LMFDB - Embedded newform 3249.2.a.be.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,0,-4,0,0,0,0,0,4,0,0,4,0,0,0,0,0,24,0,0,4,0,0,-28, 0,0,28,0,0,0,0,0,4,0,0,-24,0,0,-4,0,0,24,0,0,0,0,0,4,0,0,-24,0,0,48,0, 0,20,0,0,-20,0,0,28,0,0,24,0,0,20,0,0,0,0,0,28,0,0,48,0,0,-48,0,0,48,0, 0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.27648.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 3 $ x^4 - 6*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 171)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.741964$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.741964 q^{2} -1.44949 q^{4} +3.30136 q^{5} +1.44949 q^{7} -2.55940 q^{8} +2.44949 q^{10} -1.81743 q^{11} +1.00000 q^{13} +1.07547 q^{14} +1.00000 q^{16} -6.60272 q^{17} -4.78529 q^{20} -1.34847 q^{22} +4.78529 q^{23} +5.89898 q^{25} +0.741964 q^{26} -2.10102 q^{28} +9.57058 q^{29} +4.55051 q^{31} +5.86076 q^{32} -4.89898 q^{34} +4.78529 q^{35} +5.89898 q^{37} -8.44949 q^{40} +2.96786 q^{41} -8.34847 q^{43} +2.63435 q^{44} +3.55051 q^{46} +2.96786 q^{47} -4.89898 q^{49} +4.37683 q^{50} -1.44949 q^{52} +3.30136 q^{53} -6.00000 q^{55} -3.70982 q^{56} +7.10102 q^{58} +8.42015 q^{59} +5.00000 q^{61} +3.37631 q^{62} +2.34847 q^{64} +3.30136 q^{65} +14.3485 q^{67} +9.57058 q^{68} +3.55051 q^{70} -9.57058 q^{71} +5.00000 q^{73} +4.37683 q^{74} -2.63435 q^{77} +14.3485 q^{79} +3.30136 q^{80} +2.20204 q^{82} +3.63487 q^{83} -21.7980 q^{85} -6.19426 q^{86} +4.65153 q^{88} -16.5068 q^{89} +1.44949 q^{91} -6.93623 q^{92} +2.20204 q^{94} -12.8990 q^{97} -3.63487 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{7} + 4 q^{13} + 4 q^{16} + 24 q^{22} + 4 q^{25} - 28 q^{28} + 28 q^{31} + 4 q^{37} - 24 q^{40} - 4 q^{43} + 24 q^{46} + 4 q^{52} - 24 q^{55} + 48 q^{58} + 20 q^{61} - 20 q^{64} + 28 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^7 + 4 * q^13 + 4 * q^16 + 24 * q^22 + 4 * q^25 - 28 * q^28 + 28 * q^31 + 4 * q^37 - 24 * q^40 - 4 * q^43 + 24 * q^46 + 4 * q^52 - 24 * q^55 + 48 * q^58 + 20 * q^61 - 20 * q^64 + 28 * q^67 + 24 * q^70 + 20 * q^73 + 28 * q^79 + 48 * q^82 - 48 * q^85 + 48 * q^88 - 4 * q^91 + 48 * q^94 - 32 * q^97

Coefficient data

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

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

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

$2$	0.741964	0.524648	0.262324	−	0.964980i	$-0.415511\pi$
$2$	0.741964	0.524648	0.262324	+	0.964980i	$0.415511\pi$
$3$	0	0
$3$	0	0
$4$	−1.44949	−0.724745
$5$	3.30136	1.47641	0.738207	−	0.674575i	$-0.235673\pi$
$5$	3.30136	1.47641	0.738207	+	0.674575i	$0.235673\pi$
$6$	0	0
$7$	1.44949	0.547856	0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949	0.547856	0.273928	+	0.961750i	$0.411677\pi$
$8$	−2.55940	−0.904883
$9$	0	0
$10$	2.44949	0.774597
$11$	−1.81743	−0.547977	−0.273988	−	0.961733i	$-0.588343\pi$
$11$	−1.81743	−0.547977	−0.273988	+	0.961733i	$0.588343\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	1.07547	0.287431
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.60272	−1.60139	−0.800697	−	0.599069i	$-0.795538\pi$
$17$	−6.60272	−1.60139	−0.800697	+	0.599069i	$0.795538\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−4.78529	−1.07002
$21$	0	0
$22$	−1.34847	−0.287495
$23$	4.78529	0.997801	0.498901	−	0.866659i	$-0.333737\pi$
$23$	4.78529	0.997801	0.498901	+	0.866659i	$0.333737\pi$
$24$	0	0
$25$	5.89898	1.17980
$26$	0.741964	0.145511
$27$	0	0
$28$	−2.10102	−0.397056
$29$	9.57058	1.77721	0.888606	−	0.458672i	$-0.151675\pi$
$29$	9.57058	1.77721	0.888606	+	0.458672i	$0.151675\pi$
$30$	0	0
$31$	4.55051	0.817296	0.408648	−	0.912692i	$-0.366000\pi$
$31$	4.55051	0.817296	0.408648	+	0.912692i	$0.366000\pi$
$32$	5.86076	1.03605
$33$	0	0
$34$	−4.89898	−0.840168
$35$	4.78529	0.808861
$36$	0	0
$37$	5.89898	0.969786	0.484893	−	0.874573i	$-0.338858\pi$
$37$	5.89898	0.969786	0.484893	+	0.874573i	$0.338858\pi$
$38$	0	0
$39$	0	0
$40$	−8.44949	−1.33598
$41$	2.96786	0.463501	0.231751	−	0.972775i	$-0.425555\pi$
$41$	2.96786	0.463501	0.231751	+	0.972775i	$0.425555\pi$
$42$	0	0
$43$	−8.34847	−1.27313	−0.636565	−	0.771223i	$-0.719645\pi$
$43$	−8.34847	−1.27313	−0.636565	+	0.771223i	$0.719645\pi$
$44$	2.63435	0.397143
$45$	0	0
$46$	3.55051	0.523494
$47$	2.96786	0.432906	0.216453	−	0.976293i	$-0.430551\pi$
$47$	2.96786	0.432906	0.216453	+	0.976293i	$0.430551\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	4.37683	0.618977
$51$	0	0
$52$	−1.44949	−0.201008
$53$	3.30136	0.453477	0.226738	−	0.973956i	$-0.427194\pi$
$53$	3.30136	0.453477	0.226738	+	0.973956i	$0.427194\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	−3.70982	−0.495745
$57$	0	0
$58$	7.10102	0.932410
$59$	8.42015	1.09621	0.548105	−	0.836409i	$-0.315349\pi$
$59$	8.42015	1.09621	0.548105	+	0.836409i	$0.315349\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	3.37631	0.428792
$63$	0	0
$64$	2.34847	0.293559
$65$	3.30136	0.409483
$66$	0	0
