LMFDB - Embedded newform 3249.2.a.be.1.1

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			1.1
Root			$-2.33441$ 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)

$f(q)$	$=$	$q-2.33441 q^{2} +3.44949 q^{4} +1.04930 q^{5} -3.44949 q^{7} -3.38371 q^{8} -2.44949 q^{10} -5.71812 q^{11} +1.00000 q^{13} +8.05254 q^{14} +1.00000 q^{16} -2.09859 q^{17} +3.61953 q^{20} +13.3485 q^{22} -3.61953 q^{23} -3.89898 q^{25} -2.33441 q^{26} -11.8990 q^{28} -7.23907 q^{29} +9.44949 q^{31} +4.43300 q^{32} +4.89898 q^{34} -3.61953 q^{35} -3.89898 q^{37} -3.55051 q^{40} -9.33766 q^{41} +6.34847 q^{43} -19.7246 q^{44} +8.44949 q^{46} -9.33766 q^{47} +4.89898 q^{49} +9.10183 q^{50} +3.44949 q^{52} +1.04930 q^{53} -6.00000 q^{55} +11.6721 q^{56} +16.8990 q^{58} +7.81671 q^{59} +5.00000 q^{61} -22.0590 q^{62} -12.3485 q^{64} +1.04930 q^{65} -0.348469 q^{67} -7.23907 q^{68} +8.44949 q^{70} +7.23907 q^{71} +5.00000 q^{73} +9.10183 q^{74} +19.7246 q^{77} -0.348469 q^{79} +1.04930 q^{80} +21.7980 q^{82} +11.4362 q^{83} -2.20204 q^{85} -14.8200 q^{86} +19.3485 q^{88} -5.24648 q^{89} -3.44949 q^{91} -12.4855 q^{92} +21.7980 q^{94} -3.10102 q^{97} -11.4362 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.33441	−1.65068	−0.825340	−	0.564636i	$-0.809017\pi$
$2$	−2.33441	−1.65068	−0.825340	+	0.564636i	$0.809017\pi$
$3$	0	0
$3$	0	0
$4$	3.44949	1.72474
$5$	1.04930	0.469259	0.234630	−	0.972085i	$-0.424612\pi$
$5$	1.04930	0.469259	0.234630	+	0.972085i	$0.424612\pi$
$6$	0	0
$7$	−3.44949	−1.30378	−0.651892	−	0.758312i	$-0.726025\pi$
$7$	−3.44949	−1.30378	−0.651892	+	0.758312i	$0.726025\pi$
$8$	−3.38371	−1.19632
$9$	0	0
$10$	−2.44949	−0.774597
$11$	−5.71812	−1.72408	−0.862040	−	0.506841i	$-0.830813\pi$
$11$	−5.71812	−1.72408	−0.862040	+	0.506841i	$0.830813\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	8.05254	2.15213
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.09859	−0.508983	−0.254491	−	0.967075i	$-0.581908\pi$
$17$	−2.09859	−0.508983	−0.254491	+	0.967075i	$0.581908\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.61953	0.809352
$21$	0	0
$22$	13.3485	2.84590
$23$	−3.61953	−0.754725	−0.377362	−	0.926066i	$-0.623169\pi$
$23$	−3.61953	−0.754725	−0.377362	+	0.926066i	$0.623169\pi$
$24$	0	0
$25$	−3.89898	−0.779796
$26$	−2.33441	−0.457816
$27$	0	0
$28$	−11.8990	−2.24870
$29$	−7.23907	−1.34426	−0.672130	−	0.740433i	$-0.734621\pi$
$29$	−7.23907	−1.34426	−0.672130	+	0.740433i	$0.734621\pi$
$30$	0	0
$31$	9.44949	1.69718	0.848589	−	0.529052i	$-0.177452\pi$
$31$	9.44949	1.69718	0.848589	+	0.529052i	$0.177452\pi$
$32$	4.43300	0.783652
$33$	0	0
$34$	4.89898	0.840168
$35$	−3.61953	−0.611813
$36$	0	0
$37$	−3.89898	−0.640988	−0.320494	−	0.947250i	$-0.603849\pi$
$37$	−3.89898	−0.640988	−0.320494	+	0.947250i	$0.603849\pi$
$38$	0	0
$39$	0	0
$40$	−3.55051	−0.561385
$41$	−9.33766	−1.45830	−0.729149	−	0.684356i	$-0.760084\pi$
$41$	−9.33766	−1.45830	−0.729149	+	0.684356i	$0.760084\pi$
$42$	0	0
$43$	6.34847	0.968132	0.484066	−	0.875031i	$-0.339159\pi$
$43$	6.34847	0.968132	0.484066	+	0.875031i	$0.339159\pi$
$44$	−19.7246	−2.97360
$45$	0	0
$46$	8.44949	1.24581
$47$	−9.33766	−1.36204	−0.681019	−	0.732266i	$-0.738463\pi$
$47$	−9.33766	−1.36204	−0.681019	+	0.732266i	$0.738463\pi$
$48$	0	0
$49$	4.89898	0.699854
$50$	9.10183	1.28719
$51$	0	0
$52$	3.44949	0.478358
$53$	1.04930	0.144132	0.0720659	−	0.997400i	$-0.477041\pi$
$53$	1.04930	0.144132	0.0720659	+	0.997400i	$0.477041\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	11.6721	1.55975
$57$	0	0
$58$	16.8990	2.21894
$59$	7.81671	1.01765	0.508825	−	0.860870i	$-0.330080\pi$
$59$	7.81671	1.01765	0.508825	+	0.860870i	$0.330080\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−22.0590	−2.80150
$63$	0	0
$64$	−12.3485	−1.54356
$65$	1.04930	0.130149
$66$	0	0
$67$	−0.348469	−0.0425723	−0.0212861	−	0.999773i	$-0.506776\pi$
$67$	−0.348469	−0.0425723	−0.0212861	+	0.999773i	$0.506776\pi$
$68$	−7.23907	−0.877866
$69$	0	0
$70$	8.44949	1.00991
$71$	7.23907	0.859119	0.429560	−	0.903039i	$-0.358669\pi$
$71$	7.23907	0.859119	0.429560	+	0.903039i	$0.358669\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	9.10183	1.05807
$75$	0	0
$76$	0	0
$77$	19.7246	2.24783
$78$	0	0
$79$	−0.348469	−0.0392059	−0.0196029	−	0.999808i	$-0.506240\pi$
$79$	−0.348469	−0.0392059	−0.0196029	+	0.999808i	$0.506240\pi$
$80$	1.04930	0.117315
$81$	0	0
