i	$-0.400158\pi$
$109$	6.44264	0.617093	0.308546	+	0.951209i	$0.400158\pi$
$110$	0	0
$111$	4.94950	0.469786
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	20.5804	1.91913
$116$	0	0
$117$	−6.03677	−0.558100
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.61718	−0.783380
$122$	0	0
$123$	5.08727	0.458704
$124$	0	0
$125$	6.74657	0.603431
$126$	0	0
$127$	3.79707	0.336935	0.168468	−	0.985707i	$-0.446118\pi$
$127$	3.79707	0.336935	0.168468	+	0.985707i	$0.446118\pi$
$128$	0	0
$129$	3.45636	0.304316
$130$	0	0
$131$	−1.54364	−0.134868	−0.0674341	−	0.997724i	$-0.521481\pi$
$131$	−1.54364	−0.134868	−0.0674341	+	0.997724i	$0.521481\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.74657	0.236387
$136$	0	0
$137$	1.49314	0.127567	0.0637836	−	0.997964i	$-0.479683\pi$
$137$	1.49314	0.127567	0.0637836	+	0.997964i	$0.479683\pi$
$138$	0	0
$139$	2.94950	0.250173	0.125087	−	0.992146i	$-0.460079\pi$
$139$	2.94950	0.250173	0.125087	+	0.992146i	$0.460079\pi$
$140$	0	0
$141$	−9.49314	−0.799466
$142$	0	0
$143$	9.31859	0.779259
$144$	0	0
$145$	3.44264	0.285895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.49314	0.122323	0.0611613	−	0.998128i	$-0.480520\pi$
$149$	1.49314	0.122323	0.0611613	+	0.998128i	$0.480520\pi$
$150$	0	0
$151$	9.73284	0.792047	0.396024	−	0.918240i	$-0.370390\pi$
$151$	9.73284	0.792047	0.396024	+	0.918240i	$0.370390\pi$
$152$	0	0
$153$	7.49314	0.605784
$154$	0	0
$155$	−14.5299	−1.16707
$156$	0	0
$157$	−9.08727	−0.725243	−0.362622	−	0.931936i	$-0.618118\pi$
$157$	−9.08727	−0.725243	−0.362622	+	0.931936i	$0.618118\pi$
$158$	0	0
$159$	−3.83384	−0.304043
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.5667	−1.53258	−0.766290	−	0.642494i	$-0.777900\pi$
$163$	−19.5667	−1.53258	−0.766290	+	0.642494i	$0.777900\pi$
$164$	0	0
$165$	−4.23970	−0.330061
$166$	0	0
$167$	−11.8990	−0.920772	−0.460386	−	0.887719i	$-0.652289\pi$
$167$	−11.8990	−0.920772	−0.460386	+	0.887719i	$0.652289\pi$
$168$	0	0
$169$	23.4426	1.80328
$170$	0	0
$171$	6.03677	0.461644
$172$	0	0
$173$	−2.58041	−0.196185	−0.0980925	−	0.995177i	$-0.531274\pi$
$173$	−2.58041	−0.196185	−0.0980925	+	0.995177i	$0.531274\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.54364	0.416685
$178$	0	0
$179$	2.98627	0.223205	0.111602	−	0.993753i	$-0.464402\pi$
$179$	2.98627	0.223205	0.111602	+	0.993753i	$0.464402\pi$
$180$	0	0
$181$	10.5436	0.783702	0.391851	−	0.920029i	$-0.371835\pi$
$181$	10.5436	0.783702	0.391851	+	0.920029i	$0.371835\pi$
$182$	0	0
$183$	−14.5804	−1.07781
$184$	0	0
$185$	13.5941	0.999461
$186$	0	0
$187$	−11.5667	−0.845840
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1608	1.24171	0.620857	−	0.783924i	$-0.286785\pi$
$191$	17.1608	1.24171	0.620857	+	0.783924i	$0.286785\pi$
$192$	0	0
$193$	10.8990	0.784527	0.392264	−	0.919853i	$-0.371692\pi$
$193$	10.8990	0.784527	0.392264	+	0.919853i	$0.371692\pi$
$194$	0	0
$195$	−16.5804	−1.18735
$196$	0	0
$197$	−5.41959	−0.386130	−0.193065	−	0.981186i	$-0.561843\pi$
$197$	−5.41959	−0.386130	−0.193065	+	0.981186i	$0.561843\pi$
$198$	0	0
$199$	−22.9863	−1.62945	−0.814727	−	0.579845i	$-0.803113\pi$
$199$	−22.9863	−1.62945	−0.814727	+	0.579845i	$0.803113\pi$
$200$	0	0
$201$	−4.03677	−0.284732
$202$	0	0
$203$	0	0
$204$	0	0
$205$	13.9725	0.975885
$206$	0	0
$207$	−7.49314	−0.520809
$208$	0	0
$209$	−9.31859	−0.644580
$210$	0	0
$211$	−3.08727	−0.212537	−0.106268	−	0.994337i	$-0.533890\pi$
$211$	−3.08727	−0.212537	−0.106268	+	0.994337i	$0.533890\pi$
$212$	0	0
$213$	5.49314	0.376384
$214$	0	0
$215$	9.49314	0.647427
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.5436	−0.847620
$220$	0	0
$221$	−45.2344	−3.04279
$222$	0	0
$223$	−22.7466	−1.52322	−0.761611	−	0.648034i	$-0.775591\pi$
$223$	−22.7466	−1.52322	−0.761611	+	0.648034i	$0.775591\pi$
$224$	0	0
$225$	2.54364	0.169576
$226$	0	0
$227$	−22.6035	−1.50024	−0.750122	−	0.661299i	$-0.770005\pi$
$227$	−22.6035	−1.50024	−0.750122	+	0.661299i	$0.770005\pi$
$228$	0	0