$67$	14.3485	1.75294	0.876472	−	0.481452i	$-0.159891\pi$
$67$	14.3485	1.75294	0.876472	+	0.481452i	$0.159891\pi$
$68$	9.57058	1.16060
$69$	0	0
$70$	3.55051	0.424367
$71$	−9.57058	−1.13582	−0.567909	−	0.823091i	$-0.692248\pi$
$71$	−9.57058	−1.13582	−0.567909	+	0.823091i	$0.692248\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	4.37683	0.508796
$75$	0	0
$76$	0	0
$77$	−2.63435	−0.300212
$78$	0	0
$79$	14.3485	1.61433	0.807164	−	0.590327i	$-0.201001\pi$
$79$	14.3485	1.61433	0.807164	+	0.590327i	$0.201001\pi$
$80$	3.30136	0.369103
$81$	0	0
$82$	2.20204	0.243175
$83$	3.63487	0.398978	0.199489	−	0.979900i	$-0.436072\pi$
$83$	3.63487	0.398978	0.199489	+	0.979900i	$0.436072\pi$
$84$	0	0
$85$	−21.7980	−2.36432
$86$	−6.19426	−0.667944
$87$	0	0
$88$	4.65153	0.495855
$89$	−16.5068	−1.74972	−0.874859	−	0.484378i	$-0.839046\pi$
$89$	−16.5068	−1.74972	−0.874859	+	0.484378i	$0.839046\pi$
$90$	0	0
$91$	1.44949	0.151948
$92$	−6.93623	−0.723152
$93$	0	0
$94$	2.20204	0.227123
$95$	0	0
$96$	0	0
$97$	−12.8990	−1.30969	−0.654846	−	0.755762i	$-0.727267\pi$
$97$	−12.8990	−1.30969	−0.654846	+	0.755762i	$0.727267\pi$
$98$	−3.63487	−0.367177
$99$	0	0
$100$	−8.55051	−0.855051
$101$	6.60272	0.656995	0.328498	−	0.944505i	$-0.393458\pi$
$101$	6.60272	0.656995	0.328498	+	0.944505i	$0.393458\pi$
$102$	0	0
$103$	−1.44949	−0.142822	−0.0714112	−	0.997447i	$-0.522750\pi$
$103$	−1.44949	−0.142822	−0.0714112	+	0.997447i	$0.522750\pi$
$104$	−2.55940	−0.250969
$105$	0	0
$106$	2.44949	0.237915
$107$	−12.5384	−1.21214	−0.606068	−	0.795413i	$-0.707254\pi$
$107$	−12.5384	−1.21214	−0.606068	+	0.795413i	$0.707254\pi$
$108$	0	0
$109$	−0.898979	−0.0861066	−0.0430533	−	0.999073i	$-0.513709\pi$
$109$	−0.898979	−0.0861066	−0.0430533	+	0.999073i	$0.513709\pi$
$110$	−4.45178	−0.424461
$111$	0	0
$112$	1.44949	0.136964
$113$	6.26922	0.589758	0.294879	−	0.955535i	$-0.404721\pi$
$113$	6.26922	0.589758	0.294879	+	0.955535i	$0.404721\pi$
$114$	0	0
$115$	15.7980	1.47317
$116$	−13.8725	−1.28802
$117$	0	0
$118$	6.24745	0.575124
$119$	−9.57058	−0.877333
$120$	0	0
$121$	−7.69694	−0.699722
$122$	3.70982	0.335871
$123$	0	0
$124$	−6.59592	−0.592331
$125$	2.96786	0.265453
$126$	0	0
$127$	13.7980	1.22437	0.612185	−	0.790714i	$-0.290291\pi$
$127$	13.7980	1.22437	0.612185	+	0.790714i	$0.290291\pi$
$128$	−9.97903	−0.882030
$129$	0	0
$130$	2.44949	0.214834
$131$	16.8403	1.47134	0.735672	−	0.677338i	$-0.236866\pi$
$131$	16.8403	1.47134	0.735672	+	0.677338i	$0.236866\pi$
$132$	0	0
$133$	0	0
$134$	10.6460	0.919678
$135$	0	0
$136$	16.8990	1.44908
$137$	−16.8403	−1.43876	−0.719382	−	0.694614i	$-0.755575\pi$
$137$	−16.8403	−1.43876	−0.719382	+	0.694614i	$0.755575\pi$
$138$	0	0
$139$	12.3485	1.04738	0.523692	−	0.851908i	$-0.324554\pi$
$139$	12.3485	1.04738	0.523692	+	0.851908i	$0.324554\pi$
$140$	−6.93623	−0.586218
$141$	0	0
$142$	−7.10102	−0.595904
$143$	−1.81743	−0.151981
$144$	0	0
$145$	31.5959	2.62390
$146$	3.70982	0.307027
$147$	0	0
$148$	−8.55051	−0.702848
$149$	6.93623	0.568238	0.284119	−	0.958789i	$-0.408299\pi$
$149$	6.93623	0.568238	0.284119	+	0.958789i	$0.408299\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	0	0
$154$	−1.95459	−0.157506
$155$	15.0229	1.20667
$156$	0	0
$157$	13.6969	1.09313	0.546567	−	0.837415i	$-0.315934\pi$
$157$	13.6969	1.09313	0.546567	+	0.837415i	$0.315934\pi$
$158$	10.6460	0.846954
$159$	0	0
$160$	19.3485	1.52963
$161$	6.93623	0.546651
$162$	0	0
$163$	−4.55051	−0.356423	−0.178212	−	0.983992i	$-0.557031\pi$
$163$	−4.55051	−0.356423	−0.178212	+	0.983992i	$0.557031\pi$
$164$	−4.30188	−0.335920
$165$	0	0
$166$	2.69694	0.209323
$167$	−12.0550	−0.932845	−0.466423	−	0.884562i	$-0.654457\pi$
$167$	−12.0550	−0.932845	−0.466423	+	0.884562i	$0.654457\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−16.1733	−1.24044
$171$	0	0
$172$	12.1010	0.922694
$173$	−9.57058	−0.727637	−0.363819	−	0.931470i	$-0.618527\pi$
$173$	−9.57058	−0.727637	−0.363819	+	0.931470i	$0.618527\pi$
$174$	0	0
$175$	8.55051	0.646358
$176$	−1.81743	−0.136994
$177$	0	0
$178$	−12.2474	−0.917985
$179$	17.9907	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$179$	17.9907	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$180$	0	0
$181$	11.1010	0.825132	0.412566	−	0.910928i	$-0.364633\pi$
$181$	11.1010	0.825132	0.412566	+	0.910928i	$0.364633\pi$
$182$	1.07547	0.0797191
$183$	0	0
$184$	−12.2474	−0.902894
$185$	19.4747	1.43181
$186$	0	0
$187$	12.0000	0.877527
$188$	−4.30188	−0.313747
$189$	0	0
$190$	0	0
$191$	−1.81743	−0.131505	−0.0657524	−	0.997836i	$-0.520945\pi$
$191$	−1.81743	−0.131505	−0.0657524	+	0.997836i	$0.520945\pi$
$192$	0	0
$193$	9.69694	0.698001	0.349000	−	0.937123i	$-0.386521\pi$
$193$	9.69694	0.698001	0.349000	+	0.937123i	$0.386521\pi$
$194$	−9.57058	−0.687127
$195$	0	0
$196$	7.10102	0.507216
$197$	−13.5389	−0.964610	−0.482305	−	0.876003i	$-0.660200\pi$
$197$	−13.5389	−0.964610	−0.482305	+	0.876003i	$0.660200\pi$
$198$	0	0
$199$	−14.3485	−1.01714	−0.508568	−	0.861022i	$-0.669825\pi$