$82$	21.7980	2.40718
$83$	11.4362	1.25529	0.627646	−	0.778499i	$-0.284019\pi$
$83$	11.4362	1.25529	0.627646	+	0.778499i	$0.284019\pi$
$84$	0	0
$85$	−2.20204	−0.238845
$86$	−14.8200	−1.59808
$87$	0	0
$88$	19.3485	2.06255
$89$	−5.24648	−0.556125	−0.278063	−	0.960563i	$-0.589692\pi$
$89$	−5.24648	−0.556125	−0.278063	+	0.960563i	$0.589692\pi$
$90$	0	0
$91$	−3.44949	−0.361605
$92$	−12.4855	−1.30171
$93$	0	0
$94$	21.7980	2.24829
$95$	0	0
$96$	0	0
$97$	−3.10102	−0.314861	−0.157430	−	0.987530i	$-0.550321\pi$
$97$	−3.10102	−0.314861	−0.157430	+	0.987530i	$0.550321\pi$
$98$	−11.4362	−1.15524
$99$	0	0
$100$	−13.4495	−1.34495
$101$	2.09859	0.208818	0.104409	−	0.994534i	$-0.466705\pi$
$101$	2.09859	0.208818	0.104409	+	0.994534i	$0.466705\pi$
$102$	0	0
$103$	3.44949	0.339888	0.169944	−	0.985454i	$-0.445641\pi$
$103$	3.44949	0.339888	0.169944	+	0.985454i	$0.445641\pi$
$104$	−3.38371	−0.331800
$105$	0	0
$106$	−2.44949	−0.237915
$107$	16.5767	1.60253	0.801266	−	0.598308i	$-0.204160\pi$
$107$	16.5767	1.60253	0.801266	+	0.598308i	$0.204160\pi$
$108$	0	0
$109$	8.89898	0.852368	0.426184	−	0.904637i	$-0.359858\pi$
$109$	8.89898	0.852368	0.426184	+	0.904637i	$0.359858\pi$
$110$	14.0065	1.33547
$111$	0	0
$112$	−3.44949	−0.325946
$113$	−8.28836	−0.779703	−0.389852	−	0.920878i	$-0.627474\pi$
$113$	−8.28836	−0.779703	−0.389852	+	0.920878i	$0.627474\pi$
$114$	0	0
$115$	−3.79796	−0.354162
$116$	−24.9711	−2.31851
$117$	0	0
$118$	−18.2474	−1.67981
$119$	7.23907	0.663604
$120$	0	0
$121$	21.6969	1.97245
$122$	−11.6721	−1.05674
$123$	0	0
$124$	32.5959	2.92720
$125$	−9.33766	−0.835185
$126$	0	0
$127$	−5.79796	−0.514486	−0.257243	−	0.966347i	$-0.582814\pi$
$127$	−5.79796	−0.514486	−0.257243	+	0.966347i	$0.582814\pi$
$128$	19.9604	1.76427
$129$	0	0
$130$	−2.44949	−0.214834
$131$	15.6334	1.36590	0.682949	−	0.730466i	$-0.260697\pi$
$131$	15.6334	1.36590	0.682949	+	0.730466i	$0.260697\pi$
$132$	0	0
$133$	0	0
$134$	0.813472	0.0702732
$135$	0	0
$136$	7.10102	0.608907
$137$	−15.6334	−1.33565	−0.667827	−	0.744317i	$-0.732775\pi$
$137$	−15.6334	−1.33565	−0.667827	+	0.744317i	$0.732775\pi$
$138$	0	0
$139$	−2.34847	−0.199195	−0.0995973	−	0.995028i	$-0.531755\pi$
$139$	−2.34847	−0.199195	−0.0995973	+	0.995028i	$0.531755\pi$
$140$	−12.4855	−1.05522
$141$	0	0
$142$	−16.8990	−1.41813
$143$	−5.71812	−0.478174
$144$	0	0
$145$	−7.59592	−0.630807
$146$	−11.6721	−0.965987
$147$	0	0
$148$	−13.4495	−1.10554
$149$	12.4855	1.02286	0.511428	−	0.859326i	$-0.329117\pi$
$149$	12.4855	1.02286	0.511428	+	0.859326i	$0.329117\pi$
$150$	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
$154$	−46.0454	−3.71044
$155$	9.91530	0.796416
$156$	0	0
$157$	−15.6969	−1.25275	−0.626376	−	0.779521i	$-0.715463\pi$
$157$	−15.6969	−1.25275	−0.626376	+	0.779521i	$0.715463\pi$
$158$	0.813472	0.0647163
$159$	0	0
$160$	4.65153	0.367736
$161$	12.4855	0.983999
$162$	0	0
$163$	−9.44949	−0.740141	−0.370071	−	0.929004i	$-0.620667\pi$
$163$	−9.44949	−0.740141	−0.370071	+	0.929004i	$0.620667\pi$
$164$	−32.2102	−2.51519
$165$	0	0
$166$	−26.6969	−2.07208
$167$	−19.2530	−1.48984	−0.744919	−	0.667154i	$-0.767512\pi$
$167$	−19.2530	−1.48984	−0.744919	+	0.667154i	$0.767512\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	5.14048	0.394257
$171$	0	0
$172$	21.8990	1.66978
$173$	7.23907	0.550376	0.275188	−	0.961390i	$-0.411260\pi$
$173$	7.23907	0.550376	0.275188	+	0.961390i	$0.411260\pi$
$174$	0	0
$175$	13.4495	1.01669
$176$	−5.71812	−0.431020
$177$	0	0
$178$	12.2474	0.917985
$179$	0.577648	0.0431754	0.0215877	−	0.999767i	$-0.493128\pi$
$179$	0.577648	0.0431754	0.0215877	+	0.999767i	$0.493128\pi$
$180$	0	0
$181$	20.8990	1.55341	0.776704	−	0.629865i	$-0.216890\pi$
$181$	20.8990	1.55341	0.776704	+	0.629865i	$0.216890\pi$
$182$	8.05254	0.596894
$183$	0	0
$184$	12.2474	0.902894
$185$	−4.09118	−0.300790
$186$	0	0
$187$	12.0000	0.877527
$188$	−32.2102	−2.34917
$189$	0	0
$190$	0	0
$191$	−5.71812	−0.413749	−0.206874	−	0.978367i	$-0.566329\pi$
$191$	−5.71812	−0.413749	−0.206874	+	0.978367i	$0.566329\pi$
$192$	0	0
$193$	−19.6969	−1.41782	−0.708908	−	0.705301i	$-0.750812\pi$
$193$	−19.6969	−1.41782	−0.708908	+	0.705301i	$0.750812\pi$
$194$	7.23907	0.519735
$195$	0	0
$196$	16.8990	1.20707
$197$	−14.5841	−1.03908	−0.519538	−	0.854447i	$-0.673896\pi$
$197$	−14.5841	−1.03908	−0.519538	+	0.854447i	$0.673896\pi$
$198$	0	0
$199$	0.348469	0.0247023	0.0123512	−	0.999924i	$-0.496068\pi$
