LMFDB - Embedded newform 9648.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9648,2,Mod(1,9648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9648, 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("9648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9648 = 2^{4} \cdot 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9648.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:	$77.0396678701$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 201)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	9648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009 q^{5} -2.70928 q^{7} +O(q^{10})$	$q-2.17009 q^{5} -2.70928 q^{7} +4.63090 q^{11} -1.36910 q^{13} +6.04945 q^{17} +0.447480 q^{19} -5.58864 q^{23} -0.290725 q^{25} -8.68035 q^{29} +4.75872 q^{31} +5.87936 q^{35} -7.97107 q^{37} +1.72261 q^{41} +1.00000 q^{43} +13.1278 q^{47} +0.340173 q^{49} +4.03612 q^{53} -10.0494 q^{55} +7.69594 q^{59} -10.3896 q^{61} +2.97107 q^{65} -1.00000 q^{67} +10.9711 q^{71} -3.73820 q^{73} -12.5464 q^{77} -5.65983 q^{79} +2.95774 q^{83} -13.1278 q^{85} -0.814315 q^{89} +3.70928 q^{91} -0.971071 q^{95} +13.9649 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5} - q^{7}+O(q^{10}) $ 3 * q - q^5 - q^7	$ 3 q - q^{5} - q^{7} + 10 q^{11} - 8 q^{13} + 2 q^{19} + 3 q^{23} - 8 q^{25} - 4 q^{29} - 11 q^{31} + 5 q^{35} - 9 q^{37} - q^{41} + 3 q^{43} + 18 q^{47} - 10 q^{49} - 7 q^{53} - 12 q^{55} + 15 q^{59} - 2 q^{61} - 6 q^{65} - 3 q^{67} + 18 q^{71} - 19 q^{73} - 2 q^{77} - 28 q^{79} - 7 q^{83} - 18 q^{85} + 6 q^{89} + 4 q^{91} + 12 q^{95} - 8 q^{97}+O(q^{100}) $ 3 * q - q^5 - q^7 + 10 * q^11 - 8 * q^13 + 2 * q^19 + 3 * q^23 - 8 * q^25 - 4 * q^29 - 11 * q^31 + 5 * q^35 - 9 * q^37 - q^41 + 3 * q^43 + 18 * q^47 - 10 * q^49 - 7 * q^53 - 12 * q^55 + 15 * q^59 - 2 * q^61 - 6 * q^65 - 3 * q^67 + 18 * q^71 - 19 * q^73 - 2 * q^77 - 28 * q^79 - 7 * q^83 - 18 * q^85 + 6 * q^89 + 4 * q^91 + 12 * q^95 - 8 * 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
$5$	−2.17009	−0.970492	−0.485246	−	0.874378i	$-0.661270\pi$
$5$	−2.17009	−0.970492	−0.485246	+	0.874378i	$0.661270\pi$
$6$	0	0
$7$	−2.70928	−1.02401	−0.512005	−	0.858983i	$-0.671097\pi$
$7$	−2.70928	−1.02401	−0.512005	+	0.858983i	$0.671097\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.63090	1.39627	0.698134	−	0.715967i	$-0.254014\pi$
$11$	4.63090	1.39627	0.698134	+	0.715967i	$0.254014\pi$
$12$	0	0
$13$	−1.36910	−0.379721	−0.189860	−	0.981811i	$-0.560804\pi$
$13$	−1.36910	−0.379721	−0.189860	+	0.981811i	$0.560804\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.04945	1.46721	0.733603	−	0.679578i	$-0.237837\pi$
$17$	6.04945	1.46721	0.733603	+	0.679578i	$0.237837\pi$
$18$	0	0
$19$	0.447480	0.102659	0.0513295	−	0.998682i	$-0.483654\pi$
$19$	0.447480	0.102659	0.0513295	+	0.998682i	$0.483654\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.58864	−1.16531	−0.582656	−	0.812719i	$-0.697986\pi$
$23$	−5.58864	−1.16531	−0.582656	+	0.812719i	$0.697986\pi$
$24$	0	0
$25$	−0.290725	−0.0581449
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.68035	−1.61190	−0.805950	−	0.591984i	$-0.798345\pi$
$29$	−8.68035	−1.61190	−0.805950	+	0.591984i	$0.798345\pi$
$30$	0	0
$31$	4.75872	0.854692	0.427346	−	0.904088i	$-0.359449\pi$
$31$	4.75872	0.854692	0.427346	+	0.904088i	$0.359449\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.87936	0.993794
$36$	0	0
$37$	−7.97107	−1.31044	−0.655218	−	0.755440i	$-0.727423\pi$
$37$	−7.97107	−1.31044	−0.655218	+	0.755440i	$0.727423\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.72261	0.269026	0.134513	−	0.990912i	$-0.457053\pi$
$41$	1.72261	0.269026	0.134513	+	0.990912i	$0.457053\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.1278	1.91489	0.957445	−	0.288615i	$-0.0931949\pi$
$47$	13.1278	1.91489	0.957445	+	0.288615i	$0.0931949\pi$
$48$	0	0
$49$	0.340173	0.0485961
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.03612	0.554403	0.277202	−	0.960812i	$-0.410593\pi$
$53$	4.03612	0.554403	0.277202	+	0.960812i	$0.410593\pi$
$54$	0	0
$55$	−10.0494	−1.35507
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.69594	1.00193	0.500963	−	0.865469i	$-0.332979\pi$
$59$	7.69594	1.00193	0.500963	+	0.865469i	$0.332979\pi$
$60$	0	0
$61$	−10.3896	−1.33025	−0.665127	−	0.746730i	$-0.731623\pi$
$61$	−10.3896	−1.33025	−0.665127	+	0.746730i	$0.731623\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.97107	0.368516
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.9711	1.30203	0.651013	−	0.759066i	$-0.274344\pi$
$71$	10.9711	1.30203	0.651013	+	0.759066i	$0.274344\pi$
$72$	0	0