$229$	11.3554	0.750383	0.375192	−	0.926947i	$-0.377577\pi$
$229$	11.3554	0.750383	0.375192	+	0.926947i	$0.377577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.41959	0.224025	0.112012	−	0.993707i	$-0.464270\pi$
$233$	3.41959	0.224025	0.112012	+	0.993707i	$0.464270\pi$
$234$	0	0
$235$	−26.0735	−1.70085
$236$	0	0
$237$	7.79707	0.506474
$238$	0	0
$239$	−7.49314	−0.484691	−0.242345	−	0.970190i	$-0.577917\pi$
$239$	−7.49314	−0.484691	−0.242345	+	0.970190i	$0.577917\pi$
$240$	0	0
$241$	−18.6035	−1.19835	−0.599177	−	0.800617i	$-0.704505\pi$
$241$	−18.6035	−1.19835	−0.599177	+	0.800617i	$0.704505\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−36.4426	−2.31879
$248$	0	0
$249$	−6.52991	−0.413816
$250$	0	0
$251$	−24.5299	−1.54831	−0.774157	−	0.632994i	$-0.781826\pi$
$251$	−24.5299	−1.54831	−0.774157	+	0.632994i	$0.781826\pi$
$252$	0	0
$253$	11.5667	0.727191
$254$	0	0
$255$	20.5804	1.28880
$256$	0	0
$257$	−17.4931	−1.09119	−0.545596	−	0.838048i	$-0.683697\pi$
$257$	−17.4931	−1.09119	−0.545596	+	0.838048i	$0.683697\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.25343	−0.0775855
$262$	0	0
$263$	−13.4931	−0.832022	−0.416011	−	0.909359i	$-0.636572\pi$
$263$	−13.4931	−0.832022	−0.416011	+	0.909359i	$0.636572\pi$
$264$	0	0
$265$	−10.5299	−0.646847
$266$	0	0
$267$	9.49314	0.580971
$268$	0	0
$269$	−6.34071	−0.386600	−0.193300	−	0.981140i	$-0.561919\pi$
$269$	−6.34071	−0.386600	−0.193300	+	0.981140i	$0.561919\pi$
$270$	0	0
$271$	−20.2397	−1.22947	−0.614737	−	0.788732i	$-0.710738\pi$
$271$	−20.2397	−1.22947	−0.614737	+	0.788732i	$0.710738\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.92645	−0.236774
$276$	0	0
$277$	2.54364	0.152832	0.0764162	−	0.997076i	$-0.475652\pi$
$277$	2.54364	0.152832	0.0764162	+	0.997076i	$0.475652\pi$
$278$	0	0
$279$	5.29021	0.316716
$280$	0	0
$281$	26.5804	1.58565	0.792827	−	0.609447i	$-0.208608\pi$
$281$	26.5804	1.58565	0.792827	+	0.609447i	$0.208608\pi$
$282$	0	0
$283$	20.5436	1.22119	0.610596	−	0.791942i	$-0.290930\pi$
$283$	20.5436	1.22119	0.610596	+	0.791942i	$0.290930\pi$
$284$	0	0
$285$	16.5804	0.982139
$286$	0	0
$287$	0	0
$288$	0	0
$289$	39.1471	2.30277
$290$	0	0
$291$	−1.54364	−0.0904896
$292$	0	0
$293$	−5.25343	−0.306909	−0.153454	−	0.988156i	$-0.549040\pi$
$293$	−5.25343	−0.306909	−0.153454	+	0.988156i	$0.549040\pi$
$294$	0	0
$295$	15.2260	0.886491
$296$	0	0
$297$	1.54364	0.0895709
$298$	0	0
$299$	45.2344	2.61597
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.58041	−0.148241
$304$	0	0
$305$	−40.0461	−2.29303
$306$	0	0
$307$	−14.8485	−0.847449	−0.423724	−	0.905791i	$-0.639277\pi$
$307$	−14.8485	−0.847449	−0.423724	+	0.905791i	$0.639277\pi$
$308$	0	0
$309$	−6.54364	−0.372255
$310$	0	0
$311$	−1.89900	−0.107682	−0.0538412	−	0.998550i	$-0.517146\pi$
$311$	−1.89900	−0.107682	−0.0538412	+	0.998550i	$0.517146\pi$
$312$	0	0
$313$	−0.0872743	−0.00493303	−0.00246652	−	0.999997i	$-0.500785\pi$
$313$	−0.0872743	−0.00493303	−0.00246652	+	0.999997i	$0.500785\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.23970	−0.462788	−0.231394	−	0.972860i	$-0.574329\pi$
$317$	−8.23970	−0.462788	−0.231394	+	0.972860i	$0.574329\pi$
$318$	0	0
$319$	1.93484	0.108330
$320$	0	0
$321$	−4.52991	−0.252835
$322$	0	0
$323$	45.2344	2.51691
$324$	0	0
$325$	−15.3554	−0.851762
$326$	0	0
$327$	−6.44264	−0.356279
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.52991	0.303951	0.151976	−	0.988384i	$-0.451436\pi$
$331$	5.52991	0.303951	0.151976	+	0.988384i	$0.451436\pi$
$332$	0	0
$333$	−4.94950	−0.271231
$334$	0	0
$335$	−11.0873	−0.605763
$336$	0	0
$337$	7.17455	0.390823	0.195411	−	0.980721i	$-0.437396\pi$
$337$	7.17455	0.390823	0.195411	+	0.980721i	$0.437396\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−8.16616	−0.442222
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−20.5804	−1.10801
$346$	0	0
$347$	−24.0735	−1.29234	−0.646168	−	0.763195i	$-0.723629\pi$
$347$	−24.0735	−1.29234	−0.646168	+	0.763195i	$0.723629\pi$
$348$	0	0