$199$	0.348469	0.0247023	0.0123512	+	0.999924i	$0.496068\pi$
$200$	13.1930	0.932887
$201$	0	0
$202$	−4.89898	−0.344691
$203$	24.9711	1.75263
$204$	0	0
$205$	−9.79796	−0.684319
$206$	−8.05254	−0.561047
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	14.3485	0.987789	0.493895	−	0.869522i	$-0.335573\pi$
$211$	14.3485	0.987789	0.493895	+	0.869522i	$0.335573\pi$
$212$	3.61953	0.248591
$213$	0	0
$214$	−38.6969	−2.64527
$215$	6.66142	0.454305
$216$	0	0
$217$	−32.5959	−2.21276
$218$	−20.7739	−1.40699
$219$	0	0
$220$	−20.6969	−1.39539
$221$	−2.09859	−0.141166
$222$	0	0
$223$	−0.348469	−0.0233352	−0.0116676	−	0.999932i	$-0.503714\pi$
$223$	−0.348469	−0.0233352	−0.0116676	+	0.999932i	$0.503714\pi$
$224$	−15.2916	−1.02171
$225$	0	0
$226$	19.3485	1.28704
$227$	−10.8586	−0.720711	−0.360355	−	0.932815i	$-0.617345\pi$
$227$	−10.8586	−0.720711	−0.360355	+	0.932815i	$0.617345\pi$
$228$	0	0
$229$	−10.7980	−0.713549	−0.356775	−	0.934190i	$-0.616124\pi$
$229$	−10.7980	−0.713549	−0.356775	+	0.934190i	$0.616124\pi$
$230$	8.86601	0.584607
$231$	0	0
$232$	24.4949	1.60817
$233$	20.7739	1.36094	0.680472	−	0.732774i	$-0.261775\pi$
$233$	20.7739	1.36094	0.680472	+	0.732774i	$0.261775\pi$
$234$	0	0
$235$	−9.79796	−0.639148
$236$	26.9637	1.75519
$237$	0	0
$238$	−16.8990	−1.09540
$239$	10.8586	0.702384	0.351192	−	0.936303i	$-0.385776\pi$
$239$	10.8586	0.702384	0.351192	+	0.936303i	$0.385776\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	−50.6496	−3.25588
$243$	0	0
$244$	17.2474	1.10415
$245$	5.14048	0.328413
$246$	0	0
$247$	0	0
$248$	−31.9743	−2.03037
$249$	0	0
$250$	21.7980	1.37862
$251$	−14.4781	−0.913852	−0.456926	−	0.889505i	$-0.651050\pi$
$251$	−14.4781	−0.913852	−0.456926	+	0.889505i	$0.651050\pi$
$252$	0	0
$253$	20.6969	1.30121
$254$	13.5348	0.849251
$255$	0	0
$256$	−21.8990	−1.36869
$257$	12.4855	0.778827	0.389413	−	0.921063i	$-0.372678\pi$
$257$	12.4855	0.778827	0.389413	+	0.921063i	$0.372678\pi$
$258$	0	0
$259$	13.4495	0.835711
$260$	3.61953	0.224474
$261$	0	0
$262$	−36.4949	−2.25466
$263$	25.9144	1.59795	0.798975	−	0.601365i	$-0.205376\pi$
$263$	25.9144	1.59795	0.798975	+	0.601365i	$0.205376\pi$
$264$	0	0
$265$	1.10102	0.0676352
$266$	0	0
$267$	0	0
$268$	−1.20204	−0.0734263
$269$	15.5274	0.946724	0.473362	−	0.880868i	$-0.343040\pi$
$269$	15.5274	0.946724	0.473362	+	0.880868i	$0.343040\pi$
$270$	0	0
$271$	17.7980	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$271$	17.7980	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$272$	−2.09859	−0.127246
$273$	0	0
$274$	36.4949	2.20474
$275$	22.2948	1.34443
$276$	0	0
$277$	−18.6969	−1.12339	−0.561695	−	0.827344i	$-0.689851\pi$
$277$	−18.6969	−1.12339	−0.561695	+	0.827344i	$0.689851\pi$
$278$	5.48230	0.328807
$279$	0	0
$280$	12.2474	0.731925
$281$	12.4855	0.744825	0.372413	−	0.928067i	$-0.378531\pi$
$281$	12.4855	0.744825	0.372413	+	0.928067i	$0.378531\pi$
$282$	0	0
$283$	−6.20204	−0.368673	−0.184337	−	0.982863i	$-0.559014\pi$
$283$	−6.20204	−0.368673	−0.184337	+	0.982863i	$0.559014\pi$
$284$	24.9711	1.48176
$285$	0	0
$286$	13.3485	0.789312
$287$	32.2102	1.90131
$288$	0	0
$289$	−12.5959	−0.740936
$290$	17.7320	1.04126
$291$	0	0
$292$	17.2474	1.00933
$293$	10.2810	0.600620	0.300310	−	0.953842i	$-0.402910\pi$
$293$	10.2810	0.600620	0.300310	+	0.953842i	$0.402910\pi$
$294$	0	0
$295$	8.20204	0.477541
$296$	13.1930	0.766828
$297$	0	0
$298$	−29.1464	−1.68841
$299$	−3.61953	−0.209323
$300$	0	0
$301$	−21.8990	−1.26224
$302$	−9.33766	−0.537322
$303$	0	0
$304$	0	0
$305$	5.24648	0.300412
$306$	0	0
$307$	29.5959	1.68913	0.844564	−	0.535454i	$-0.179860\pi$
$307$	29.5959	1.68913	0.844564	+	0.535454i	$0.179860\pi$
$308$	68.0398	3.87693
$309$	0	0
$310$	−23.1464	−1.31463
$311$	23.4501	1.32974	0.664868	−	0.746961i	$-0.268488\pi$
$311$	23.4501	1.32974	0.664868	+	0.746961i	$0.268488\pi$
$312$	0	0
$313$	3.10102	0.175280	0.0876400	−	0.996152i	$-0.472067\pi$
$313$	3.10102	0.175280	0.0876400	+	0.996152i	$0.472067\pi$
$314$	36.6432	2.06789
$315$	0	0
$316$	−1.20204	−0.0676201
$317$	−10.3870	−0.583389	−0.291695	−	0.956511i	$-0.594219\pi$
$317$	−10.3870	−0.583389	−0.291695	+	0.956511i	$0.594219\pi$
$318$	0	0
$319$	41.3939	2.31761
$320$	−12.9572	−0.724329
$321$	0	0
$322$	−29.1464	−1.62427
$323$	0	0
$324$	0	0
$325$	−3.89898	−0.216276
$326$	22.0590	1.22174
$327$	0	0
$328$	31.5959	1.74459
$329$	32.2102	1.77580