$73$	−3.73820	−0.437524	−0.218762	−	0.975778i	$-0.570202\pi$
$73$	−3.73820	−0.437524	−0.218762	+	0.975778i	$0.570202\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.5464	−1.42979
$78$	0	0
$79$	−5.65983	−0.636780	−0.318390	−	0.947960i	$-0.603142\pi$
$79$	−5.65983	−0.636780	−0.318390	+	0.947960i	$0.603142\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.95774	0.324654	0.162327	−	0.986737i	$-0.448100\pi$
$83$	2.95774	0.324654	0.162327	+	0.986737i	$0.448100\pi$
$84$	0	0
$85$	−13.1278	−1.42391
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.814315	−0.0863172	−0.0431586	−	0.999068i	$-0.513742\pi$
$89$	−0.814315	−0.0863172	−0.0431586	+	0.999068i	$0.513742\pi$
$90$	0	0
$91$	3.70928	0.388838
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.971071	−0.0996297
$96$	0	0
$97$	13.9649	1.41792	0.708962	−	0.705247i	$-0.249164\pi$
$97$	13.9649	1.41792	0.708962	+	0.705247i	$0.249164\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.5669	1.74797	0.873986	−	0.485952i	$-0.161527\pi$
$101$	17.5669	1.74797	0.873986	+	0.485952i	$0.161527\pi$
$102$	0	0
$103$	10.5464	1.03917	0.519583	−	0.854420i	$-0.326087\pi$
$103$	10.5464	1.03917	0.519583	+	0.854420i	$0.326087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.630898	0.0609912	0.0304956	−	0.999535i	$-0.490291\pi$
$107$	0.630898	0.0609912	0.0304956	+	0.999535i	$0.490291\pi$
$108$	0	0
$109$	−3.86603	−0.370299	−0.185149	−	0.982710i	$-0.559277\pi$
$109$	−3.86603	−0.370299	−0.185149	+	0.982710i	$0.559277\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.5464	−0.992120	−0.496060	−	0.868288i	$-0.665220\pi$
$113$	−10.5464	−0.992120	−0.496060	+	0.868288i	$0.665220\pi$
$114$	0	0
$115$	12.1278	1.13093
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.3896	−1.50243
$120$	0	0
$121$	10.4452	0.949565
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.4813	1.02692
$126$	0	0
$127$	−19.8082	−1.75769	−0.878846	−	0.477106i	$-0.841686\pi$
$127$	−19.8082	−1.75769	−0.878846	+	0.477106i	$0.841686\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.326842	0.0285563	0.0142782	−	0.999898i	$-0.495455\pi$
$131$	0.326842	0.0285563	0.0142782	+	0.999898i	$0.495455\pi$
$132$	0	0
$133$	−1.21235	−0.105124
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.40295	−0.717913	−0.358956	−	0.933354i	$-0.616867\pi$
$137$	−8.40295	−0.717913	−0.358956	+	0.933354i	$0.616867\pi$
$138$	0	0
$139$	0.735937	0.0624214	0.0312107	−	0.999513i	$-0.490064\pi$
$139$	0.735937	0.0624214	0.0312107	+	0.999513i	$0.490064\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.34017	−0.530192
$144$	0	0
$145$	18.8371	1.56434
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.5753	−1.11213	−0.556066	−	0.831138i	$-0.687690\pi$
$149$	−13.5753	−1.11213	−0.556066	+	0.831138i	$0.687690\pi$
$150$	0	0
$151$	−21.0433	−1.71248	−0.856240	−	0.516578i	$-0.827206\pi$
$151$	−21.0433	−1.71248	−0.856240	+	0.516578i	$0.827206\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.3268	−0.829472
$156$	0	0
$157$	−7.24128	−0.577917	−0.288958	−	0.957342i	$-0.593309\pi$
$157$	−7.24128	−0.577917	−0.288958	+	0.957342i	$0.593309\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.1412	1.19329
$162$	0	0
$163$	18.3402	1.43651	0.718257	−	0.695778i	$-0.244940\pi$
$163$	18.3402	1.43651	0.718257	+	0.695778i	$0.244940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.7454	−1.37318	−0.686590	−	0.727045i	$-0.740893\pi$
$167$	−17.7454	−1.37318	−0.686590	+	0.727045i	$0.740893\pi$
$168$	0	0
$169$	−11.1256	−0.855812
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.89269	0.600070	0.300035	−	0.953928i	$-0.403002\pi$
$173$	7.89269	0.600070	0.300035	+	0.953928i	$0.403002\pi$
$174$	0	0
$175$	0.787653	0.0595410
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.55252	0.116041	0.0580204	−	0.998315i	$-0.481521\pi$
$179$	1.55252	0.116041	0.0580204	+	0.998315i	$0.481521\pi$
$180$	0	0
$181$	9.72979	0.723210	0.361605	−	0.932331i	$-0.382229\pi$
$181$	9.72979	0.723210	0.361605	+	0.932331i	$0.382229\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.2979	1.27177
$186$	0	0
$187$	28.0144	2.04861
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0989	0.875445	0.437723	−	0.899110i	$-0.355785\pi$
$191$	12.0989	0.875445	0.437723	+	0.899110i	$0.355785\pi$
$192$	0	0
$193$	12.6020	0.907110	0.453555	−	0.891228i	$-0.350156\pi$
$193$	12.6020	0.907110	0.453555	+	0.891228i	$0.350156\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.46081	0.317820	0.158910	−	0.987293i	$-0.449202\pi$
$197$	4.46081	0.317820	0.158910	+	0.987293i	$0.449202\pi$
$198$	0	0
$199$	9.07838	0.643549	0.321775	−	0.946816i	$-0.395721\pi$