$349$	31.1608	1.66800	0.834000	−	0.551764i	$-0.186045\pi$
$349$	31.1608	1.66800	0.834000	+	0.551764i	$0.186045\pi$
$350$	0	0
$351$	6.03677	0.322219
$352$	0	0
$353$	−10.1745	−0.541537	−0.270768	−	0.962645i	$-0.587278\pi$
$353$	−10.1745	−0.541537	−0.270768	+	0.962645i	$0.587278\pi$
$354$	0	0
$355$	15.0873	0.800749
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.2481	−1.70199	−0.850995	−	0.525174i	$-0.824000\pi$
$359$	−32.2481	−1.70199	−0.850995	+	0.525174i	$0.824000\pi$
$360$	0	0
$361$	17.4426	0.918033
$362$	0	0
$363$	8.61718	0.452285
$364$	0	0
$365$	−34.4520	−1.80330
$366$	0	0
$367$	3.87062	0.202045	0.101022	−	0.994884i	$-0.467789\pi$
$367$	3.87062	0.202045	0.101022	+	0.994884i	$0.467789\pi$
$368$	0	0
$369$	−5.08727	−0.264833
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.6907	−1.79622	−0.898109	−	0.439773i	$-0.855059\pi$
$373$	−34.6907	−1.79622	−0.898109	+	0.439773i	$0.855059\pi$
$374$	0	0
$375$	−6.74657	−0.348391
$376$	0	0
$377$	7.56668	0.389704
$378$	0	0
$379$	−13.4564	−0.691207	−0.345603	−	0.938381i	$-0.612326\pi$
$379$	−13.4564	−0.691207	−0.345603	+	0.938381i	$0.612326\pi$
$380$	0	0
$381$	−3.79707	−0.194530
$382$	0	0
$383$	4.17455	0.213309	0.106655	−	0.994296i	$-0.465986\pi$
$383$	4.17455	0.213309	0.106655	+	0.994296i	$0.465986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.45636	−0.175697
$388$	0	0
$389$	18.6540	0.945793	0.472897	−	0.881118i	$-0.343208\pi$
$389$	18.6540	0.945793	0.472897	+	0.881118i	$0.343208\pi$
$390$	0	0
$391$	−56.1471	−2.83948
$392$	0	0
$393$	1.54364	0.0778662
$394$	0	0
$395$	21.4152	1.07751
$396$	0	0
$397$	1.45636	0.0730928	0.0365464	−	0.999332i	$-0.488364\pi$
$397$	1.45636	0.0730928	0.0365464	+	0.999332i	$0.488364\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.2344	−1.45989	−0.729947	−	0.683503i	$-0.760455\pi$
$401$	−29.2344	−1.45989	−0.729947	+	0.683503i	$0.760455\pi$
$402$	0	0
$403$	−31.9358	−1.59083
$404$	0	0
$405$	−2.74657	−0.136478
$406$	0	0
$407$	7.64023	0.378712
$408$	0	0
$409$	26.0598	1.28858	0.644288	−	0.764783i	$-0.277154\pi$
$409$	26.0598	1.28858	0.644288	+	0.764783i	$0.277154\pi$
$410$	0	0
$411$	−1.49314	−0.0736510
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−17.9348	−0.880387
$416$	0	0
$417$	−2.94950	−0.144438
$418$	0	0
$419$	−32.8853	−1.60655	−0.803275	−	0.595608i	$-0.796911\pi$
$419$	−32.8853	−1.60655	−0.803275	+	0.595608i	$0.796911\pi$
$420$	0	0
$421$	17.1976	0.838160	0.419080	−	0.907949i	$-0.362353\pi$
$421$	17.1976	0.838160	0.419080	+	0.907949i	$0.362353\pi$
$422$	0	0
$423$	9.49314	0.461572
$424$	0	0
$425$	19.0598	0.924537
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.31859	−0.449906
$430$	0	0
$431$	2.98627	0.143844	0.0719219	−	0.997410i	$-0.477087\pi$
$431$	2.98627	0.143844	0.0719219	+	0.997410i	$0.477087\pi$
$432$	0	0
$433$	18.4426	0.886297	0.443148	−	0.896448i	$-0.353862\pi$
$433$	18.4426	0.886297	0.443148	+	0.896448i	$0.353862\pi$
$434$	0	0
$435$	−3.44264	−0.165062
$436$	0	0
$437$	−45.2344	−2.16385
$438$	0	0
$439$	0.572020	0.0273010	0.0136505	−	0.999907i	$-0.495655\pi$
$439$	0.572020	0.0273010	0.0136505	+	0.999907i	$0.495655\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.5299	−1.73559	−0.867794	−	0.496924i	$-0.834463\pi$
$443$	−36.5299	−1.73559	−0.867794	+	0.496924i	$0.834463\pi$
$444$	0	0
$445$	26.0735	1.23600
$446$	0	0
$447$	−1.49314	−0.0706229
$448$	0	0
$449$	−22.5530	−1.06434	−0.532170	−	0.846638i	$-0.678623\pi$
$449$	−22.5530	−1.06434	−0.532170	+	0.846638i	$0.678623\pi$
$450$	0	0
$451$	7.85291	0.369779
$452$	0	0
$453$	−9.73284	−0.457289
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.2344	0.946524	0.473262	−	0.880922i	$-0.343076\pi$
$457$	20.2344	0.946524	0.473262	+	0.880922i	$0.343076\pi$
$458$	0	0
$459$	−7.49314	−0.349750
$460$	0	0
$461$	−25.4931	−1.18733	−0.593667	−	0.804711i	$-0.702320\pi$
$461$	−25.4931	−1.18733	−0.593667	+	0.804711i	$0.702320\pi$
$462$	0	0
$463$	−7.70446	−0.358057	−0.179028	−	0.983844i	$-0.557295\pi$