$330$	0	0
$331$	3.44949	0.189601	0.0948006	−	0.995496i	$-0.469779\pi$
$331$	3.44949	0.189601	0.0948006	+	0.995496i	$0.469779\pi$
$332$	39.4492	2.16506
$333$	0	0
$334$	44.9444	2.45925
$335$	−0.365647	−0.0199774
$336$	0	0
$337$	−25.6969	−1.39980	−0.699901	−	0.714240i	$-0.746772\pi$
$337$	−25.6969	−1.39980	−0.699901	+	0.714240i	$0.746772\pi$
$338$	28.0130	1.52370
$339$	0	0
$340$	−7.59592	−0.411946
$341$	−54.0334	−2.92607
$342$	0	0
$343$	7.24745	0.391325
$344$	−21.4814	−1.15820
$345$	0	0
$346$	−16.8990	−0.908495
$347$	8.76001	0.470262	0.235131	−	0.971964i	$-0.424448\pi$
$347$	8.76001	0.470262	0.235131	+	0.971964i	$0.424448\pi$
$348$	0	0
$349$	−16.7980	−0.899174	−0.449587	−	0.893237i	$-0.648429\pi$
$349$	−16.7980	−0.899174	−0.449587	+	0.893237i	$0.648429\pi$
$350$	−31.3967	−1.67822
$351$	0	0
$352$	−25.3485	−1.35108
$353$	19.7246	1.04984	0.524918	−	0.851153i	$-0.324096\pi$
$353$	19.7246	1.04984	0.524918	+	0.851153i	$0.324096\pi$
$354$	0	0
$355$	7.59592	0.403149
$356$	−18.0977	−0.959174
$357$	0	0
$358$	−1.34847	−0.0712688
$359$	14.4781	0.764127	0.382063	−	0.924136i	$-0.375214\pi$
$359$	14.4781	0.764127	0.382063	+	0.924136i	$0.375214\pi$
$360$	0	0
$361$	0	0
$362$	−48.7869	−2.56418
$363$	0	0
$364$	−11.8990	−0.623676
$365$	5.24648	0.274613
$366$	0	0
$367$	−20.3485	−1.06218	−0.531091	−	0.847315i	$-0.678218\pi$
$367$	−20.3485	−1.06218	−0.531091	+	0.847315i	$0.678218\pi$
$368$	−3.61953	−0.188681
$369$	0	0
$370$	9.55051	0.496507
$371$	−3.61953	−0.187917
$372$	0	0
$373$	30.6969	1.58943	0.794714	−	0.606985i	$-0.207621\pi$
$373$	30.6969	1.58943	0.794714	+	0.606985i	$0.207621\pi$
$374$	−28.0130	−1.44852
$375$	0	0
$376$	31.5959	1.62944
$377$	−7.23907	−0.372831
$378$	0	0
$379$	3.44949	0.177188	0.0885942	−	0.996068i	$-0.471763\pi$
$379$	3.44949	0.177188	0.0885942	+	0.996068i	$0.471763\pi$
$380$	0	0
$381$	0	0
$382$	13.3485	0.682967
$383$	−9.91530	−0.506648	−0.253324	−	0.967381i	$-0.581524\pi$
$383$	−9.91530	−0.506648	−0.253324	+	0.967381i	$0.581524\pi$
$384$	0	0
$385$	20.6969	1.05481
$386$	45.9808	2.34036
$387$	0	0
$388$	−10.6969	−0.543055
$389$	−12.4855	−0.633042	−0.316521	−	0.948585i	$-0.602515\pi$
$389$	−12.4855	−0.633042	−0.316521	+	0.948585i	$0.602515\pi$
$390$	0	0
$391$	7.59592	0.384142
$392$	−16.5767	−0.837251
$393$	0	0
$394$	34.0454	1.71518
$395$	−0.365647	−0.0183977
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	−0.813472	−0.0407756
$399$	0	0
$400$	−3.89898	−0.194949
$401$	−7.34507	−0.366795	−0.183398	−	0.983039i	$-0.558710\pi$
$401$	−7.34507	−0.366795	−0.183398	+	0.983039i	$0.558710\pi$
$402$	0	0
$403$	9.44949	0.470713
$404$	7.23907	0.360157
$405$	0	0
$406$	−58.2929	−2.89303
$407$	22.2948	1.10511
$408$	0	0
$409$	−3.10102	−0.153336	−0.0766678	−	0.997057i	$-0.524428\pi$
$409$	−3.10102	−0.153336	−0.0766678	+	0.997057i	$0.524428\pi$
$410$	22.8725	1.12959
$411$	0	0
$412$	11.8990	0.586221
$413$	−26.9637	−1.32680
$414$	0	0
$415$	12.0000	0.589057
$416$	4.43300	0.217346
$417$	0	0
$418$	0	0
$419$	−0.577648	−0.0282199	−0.0141100	−	0.999900i	$-0.504491\pi$
$419$	−0.577648	−0.0282199	−0.0141100	+	0.999900i	$0.504491\pi$
$420$	0	0
$421$	−3.10102	−0.151134	−0.0755672	−	0.997141i	$-0.524077\pi$
$421$	−3.10102	−0.151134	−0.0755672	+	0.997141i	$0.524077\pi$
$422$	−33.4953	−1.63052
$423$	0	0
$424$	−3.55051	−0.172428
$425$	8.18236	0.396903
$426$	0	0
$427$	−17.2474	−0.834663
$428$	57.1812	2.76396
$429$	0	0
$430$	−15.5505	−0.749912
$431$	9.33766	0.449779	0.224890	−	0.974384i	$-0.427798\pi$
$431$	9.33766	0.449779	0.224890	+	0.974384i	$0.427798\pi$
$432$	0	0
$433$	21.6969	1.04269	0.521344	−	0.853347i	$-0.325431\pi$
$433$	21.6969	1.04269	0.521344	+	0.853347i	$0.325431\pi$
$434$	76.0924	3.65255
$435$	0	0
$436$	30.6969	1.47012
$437$	0	0
$438$	0	0
$439$	38.3485	1.83027	0.915136	−	0.403145i	$-0.132083\pi$
$439$	38.3485	1.83027	0.915136	+	0.403145i	$0.132083\pi$
$440$	20.3023	0.967872
$441$	0	0
$442$	4.89898	0.233021
$443$	35.2520	1.67487	0.837437	−	0.546533i	$-0.184053\pi$
$443$	35.2520	1.67487	0.837437	+	0.546533i	$0.184053\pi$
$444$	0	0
$445$	−5.50510	−0.260967
$446$	0.813472	0.0385190
$447$	0	0
$448$	42.5959	2.01247
$449$	21.7172	1.02490	0.512449	−	0.858718i	$-0.328738\pi$
$449$	21.7172	1.02490	0.512449	+	0.858718i	$0.328738\pi$
$450$	0	0
$451$	53.3939	2.51422
$452$	−28.5906	−1.34479
$453$	0	0
$454$	25.3485	1.18966
$455$	−3.61953	−0.169686
$456$	0	0