$199$	9.07838	0.643549	0.321775	+	0.946816i	$0.395721\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	23.5174	1.65060
$204$	0	0
$205$	−3.73820	−0.261088
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.07223	0.143339
$210$	0	0
$211$	−15.3919	−1.05962	−0.529811	−	0.848116i	$-0.677737\pi$
$211$	−15.3919	−1.05962	−0.529811	+	0.848116i	$0.677737\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.17009	−0.147999
$216$	0	0
$217$	−12.8927	−0.875213
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.28231	−0.557129
$222$	0	0
$223$	25.5174	1.70877	0.854387	−	0.519637i	$-0.173933\pi$
$223$	25.5174	1.70877	0.854387	+	0.519637i	$0.173933\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.6381	−1.56891	−0.784457	−	0.620183i	$-0.787058\pi$
$227$	−23.6381	−1.56891	−0.784457	+	0.620183i	$0.787058\pi$
$228$	0	0
$229$	3.75872	0.248383	0.124192	−	0.992258i	$-0.460366\pi$
$229$	3.75872	0.248383	0.124192	+	0.992258i	$0.460366\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.6670	−1.61599	−0.807995	−	0.589189i	$-0.799447\pi$
$233$	−24.6670	−1.61599	−0.807995	+	0.589189i	$0.799447\pi$
$234$	0	0
$235$	−28.4885	−1.85839
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.08452	0.134836	0.0674182	−	0.997725i	$-0.478524\pi$
$239$	2.08452	0.134836	0.0674182	+	0.997725i	$0.478524\pi$
$240$	0	0
$241$	−10.1012	−0.650673	−0.325337	−	0.945598i	$-0.605478\pi$
$241$	−10.1012	−0.650673	−0.325337	+	0.945598i	$0.605478\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.738205	−0.0471622
$246$	0	0
$247$	−0.612646	−0.0389817
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.8371	−0.684032	−0.342016	−	0.939694i	$-0.611110\pi$
$251$	−10.8371	−0.684032	−0.342016	+	0.939694i	$0.611110\pi$
$252$	0	0
$253$	−25.8804	−1.62709
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.3607	−1.58196	−0.790978	−	0.611844i	$-0.790428\pi$
$257$	−25.3607	−1.58196	−0.790978	+	0.611844i	$0.790428\pi$
$258$	0	0
$259$	21.5958	1.34190
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.9311	−0.797364	−0.398682	−	0.917089i	$-0.630532\pi$
$263$	−12.9311	−0.797364	−0.398682	+	0.917089i	$0.630532\pi$
$264$	0	0
$265$	−8.75872	−0.538044
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.0638	−1.34526	−0.672628	−	0.739981i	$-0.734834\pi$
$269$	−22.0638	−1.34526	−0.672628	+	0.739981i	$0.734834\pi$
$270$	0	0
$271$	−16.1834	−0.983073	−0.491536	−	0.870857i	$-0.663564\pi$
$271$	−16.1834	−0.983073	−0.491536	+	0.870857i	$0.663564\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.34632	−0.0811859
$276$	0	0
$277$	−2.86991	−0.172436	−0.0862180	−	0.996276i	$-0.527478\pi$
$277$	−2.86991	−0.172436	−0.0862180	+	0.996276i	$0.527478\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.63090	0.395566	0.197783	−	0.980246i	$-0.436626\pi$
$281$	6.63090	0.395566	0.197783	+	0.980246i	$0.436626\pi$
$282$	0	0
$283$	−16.2329	−0.964944	−0.482472	−	0.875911i	$-0.660261\pi$
$283$	−16.2329	−0.964944	−0.482472	+	0.875911i	$0.660261\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.66701	−0.275485
$288$	0	0
$289$	19.5958	1.15270
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−33.3340	−1.94739	−0.973697	−	0.227845i	$-0.926832\pi$
$293$	−33.3340	−1.94739	−0.973697	+	0.227845i	$0.926832\pi$
$294$	0	0
$295$	−16.7009	−0.972362
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.65142	0.442493
$300$	0	0
$301$	−2.70928	−0.156160
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.5464	1.29100
$306$	0	0
$307$	−24.5113	−1.39893	−0.699467	−	0.714665i	$-0.746579\pi$
$307$	−24.5113	−1.39893	−0.699467	+	0.714665i	$0.746579\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.21235	−0.182155	−0.0910777	−	0.995844i	$-0.529031\pi$
$311$	−3.21235	−0.182155	−0.0910777	+	0.995844i	$0.529031\pi$
$312$	0	0
$313$	7.55252	0.426894	0.213447	−	0.976955i	$-0.431531\pi$
$313$	7.55252	0.426894	0.213447	+	0.976955i	$0.431531\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.72979	0.377983	0.188991	−	0.981979i	$-0.439478\pi$
$317$	6.72979	0.377983	0.188991	+	0.981979i	$0.439478\pi$
$318$	0	0
$319$	−40.1978	−2.25064
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.70701	0.150622
$324$	0	0
$325$	0.398032	0.0220788
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−35.5669	−1.96087
$330$	0	0
$331$	−1.49693	−0.0822786	−0.0411393	−	0.999153i	$-0.513099\pi$
$331$	−1.49693	−0.0822786	−0.0411393	+	0.999153i	$0.513099\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.17009	0.118564
$336$	0	0
$337$	−19.7503	−1.07587	−0.537934	−	0.842987i	$-0.680795\pi$
$337$	−19.7503	−1.07587	−0.537934	+	0.842987i	$0.680795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.0372	1.19338