$199$	−14.3485	−1.01714	−0.508568	+	0.861022i	$0.669825\pi$
$200$	−15.0978	−1.06758
$201$	0	0
$202$	4.89898	0.344691
$203$	13.8725	0.973655
$204$	0	0
$205$	9.79796	0.684319
$206$	−1.07547	−0.0749315
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−0.348469	−0.0239896	−0.0119948	−	0.999928i	$-0.503818\pi$
$211$	−0.348469	−0.0239896	−0.0119948	+	0.999928i	$0.503818\pi$
$212$	−4.78529	−0.328655
$213$	0	0
$214$	−9.30306	−0.635944
$215$	−27.5613	−1.87967
$216$	0	0
$217$	6.59592	0.447760
$218$	−0.667010	−0.0451756
$219$	0	0
$220$	8.69694	0.586347
$221$	−6.60272	−0.444147
$222$	0	0
$223$	14.3485	0.960845	0.480422	−	0.877037i	$-0.340483\pi$
$223$	14.3485	0.960845	0.480422	+	0.877037i	$0.340483\pi$
$224$	8.49511	0.567603
$225$	0	0
$226$	4.65153	0.309415
$227$	14.3559	0.952832	0.476416	−	0.879220i	$-0.341936\pi$
$227$	14.3559	0.952832	0.476416	+	0.879220i	$0.341936\pi$
$228$	0	0
$229$	8.79796	0.581385	0.290693	−	0.956816i	$-0.406114\pi$
$229$	8.79796	0.581385	0.290693	+	0.956816i	$0.406114\pi$
$230$	11.7215	0.772894
$231$	0	0
$232$	−24.4949	−1.60817
$233$	0.667010	0.0436973	0.0218486	−	0.999761i	$-0.493045\pi$
$233$	0.667010	0.0436973	0.0218486	+	0.999761i	$0.493045\pi$
$234$	0	0
$235$	9.79796	0.639148
$236$	−12.2049	−0.794473
$237$	0	0
$238$	−7.10102	−0.460291
$239$	−14.3559	−0.928604	−0.464302	−	0.885677i	$-0.653695\pi$
$239$	−14.3559	−0.928604	−0.464302	+	0.885677i	$0.653695\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	−5.71085	−0.367107
$243$	0	0
$244$	−7.24745	−0.463970
$245$	−16.1733	−1.03327
$246$	0	0
$247$	0	0
$248$	−11.6466	−0.739557
$249$	0	0
$250$	2.20204	0.139269
$251$	19.1412	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$251$	19.1412	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$252$	0	0
$253$	−8.69694	−0.546772
$254$	10.2376	0.642363
$255$	0	0
$256$	−12.1010	−0.756314
$257$	6.93623	0.432670	0.216335	−	0.976319i	$-0.430590\pi$
$257$	6.93623	0.432670	0.216335	+	0.976319i	$0.430590\pi$
$258$	0	0
$259$	8.55051	0.531303
$260$	−4.78529	−0.296771
$261$	0	0
$262$	12.4949	0.771937
$263$	−15.5063	−0.956159	−0.478079	−	0.878317i	$-0.658667\pi$
$263$	−15.5063	−0.956159	−0.478079	+	0.878317i	$0.658667\pi$
$264$	0	0
$265$	10.8990	0.669519
$266$	0	0
$267$	0	0
$268$	−20.7980	−1.27044
$269$	−15.8398	−0.965769	−0.482885	−	0.875684i	$-0.660411\pi$
$269$	−15.8398	−0.965769	−0.482885	+	0.875684i	$0.660411\pi$
$270$	0	0
$271$	−1.79796	−0.109218	−0.0546091	−	0.998508i	$-0.517391\pi$
$271$	−1.79796	−0.109218	−0.0546091	+	0.998508i	$0.517391\pi$
$272$	−6.60272	−0.400349
$273$	0	0
$274$	−12.4949	−0.754844
$275$	−10.7210	−0.646501
$276$	0	0
$277$	10.6969	0.642717	0.321358	−	0.946958i	$-0.395861\pi$
$277$	10.6969	0.642717	0.321358	+	0.946958i	$0.395861\pi$
$278$	9.16212	0.549507
$279$	0	0
$280$	−12.2474	−0.731925
$281$	6.93623	0.413781	0.206890	−	0.978364i	$-0.433666\pi$
$281$	6.93623	0.413781	0.206890	+	0.978364i	$0.433666\pi$
$282$	0	0
$283$	−25.7980	−1.53353	−0.766765	−	0.641928i	$-0.778135\pi$
$283$	−25.7980	−1.53353	−0.766765	+	0.641928i	$0.778135\pi$
$284$	13.8725	0.823179
$285$	0	0
$286$	−1.34847	−0.0797367
$287$	4.30188	0.253932
$288$	0	0
$289$	26.5959	1.56447
$290$	23.4430	1.37662
$291$	0	0
$292$	−7.24745	−0.424125
$293$	−32.3466	−1.88971	−0.944854	−	0.327492i	$-0.893797\pi$
$293$	−32.3466	−1.88971	−0.944854	+	0.327492i	$0.893797\pi$
$294$	0	0
$295$	27.7980	1.61846
$296$	−15.0978	−0.877543
$297$	0	0
$298$	5.14643	0.298125
$299$	4.78529	0.276740
$300$	0	0
$301$	−12.1010	−0.697491
$302$	2.96786	0.170781
$303$	0	0
$304$	0	0
$305$	16.5068	0.945177
$306$	0	0
$307$	−9.59592	−0.547668	−0.273834	−	0.961777i	$-0.588292\pi$
$307$	−9.59592	−0.547668	−0.273834	+	0.961777i	$0.588292\pi$
$308$	3.81846	0.217577
$309$	0	0
$310$	11.1464	0.633075
$311$	25.2605	1.43239	0.716195	−	0.697901i	$-0.245882\pi$
$311$	25.2605	1.43239	0.716195	+	0.697901i	$0.245882\pi$
$312$	0	0
$313$	12.8990	0.729093	0.364547	−	0.931185i	$-0.381224\pi$
$313$	12.8990	0.729093	0.364547	+	0.931185i	$0.381224\pi$
$314$	10.1626	0.573511
$315$	0	0
$316$	−20.7980	−1.16998
$317$	−0.333505	−0.0187315	−0.00936576	−	0.999956i	$-0.502981\pi$
$317$	−0.333505	−0.0187315	−0.00936576	+	0.999956i	$0.502981\pi$
$318$	0	0
$319$	−17.3939	−0.973870
$320$	7.75314	0.433414
$321$	0	0
$322$	5.14643	0.286799
$323$	0	0
$324$	0	0
$325$	5.89898	0.327217
$326$	−3.37631	−0.186997
$327$	0	0
$328$	−7.59592	−0.419414
$329$	4.30188	0.237170
$330$	0	0
$331$	−1.44949	−0.0796712	−0.0398356	−	0.999206i	$-0.512683\pi$
$331$	−1.44949	−0.0796712	−0.0398356	+	0.999206i	$0.512683\pi$
$332$	−5.26870	−0.289157
$333$	0	0
$334$	−8.94439	−0.489415
$335$	47.3695	2.58807
$336$	0	0
$337$	3.69694	0.201385	0.100693	−	0.994918i	$-0.467894\pi$
$337$	3.69694	0.201385	0.100693	+	0.994918i	$0.467894\pi$
$338$	−8.90357	−0.484290
$339$	0	0
$340$	31.5959	1.71353
$341$	−8.27025	−0.447859
$342$	0	0
$343$	−17.2474	−0.931275
$344$	21.3670	1.15203