$463$	−7.70446	−0.358057	−0.179028	+	0.983844i	$0.557295\pi$
$464$	0	0
$465$	14.5299	0.673808
$466$	0	0
$467$	−36.1471	−1.67269	−0.836344	−	0.548205i	$-0.815311\pi$
$467$	−36.1471	−1.67269	−0.836344	+	0.548205i	$0.815311\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.08727	0.418719
$472$	0	0
$473$	5.33537	0.245321
$474$	0	0
$475$	15.3554	0.704552
$476$	0	0
$477$	3.83384	0.175540
$478$	0	0
$479$	26.5804	1.21449	0.607245	−	0.794515i	$-0.292275\pi$
$479$	26.5804	1.21449	0.607245	+	0.794515i	$0.292275\pi$
$480$	0	0
$481$	29.8790	1.36237
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.23970	−0.192515
$486$	0	0
$487$	−40.4236	−1.83177	−0.915883	−	0.401444i	$-0.868508\pi$
$487$	−40.4236	−1.83177	−0.915883	+	0.401444i	$0.868508\pi$
$488$	0	0
$489$	19.5667	0.884836
$490$	0	0
$491$	−14.6309	−0.660284	−0.330142	−	0.943931i	$-0.607097\pi$
$491$	−14.6309	−0.660284	−0.330142	+	0.943931i	$0.607097\pi$
$492$	0	0
$493$	−9.39214	−0.423000
$494$	0	0
$495$	4.23970	0.190561
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.1103	1.61652	0.808260	−	0.588826i	$-0.200410\pi$
$499$	36.1103	1.61652	0.808260	+	0.588826i	$0.200410\pi$
$500$	0	0
$501$	11.8990	0.531608
$502$	0	0
$503$	19.4931	0.869156	0.434578	−	0.900634i	$-0.356898\pi$
$503$	19.4931	0.869156	0.434578	+	0.900634i	$0.356898\pi$
$504$	0	0
$505$	−7.08727	−0.315380
$506$	0	0
$507$	−23.4426	−1.04112
$508$	0	0
$509$	23.9074	1.05968	0.529838	−	0.848099i	$-0.322253\pi$
$509$	23.9074	1.05968	0.529838	+	0.848099i	$0.322253\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.03677	−0.266530
$514$	0	0
$515$	−17.9725	−0.791965
$516$	0	0
$517$	−14.6540	−0.644480
$518$	0	0
$519$	2.58041	0.113267
$520$	0	0
$521$	−7.08727	−0.310499	−0.155250	−	0.987875i	$-0.549618\pi$
$521$	−7.08727	−0.310499	−0.155250	+	0.987875i	$0.549618\pi$
$522$	0	0
$523$	28.7475	1.25704	0.628520	−	0.777793i	$-0.283661\pi$
$523$	28.7475	1.25704	0.628520	+	0.777793i	$0.283661\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	39.6402	1.72676
$528$	0	0
$529$	33.1471	1.44118
$530$	0	0
$531$	−5.54364	−0.240573
$532$	0	0
$533$	30.7107	1.33023
$534$	0	0
$535$	−12.4417	−0.537902
$536$	0	0
$537$	−2.98627	−0.128867
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.02305	0.301944	0.150972	−	0.988538i	$-0.451760\pi$
$541$	7.02305	0.301944	0.150972	+	0.988538i	$0.451760\pi$
$542$	0	0
$543$	−10.5436	−0.452471
$544$	0	0
$545$	−17.6951	−0.757976
$546$	0	0
$547$	−5.59414	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$547$	−5.59414	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$548$	0	0
$549$	14.5804	0.622277
$550$	0	0
$551$	−7.56668	−0.322352
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13.5941	−0.577039
$556$	0	0
$557$	−28.6456	−1.21375	−0.606876	−	0.794797i	$-0.707577\pi$
$557$	−28.6456	−1.21375	−0.606876	+	0.794797i	$0.707577\pi$
$558$	0	0
$559$	20.8653	0.882507
$560$	0	0
$561$	11.5667	0.488346
$562$	0	0
$563$	19.8927	0.838379	0.419189	−	0.907899i	$-0.362314\pi$
$563$	19.8927	0.838379	0.419189	+	0.907899i	$0.362314\pi$
$564$	0	0
$565$	−21.9725	−0.924392
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.6402	1.49412	0.747058	−	0.664759i	$-0.231466\pi$
$569$	35.6402	1.49412	0.747058	+	0.664759i	$0.231466\pi$
$570$	0	0
$571$	27.6770	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$571$	27.6770	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$572$	0	0
$573$	−17.1608	−0.716904
$574$	0	0
$575$	−19.0598	−0.794849
$576$	0	0
$577$	−6.01373	−0.250355	−0.125177	−	0.992134i	$-0.539950\pi$
$577$	−6.01373	−0.250355	−0.125177	+	0.992134i	$0.539950\pi$
$578$	0	0
$579$	−10.8990	−0.452947
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.91806	−0.245101
$584$	0	0
$585$	16.5804	0.685516
$586$	0	0
$587$	−15.5436	−0.641555	−0.320777	−	0.947155i	$-0.603944\pi$
$587$	−15.5436	−0.641555	−0.320777	+	0.947155i	$0.603944\pi$
$588$	0	0
$589$	31.9358	1.31589
$590$	0	0
$591$	5.41959	0.222932
$592$	0	0