$342$	0	0
$343$	18.0433	0.974247
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.2495	−1.24810	−0.624050	−	0.781385i	$-0.714514\pi$
$347$	−23.2495	−1.24810	−0.624050	+	0.781385i	$0.714514\pi$
$348$	0	0
$349$	−9.25953	−0.495651	−0.247826	−	0.968805i	$-0.579716\pi$
$349$	−9.25953	−0.495651	−0.247826	+	0.968805i	$0.579716\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.1701	0.860647	0.430323	−	0.902675i	$-0.358400\pi$
$353$	16.1701	0.860647	0.430323	+	0.902675i	$0.358400\pi$
$354$	0	0
$355$	−23.8082	−1.26361
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.30018	0.226955	0.113477	−	0.993541i	$-0.463801\pi$
$359$	4.30018	0.226955	0.113477	+	0.993541i	$0.463801\pi$
$360$	0	0
$361$	−18.7998	−0.989461
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.11223	0.424613
$366$	0	0
$367$	−35.4329	−1.84958	−0.924792	−	0.380473i	$-0.875761\pi$
$367$	−35.4329	−1.84958	−0.924792	+	0.380473i	$0.875761\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.9350	−0.567714
$372$	0	0
$373$	−15.2846	−0.791406	−0.395703	−	0.918379i	$-0.629499\pi$
$373$	−15.2846	−0.791406	−0.395703	+	0.918379i	$0.629499\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.8843	0.612072
$378$	0	0
$379$	14.5402	0.746882	0.373441	−	0.927654i	$-0.378178\pi$
$379$	14.5402	0.746882	0.373441	+	0.927654i	$0.378178\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.13624	0.160254	0.0801271	−	0.996785i	$-0.474467\pi$
$383$	3.13624	0.160254	0.0801271	+	0.996785i	$0.474467\pi$
$384$	0	0
$385$	27.2267	1.38760
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.9399	−1.36591	−0.682953	−	0.730462i	$-0.739305\pi$
$389$	−26.9399	−1.36591	−0.682953	+	0.730462i	$0.739305\pi$
$390$	0	0
$391$	−33.8082	−1.70975
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.2823	0.617990
$396$	0	0
$397$	−17.7854	−0.892623	−0.446311	−	0.894878i	$-0.647263\pi$
$397$	−17.7854	−0.892623	−0.446311	+	0.894878i	$0.647263\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.93722	−0.496241	−0.248121	−	0.968729i	$-0.579813\pi$
$401$	−9.93722	−0.496241	−0.248121	+	0.968729i	$0.579813\pi$
$402$	0	0
$403$	−6.51518	−0.324544
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−36.9132	−1.82972
$408$	0	0
$409$	−18.3090	−0.905321	−0.452660	−	0.891683i	$-0.649525\pi$
$409$	−18.3090	−0.905321	−0.452660	+	0.891683i	$0.649525\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−20.8504	−1.02598
$414$	0	0
$415$	−6.41855	−0.315074
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.9821	−0.780778	−0.390389	−	0.920650i	$-0.627660\pi$
$419$	−15.9821	−0.780778	−0.390389	+	0.920650i	$0.627660\pi$
$420$	0	0
$421$	36.7731	1.79221	0.896106	−	0.443841i	$-0.146384\pi$
$421$	36.7731	1.79221	0.896106	+	0.443841i	$0.146384\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.75872	−0.0853106
$426$	0	0
$427$	28.1483	1.36219
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.1412	1.01833	0.509167	−	0.860668i	$-0.329954\pi$
$431$	21.1412	1.01833	0.509167	+	0.860668i	$0.329954\pi$
$432$	0	0
$433$	13.3607	0.642074	0.321037	−	0.947067i	$-0.395969\pi$
$433$	13.3607	0.642074	0.321037	+	0.947067i	$0.395969\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.50080	−0.119630
$438$	0	0
$439$	25.5441	1.21915	0.609577	−	0.792727i	$-0.291339\pi$
$439$	25.5441	1.21915	0.609577	+	0.792727i	$0.291339\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.3958	−0.921521	−0.460760	−	0.887525i	$-0.652423\pi$
$443$	−19.3958	−0.921521	−0.460760	+	0.887525i	$0.652423\pi$
$444$	0	0
$445$	1.76713	0.0837702
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.3112	1.00574	0.502870	−	0.864362i	$-0.332277\pi$
$449$	21.3112	1.00574	0.502870	+	0.864362i	$0.332277\pi$
$450$	0	0
$451$	7.97721	0.375632
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.04945	−0.377364
$456$	0	0
$457$	−24.6681	−1.15392	−0.576962	−	0.816771i	$-0.695762\pi$
$457$	−24.6681	−1.15392	−0.576962	+	0.816771i	$0.695762\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.5958	−1.14554	−0.572771	−	0.819716i	$-0.694132\pi$
$461$	−24.5958	−1.14554	−0.572771	+	0.819716i	$0.694132\pi$
$462$	0	0
$463$	−0.793796	−0.0368908	−0.0184454	−	0.999830i	$-0.505872\pi$
$463$	−0.793796	−0.0368908	−0.0184454	+	0.999830i	$0.505872\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.80817	0.268770	0.134385	−	0.990929i	$-0.457094\pi$
$467$	5.80817	0.268770	0.134385	+	0.990929i	$0.457094\pi$
$468$	0	0
$469$	2.70928	0.125103
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.63090	0.212929
$474$	0	0
$475$	−0.130094	−0.00596910