$345$	0	0
$346$	−7.10102	−0.381753
$347$	−20.9586	−1.12512	−0.562558	−	0.826758i	$-0.690183\pi$
$347$	−20.9586	−1.12512	−0.562558	+	0.826758i	$0.690183\pi$
$348$	0	0
$349$	2.79796	0.149771	0.0748857	−	0.997192i	$-0.476141\pi$
$349$	2.79796	0.149771	0.0748857	+	0.997192i	$0.476141\pi$
$350$	6.34417	0.339110
$351$	0	0
$352$	−10.6515	−0.567728
$353$	−2.63435	−0.140212	−0.0701062	−	0.997540i	$-0.522334\pi$
$353$	−2.63435	−0.140212	−0.0701062	+	0.997540i	$0.522334\pi$
$354$	0	0
$355$	−31.5959	−1.67694
$356$	23.9264	1.26810
$357$	0	0
$358$	13.3485	0.705489
$359$	−19.1412	−1.01023	−0.505116	−	0.863052i	$-0.668550\pi$
$359$	−19.1412	−1.01023	−0.505116	+	0.863052i	$0.668550\pi$
$360$	0	0
$361$	0	0
$362$	8.23656	0.432904
$363$	0	0
$364$	−2.10102	−0.110123
$365$	16.5068	0.864005
$366$	0	0
$367$	−5.65153	−0.295008	−0.147504	−	0.989061i	$-0.547124\pi$
$367$	−5.65153	−0.295008	−0.147504	+	0.989061i	$0.547124\pi$
$368$	4.78529	0.249450
$369$	0	0
$370$	14.4495	0.751193
$371$	4.78529	0.248440
$372$	0	0
$373$	1.30306	0.0674700	0.0337350	−	0.999431i	$-0.489260\pi$
$373$	1.30306	0.0674700	0.0337350	+	0.999431i	$0.489260\pi$
$374$	8.90357	0.460392
$375$	0	0
$376$	−7.59592	−0.391730
$377$	9.57058	0.492910
$378$	0	0
$379$	−1.44949	−0.0744553	−0.0372276	−	0.999307i	$-0.511853\pi$
$379$	−1.44949	−0.0744553	−0.0372276	+	0.999307i	$0.511853\pi$
$380$	0	0
$381$	0	0
$382$	−1.34847	−0.0689937
$383$	−15.0229	−0.767633	−0.383816	−	0.923409i	$-0.625390\pi$
$383$	−15.0229	−0.767633	−0.383816	+	0.923409i	$0.625390\pi$
$384$	0	0
$385$	−8.69694	−0.443237
$386$	7.19478	0.366205
$387$	0	0
$388$	18.6969	0.949193
$389$	−6.93623	−0.351681	−0.175840	−	0.984419i	$-0.556264\pi$
$389$	−6.93623	−0.351681	−0.175840	+	0.984419i	$0.556264\pi$
$390$	0	0
$391$	−31.5959	−1.59787
$392$	12.5384	0.633286
$393$	0	0
$394$	−10.0454	−0.506080
$395$	47.3695	2.38342
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	−10.6460	−0.533638
$399$	0	0
$400$	5.89898	0.294949
$401$	−23.1095	−1.15403	−0.577017	−	0.816732i	$-0.695783\pi$
$401$	−23.1095	−1.15403	−0.577017	+	0.816732i	$0.695783\pi$
$402$	0	0
$403$	4.55051	0.226677
$404$	−9.57058	−0.476154
$405$	0	0
$406$	10.2929	0.510826
$407$	−10.7210	−0.531420
$408$	0	0
$409$	−12.8990	−0.637813	−0.318907	−	0.947786i	$-0.603316\pi$
$409$	−12.8990	−0.637813	−0.318907	+	0.947786i	$0.603316\pi$
$410$	7.26973	0.359026
$411$	0	0
$412$	2.10102	0.103510
$413$	12.2049	0.600565
$414$	0	0
$415$	12.0000	0.589057
$416$	5.86076	0.287347
$417$	0	0
$418$	0	0
$419$	−17.9907	−0.878905	−0.439452	−	0.898266i	$-0.644827\pi$
$419$	−17.9907	−0.878905	−0.439452	+	0.898266i	$0.644827\pi$
$420$	0	0
$421$	−12.8990	−0.628658	−0.314329	−	0.949314i	$-0.601779\pi$
$421$	−12.8990	−0.628658	−0.314329	+	0.949314i	$0.601779\pi$
$422$	−0.258552	−0.0125861
$423$	0	0
$424$	−8.44949	−0.410343
$425$	−38.9493	−1.88932
$426$	0	0
$427$	7.24745	0.350729
$428$	18.1743	0.878489
$429$	0	0
$430$	−20.4495	−0.986162
$431$	−2.96786	−0.142957	−0.0714783	−	0.997442i	$-0.522772\pi$
$431$	−2.96786	−0.142957	−0.0714783	+	0.997442i	$0.522772\pi$
$432$	0	0
$433$	−7.69694	−0.369891	−0.184946	−	0.982749i	$-0.559211\pi$
$433$	−7.69694	−0.369891	−0.184946	+	0.982749i	$0.559211\pi$
$434$	4.89393	0.234916
$435$	0	0
$436$	1.30306	0.0624053
$437$	0	0
$438$	0	0
$439$	23.6515	1.12883	0.564413	−	0.825493i	$-0.309103\pi$
$439$	23.6515	1.12883	0.564413	+	0.825493i	$0.309103\pi$
$440$	15.3564	0.732087
$441$	0	0
$442$	−4.89898	−0.233021
$443$	−18.4741	−0.877733	−0.438866	−	0.898552i	$-0.644620\pi$
$443$	−18.4741	−0.877733	−0.438866	+	0.898552i	$0.644620\pi$
$444$	0	0
$445$	−54.4949	−2.58331
$446$	10.6460	0.504105
$447$	0	0
$448$	3.40408	0.160828
$449$	−28.7117	−1.35499	−0.677495	−	0.735527i	$-0.736934\pi$
$449$	−28.7117	−1.35499	−0.677495	+	0.735527i	$0.736934\pi$
$450$	0	0
$451$	−5.39388	−0.253988
$452$	−9.08716	−0.427424
$453$	0	0
$454$	10.6515	0.499901
$455$	4.78529	0.224338
$456$	0	0
$457$	−29.8990	−1.39862	−0.699308	−	0.714821i	$-0.746508\pi$
$457$	−29.8990	−1.39862	−0.699308	+	0.714821i	$0.746508\pi$
$458$	6.52777	0.305023
$459$	0	0
$460$	−22.8990	−1.06767
$461$	−9.23707	−0.430213	−0.215107	−	0.976591i	$-0.569010\pi$
$461$	−9.23707	−0.430213	−0.215107	+	0.976591i	$0.569010\pi$
$462$	0	0
$463$	34.1464	1.58692	0.793460	−	0.608623i	$-0.208278\pi$
$463$	34.1464	1.58692	0.793460	+	0.608623i	$0.208278\pi$
$464$	9.57058	0.444303
$465$	0	0
$466$	0.494897	0.0229257
$467$	16.1733	0.748411	0.374205	−	0.927346i	$-0.377915\pi$
$467$	16.1733	0.748411	0.374205	+	0.927346i	$0.377915\pi$
$468$	0	0
$469$	20.7980	0.960361
$470$	7.26973	0.335328
$471$	0	0
$472$	−21.5505	−0.991943
$473$	15.1728	0.697645
$474$	0	0
$475$	0	0
$476$	13.8725	0.635843
$477$	0	0
$478$	−10.6515	−0.487190
$479$	−9.57058	−0.437291	−0.218645	−	0.975804i	$-0.570164\pi$
$479$	−9.57058	−0.437291	−0.218645	+	0.975804i	$0.570164\pi$
$480$	0	0
$481$	5.89898	0.268970
$482$	14.0973	0.642115
$483$	0	0
$484$	11.1566	0.507120