$593$	−48.6540	−1.99798	−0.998989	−	0.0449488i	$-0.985688\pi$
$593$	−48.6540	−1.99798	−0.998989	+	0.0449488i	$0.985688\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.9863	0.940766
$598$	0	0
$599$	−20.9127	−0.854471	−0.427235	−	0.904140i	$-0.640512\pi$
$599$	−20.9127	−0.854471	−0.427235	+	0.904140i	$0.640512\pi$
$600$	0	0
$601$	−7.98627	−0.325767	−0.162883	−	0.986645i	$-0.552079\pi$
$601$	−7.98627	−0.325767	−0.162883	+	0.986645i	$0.552079\pi$
$602$	0	0
$603$	4.03677	0.164390
$604$	0	0
$605$	23.6677	0.962228
$606$	0	0
$607$	2.70979	0.109987	0.0549936	−	0.998487i	$-0.482486\pi$
$607$	2.70979	0.109987	0.0549936	+	0.998487i	$0.482486\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−57.3079	−2.31843
$612$	0	0
$613$	16.5069	0.666706	0.333353	−	0.942802i	$-0.391820\pi$
$613$	16.5069	0.666706	0.333353	+	0.942802i	$0.391820\pi$
$614$	0	0
$615$	−13.9725	−0.563427
$616$	0	0
$617$	−23.1608	−0.932420	−0.466210	−	0.884674i	$-0.654381\pi$
$617$	−23.1608	−0.932420	−0.466210	+	0.884674i	$0.654381\pi$
$618$	0	0
$619$	29.4289	1.18285	0.591424	−	0.806361i	$-0.298566\pi$
$619$	29.4289	1.18285	0.591424	+	0.806361i	$0.298566\pi$
$620$	0	0
$621$	7.49314	0.300689
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.2481	−1.24992
$626$	0	0
$627$	9.31859	0.372149
$628$	0	0
$629$	−37.0873	−1.47877
$630$	0	0
$631$	−35.7054	−1.42141	−0.710705	−	0.703491i	$-0.751624\pi$
$631$	−35.7054	−1.42141	−0.710705	+	0.703491i	$0.751624\pi$
$632$	0	0
$633$	3.08727	0.122708
$634$	0	0
$635$	−10.4289	−0.413859
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.49314	−0.217305
$640$	0	0
$641$	−3.36282	−0.132824	−0.0664118	−	0.997792i	$-0.521155\pi$
$641$	−3.36282	−0.132824	−0.0664118	+	0.997792i	$0.521155\pi$
$642$	0	0
$643$	−10.2113	−0.402695	−0.201348	−	0.979520i	$-0.564532\pi$
$643$	−10.2113	−0.402695	−0.201348	+	0.979520i	$0.564532\pi$
$644$	0	0
$645$	−9.49314	−0.373792
$646$	0	0
$647$	−20.5530	−0.808020	−0.404010	−	0.914755i	$-0.632384\pi$
$647$	−20.5530	−0.808020	−0.404010	+	0.914755i	$0.632384\pi$
$648$	0	0
$649$	8.55736	0.335906
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.3544	0.444333	0.222167	−	0.975009i	$-0.428687\pi$
$653$	11.3544	0.444333	0.222167	+	0.975009i	$0.428687\pi$
$654$	0	0
$655$	4.23970	0.165659
$656$	0	0
$657$	12.5436	0.489374
$658$	0	0
$659$	−21.2344	−0.827174	−0.413587	−	0.910465i	$-0.635724\pi$
$659$	−21.2344	−0.827174	−0.413587	+	0.910465i	$0.635724\pi$
$660$	0	0
$661$	28.9220	1.12494	0.562469	−	0.826819i	$-0.309852\pi$
$661$	28.9220	1.12494	0.562469	+	0.826819i	$0.309852\pi$
$662$	0	0
$663$	45.2344	1.75676
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.39214	0.363665
$668$	0	0
$669$	22.7466	0.879433
$670$	0	0
$671$	−22.5069	−0.868868
$672$	0	0
$673$	−34.2618	−1.32070	−0.660348	−	0.750960i	$-0.729591\pi$
$673$	−34.2618	−1.32070	−0.660348	+	0.750960i	$0.729591\pi$
$674$	0	0
$675$	−2.54364	−0.0979046
$676$	0	0
$677$	−34.7466	−1.33542	−0.667710	−	0.744422i	$-0.732725\pi$
$677$	−34.7466	−1.33542	−0.667710	+	0.744422i	$0.732725\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	22.6035	0.866166
$682$	0	0
$683$	−0.281814	−0.0107833	−0.00539166	−	0.999985i	$-0.501716\pi$
$683$	−0.281814	−0.0107833	−0.00539166	+	0.999985i	$0.501716\pi$
$684$	0	0
$685$	−4.10100	−0.156691
$686$	0	0
$687$	−11.3554	−0.433234
$688$	0	0
$689$	−23.1440	−0.881718
$690$	0	0
$691$	−34.4152	−1.30922	−0.654608	−	0.755969i	$-0.727166\pi$
$691$	−34.4152	−1.30922	−0.654608	+	0.755969i	$0.727166\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.10100	−0.307288
$696$	0	0
$697$	−38.1196	−1.44388
$698$	0	0
$699$	−3.41959	−0.129341
$700$	0	0
$701$	0.498472	0.0188270	0.00941352	−	0.999956i	$-0.497004\pi$
$701$	0.498472	0.0188270	0.00941352	+	0.999956i	$0.497004\pi$
$702$	0	0
$703$	−29.8790	−1.12691
$704$	0	0
$705$	26.0735	0.981987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.17455	0.156778	0.0783892	−	0.996923i	$-0.475022\pi$
$709$	4.17455	0.156778	0.0783892	+	0.996923i	$0.475022\pi$