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.27739	0.286821	0.143411	−	0.989663i	$-0.454193\pi$
$479$	6.27739	0.286821	0.143411	+	0.989663i	$0.454193\pi$
$480$	0	0
$481$	10.9132	0.497600
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30.3051	−1.37608
$486$	0	0
$487$	−9.26794	−0.419970	−0.209985	−	0.977705i	$-0.567342\pi$
$487$	−9.26794	−0.419970	−0.209985	+	0.977705i	$0.567342\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.4608	−1.10390	−0.551950	−	0.833877i	$-0.686116\pi$
$491$	−24.4608	−1.10390	−0.551950	+	0.833877i	$0.686116\pi$
$492$	0	0
$493$	−52.5113	−2.36499
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29.7237	−1.33329
$498$	0	0
$499$	−23.4329	−1.04900	−0.524501	−	0.851410i	$-0.675748\pi$
$499$	−23.4329	−1.04900	−0.524501	+	0.851410i	$0.675748\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.9337	0.710450	0.355225	−	0.934781i	$-0.384404\pi$
$503$	15.9337	0.710450	0.355225	+	0.934781i	$0.384404\pi$
$504$	0	0
$505$	−38.1217	−1.69639
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.9071	1.23696	0.618480	−	0.785801i	$-0.287749\pi$
$509$	27.9071	1.23696	0.618480	+	0.785801i	$0.287749\pi$
$510$	0	0
$511$	10.1278	0.448029
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.8865	−1.00850
$516$	0	0
$517$	60.7936	2.67370
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.7237	−0.864109	−0.432054	−	0.901848i	$-0.642211\pi$
$521$	−19.7237	−0.864109	−0.432054	+	0.901848i	$0.642211\pi$
$522$	0	0
$523$	24.2739	1.06142	0.530712	−	0.847552i	$-0.321925\pi$
$523$	24.2739	1.06142	0.530712	+	0.847552i	$0.321925\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.7877	1.25401
$528$	0	0
$529$	8.23287	0.357951
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.35842	−0.102155
$534$	0	0
$535$	−1.36910	−0.0591915
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.57531	0.0678532
$540$	0	0
$541$	37.1194	1.59589	0.797944	−	0.602731i	$-0.205921\pi$
$541$	37.1194	1.59589	0.797944	+	0.602731i	$0.205921\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.38962	0.359372
$546$	0	0
$547$	−25.0494	−1.07104	−0.535519	−	0.844523i	$-0.679884\pi$
$547$	−25.0494	−1.07104	−0.535519	+	0.844523i	$0.679884\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.88428	−0.165476
$552$	0	0
$553$	15.3340	0.652069
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.0121	0.975054	0.487527	−	0.873108i	$-0.337899\pi$
$557$	23.0121	0.975054	0.487527	+	0.873108i	$0.337899\pi$
$558$	0	0
$559$	−1.36910	−0.0579069
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.3835	−1.49124	−0.745618	−	0.666374i	$-0.767846\pi$
$563$	−35.3835	−1.49124	−0.745618	+	0.666374i	$0.767846\pi$
$564$	0	0
$565$	22.8865	0.962844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.5136	1.15343	0.576714	−	0.816946i	$-0.304335\pi$
$569$	27.5136	1.15343	0.576714	+	0.816946i	$0.304335\pi$
$570$	0	0
$571$	−25.2800	−1.05794	−0.528969	−	0.848641i	$-0.677421\pi$
$571$	−25.2800	−1.05794	−0.528969	+	0.848641i	$0.677421\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.62475	0.0677569
$576$	0	0
$577$	−33.6475	−1.40077	−0.700383	−	0.713767i	$-0.746987\pi$
$577$	−33.6475	−1.40077	−0.700383	+	0.713767i	$0.746987\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.01333	−0.332449
$582$	0	0
$583$	18.6908	0.774096
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.7838	−0.940387	−0.470194	−	0.882563i	$-0.655816\pi$
$587$	−22.7838	−0.940387	−0.470194	+	0.882563i	$0.655816\pi$
$588$	0	0
$589$	2.12943	0.0877418
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.6647	1.05392	0.526962	−	0.849889i	$-0.323331\pi$
$593$	25.6647	1.05392	0.526962	+	0.849889i	$0.323331\pi$
$594$	0	0
$595$	35.5669	1.45810
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.44134	0.0588914	0.0294457	−	0.999566i	$-0.490626\pi$
$599$	1.44134	0.0588914	0.0294457	+	0.999566i	$0.490626\pi$
$600$	0	0
$601$	28.4391	1.16005	0.580027	−	0.814597i	$-0.303042\pi$
$601$	28.4391	1.16005	0.580027	+	0.814597i	$0.303042\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.6670	−0.921545
$606$	0	0
$607$	−1.97721	−0.0802526	−0.0401263	−	0.999195i	$-0.512776\pi$
$607$	−1.97721	−0.0802526	−0.0401263	+	0.999195i	$0.512776\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.9733	−0.727123
$612$	0	0
$613$	−30.6020	−1.23600	−0.618001	−	0.786177i	$-0.712057\pi$
$613$	−30.6020	−1.23600	−0.618001	+	0.786177i	$0.712057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.5981	1.51364	0.756821	−	0.653622i	$-0.226751\pi$
$617$	37.5981	1.51364	0.756821	+	0.653622i	$0.226751\pi$
$618$	0	0
$619$	31.3568	1.26034	0.630168	−	0.776458i	$-0.282986\pi$