$485$	−42.5842	−1.93365
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−12.7970	−0.579292
$489$	0	0
$490$	−12.0000	−0.542105
$491$	−30.0457	−1.35595	−0.677973	−	0.735087i	$-0.737141\pi$
$491$	−30.0457	−1.35595	−0.677973	+	0.735087i	$0.737141\pi$
$492$	0	0
$493$	−63.1918	−2.84602
$494$	0	0
$495$	0	0
$496$	4.55051	0.204324
$497$	−13.8725	−0.622264
$498$	0	0
$499$	42.3485	1.89578	0.947889	−	0.318601i	$-0.103213\pi$
$499$	42.3485	1.89578	0.947889	+	0.318601i	$0.103213\pi$
$500$	−4.30188	−0.192386
$501$	0	0
$502$	14.2020	0.633868
$503$	−0.667010	−0.0297405	−0.0148703	−	0.999889i	$-0.504734\pi$
$503$	−0.667010	−0.0297405	−0.0148703	+	0.999889i	$0.504734\pi$
$504$	0	0
$505$	21.7980	0.969996
$506$	−6.45281	−0.286863
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−33.6806	−1.49287	−0.746433	−	0.665460i	$-0.768235\pi$
$509$	−33.6806	−1.49287	−0.746433	+	0.665460i	$0.768235\pi$
$510$	0	0
$511$	7.24745	0.320608
$512$	10.9795	0.485232
$513$	0	0
$514$	5.14643	0.226999
$515$	−4.78529	−0.210865
$516$	0	0
$517$	−5.39388	−0.237222
$518$	6.34417	0.278747
$519$	0	0
$520$	−8.44949	−0.370535
$521$	−9.90408	−0.433906	−0.216953	−	0.976182i	$-0.569612\pi$
$521$	−9.90408	−0.433906	−0.216953	+	0.976182i	$0.569612\pi$
$522$	0	0
$523$	5.65153	0.247124	0.123562	−	0.992337i	$-0.460568\pi$
$523$	5.65153	0.247124	0.123562	+	0.992337i	$0.460568\pi$
$524$	−24.4099	−1.06635
$525$	0	0
$526$	−11.5051	−0.501646
$527$	−30.0457	−1.30881
$528$	0	0
$529$	−0.101021	−0.00439220
$530$	8.08665	0.351262
$531$	0	0
$532$	0	0
$533$	2.96786	0.128552
$534$	0	0
$535$	−41.3939	−1.78961
$536$	−36.7234	−1.58621
$537$	0	0
$538$	−11.7526	−0.506688
$539$	8.90357	0.383504
$540$	0	0
$541$	19.6969	0.846838	0.423419	−	0.905934i	$-0.360830\pi$
$541$	19.6969	0.846838	0.423419	+	0.905934i	$0.360830\pi$
$542$	−1.33402	−0.0573011
$543$	0	0
$544$	−38.6969	−1.65912
$545$	−2.96786	−0.127129
$546$	0	0
$547$	32.3485	1.38312	0.691560	−	0.722319i	$-0.256924\pi$
$547$	32.3485	1.38312	0.691560	+	0.722319i	$0.256924\pi$
$548$	24.4099	1.04274
$549$	0	0
$550$	−7.95459	−0.339185
$551$	0	0
$552$	0	0
$553$	20.7980	0.884419
$554$	7.93674	0.337200
$555$	0	0
$556$	−17.8990	−0.759086
$557$	22.7760	0.965051	0.482525	−	0.875882i	$-0.339720\pi$
$557$	22.7760	0.965051	0.482525	+	0.875882i	$0.339720\pi$
$558$	0	0
$559$	−8.34847	−0.353103
$560$	4.78529	0.202215
$561$	0	0
$562$	5.14643	0.217089
$563$	−19.8082	−0.834814	−0.417407	−	0.908720i	$-0.637061\pi$
$563$	−19.8082	−0.834814	−0.417407	+	0.908720i	$0.637061\pi$
$564$	0	0
$565$	20.6969	0.870727
$566$	−19.1412	−0.804563
$567$	0	0
$568$	24.4949	1.02778
$569$	−8.90357	−0.373257	−0.186628	−	0.982431i	$-0.559756\pi$
$569$	−8.90357	−0.373257	−0.186628	+	0.982431i	$0.559756\pi$
$570$	0	0
$571$	−13.2474	−0.554388	−0.277194	−	0.960814i	$-0.589405\pi$
$571$	−13.2474	−0.554388	−0.277194	+	0.960814i	$0.589405\pi$
$572$	2.63435	0.110148
$573$	0	0
$574$	3.19184	0.133225
$575$	28.2283	1.17720
$576$	0	0
$577$	−6.69694	−0.278797	−0.139399	−	0.990236i	$-0.544517\pi$
$577$	−6.69694	−0.278797	−0.139399	+	0.990236i	$0.544517\pi$
$578$	19.7332	0.820793
$579$	0	0
$580$	−45.7980	−1.90166
$581$	5.26870	0.218583
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−12.7970	−0.529543
$585$	0	0
$586$	−24.0000	−0.991431
$587$	11.3880	0.470033	0.235017	−	0.971991i	$-0.424486\pi$
$587$	11.3880	0.470033	0.235017	+	0.971991i	$0.424486\pi$
$588$	0	0
$589$	0	0
$590$	20.6251	0.849121
$591$	0	0
$592$	5.89898	0.242447
$593$	−10.5711	−0.434103	−0.217051	−	0.976160i	$-0.569644\pi$
$593$	−10.5711	−0.434103	−0.217051	+	0.976160i	$0.569644\pi$
$594$	0	0
$595$	−31.5959	−1.29531
$596$	−10.0540	−0.411827
$597$	0	0
$598$	3.55051	0.145191
$599$	−31.8632	−1.30189	−0.650947	−	0.759123i	$-0.725628\pi$
$599$	−31.8632	−1.30189	−0.650947	+	0.759123i	$0.725628\pi$
$600$	0	0
$601$	−12.1010	−0.493611	−0.246805	−	0.969065i	$-0.579381\pi$
$601$	−12.1010	−0.493611	−0.246805	+	0.969065i	$0.579381\pi$
$602$	−8.97852	−0.365937
$603$	0	0
$604$	−5.79796	−0.235916
$605$	−25.4104	−1.03308
$606$	0	0
$607$	45.9444	1.86483	0.932413	−	0.361396i	$-0.117700\pi$
$607$	45.9444	1.86483	0.932413	+	0.361396i	$0.117700\pi$
$608$	0	0
$609$	0	0
$610$	12.2474	0.495885
$611$	2.96786	0.120067
$612$	0	0
$613$	12.8990	0.520985	0.260492	−	0.965476i	$-0.416115\pi$
$613$	12.8990	0.520985	0.260492	+	0.965476i	$0.416115\pi$
$614$	−7.11982	−0.287333
$615$	0	0
$616$	6.74235	0.271657
$617$	−23.1095	−0.930354	−0.465177	−	0.885218i	$-0.654009\pi$
$617$	−23.1095	−0.930354	−0.465177	+	0.885218i	$0.654009\pi$
$618$	0	0
$619$	−25.2474	−1.01478	−0.507390	−	0.861716i	$-0.669390\pi$
$619$	−25.2474	−1.01478	−0.507390	+	0.861716i	$0.669390\pi$
$620$	−21.7755	−0.874525
$621$	0	0
$622$	18.7423	0.751500
$623$	−23.9264	−0.958593
$624$	0	0
$625$	−19.6969	−0.787878
$626$	9.57058	0.382517
$627$	0	0
$628$	−19.8536	−0.792244
$629$	−38.9493	−1.55301
$630$	0	0
$631$	−8.34847	−0.332347	−0.166174	−	0.986097i	$-0.553141\pi$