$710$	0	0
$711$	−7.79707	−0.292413
$712$	0	0
$713$	−39.6402	−1.48454
$714$	0	0
$715$	−25.5941	−0.957166
$716$	0	0
$717$	7.49314	0.279836
$718$	0	0
$719$	−0.986273	−0.0367818	−0.0183909	−	0.999831i	$-0.505854\pi$
$719$	−0.986273	−0.0367818	−0.0183909	+	0.999831i	$0.505854\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.6035	0.691870
$724$	0	0
$725$	−3.18828	−0.118410
$726$	0	0
$727$	−34.9304	−1.29550	−0.647749	−	0.761854i	$-0.724289\pi$
$727$	−34.9304	−1.29550	−0.647749	+	0.761854i	$0.724289\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.8990	−0.957909
$732$	0	0
$733$	−5.12405	−0.189261	−0.0946305	−	0.995512i	$-0.530167\pi$
$733$	−5.12405	−0.189261	−0.0946305	+	0.995512i	$0.530167\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.23131	−0.229533
$738$	0	0
$739$	16.9220	0.622488	0.311244	−	0.950330i	$-0.399254\pi$
$739$	16.9220	0.622488	0.311244	+	0.950330i	$0.399254\pi$
$740$	0	0
$741$	36.4426	1.33875
$742$	0	0
$743$	6.81172	0.249898	0.124949	−	0.992163i	$-0.460123\pi$
$743$	6.81172	0.249898	0.124949	+	0.992163i	$0.460123\pi$
$744$	0	0
$745$	−4.10100	−0.150249
$746$	0	0
$747$	6.52991	0.238917
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.1755	−0.955157	−0.477578	−	0.878589i	$-0.658485\pi$
$751$	−26.1755	−0.955157	−0.477578	+	0.878589i	$0.658485\pi$
$752$	0	0
$753$	24.5299	0.893920
$754$	0	0
$755$	−26.7319	−0.972874
$756$	0	0
$757$	−49.7138	−1.80688	−0.903439	−	0.428717i	$-0.858966\pi$
$757$	−49.7138	−1.80688	−0.903439	+	0.428717i	$0.858966\pi$
$758$	0	0
$759$	−11.5667	−0.419844
$760$	0	0
$761$	−12.6540	−0.458706	−0.229353	−	0.973343i	$-0.573661\pi$
$761$	−12.6540	−0.458706	−0.229353	+	0.973343i	$0.573661\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−20.5804	−0.744086
$766$	0	0
$767$	33.4657	1.20838
$768$	0	0
$769$	−20.2344	−0.729670	−0.364835	−	0.931072i	$-0.618875\pi$
$769$	−20.2344	−0.729670	−0.364835	+	0.931072i	$0.618875\pi$
$770$	0	0
$771$	17.4931	0.630000
$772$	0	0
$773$	19.7412	0.710043	0.355021	−	0.934858i	$-0.384474\pi$
$773$	19.7412	0.710043	0.355021	+	0.934858i	$0.384474\pi$
$774$	0	0
$775$	13.4564	0.483367
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.7107	−1.10033
$780$	0	0
$781$	8.47941	0.303417
$782$	0	0
$783$	1.25343	0.0447940
$784$	0	0
$785$	24.9588	0.890818
$786$	0	0
$787$	16.9127	0.602874	0.301437	−	0.953486i	$-0.402534\pi$
$787$	16.9127	0.602874	0.301437	+	0.953486i	$0.402534\pi$
$788$	0	0
$789$	13.4931	0.480368
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−88.0186	−3.12563
$794$	0	0
$795$	10.5299	0.373457
$796$	0	0
$797$	−3.48475	−0.123436	−0.0617180	−	0.998094i	$-0.519658\pi$
$797$	−3.48475	−0.123436	−0.0617180	+	0.998094i	$0.519658\pi$
$798$	0	0
$799$	71.1334	2.51652
$800$	0	0
$801$	−9.49314	−0.335423
$802$	0	0
$803$	−19.3628	−0.683299
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.34071	0.223203
$808$	0	0
$809$	−19.1883	−0.674624	−0.337312	−	0.941393i	$-0.609518\pi$
$809$	−19.1883	−0.674624	−0.337312	+	0.941393i	$0.609518\pi$
$810$	0	0
$811$	48.8285	1.71460	0.857300	−	0.514817i	$-0.172140\pi$
$811$	48.8285	1.71460	0.857300	+	0.514817i	$0.172140\pi$
$812$	0	0
$813$	20.2397	0.709837
$814$	0	0
$815$	53.7412	1.88247
$816$	0	0
$817$	−20.8653	−0.729984
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.5446	0.926412	0.463206	−	0.886251i	$-0.346699\pi$
$821$	26.5446	0.926412	0.463206	+	0.886251i	$0.346699\pi$
$822$	0	0
$823$	16.9588	0.591147	0.295574	−	0.955320i	$-0.404489\pi$
$823$	16.9588	0.591147	0.295574	+	0.955320i	$0.404489\pi$
$824$	0	0
$825$	3.92645	0.136702
$826$	0	0
$827$	−42.4289	−1.47540	−0.737699	−	0.675130i	$-0.764088\pi$
$827$	−42.4289	−1.47540	−0.737699	+	0.675130i	$0.764088\pi$
$828$	0	0
$829$	31.9358	1.10918	0.554588	−	0.832125i	$-0.312876\pi$
$829$	31.9358	1.10918	0.554588	+	0.832125i	$0.312876\pi$
$830$	0	0
$831$	−2.54364	−0.0882378
$832$	0	0
$833$	0	0
$834$	0	0
$835$	32.6814	1.13099
$836$	0	0
$837$	−5.29021	−0.182856
$838$	0	0