$619$	31.3568	1.26034	0.630168	+	0.776458i	$0.282986\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.20620	0.0883897
$624$	0	0
$625$	−23.4619	−0.938474
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.2206	−1.92268
$630$	0	0
$631$	16.5814	0.660097	0.330049	−	0.943964i	$-0.392935\pi$
$631$	16.5814	0.660097	0.330049	+	0.943964i	$0.392935\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.9854	1.70583
$636$	0	0
$637$	−0.465732	−0.0184530
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.4524	1.36079	0.680394	−	0.732847i	$-0.261809\pi$
$641$	34.4524	1.36079	0.680394	+	0.732847i	$0.261809\pi$
$642$	0	0
$643$	22.6765	0.894273	0.447136	−	0.894466i	$-0.352444\pi$
$643$	22.6765	0.894273	0.447136	+	0.894466i	$0.352444\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.1399	0.634526	0.317263	−	0.948338i	$-0.397236\pi$
$647$	16.1399	0.634526	0.317263	+	0.948338i	$0.397236\pi$
$648$	0	0
$649$	35.6391	1.39896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.6358	0.572744	0.286372	−	0.958119i	$-0.407551\pi$
$653$	14.6358	0.572744	0.286372	+	0.958119i	$0.407551\pi$
$654$	0	0
$655$	−0.709275	−0.0277137
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−46.4934	−1.81113	−0.905564	−	0.424210i	$-0.860552\pi$
$659$	−46.4934	−1.81113	−0.905564	+	0.424210i	$0.860552\pi$
$660$	0	0
$661$	−38.3812	−1.49286	−0.746428	−	0.665466i	$-0.768233\pi$
$661$	−38.3812	−1.49286	−0.746428	+	0.665466i	$0.768233\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.63090	0.102022
$666$	0	0
$667$	48.5113	1.87837
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−48.1133	−1.85739
$672$	0	0
$673$	41.6658	1.60610	0.803049	−	0.595913i	$-0.203210\pi$
$673$	41.6658	1.60610	0.803049	+	0.595913i	$0.203210\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.6875	−1.67905	−0.839524	−	0.543322i	$-0.817166\pi$
$677$	−43.6875	−1.67905	−0.839524	+	0.543322i	$0.817166\pi$
$678$	0	0
$679$	−37.8348	−1.45197
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.3584	−1.54427	−0.772136	−	0.635457i	$-0.780812\pi$
$683$	−40.3584	−1.54427	−0.772136	+	0.635457i	$0.780812\pi$
$684$	0	0
$685$	18.2351	0.696729
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.52586	−0.210518
$690$	0	0
$691$	0.255652	0.00972547	0.00486273	−	0.999988i	$-0.498452\pi$
$691$	0.255652	0.00972547	0.00486273	+	0.999988i	$0.498452\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.59705	−0.0605795
$696$	0	0
$697$	10.4208	0.394717
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.2218	−0.574920	−0.287460	−	0.957793i	$-0.592811\pi$
$701$	−15.2218	−0.574920	−0.287460	+	0.957793i	$0.592811\pi$
$702$	0	0
$703$	−3.56690	−0.134528
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−47.5936	−1.78994
$708$	0	0
$709$	−9.56916	−0.359377	−0.179689	−	0.983724i	$-0.557509\pi$
$709$	−9.56916	−0.359377	−0.179689	+	0.983724i	$0.557509\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−26.5948	−0.995982
$714$	0	0
$715$	13.7587	0.514547
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.9516	0.818656	0.409328	−	0.912387i	$-0.365763\pi$
$719$	21.9516	0.818656	0.409328	+	0.912387i	$0.365763\pi$
$720$	0	0
$721$	−28.5730	−1.06412
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.52359	0.0937238
$726$	0	0
$727$	−6.70540	−0.248690	−0.124345	−	0.992239i	$-0.539683\pi$
$727$	−6.70540	−0.248690	−0.124345	+	0.992239i	$0.539683\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.04945	0.223747
$732$	0	0
$733$	−38.9939	−1.44027	−0.720135	−	0.693833i	$-0.755920\pi$
$733$	−38.9939	−1.44027	−0.720135	+	0.693833i	$0.755920\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.63090	−0.170581
$738$	0	0
$739$	−26.9565	−0.991612	−0.495806	−	0.868433i	$-0.665127\pi$
$739$	−26.9565	−0.991612	−0.495806	+	0.868433i	$0.665127\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.3428	1.14986	0.574928	−	0.818204i	$-0.305030\pi$
$743$	31.3428	1.14986	0.574928	+	0.818204i	$0.305030\pi$
$744$	0	0
$745$	29.4596	1.07932
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.70928	−0.0624556
$750$	0	0
$751$	−6.01438	−0.219468	−0.109734	−	0.993961i	$-0.535000\pi$
$751$	−6.01438	−0.219468	−0.109734	+	0.993961i	$0.535000\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.6658	1.66195
$756$	0	0
$757$	−3.57531	−0.129947	−0.0649734	−	0.997887i	$-0.520696\pi$
$757$	−3.57531	−0.129947	−0.0649734	+	0.997887i	$0.520696\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.81205	0.283187	0.141593	−	0.989925i	$-0.454777\pi$
$761$	7.81205	0.283187	0.141593	+	0.989925i	$0.454777\pi$
$762$	0	0
$763$	10.4741	0.379189
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.5365	−0.380452