$631$	−8.34847	−0.332347	−0.166174	+	0.986097i	$0.553141\pi$
$632$	−36.7234	−1.46078
$633$	0	0
$634$	−0.247449	−0.00982744
$635$	45.5520	1.80768
$636$	0	0
$637$	−4.89898	−0.194105
$638$	−12.9056	−0.510939
$639$	0	0
$640$	−32.9444	−1.30224
$641$	−14.8393	−0.586116	−0.293058	−	0.956095i	$-0.594673\pi$
$641$	−14.8393	−0.586116	−0.293058	+	0.956095i	$0.594673\pi$
$642$	0	0
$643$	18.3485	0.723593	0.361796	−	0.932257i	$-0.382164\pi$
$643$	18.3485	0.723593	0.361796	+	0.932257i	$0.382164\pi$
$644$	−10.0540	−0.396183
$645$	0	0
$646$	0	0
$647$	−12.5384	−0.492937	−0.246468	−	0.969151i	$-0.579270\pi$
$647$	−12.5384	−0.492937	−0.246468	+	0.969151i	$0.579270\pi$
$648$	0	0
$649$	−15.3031	−0.600698
$650$	4.37683	0.171673
$651$	0	0
$652$	6.59592	0.258316
$653$	−5.26870	−0.206180	−0.103090	−	0.994672i	$-0.532873\pi$
$653$	−5.26870	−0.206180	−0.103090	+	0.994672i	$0.532873\pi$
$654$	0	0
$655$	55.5959	2.17231
$656$	2.96786	0.115875
$657$	0	0
$658$	3.19184	0.124431
$659$	43.7346	1.70366	0.851829	−	0.523820i	$-0.175493\pi$
$659$	43.7346	1.70366	0.851829	+	0.523820i	$0.175493\pi$
$660$	0	0
$661$	37.1918	1.44659	0.723297	−	0.690537i	$-0.242626\pi$
$661$	37.1918	1.44659	0.723297	+	0.690537i	$0.242626\pi$
$662$	−1.07547	−0.0417993
$663$	0	0
$664$	−9.30306	−0.361029
$665$	0	0
$666$	0	0
$667$	45.7980	1.77330
$668$	17.4736	0.676075
$669$	0	0
$670$	35.1464	1.35782
$671$	−9.08716	−0.350806
$672$	0	0
$673$	−12.1010	−0.466460	−0.233230	−	0.972422i	$-0.574930\pi$
$673$	−12.1010	−0.466460	−0.233230	+	0.972422i	$0.574930\pi$
$674$	2.74299	0.105656
$675$	0	0
$676$	17.3939	0.668995
$677$	−7.26973	−0.279398	−0.139699	−	0.990194i	$-0.544614\pi$
$677$	−7.26973	−0.279398	−0.139699	+	0.990194i	$0.544614\pi$
$678$	0	0
$679$	−18.6969	−0.717523
$680$	55.7896	2.13943
$681$	0	0
$682$	−6.13622	−0.234968
$683$	17.9907	0.688396	0.344198	−	0.938897i	$-0.388151\pi$
$683$	17.9907	0.688396	0.344198	+	0.938897i	$0.388151\pi$
$684$	0	0
$685$	−55.5959	−2.12421
$686$	−12.7970	−0.488591
$687$	0	0
$688$	−8.34847	−0.318282
$689$	3.30136	0.125772
$690$	0	0
$691$	−33.3939	−1.27036	−0.635181	−	0.772363i	$-0.719075\pi$
$691$	−33.3939	−1.27036	−0.635181	+	0.772363i	$0.719075\pi$
$692$	13.8725	0.527351
$693$	0	0
$694$	−15.5505	−0.590289
$695$	40.7667	1.54637
$696$	0	0
$697$	−19.5959	−0.742248
$698$	2.07598	0.0785772
$699$	0	0
$700$	−12.3939	−0.468445
$701$	30.3793	1.14741	0.573704	−	0.819063i	$-0.305506\pi$
$701$	30.3793	1.14741	0.573704	+	0.819063i	$0.305506\pi$
$702$	0	0
$703$	0	0
$704$	−4.26818	−0.160863
$705$	0	0
$706$	−1.95459	−0.0735621
$707$	9.57058	0.359939
$708$	0	0
$709$	−30.3939	−1.14147	−0.570733	−	0.821136i	$-0.693341\pi$
$709$	−30.3939	−1.14147	−0.570733	+	0.821136i	$0.693341\pi$
$710$	−23.4430	−0.879801
$711$	0	0
$712$	42.2474	1.58329
$713$	21.7755	0.815499
$714$	0	0
$715$	−6.00000	−0.224387
$716$	−26.0774	−0.974557
$717$	0	0
$718$	−14.2020	−0.530015
$719$	−3.15145	−0.117529	−0.0587647	−	0.998272i	$-0.518716\pi$
$719$	−3.15145	−0.117529	−0.0587647	+	0.998272i	$0.518716\pi$
$720$	0	0
$721$	−2.10102	−0.0782461
$722$	0	0
$723$	0	0
$724$	−16.0908	−0.598010
$725$	56.4566	2.09675
$726$	0	0
$727$	3.04541	0.112948	0.0564740	−	0.998404i	$-0.482014\pi$
$727$	3.04541	0.112948	0.0564740	+	0.998404i	$0.482014\pi$
$728$	−3.70982	−0.137495
$729$	0	0
$730$	12.2474	0.453298
$731$	55.1226	2.03878
$732$	0	0
$733$	34.6969	1.28156	0.640780	−	0.767724i	$-0.278611\pi$
$733$	34.6969	1.28156	0.640780	+	0.767724i	$0.278611\pi$
$734$	−4.19323	−0.154775
$735$	0	0
$736$	28.0454	1.03377
$737$	−26.0774	−0.960573
$738$	0	0
$739$	15.0454	0.553454	0.276727	−	0.960949i	$-0.410750\pi$
$739$	15.0454	0.553454	0.276727	+	0.960949i	$0.410750\pi$
$740$	−28.2283	−1.03769
$741$	0	0
$742$	3.55051	0.130343
$743$	13.6889	0.502195	0.251098	−	0.967962i	$-0.419208\pi$
$743$	13.6889	0.502195	0.251098	+	0.967962i	$0.419208\pi$
$744$	0	0
$745$	22.8990	0.838954
$746$	0.966824	0.0353980
$747$	0	0
$748$	−17.3939	−0.635983
$749$	−18.1743	−0.664075
$750$	0	0
$751$	−9.04541	−0.330072	−0.165036	−	0.986288i	$-0.552774\pi$
$751$	−9.04541	−0.330072	−0.165036	+	0.986288i	$0.552774\pi$
$752$	2.96786	0.108227
$753$	0	0
$754$	7.10102	0.258604
$755$	13.2054	0.480595
$756$	0	0
$757$	−42.3939	−1.54083	−0.770416	−	0.637542i	$-0.779951\pi$
$757$	−42.3939	−1.54083	−0.770416	+	0.637542i	$0.779951\pi$
$758$	−1.07547	−0.0390628
$759$	0	0
$760$	0	0
$761$	−26.0774	−0.945304	−0.472652	−	0.881249i	$-0.656703\pi$
$761$	−26.0774	−0.945304	−0.472652	+	0.881249i	$0.656703\pi$
$762$	0	0
$763$	−1.30306	−0.0471740
$764$	2.63435	0.0953074
$765$	0	0
$766$	−11.1464	−0.402737
$767$	8.42015	0.304034
$768$	0	0
$769$	−30.3939	−1.09603	−0.548016	−	0.836468i	$-0.684617\pi$
$769$	−30.3939	−1.09603	−0.548016	+	0.836468i	$0.684617\pi$
$770$	−6.45281	−0.232543
$771$	0	0
$772$	−14.0556	−0.505873
$773$	−2.96786	−0.106746	−0.0533732	−	0.998575i	$-0.516997\pi$
$773$	−2.96786	−0.106746	−0.0533732	+	0.998575i	$0.516997\pi$
$774$	0	0
$775$	26.8434	0.964242