$839$	49.5392	1.71028	0.855142	−	0.518394i	$-0.173470\pi$
$839$	49.5392	1.71028	0.855142	+	0.518394i	$0.173470\pi$
$840$	0	0
$841$	−27.4289	−0.945824
$842$	0	0
$843$	−26.5804	−0.915478
$844$	0	0
$845$	−64.3868	−2.21497
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.5436	−0.705056
$850$	0	0
$851$	37.0873	1.27134
$852$	0	0
$853$	−11.4564	−0.392258	−0.196129	−	0.980578i	$-0.562837\pi$
$853$	−11.4564	−0.392258	−0.196129	+	0.980578i	$0.562837\pi$
$854$	0	0
$855$	−16.5804	−0.567038
$856$	0	0
$857$	−32.2481	−1.10157	−0.550787	−	0.834646i	$-0.685672\pi$
$857$	−32.2481	−1.10157	−0.550787	+	0.834646i	$0.685672\pi$
$858$	0	0
$859$	3.08727	0.105336	0.0526682	−	0.998612i	$-0.483227\pi$
$859$	3.08727	0.105336	0.0526682	+	0.998612i	$0.483227\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.1471	−0.617734	−0.308867	−	0.951105i	$-0.599950\pi$
$863$	−18.1471	−0.617734	−0.308867	+	0.951105i	$0.599950\pi$
$864$	0	0
$865$	7.08727	0.240975
$866$	0	0
$867$	−39.1471	−1.32951
$868$	0	0
$869$	12.0358	0.408288
$870$	0	0
$871$	−24.3691	−0.825715
$872$	0	0
$873$	1.54364	0.0522442
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.88527	−0.300034	−0.150017	−	0.988683i	$-0.547933\pi$
$877$	−8.88527	−0.300034	−0.150017	+	0.988683i	$0.547933\pi$
$878$	0	0
$879$	5.25343	0.177194
$880$	0	0
$881$	−41.5667	−1.40042	−0.700209	−	0.713938i	$-0.746910\pi$
$881$	−41.5667	−1.40042	−0.700209	+	0.713938i	$0.746910\pi$
$882$	0	0
$883$	28.7182	0.966444	0.483222	−	0.875498i	$-0.339466\pi$
$883$	28.7182	0.966444	0.483222	+	0.875498i	$0.339466\pi$
$884$	0	0
$885$	−15.2260	−0.511816
$886$	0	0
$887$	−41.8148	−1.40400	−0.702001	−	0.712176i	$-0.747710\pi$
$887$	−41.8148	−1.40400	−0.702001	+	0.712176i	$0.747710\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.54364	−0.0517138
$892$	0	0
$893$	57.3079	1.91774
$894$	0	0
$895$	−8.20200	−0.274163
$896$	0	0
$897$	−45.2344	−1.51033
$898$	0	0
$899$	−6.63091	−0.221153
$900$	0	0
$901$	28.7275	0.957052
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−28.9588	−0.962624
$906$	0	0
$907$	−57.1701	−1.89830	−0.949152	−	0.314819i	$-0.898056\pi$
$907$	−57.1701	−1.89830	−0.949152	+	0.314819i	$0.898056\pi$
$908$	0	0
$909$	2.58041	0.0855868
$910$	0	0
$911$	24.4059	0.808602	0.404301	−	0.914626i	$-0.367515\pi$
$911$	24.4059	0.808602	0.404301	+	0.914626i	$0.367515\pi$
$912$	0	0
$913$	−10.0798	−0.333593
$914$	0	0
$915$	40.0461	1.32388
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47.7045	−1.57362	−0.786812	−	0.617192i	$-0.788270\pi$
$919$	−47.7045	−1.57362	−0.786812	+	0.617192i	$0.788270\pi$
$920$	0	0
$921$	14.8485	0.489275
$922$	0	0
$923$	33.1608	1.09150
$924$	0	0
$925$	−12.5897	−0.413948
$926$	0	0
$927$	6.54364	0.214921
$928$	0	0
$929$	39.2344	1.28724	0.643619	−	0.765346i	$-0.277432\pi$
$929$	39.2344	1.28724	0.643619	+	0.765346i	$0.277432\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.89900	0.0621704
$934$	0	0
$935$	31.7687	1.03895
$936$	0	0
$937$	15.3491	0.501433	0.250717	−	0.968061i	$-0.419334\pi$
$937$	15.3491	0.501433	0.250717	+	0.968061i	$0.419334\pi$
$938$	0	0
$939$	0.0872743	0.00284809
$940$	0	0
$941$	12.1662	0.396605	0.198303	−	0.980141i	$-0.436457\pi$
$941$	12.1662	0.396605	0.198303	+	0.980141i	$0.436457\pi$
$942$	0	0
$943$	38.1196	1.24135
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.0598	−0.619361	−0.309680	−	0.950841i	$-0.600222\pi$
$947$	−19.0598	−0.619361	−0.309680	+	0.950841i	$0.600222\pi$
$948$	0	0
$949$	−75.7231	−2.45808
$950$	0	0
$951$	8.23970	0.267191
$952$	0	0
$953$	17.1334	0.555004	0.277502	−	0.960725i	$-0.410493\pi$
$953$	17.1334	0.555004	0.277502	+	0.960725i	$0.410493\pi$
$954$	0	0
$955$	−47.1334	−1.52520
$956$	0	0
$957$	−1.93484	−0.0625446
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.01373	−0.0972170
$962$	0	0
$963$	4.52991	0.145974
$964$	0	0
$965$	−29.9348	−0.963637
$966$	0	0
$967$	−26.7833	−0.861294	−0.430647	−	0.902520i	$-0.641715\pi$
$967$	−26.7833	−0.861294	−0.430647	+	0.902520i	$0.641715\pi$