$768$	0	0
$769$	−21.2990	−0.768060	−0.384030	−	0.923321i	$-0.625464\pi$
$769$	−21.2990	−0.768060	−0.384030	+	0.923321i	$0.625464\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.1522	0.724825	0.362412	−	0.932018i	$-0.381953\pi$
$773$	20.1522	0.724825	0.362412	+	0.932018i	$0.381953\pi$
$774$	0	0
$775$	−1.38348	−0.0496960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.770832	0.0276179
$780$	0	0
$781$	50.8059	1.81798
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.7142	0.560864
$786$	0	0
$787$	−17.9048	−0.638237	−0.319119	−	0.947715i	$-0.603387\pi$
$787$	−17.9048	−0.638237	−0.319119	+	0.947715i	$0.603387\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	28.5730	1.01594
$792$	0	0
$793$	14.2245	0.505125
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.3958	−0.899564	−0.449782	−	0.893138i	$-0.648498\pi$
$797$	−25.3958	−0.899564	−0.449782	+	0.893138i	$0.648498\pi$
$798$	0	0
$799$	79.4161	2.80954
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.3112	−0.610901
$804$	0	0
$805$	−32.8576	−1.15808
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13.8482	0.486876	0.243438	−	0.969917i	$-0.421725\pi$
$809$	13.8482	0.486876	0.243438	+	0.969917i	$0.421725\pi$
$810$	0	0
$811$	15.4924	0.544012	0.272006	−	0.962296i	$-0.412313\pi$
$811$	15.4924	0.544012	0.272006	+	0.962296i	$0.412313\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.7998	−1.39412
$816$	0	0
$817$	0.447480	0.0156553
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.7670	1.21338	0.606688	−	0.794940i	$-0.292498\pi$
$821$	34.7670	1.21338	0.606688	+	0.794940i	$0.292498\pi$
$822$	0	0
$823$	42.4573	1.47997	0.739985	−	0.672624i	$-0.234833\pi$
$823$	42.4573	1.47997	0.739985	+	0.672624i	$0.234833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.6802	−1.24072	−0.620361	−	0.784317i	$-0.713014\pi$
$827$	−35.6802	−1.24072	−0.620361	+	0.784317i	$0.713014\pi$
$828$	0	0
$829$	−26.2823	−0.912823	−0.456411	−	0.889769i	$-0.650865\pi$
$829$	−26.2823	−0.912823	−0.456411	+	0.889769i	$0.650865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.05786	0.0713006
$834$	0	0
$835$	38.5090	1.33266
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.0806	0.417070	0.208535	−	0.978015i	$-0.433130\pi$
$839$	12.0806	0.417070	0.208535	+	0.978015i	$0.433130\pi$
$840$	0	0
$841$	46.3484	1.59822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24.1434	0.830559
$846$	0	0
$847$	−28.2990	−0.972364
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.5474	1.52707
$852$	0	0
$853$	−27.0472	−0.926078	−0.463039	−	0.886338i	$-0.653241\pi$
$853$	−27.0472	−0.926078	−0.463039	+	0.886338i	$0.653241\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.2956	−1.51311	−0.756555	−	0.653930i	$-0.773119\pi$
$857$	−44.2956	−1.51311	−0.756555	+	0.653930i	$0.773119\pi$
$858$	0	0
$859$	−27.2990	−0.931428	−0.465714	−	0.884935i	$-0.654203\pi$
$859$	−27.2990	−0.931428	−0.465714	+	0.884935i	$0.654203\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.7477	0.365855	0.182927	−	0.983126i	$-0.441443\pi$
$863$	10.7477	0.365855	0.182927	+	0.983126i	$0.441443\pi$
$864$	0	0
$865$	−17.1278	−0.582364
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−26.2101	−0.889116
$870$	0	0
$871$	1.36910	0.0463903
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−31.1061	−1.05158
$876$	0	0
$877$	45.0349	1.52072	0.760360	−	0.649502i	$-0.225022\pi$
$877$	45.0349	1.52072	0.760360	+	0.649502i	$0.225022\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.9155	1.34479	0.672393	−	0.740194i	$-0.265266\pi$
$881$	39.9155	1.34479	0.672393	+	0.740194i	$0.265266\pi$
$882$	0	0
$883$	7.94828	0.267481	0.133741	−	0.991016i	$-0.457301\pi$
$883$	7.94828	0.267481	0.133741	+	0.991016i	$0.457301\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.2934	1.05073	0.525364	−	0.850877i	$-0.323929\pi$
$887$	31.2934	1.05073	0.525364	+	0.850877i	$0.323929\pi$
$888$	0	0
$889$	53.6658	1.79989
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.87444	0.196581
$894$	0	0
$895$	−3.36910	−0.112617
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.3074	−1.37768
$900$	0	0
$901$	24.4163	0.813424
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1145	−0.701870
$906$	0	0
$907$	17.9733	0.596795	0.298397	−	0.954442i	$-0.403548\pi$
$907$	17.9733	0.596795	0.298397	+	0.954442i	$0.403548\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	49.0121	1.62384	0.811922	−	0.583766i	$-0.198422\pi$
$911$	49.0121	1.62384	0.811922	+	0.583766i	$0.198422\pi$
$912$	0	0
$913$	13.6970	0.453304
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.885505	−0.0292419