$457$	−20.1010	−0.940286	−0.470143	−	0.882590i	$-0.655798\pi$
$457$	−20.1010	−0.940286	−0.470143	+	0.882590i	$0.655798\pi$
$458$	25.2069	1.17784
$459$	0	0
$460$	−13.1010	−0.610838
$461$	17.6260	0.820926	0.410463	−	0.911877i	$-0.365367\pi$
$461$	17.6260	0.820926	0.410463	+	0.911877i	$0.365367\pi$
$462$	0	0
$463$	−0.146428	−0.00680510	−0.00340255	−	0.999994i	$-0.501083\pi$
$463$	−0.146428	−0.00680510	−0.00340255	+	0.999994i	$0.501083\pi$
$464$	−7.23907	−0.336065
$465$	0	0
$466$	−48.4949	−2.24648
$467$	−5.14048	−0.237873	−0.118936	−	0.992902i	$-0.537948\pi$
$467$	−5.14048	−0.237873	−0.118936	+	0.992902i	$0.537948\pi$
$468$	0	0
$469$	1.20204	0.0555051
$470$	22.8725	1.05503
$471$	0	0
$472$	−26.4495	−1.21744
$473$	−36.3013	−1.66914
$474$	0	0
$475$	0	0
$476$	24.9711	1.14455
$477$	0	0
$478$	−25.3485	−1.15941
$479$	7.23907	0.330761	0.165381	−	0.986230i	$-0.447115\pi$
$479$	7.23907	0.330761	0.165381	+	0.986230i	$0.447115\pi$
$480$	0	0
$481$	−3.89898	−0.177778
$482$	−44.3539	−2.02026
$483$	0	0
$484$	74.8434	3.40197
$485$	−3.25389	−0.147751
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−16.9185	−0.765867
$489$	0	0
$490$	−12.0000	−0.542105
$491$	−19.8306	−0.894943	−0.447471	−	0.894298i	$-0.647675\pi$
$491$	−19.8306	−0.894943	−0.447471	+	0.894298i	$0.647675\pi$
$492$	0	0
$493$	15.1918	0.684206
$494$	0	0
$495$	0	0
$496$	9.44949	0.424295
$497$	−24.9711	−1.12011
$498$	0	0
$499$	27.6515	1.23785	0.618926	−	0.785449i	$-0.287568\pi$
$499$	27.6515	1.23785	0.618926	+	0.785449i	$0.287568\pi$
$500$	−32.2102	−1.44048
$501$	0	0
$502$	33.7980	1.50848
$503$	−20.7739	−0.926263	−0.463131	−	0.886290i	$-0.653274\pi$
$503$	−20.7739	−0.926263	−0.463131	+	0.886290i	$0.653274\pi$
$504$	0	0
$505$	2.20204	0.0979895
$506$	−48.3152	−2.14787
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−31.2669	−1.38588	−0.692940	−	0.720996i	$-0.743685\pi$
$509$	−31.2669	−1.38588	−0.692940	+	0.720996i	$0.743685\pi$
$510$	0	0
$511$	−17.2474	−0.762982
$512$	11.2004	0.494993
$513$	0	0
$514$	−29.1464	−1.28559
$515$	3.61953	0.159496
$516$	0	0
$517$	53.3939	2.34826
$518$	−31.3967	−1.37949
$519$	0	0
$520$	−3.55051	−0.155700
$521$	−3.14789	−0.137911	−0.0689557	−	0.997620i	$-0.521967\pi$
$521$	−3.14789	−0.137911	−0.0689557	+	0.997620i	$0.521967\pi$
$522$	0	0
$523$	20.3485	0.889776	0.444888	−	0.895586i	$-0.353243\pi$
$523$	20.3485	0.889776	0.444888	+	0.895586i	$0.353243\pi$
$524$	53.9274	2.35583
$525$	0	0
$526$	−60.4949	−2.63770
$527$	−19.8306	−0.863835
$528$	0	0
$529$	−9.89898	−0.430390
$530$	−2.57024	−0.111644
$531$	0	0
$532$	0	0
$533$	−9.33766	−0.404459
$534$	0	0
$535$	17.3939	0.752003
$536$	1.17912	0.0509302
$537$	0	0
$538$	−36.2474	−1.56274
$539$	−28.0130	−1.20660
$540$	0	0
$541$	−9.69694	−0.416904	−0.208452	−	0.978033i	$-0.566842\pi$
$541$	−9.69694	−0.416904	−0.208452	+	0.978033i	$0.566842\pi$
$542$	−41.5478	−1.78463
$543$	0	0
$544$	−9.30306	−0.398865
$545$	9.33766	0.399981
$546$	0	0
$547$	17.6515	0.754725	0.377362	−	0.926066i	$-0.376831\pi$
$547$	17.6515	0.754725	0.377362	+	0.926066i	$0.376831\pi$
$548$	−53.9274	−2.30366
$549$	0	0
$550$	−52.0454	−2.21922
$551$	0	0
$552$	0	0
$553$	1.20204	0.0511160
$554$	43.6464	1.85436
$555$	0	0
$556$	−8.10102	−0.343560
$557$	−3.04189	−0.128889	−0.0644444	−	0.997921i	$-0.520528\pi$
$557$	−3.04189	−0.128889	−0.0644444	+	0.997921i	$0.520528\pi$
$558$	0	0
$559$	6.34847	0.268512
$560$	−3.61953	−0.152953
$561$	0	0
$562$	−29.1464	−1.22947
$563$	−6.29577	−0.265335	−0.132668	−	0.991161i	$-0.542354\pi$
$563$	−6.29577	−0.265335	−0.132668	+	0.991161i	$0.542354\pi$
$564$	0	0
$565$	−8.69694	−0.365883
$566$	14.4781	0.608561
$567$	0	0
$568$	−24.4949	−1.02778
$569$	28.0130	1.17436	0.587182	−	0.809455i	$-0.300237\pi$
$569$	28.0130	1.17436	0.587182	+	0.809455i	$0.300237\pi$
$570$	0	0
$571$	11.2474	0.470691	0.235346	−	0.971912i	$-0.424378\pi$
$571$	11.2474	0.470691	0.235346	+	0.971912i	$0.424378\pi$
$572$	−19.7246	−0.824727
$573$	0	0
$574$	−75.1918	−3.13845
$575$	14.1125	0.588531
$576$	0	0
$577$	22.6969	0.944886	0.472443	−	0.881361i	$-0.343372\pi$
$577$	22.6969	0.944886	0.472443	+	0.881361i	$0.343372\pi$
$578$	29.4041	1.22305
$579$	0	0
$580$	−26.2020	−1.08798
$581$	−39.4492	−1.63663
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−16.9185	−0.700094
$585$	0	0
$586$	−24.0000	−0.991431
$587$	−1.52094	−0.0627760	−0.0313880	−	0.999507i	$-0.509993\pi$
$587$	−1.52094	−0.0627760	−0.0313880	+	0.999507i	$0.509993\pi$