$776$	33.0136	1.18512
$777$	0	0
$778$	−5.14643	−0.184508
$779$	0	0
$780$	0	0
$781$	17.3939	0.622402
$782$	−23.4430	−0.838321
$783$	0	0
$784$	−4.89898	−0.174964
$785$	45.2185	1.61392
$786$	0	0
$787$	−22.7526	−0.811041	−0.405520	−	0.914086i	$-0.632910\pi$
$787$	−22.7526	−0.811041	−0.405520	+	0.914086i	$0.632910\pi$
$788$	19.6246	0.699096
$789$	0	0
$790$	35.1464	1.25045
$791$	9.08716	0.323102
$792$	0	0
$793$	5.00000	0.177555
$794$	−5.19375	−0.184319
$795$	0	0
$796$	20.7980	0.737164
$797$	−44.8850	−1.58991	−0.794955	−	0.606669i	$-0.792505\pi$
$797$	−44.8850	−1.58991	−0.794955	+	0.606669i	$0.792505\pi$
$798$	0	0
$799$	−19.5959	−0.693254
$800$	34.5725	1.22232
$801$	0	0
$802$	−17.1464	−0.605461
$803$	−9.08716	−0.320679
$804$	0	0
$805$	22.8990	0.807083
$806$	3.37631	0.118926
$807$	0	0
$808$	−16.8990	−0.594504
$809$	20.8087	0.731594	0.365797	−	0.930695i	$-0.380796\pi$
$809$	20.8087	0.731594	0.365797	+	0.930695i	$0.380796\pi$
$810$	0	0
$811$	1.79796	0.0631349	0.0315674	−	0.999502i	$-0.489950\pi$
$811$	1.79796	0.0631349	0.0315674	+	0.999502i	$0.489950\pi$
$812$	−20.1080	−0.705652
$813$	0	0
$814$	−7.95459	−0.278808
$815$	−15.0229	−0.526228
$816$	0	0
$817$	0	0
$818$	−9.57058	−0.334627
$819$	0	0
$820$	−14.2020	−0.495957
$821$	−2.30084	−0.0803000	−0.0401500	−	0.999194i	$-0.512784\pi$
$821$	−2.30084	−0.0803000	−0.0401500	+	0.999194i	$0.512784\pi$
$822$	0	0
$823$	−49.7980	−1.73585	−0.867924	−	0.496697i	$-0.834546\pi$
$823$	−49.7980	−1.73585	−0.867924	+	0.496697i	$0.834546\pi$
$824$	3.70982	0.129238
$825$	0	0
$826$	9.05561	0.315085
$827$	40.2833	1.40079	0.700394	−	0.713756i	$-0.253007\pi$
$827$	40.2833	1.40079	0.700394	+	0.713756i	$0.253007\pi$
$828$	0	0
$829$	−26.1918	−0.909680	−0.454840	−	0.890573i	$-0.650304\pi$
$829$	−26.1918	−0.909680	−0.454840	+	0.890573i	$0.650304\pi$
$830$	8.90357	0.309047
$831$	0	0
$832$	2.34847	0.0814185
$833$	32.3466	1.12074
$834$	0	0
$835$	−39.7980	−1.37727
$836$	0	0
$837$	0	0
$838$	−13.3485	−0.461115
$839$	−51.0043	−1.76087	−0.880433	−	0.474171i	$-0.842748\pi$
$839$	−51.0043	−1.76087	−0.880433	+	0.474171i	$0.842748\pi$
$840$	0	0
$841$	62.5959	2.15848
$842$	−9.57058	−0.329824
$843$	0	0
$844$	0.505103	0.0173863
$845$	−39.6163	−1.36284
$846$	0	0
$847$	−11.1566	−0.383346
$848$	3.30136	0.113369
$849$	0	0
$850$	−28.8990	−0.991227
$851$	28.2283	0.967654
$852$	0	0
$853$	−10.3031	−0.352770	−0.176385	−	0.984321i	$-0.556440\pi$
$853$	−10.3031	−0.352770	−0.176385	+	0.984321i	$0.556440\pi$
$854$	5.37734	0.184009
$855$	0	0
$856$	32.0908	1.09684
$857$	26.4109	0.902179	0.451089	−	0.892479i	$-0.351036\pi$
$857$	26.4109	0.902179	0.451089	+	0.892479i	$0.351036\pi$
$858$	0	0
$859$	−8.95459	−0.305527	−0.152763	−	0.988263i	$-0.548817\pi$
$859$	−8.95459	−0.305527	−0.152763	+	0.988263i	$0.548817\pi$
$860$	39.9498	1.36228
$861$	0	0
$862$	−2.20204	−0.0750018
$863$	−23.4430	−0.798010	−0.399005	−	0.916949i	$-0.630644\pi$
$863$	−23.4430	−0.798010	−0.399005	+	0.916949i	$0.630644\pi$
$864$	0	0
$865$	−31.5959	−1.07429
$866$	−5.71085	−0.194063
$867$	0	0
$868$	−9.56072	−0.324512
$869$	−26.0774	−0.884614
$870$	0	0
$871$	14.3485	0.486179
$872$	2.30084	0.0779164
$873$	0	0
$874$	0	0
$875$	4.30188	0.145430
$876$	0	0
$877$	−25.6969	−0.867724	−0.433862	−	0.900979i	$-0.642849\pi$
$877$	−25.6969	−0.867724	−0.433862	+	0.900979i	$0.642849\pi$
$878$	17.5486	0.592236
$879$	0	0
$880$	−6.00000	−0.202260
$881$	43.8845	1.47851	0.739253	−	0.673427i	$-0.235179\pi$
$881$	43.8845	1.47851	0.739253	+	0.673427i	$0.235179\pi$
$882$	0	0
$883$	15.6515	0.526716	0.263358	−	0.964698i	$-0.415170\pi$
$883$	15.6515	0.526716	0.263358	+	0.964698i	$0.415170\pi$
$884$	9.57058	0.321893
$885$	0	0
$886$	−13.7071	−0.460500
$887$	−39.1329	−1.31395	−0.656977	−	0.753910i	$-0.728165\pi$
$887$	−39.1329	−1.31395	−0.656977	+	0.753910i	$0.728165\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	−40.4332	−1.35533
$891$	0	0
$892$	−20.7980	−0.696367
$893$	0	0
$894$	0	0
$895$	59.3939	1.98532
$896$	−14.4645	−0.483225
$897$	0	0
$898$	−21.3031	−0.710892
$899$	43.5510	1.45251
$900$	0	0
$901$	−21.7980	−0.726195
$902$	−4.00206	−0.133254
$903$	0	0
$904$	−16.0454	−0.533662
$905$	36.6485	1.21824
$906$	0	0
$907$	43.1918	1.43416	0.717081	−	0.696990i	$-0.245478\pi$
$907$	43.1918	1.43416	0.717081	+	0.696990i	$0.245478\pi$
$908$	−20.8087	−0.690560
$909$	0	0
$910$	3.55051	0.117698
$911$	23.4430	0.776702	0.388351	−	0.921512i	$-0.373045\pi$
$911$	23.4430	0.776702	0.388351	+	0.921512i	$0.373045\pi$
$912$	0	0
$913$	−6.60612	−0.218631
$914$	−22.1840	−0.733780
$915$	0	0
$916$	−12.7526	−0.421356
$917$	24.4099	0.806084
$918$	0	0
$919$	4.75255	0.156772	0.0783861	−	0.996923i	$-0.475023\pi$
$919$	4.75255	0.156772	0.0783861	+	0.996923i	$0.475023\pi$
$920$	−40.4332	−1.33304
$921$	0	0
$922$	−6.85357	−0.225710
$923$	−9.57058	−0.315019
$924$	0	0
$925$	34.7980	1.14415
$926$	25.3354	0.832573
$927$	0	0
$928$	56.0908	1.84127
$929$	−50.4872	−1.65643	−0.828216	−	0.560409i	$-0.810644\pi$