$968$	0	0
$969$	−45.2344	−1.45314
$970$	0	0
$971$	11.5162	0.369572	0.184786	−	0.982779i	$-0.440841\pi$
$971$	11.5162	0.369572	0.184786	+	0.982779i	$0.440841\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	15.3554	0.491765
$976$	0	0
$977$	21.3354	0.682579	0.341289	−	0.939958i	$-0.389136\pi$
$977$	21.3354	0.682579	0.341289	+	0.939958i	$0.389136\pi$
$978$	0	0
$979$	14.6540	0.468343
$980$	0	0
$981$	6.44264	0.205698
$982$	0	0
$983$	50.3952	1.60736	0.803678	−	0.595064i	$-0.202873\pi$
$983$	50.3952	1.60736	0.803678	+	0.595064i	$0.202873\pi$
$984$	0	0
$985$	14.8853	0.474284
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.8990	0.823540
$990$	0	0
$991$	−12.6823	−0.402868	−0.201434	−	0.979502i	$-0.564560\pi$
$991$	−12.6823	−0.402868	−0.201434	+	0.979502i	$0.564560\pi$
$992$	0	0
$993$	−5.52991	−0.175486
$994$	0	0
$995$	63.1334	2.00146
$996$	0	0
$997$	40.3416	1.27763	0.638816	−	0.769359i	$-0.279424\pi$
$997$	40.3416	1.27763	0.638816	+	0.769359i	$0.279424\pi$
$998$	0	0
$999$	4.94950	0.156595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.eh.1.1		3
4.3	odd	2		9408.2.a.ej.1.1		3
7.2	even	3		1344.2.q.z.193.3		6
7.4	even	3		1344.2.q.z.961.3		6
7.6	odd	2		9408.2.a.ei.1.3		3
8.3	odd	2		4704.2.a.bs.1.3		3
8.5	even	2		4704.2.a.bu.1.3		3
28.11	odd	6		1344.2.q.y.961.3		6
28.23	odd	6		1344.2.q.y.193.3		6
28.27	even	2		9408.2.a.eg.1.3		3
56.11	odd	6		672.2.q.l.289.1	yes	6
56.13	odd	2		4704.2.a.bt.1.1		3
56.27	even	2		4704.2.a.bv.1.1		3
56.37	even	6		672.2.q.k.193.1	✓	6
56.51	odd	6		672.2.q.l.193.1	yes	6
56.53	even	6		672.2.q.k.289.1	yes	6
168.11	even	6		2016.2.s.v.289.3		6
168.53	odd	6		2016.2.s.u.289.3		6
168.107	even	6		2016.2.s.v.865.3		6
168.149	odd	6		2016.2.s.u.865.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.k.193.1	✓	6	56.37	even	6
672.2.q.k.289.1	yes	6	56.53	even	6
672.2.q.l.193.1	yes	6	56.51	odd	6
672.2.q.l.289.1	yes	6	56.11	odd	6
1344.2.q.y.193.3		6	28.23	odd	6
1344.2.q.y.961.3		6	28.11	odd	6
1344.2.q.z.193.3		6	7.2	even	3
1344.2.q.z.961.3		6	7.4	even	3
2016.2.s.u.289.3		6	168.53	odd	6
2016.2.s.u.865.3		6	168.149	odd	6
2016.2.s.v.289.3		6	168.11	even	6
2016.2.s.v.865.3		6	168.107	even	6
4704.2.a.bs.1.3		3	8.3	odd	2
4704.2.a.bt.1.1		3	56.13	odd	2
4704.2.a.bu.1.3		3	8.5	even	2
4704.2.a.bv.1.1		3	56.27	even	2
9408.2.a.eg.1.3		3	28.27	even	2
9408.2.a.eh.1.1		3	1.1	even	1	trivial
9408.2.a.ei.1.3		3	7.6	odd	2
9408.2.a.ej.1.1		3	4.3	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:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.74657 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.74657 q^{5} +1.00000 q^{9} -1.54364 q^{11} -6.03677 q^{13} +2.74657 q^{15} +7.49314 q^{17} +6.03677 q^{19} -7.49314 q^{23} +2.54364 q^{25} -1.00000 q^{27} -1.25343 q^{29} +5.29021 q^{31} +1.54364 q^{33} -4.94950 q^{37} +6.03677 q^{39} -5.08727 q^{41} -3.45636 q^{43} -2.74657 q^{45} +9.49314 q^{47} -7.49314 q^{51} +3.83384 q^{53} +4.23970 q^{55} -6.03677 q^{57} -5.54364 q^{59} +14.5804 q^{61} +16.5804 q^{65} +4.03677 q^{67} +7.49314 q^{69} -5.49314 q^{71} +12.5436 q^{73} -2.54364 q^{75} -7.79707 q^{79} +1.00000 q^{81} +6.52991 q^{83} -20.5804 q^{85} +1.25343 q^{87} -9.49314 q^{89} -5.29021 q^{93} -16.5804 q^{95} +1.54364 q^{97} -1.54364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} - 3 q^{37} - 3 q^{39} - 6 q^{41} - 15 q^{43} + 12 q^{47} - 6 q^{51} - 6 q^{53} - 12 q^{55} + 3 q^{57} - 12 q^{59} + 18 q^{61} + 24 q^{65} - 9 q^{67} + 6 q^{69} + 33 q^{73} - 3 q^{75} - 27 q^{79} + 3 q^{81} - 18 q^{83} - 36 q^{85} + 12 q^{87} - 12 q^{89} - 3 q^{93} - 24 q^{95}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 - 3 * q^37 - 3 * q^39 - 6 * q^41 - 15 * q^43 + 12 * q^47 - 6 * q^51 - 6 * q^53 - 12 * q^55 + 3 * q^57 - 12 * q^59 + 18 * q^61 + 24 * q^65 - 9 * q^67 + 6 * q^69 + 33 * q^73 - 3 * q^75 - 27 * q^79 + 3 * q^81 - 18 * q^83 - 36 * q^85 + 12 * q^87 - 12 * q^89 - 3 * q^93 - 24 * q^95

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