$588$	0	0
$589$	0	0
$590$	−19.1470	−0.788268
$591$	0	0
$592$	−3.89898	−0.160247
$593$	−23.9218	−0.982350	−0.491175	−	0.871061i	$-0.663432\pi$
$593$	−23.9218	−0.982350	−0.491175	+	0.871061i	$0.663432\pi$
$594$	0	0
$595$	7.59592	0.311402
$596$	43.0688	1.76416
$597$	0	0
$598$	8.44949	0.345525
$599$	−25.5487	−1.04389	−0.521946	−	0.852978i	$-0.674794\pi$
$599$	−25.5487	−1.04389	−0.521946	+	0.852978i	$0.674794\pi$
$600$	0	0
$601$	−21.8990	−0.893278	−0.446639	−	0.894714i	$-0.647379\pi$
$601$	−21.8990	−0.893278	−0.446639	+	0.894714i	$0.647379\pi$
$602$	51.1213	2.08355
$603$	0	0
$604$	13.7980	0.561431
$605$	22.7665	0.925590
$606$	0	0
$607$	−7.94439	−0.322453	−0.161226	−	0.986917i	$-0.551545\pi$
$607$	−7.94439	−0.322453	−0.161226	+	0.986917i	$0.551545\pi$
$608$	0	0
$609$	0	0
$610$	−12.2474	−0.495885
$611$	−9.33766	−0.377761
$612$	0	0
$613$	3.10102	0.125249	0.0626245	−	0.998037i	$-0.480053\pi$
$613$	3.10102	0.125249	0.0626245	+	0.998037i	$0.480053\pi$
$614$	−69.0891	−2.78821
$615$	0	0
$616$	−66.7423	−2.68913
$617$	−7.34507	−0.295701	−0.147851	−	0.989010i	$-0.547235\pi$
$617$	−7.34507	−0.295701	−0.147851	+	0.989010i	$0.547235\pi$
$618$	0	0
$619$	−0.752551	−0.0302476	−0.0151238	−	0.999886i	$-0.504814\pi$
$619$	−0.752551	−0.0302476	−0.0151238	+	0.999886i	$0.504814\pi$
$620$	34.2027	1.37362
$621$	0	0
$622$	−54.7423	−2.19497
$623$	18.0977	0.725068
$624$	0	0
$625$	9.69694	0.387878
$626$	−7.23907	−0.289331
$627$	0	0
$628$	−54.1464	−2.16068
$629$	8.18236	0.326252
$630$	0	0
$631$	6.34847	0.252729	0.126364	−	0.991984i	$-0.459669\pi$
$631$	6.34847	0.252729	0.126364	+	0.991984i	$0.459669\pi$
$632$	1.17912	0.0469028
$633$	0	0
$634$	24.2474	0.962989
$635$	−6.08377	−0.241427
$636$	0	0
$637$	4.89898	0.194105
$638$	−96.6305	−3.82564
$639$	0	0
$640$	20.9444	0.827900
$641$	46.6883	1.84408	0.922038	−	0.387099i	$-0.126523\pi$
$641$	46.6883	1.84408	0.922038	+	0.387099i	$0.126523\pi$
$642$	0	0
$643$	3.65153	0.144002	0.0720012	−	0.997405i	$-0.477061\pi$
$643$	3.65153	0.144002	0.0720012	+	0.997405i	$0.477061\pi$
$644$	43.0688	1.69715
$645$	0	0
$646$	0	0
$647$	16.5767	0.651698	0.325849	−	0.945422i	$-0.394350\pi$
$647$	16.5767	0.651698	0.325849	+	0.945422i	$0.394350\pi$
$648$	0	0
$649$	−44.6969	−1.75451
$650$	9.10183	0.357003
$651$	0	0
$652$	−32.5959	−1.27655
$653$	39.4492	1.54377	0.771884	−	0.635764i	$-0.219315\pi$
$653$	39.4492	1.54377	0.771884	+	0.635764i	$0.219315\pi$
$654$	0	0
$655$	16.4041	0.640961
$656$	−9.33766	−0.364574
$657$	0	0
$658$	−75.1918	−2.93128
$659$	−11.8019	−0.459737	−0.229868	−	0.973222i	$-0.573830\pi$
$659$	−11.8019	−0.459737	−0.229868	+	0.973222i	$0.573830\pi$
$660$	0	0
$661$	−41.1918	−1.60218	−0.801088	−	0.598546i	$-0.795745\pi$
$661$	−41.1918	−1.60218	−0.801088	+	0.598546i	$0.795745\pi$
$662$	−8.05254	−0.312971
$663$	0	0
$664$	−38.6969	−1.50173
$665$	0	0
$666$	0	0
$667$	26.2020	1.01455
$668$	−66.4129	−2.56959
$669$	0	0
$670$	0.853572	0.0329764
$671$	−28.5906	−1.10373
$672$	0	0
$673$	−21.8990	−0.844144	−0.422072	−	0.906562i	$-0.638697\pi$
$673$	−21.8990	−0.844144	−0.422072	+	0.906562i	$0.638697\pi$
$674$	59.9873	2.31062
$675$	0	0
$676$	−41.3939	−1.59207
$677$	−22.8725	−0.879061	−0.439531	−	0.898228i	$-0.644855\pi$
$677$	−22.8725	−0.879061	−0.439531	+	0.898228i	$0.644855\pi$
$678$	0	0
$679$	10.6969	0.410511
$680$	7.45107	0.285735
$681$	0	0
$682$	126.136	4.83001
$683$	0.577648	0.0221031	0.0110515	−	0.999939i	$-0.496482\pi$
$683$	0.577648	0.0221031	0.0110515	+	0.999939i	$0.496482\pi$
$684$	0	0
$685$	−16.4041	−0.626768
$686$	−16.9185	−0.645953
$687$	0	0
$688$	6.34847	0.242033
$689$	1.04930	0.0399750
$690$	0	0
$691$	25.3939	0.966029	0.483014	−	0.875612i	$-0.339542\pi$
$691$	25.3939	0.966029	0.483014	+	0.875612i	$0.339542\pi$
$692$	24.9711	0.949258
$693$	0	0
$694$	−20.4495	−0.776252
$695$	−2.46424	−0.0934739
$696$	0	0
$697$	19.5959	0.742248
$698$	39.2134	1.48425
$699$	0	0
$700$	46.3939	1.75352
$701$	30.2176	1.14130	0.570651	−	0.821193i	$-0.306691\pi$
$701$	30.2176	1.14130	0.570651	+	0.821193i	$0.306691\pi$
$702$	0	0
$703$	0	0
$704$	70.6101	2.66122
$705$	0	0
$706$	−46.0454	−1.73294
$707$	−7.23907	−0.272253
$708$	0	0
$709$	28.3939	1.06635	0.533177	−	0.846004i	$-0.320998\pi$
$709$	28.3939	1.06635	0.533177	+	0.846004i	$0.320998\pi$
$710$	−17.7320	−0.665471
$711$	0	0
$712$	17.7526	0.665305
$713$	−34.2027	−1.28090
$714$	0	0
$715$	−6.00000	−0.224387