$918$	0	0
$919$	−46.6307	−1.53821	−0.769103	−	0.639125i	$-0.779297\pi$
$919$	−46.6307	−1.53821	−0.769103	+	0.639125i	$0.779297\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.0205	−0.494406
$924$	0	0
$925$	2.31739	0.0761952
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.9214	−1.30978	−0.654890	−	0.755724i	$-0.727285\pi$
$929$	−39.9214	−1.30978	−0.654890	+	0.755724i	$0.727285\pi$
$930$	0	0
$931$	0.152221	0.00498883
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−60.7936	−1.98816
$936$	0	0
$937$	34.6888	1.13323	0.566616	−	0.823982i	$-0.308252\pi$
$937$	34.6888	1.13323	0.566616	+	0.823982i	$0.308252\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15.6248	0.509352	0.254676	−	0.967026i	$-0.418031\pi$
$941$	15.6248	0.509352	0.254676	+	0.967026i	$0.418031\pi$
$942$	0	0
$943$	−9.62702	−0.313499
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.9432	0.973023	0.486511	−	0.873674i	$-0.338269\pi$
$947$	29.9432	0.973023	0.486511	+	0.873674i	$0.338269\pi$
$948$	0	0
$949$	5.11799	0.166137
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.3234	−1.37099	−0.685494	−	0.728078i	$-0.740414\pi$
$953$	−42.3234	−1.37099	−0.685494	+	0.728078i	$0.740414\pi$
$954$	0	0
$955$	−26.2557	−0.849613
$956$	0	0
$957$	0	0
$958$	0	0
$959$	22.7659	0.735150
$960$	0	0
$961$	−8.35455	−0.269502
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.3474	−0.880343
$966$	0	0
$967$	15.4368	0.496414	0.248207	−	0.968707i	$-0.420159\pi$
$967$	15.4368	0.496414	0.248207	+	0.968707i	$0.420159\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.8143	0.924695	0.462348	−	0.886699i	$-0.347007\pi$
$971$	28.8143	0.924695	0.462348	+	0.886699i	$0.347007\pi$
$972$	0	0
$973$	−1.99386	−0.0639201
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.78539	−0.313062	−0.156531	−	0.987673i	$-0.550031\pi$
$977$	−9.78539	−0.313062	−0.156531	+	0.987673i	$0.550031\pi$
$978$	0	0
$979$	−3.77101	−0.120522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.2495	−0.869124	−0.434562	−	0.900642i	$-0.643097\pi$
$983$	−27.2495	−0.869124	−0.434562	+	0.900642i	$0.643097\pi$
$984$	0	0
$985$	−9.68035	−0.308441
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.58864	−0.177708
$990$	0	0
$991$	41.6186	1.32206	0.661029	−	0.750360i	$-0.270120\pi$
$991$	41.6186	1.32206	0.661029	+	0.750360i	$0.270120\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−19.7009	−0.624559
$996$	0	0
$997$	1.03507	0.0327811	0.0163905	−	0.999866i	$-0.494782\pi$
$997$	1.03507	0.0327811	0.0163905	+	0.999866i	$0.494782\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9648.2.a.bn.1.1		3
3.2	odd	2		3216.2.a.u.1.3		3
4.3	odd	2		603.2.a.i.1.2		3
12.11	even	2		201.2.a.d.1.2	✓	3
60.59	even	2		5025.2.a.m.1.2		3
84.83	odd	2		9849.2.a.ba.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.a.d.1.2	✓	3	12.11	even	2
603.2.a.i.1.2		3	4.3	odd	2
3216.2.a.u.1.3		3	3.2	odd	2
5025.2.a.m.1.2		3	60.59	even	2
9648.2.a.bn.1.1		3	1.1	even	1	trivial
9849.2.a.ba.1.2		3	84.83	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:	\( N \)	\(=\)	\( 9648 = 2^{4} \cdot 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9648.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{5} -2.70928 q^{7} +O(q^{10})\)	\(q-2.17009 q^{5} -2.70928 q^{7} +4.63090 q^{11} -1.36910 q^{13} +6.04945 q^{17} +0.447480 q^{19} -5.58864 q^{23} -0.290725 q^{25} -8.68035 q^{29} +4.75872 q^{31} +5.87936 q^{35} -7.97107 q^{37} +1.72261 q^{41} +1.00000 q^{43} +13.1278 q^{47} +0.340173 q^{49} +4.03612 q^{53} -10.0494 q^{55} +7.69594 q^{59} -10.3896 q^{61} +2.97107 q^{65} -1.00000 q^{67} +10.9711 q^{71} -3.73820 q^{73} -12.5464 q^{77} -5.65983 q^{79} +2.95774 q^{83} -13.1278 q^{85} -0.814315 q^{89} +3.70928 q^{91} -0.971071 q^{95} +13.9649 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5} - q^{7}+O(q^{10}) \) 3 * q - q^5 - q^7	\( 3 q - q^{5} - q^{7} + 10 q^{11} - 8 q^{13} + 2 q^{19} + 3 q^{23} - 8 q^{25} - 4 q^{29} - 11 q^{31} + 5 q^{35} - 9 q^{37} - q^{41} + 3 q^{43} + 18 q^{47} - 10 q^{49} - 7 q^{53} - 12 q^{55} + 15 q^{59} - 2 q^{61} - 6 q^{65} - 3 q^{67} + 18 q^{71} - 19 q^{73} - 2 q^{77} - 28 q^{79} - 7 q^{83} - 18 q^{85} + 6 q^{89} + 4 q^{91} + 12 q^{95} - 8 q^{97}+O(q^{100}) \) 3 * q - q^5 - q^7 + 10 * q^11 - 8 * q^13 + 2 * q^19 + 3 * q^23 - 8 * q^25 - 4 * q^29 - 11 * q^31 + 5 * q^35 - 9 * q^37 - q^41 + 3 * q^43 + 18 * q^47 - 10 * q^49 - 7 * q^53 - 12 * q^55 + 15 * q^59 - 2 * q^61 - 6 * q^65 - 3 * q^67 + 18 * q^71 - 19 * q^73 - 2 * q^77 - 28 * q^79 - 7 * q^83 - 18 * q^85 + 6 * q^89 + 4 * q^91 + 12 * q^95 - 8 * q^97

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