$929$	−50.4872	−1.65643	−0.828216	+	0.560409i	$0.810644\pi$
$930$	0	0
$931$	0	0
$932$	−0.966824	−0.0316694
$933$	0	0
$934$	12.0000	0.392652
$935$	39.6163	1.29559
$936$	0	0
$937$	49.0908	1.60373	0.801864	−	0.597507i	$-0.203842\pi$
$937$	49.0908	1.60373	0.801864	+	0.597507i	$0.203842\pi$
$938$	15.4313	0.503851
$939$	0	0
$940$	−14.2020	−0.463220
$941$	45.5520	1.48495	0.742477	−	0.669872i	$-0.233651\pi$
$941$	45.5520	1.48495	0.742477	+	0.669872i	$0.233651\pi$
$942$	0	0
$943$	14.2020	0.462482
$944$	8.42015	0.274053
$945$	0	0
$946$	11.2577	0.366018
$947$	24.1100	0.783471	0.391735	−	0.920078i	$-0.371875\pi$
$947$	24.1100	0.783471	0.391735	+	0.920078i	$0.371875\pi$
$948$	0	0
$949$	5.00000	0.162307
$950$	0	0
$951$	0	0
$952$	24.4949	0.793884
$953$	25.4104	0.823123	0.411561	−	0.911382i	$-0.364984\pi$
$953$	25.4104	0.823123	0.411561	+	0.911382i	$0.364984\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	20.8087	0.673001
$957$	0	0
$958$	−7.10102	−0.229424
$959$	−24.4099	−0.788235
$960$	0	0
$961$	−10.2929	−0.332028
$962$	4.37683	0.141115
$963$	0	0
$964$	−27.5403	−0.887014
$965$	32.0131	1.03054
$966$	0	0
$967$	−5.65153	−0.181741	−0.0908705	−	0.995863i	$-0.528965\pi$
$967$	−5.65153	−0.181741	−0.0908705	+	0.995863i	$0.528965\pi$
$968$	19.6995	0.633166
$969$	0	0
$970$	−31.5959	−1.01448
$971$	35.3144	1.13329	0.566647	−	0.823960i	$-0.308240\pi$
$971$	35.3144	1.13329	0.566647	+	0.823960i	$0.308240\pi$
$972$	0	0
$973$	17.8990	0.573815
$974$	−5.93571	−0.190192
$975$	0	0
$976$	5.00000	0.160046
$977$	−16.1733	−0.517430	−0.258715	−	0.965954i	$-0.583299\pi$
$977$	−16.1733	−0.517430	−0.258715	+	0.965954i	$0.583299\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	23.4430	0.748860
$981$	0	0
$982$	−22.2929	−0.711394
$983$	−27.5613	−0.879069	−0.439535	−	0.898226i	$-0.644857\pi$
$983$	−27.5613	−0.879069	−0.439535	+	0.898226i	$0.644857\pi$
$984$	0	0
$985$	−44.6969	−1.42416
$986$	−46.8861	−1.49316
$987$	0	0
$988$	0	0
$989$	−39.9498	−1.27033
$990$	0	0
$991$	47.0454	1.49445	0.747223	−	0.664573i	$-0.231387\pi$
$991$	47.0454	1.49445	0.747223	+	0.664573i	$0.231387\pi$
$992$	26.6694	0.846755
$993$	0	0
$994$	−10.2929	−0.326470
$995$	−47.3695	−1.50171
$996$	0	0
$997$	43.0908	1.36470	0.682350	−	0.731026i	$-0.260958\pi$
$997$	43.0908	1.36470	0.682350	+	0.731026i	$0.260958\pi$
$998$	31.4210	0.994615
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.be.1.3		4
3.2	odd	2	inner	3249.2.a.be.1.2		4
19.8	odd	6		171.2.f.c.64.3	yes	8
19.12	odd	6		171.2.f.c.163.3	yes	8
19.18	odd	2		3249.2.a.bd.1.2		4
57.8	even	6		171.2.f.c.64.2	✓	8
57.50	even	6		171.2.f.c.163.2	yes	8
57.56	even	2		3249.2.a.bd.1.3		4
76.27	even	6		2736.2.s.bb.577.1		8
76.31	even	6		2736.2.s.bb.1873.1		8
228.107	odd	6		2736.2.s.bb.1873.4		8
228.179	odd	6		2736.2.s.bb.577.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.f.c.64.2	✓	8	57.8	even	6
171.2.f.c.64.3	yes	8	19.8	odd	6
171.2.f.c.163.2	yes	8	57.50	even	6
171.2.f.c.163.3	yes	8	19.12	odd	6
2736.2.s.bb.577.1		8	76.27	even	6
2736.2.s.bb.577.4		8	228.179	odd	6
2736.2.s.bb.1873.1		8	76.31	even	6
2736.2.s.bb.1873.4		8	228.107	odd	6
3249.2.a.bd.1.2		4	19.18	odd	2
3249.2.a.bd.1.3		4	57.56	even	2
3249.2.a.be.1.2		4	3.2	odd	2	inner
3249.2.a.be.1.3		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q+0.741964 q^{2} -1.44949 q^{4} +3.30136 q^{5} +1.44949 q^{7} -2.55940 q^{8} +2.44949 q^{10} -1.81743 q^{11} +1.00000 q^{13} +1.07547 q^{14} +1.00000 q^{16} -6.60272 q^{17} -4.78529 q^{20} -1.34847 q^{22} +4.78529 q^{23} +5.89898 q^{25} +0.741964 q^{26} -2.10102 q^{28} +9.57058 q^{29} +4.55051 q^{31} +5.86076 q^{32} -4.89898 q^{34} +4.78529 q^{35} +5.89898 q^{37} -8.44949 q^{40} +2.96786 q^{41} -8.34847 q^{43} +2.63435 q^{44} +3.55051 q^{46} +2.96786 q^{47} -4.89898 q^{49} +4.37683 q^{50} -1.44949 q^{52} +3.30136 q^{53} -6.00000 q^{55} -3.70982 q^{56} +7.10102 q^{58} +8.42015 q^{59} +5.00000 q^{61} +3.37631 q^{62} +2.34847 q^{64} +3.30136 q^{65} +14.3485 q^{67} +9.57058 q^{68} +3.55051 q^{70} -9.57058 q^{71} +5.00000 q^{73} +4.37683 q^{74} -2.63435 q^{77} +14.3485 q^{79} +3.30136 q^{80} +2.20204 q^{82} +3.63487 q^{83} -21.7980 q^{85} -6.19426 q^{86} +4.65153 q^{88} -16.5068 q^{89} +1.44949 q^{91} -6.93623 q^{92} +2.20204 q^{94} -12.8990 q^{97} -3.63487 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{7} + 4 q^{13} + 4 q^{16} + 24 q^{22} + 4 q^{25} - 28 q^{28} + 28 q^{31} + 4 q^{37} - 24 q^{40} - 4 q^{43} + 24 q^{46} + 4 q^{52} - 24 q^{55} + 48 q^{58} + 20 q^{61} - 20 q^{64} + 28 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^7 + 4 * q^13 + 4 * q^16 + 24 * q^22 + 4 * q^25 - 28 * q^28 + 28 * q^31 + 4 * q^37 - 24 * q^40 - 4 * q^43 + 24 * q^46 + 4 * q^52 - 24 * q^55 + 48 * q^58 + 20 * q^61 - 20 * q^64 + 28 * q^67 + 24 * q^70 + 20 * q^73 + 28 * q^79 + 48 * q^82 - 48 * q^85 + 48 * q^88 - 4 * q^91 + 48 * q^94 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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