$716$	1.99259	0.0744666
$717$	0	0
$718$	−33.7980	−1.26133
$719$	−47.2659	−1.76272	−0.881361	−	0.472444i	$-0.843372\pi$
$719$	−47.2659	−1.76272	−0.881361	+	0.472444i	$0.843372\pi$
$720$	0	0
$721$	−11.8990	−0.443141
$722$	0	0
$723$	0	0
$724$	72.0908	2.67923
$725$	28.2250	1.04825
$726$	0	0
$727$	−41.0454	−1.52229	−0.761145	−	0.648582i	$-0.775362\pi$
$727$	−41.0454	−1.52229	−0.761145	+	0.648582i	$0.775362\pi$
$728$	11.6721	0.432596
$729$	0	0
$730$	−12.2474	−0.453298
$731$	−13.3228	−0.492763
$732$	0	0
$733$	5.30306	0.195873	0.0979365	−	0.995193i	$-0.468776\pi$
$733$	5.30306	0.195873	0.0979365	+	0.995193i	$0.468776\pi$
$734$	47.5018	1.75332
$735$	0	0
$736$	−16.0454	−0.591442
$737$	1.99259	0.0733980
$738$	0	0
$739$	−29.0454	−1.06845	−0.534226	−	0.845342i	$-0.679397\pi$
$739$	−29.0454	−1.06845	−0.534226	+	0.845342i	$0.679397\pi$
$740$	−14.1125	−0.518785
$741$	0	0
$742$	8.44949	0.310191
$743$	−31.6325	−1.16048	−0.580242	−	0.814444i	$-0.697042\pi$
$743$	−31.6325	−1.16048	−0.580242	+	0.814444i	$0.697042\pi$
$744$	0	0
$745$	13.1010	0.479984
$746$	−71.6594	−2.62364
$747$	0	0
$748$	41.3939	1.51351
$749$	−57.1812	−2.08936
$750$	0	0
$751$	35.0454	1.27883	0.639413	−	0.768864i	$-0.279178\pi$
$751$	35.0454	1.27883	0.639413	+	0.768864i	$0.279178\pi$
$752$	−9.33766	−0.340509
$753$	0	0
$754$	16.8990	0.615425
$755$	4.19718	0.152751
$756$	0	0
$757$	16.3939	0.595846	0.297923	−	0.954590i	$-0.403706\pi$
$757$	16.3939	0.595846	0.297923	+	0.954590i	$0.403706\pi$
$758$	−8.05254	−0.292481
$759$	0	0
$760$	0	0
$761$	1.99259	0.0722313	0.0361157	−	0.999348i	$-0.488502\pi$
$761$	1.99259	0.0722313	0.0361157	+	0.999348i	$0.488502\pi$
$762$	0	0
$763$	−30.6969	−1.11130
$764$	−19.7246	−0.713611
$765$	0	0
$766$	23.1464	0.836314
$767$	7.81671	0.282245
$768$	0	0
$769$	28.3939	1.02391	0.511955	−	0.859012i	$-0.328922\pi$
$769$	28.3939	1.02391	0.511955	+	0.859012i	$0.328922\pi$
$770$	−48.3152	−1.74116
$771$	0	0
$772$	−67.9444	−2.44537
$773$	9.33766	0.335852	0.167926	−	0.985800i	$-0.446293\pi$
$773$	9.33766	0.335852	0.167926	+	0.985800i	$0.446293\pi$
$774$	0	0
$775$	−36.8434	−1.32345
$776$	10.4930	0.376675
$777$	0	0
$778$	29.1464	1.04495
$779$	0	0
$780$	0	0
$781$	−41.3939	−1.48119
$782$	−17.7320	−0.634096
$783$	0	0
$784$	4.89898	0.174964
$785$	−16.4707	−0.587865
$786$	0	0
$787$	−47.2474	−1.68419	−0.842095	−	0.539329i	$-0.818678\pi$
$787$	−47.2474	−1.68419	−0.842095	+	0.539329i	$0.818678\pi$
$788$	−50.3078	−1.79214
$789$	0	0
$790$	0.853572	0.0303687
$791$	28.5906	1.01657
$792$	0	0
$793$	5.00000	0.177555
$794$	16.3409	0.579917
$795$	0	0
$796$	1.20204	0.0426052
$797$	26.8577	0.951348	0.475674	−	0.879622i	$-0.342204\pi$
$797$	26.8577	0.951348	0.475674	+	0.879622i	$0.342204\pi$
$798$	0	0
$799$	19.5959	0.693254
$800$	−17.2842	−0.611089
$801$	0	0
$802$	17.1464	0.605461
$803$	−28.5906	−1.00894
$804$	0	0
$805$	13.1010	0.461750
$806$	−22.0590	−0.776996
$807$	0	0
$808$	−7.10102	−0.249813
$809$	37.4566	1.31690	0.658452	−	0.752622i	$-0.271211\pi$
$809$	37.4566	1.31690	0.658452	+	0.752622i	$0.271211\pi$
$810$	0	0
$811$	−17.7980	−0.624971	−0.312485	−	0.949923i	$-0.601162\pi$
$811$	−17.7980	−0.624971	−0.312485	+	0.949923i	$0.601162\pi$
$812$	86.1375	3.02283
$813$	0	0
$814$	−52.0454	−1.82419
$815$	−9.91530	−0.347318
$816$	0	0
$817$	0	0
$818$	7.23907	0.253108
$819$	0	0
$820$	−33.7980	−1.18028
$821$	30.1116	1.05090	0.525450	−	0.850824i	$-0.323897\pi$
$821$	30.1116	1.05090	0.525450	+	0.850824i	$0.323897\pi$
$822$	0	0
$823$	−30.2020	−1.05278	−0.526388	−	0.850244i	$-0.676454\pi$
$823$	−30.2020	−1.05278	−0.526388	+	0.850244i	$0.676454\pi$
$824$	−11.6721	−0.406616
$825$	0	0
$826$	62.9444	2.19012
$827$	33.3654	1.16023	0.580115	−	0.814534i	$-0.303008\pi$
$827$	33.3654	1.16023	0.580115	+	0.814534i	$0.303008\pi$
$828$	0	0
$829$	52.1918	1.81270	0.906349	−	0.422531i	$-0.138858\pi$
$829$	52.1918	1.81270	0.906349	+	0.422531i	$0.138858\pi$
$830$	−28.0130	−0.972344
$831$	0	0
$832$	−12.3485	−0.428106
$833$	−10.2810	−0.356214
$834$	0	0
$835$	−20.2020	−0.699120
$836$	0	0
$837$	0	0
$838$	1.34847	0.0465821
$839$	−11.0706	−0.382200	−0.191100	−	0.981571i	$-0.561205\pi$
$839$	−11.0706	−0.382200	−0.191100	+	0.981571i	$0.561205\pi$
$840$	0	0
$841$	23.4041	0.807037
$842$	7.23907	0.249475
$843$	0	0
$844$	49.4949	1.70368
$845$	−12.5915	−0.433162
$846$	0	0
$847$	−74.8434	−2.57165
$848$	1.04930	0.0360329
$849$	0	0
$850$	−19.1010	−0.655160
$851$	14.1125	0.483770
$852$	0	0
$853$	−39.6969	−1.35920	−0.679599	−	0.733584i	$-0.737846\pi$
$853$	−39.6969	−1.35920	−0.679599	+	0.733584i	$0.737846\pi$
$854$	40.2627	1.37776
$855$	0	0
$856$	−56.0908	−1.91714
$857$	8.39436	0.286746	0.143373	−	0.989669i	$-0.454205\pi$
$857$	8.39436	0.286746	0.143373	+	0.989669i	$0.454205\pi$
$858$	0	0
$859$	−53.0454	−1.80989	−0.904943	−	0.425533i	$-0.860087\pi$
$859$	−53.0454	−1.80989	−0.904943	+	0.425533i	$0.860087\pi$
$860$	22.9785	0.783560
$861$	0	0
$862$	−21.7980	−0.742441
$863$	−17.7320	−0.603605	−0.301802	−	0.953370i	$-0.597588\pi$
$863$	−17.7320	−0.603605	−0.301802	+	0.953370i	$0.597588\pi$
$864$	0	0
$865$	7.59592	0.258269
$866$	−50.6496	−1.72114
$867$	0	0
$868$	−112.439	−3.81644
$869$	1.99259	0.0675940
$870$	0	0
$871$	−0.348469	−0.0118074
$872$	−30.1116	−1.01971
$873$	0	0
$874$	0	0
$875$	32.2102	1.08890
$876$	0	0
$877$	3.69694	0.124837	0.0624184	−	0.998050i	$-0.480119\pi$
$877$	3.69694	0.124837	0.0624184	+	0.998050i	$0.480119\pi$
$878$	−89.5212	−3.02119
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−58.0185	−1.95469	−0.977347	−	0.211643i	$-0.932119\pi$
$881$	−58.0185	−1.95469	−0.977347	+	0.211643i	$0.932119\pi$
$882$	0	0
$883$	30.3485	1.02131	0.510654	−	0.859787i	$-0.329403\pi$
$883$	30.3485	1.02131	0.510654	+	0.859787i	$0.329403\pi$
$884$	−7.23907	−0.243476
$885$	0	0
$886$	−82.2929	−2.76468
$887$	−48.4212	−1.62583	−0.812913	−	0.582385i	$-0.802119\pi$
$887$	−48.4212	−1.62583	−0.812913	+	0.582385i	$0.802119\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	12.8512	0.430773
$891$	0	0
$892$	−1.20204	−0.0402473
$893$	0	0
$894$	0	0
$895$	0.606123	0.0202605
$896$	−68.8533	−2.30023
$897$	0	0
$898$	−50.6969	−1.69178
$899$	−68.4055	−2.28145
$900$	0	0
$901$	−2.20204	−0.0733606
$902$	−124.643	−4.15017
$903$	0	0
$904$	28.0454	0.932776
$905$	21.9292	0.728951
$906$	0	0
$907$	−35.1918	−1.16853	−0.584263	−	0.811564i	$-0.698616\pi$
$907$	−35.1918	−1.16853	−0.584263	+	0.811564i	$0.698616\pi$
$908$	−37.4566	−1.24304
$909$	0	0
$910$	8.44949	0.280098
$911$	17.7320	0.587488	0.293744	−	0.955884i	$-0.405099\pi$
$911$	17.7320	0.587488	0.293744	+	0.955884i	$0.405099\pi$
$912$	0	0
$913$	−65.3939	−2.16422
$914$	46.9241	1.55211
$915$	0	0
$916$	−37.2474	−1.23069
$917$	−53.9274	−1.78084
$918$	0	0
$919$	29.2474	0.964784	0.482392	−	0.875955i	$-0.339768\pi$
$919$	29.2474	0.964784	0.482392	+	0.875955i	$0.339768\pi$
$920$	12.8512	0.423691
$921$	0	0
$922$	−41.1464	−1.35509
$923$	7.23907	0.238277
$924$	0	0
$925$	15.2020	0.499840
$926$	0.341824	0.0112330
$927$	0	0
$928$	−32.0908	−1.05343
$929$	55.9199	1.83467	0.917337	−	0.398112i	$-0.130334\pi$
$929$	55.9199	1.83467	0.917337	+	0.398112i	$0.130334\pi$
$930$	0	0
$931$	0	0
$932$	71.6594	2.34728
$933$	0	0
$934$	12.0000	0.392652
$935$	12.5915	0.411787
$936$	0	0
$937$	−39.0908	−1.27704	−0.638521	−	0.769605i	$-0.720453\pi$
$937$	−39.0908	−1.27704	−0.638521	+	0.769605i	$0.720453\pi$
$938$	−2.80606	−0.0916212
$939$	0	0
$940$	−33.7980	−1.10237
$941$	−6.08377	−0.198325	−0.0991626	−	0.995071i	$-0.531616\pi$
$941$	−6.08377	−0.198325	−0.0991626	+	0.995071i	$0.531616\pi$
$942$	0	0
$943$	33.7980	1.10061
$944$	7.81671	0.254412
$945$	0	0
$946$	84.7423	2.75521
$947$	38.5059	1.25127	0.625637	−	0.780114i	$-0.284839\pi$
$947$	38.5059	1.25127	0.625637	+	0.780114i	$0.284839\pi$
$948$	0	0
$949$	5.00000	0.162307
$950$	0	0
$951$	0	0
$952$	−24.4949	−0.793884
$953$	−22.7665	−0.737479	−0.368740	−	0.929533i	$-0.620211\pi$
$953$	−22.7665	−0.737479	−0.368740	+	0.929533i	$0.620211\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	37.4566	1.21143
$957$	0	0
$958$	−16.8990	−0.545981
$959$	53.9274	1.74140
$960$	0	0
$961$	58.2929	1.88041
$962$	9.10183	0.293455
$963$	0	0
$964$	65.5403	2.11091
$965$	−20.6679	−0.665323
$966$	0	0
$967$	−20.3485	−0.654363	−0.327181	−	0.944962i	$-0.606099\pi$
$967$	−20.3485	−0.654363	−0.327181	+	0.944962i	$0.606099\pi$
$968$	−73.4161	−2.35968
$969$	0	0
$970$	7.59592	0.243890
$971$	−19.6186	−0.629591	−0.314796	−	0.949160i	$-0.601936\pi$
$971$	−19.6186	−0.629591	−0.314796	+	0.949160i	$0.601936\pi$
$972$	0	0
$973$	8.10102	0.259707
$974$	18.6753	0.598396
$975$	0	0
$976$	5.00000	0.160046
$977$	5.14048	0.164458	0.0822292	−	0.996613i	$-0.473796\pi$
$977$	5.14048	0.164458	0.0822292	+	0.996613i	$0.473796\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	17.7320	0.566429
$981$	0	0
$982$	46.2929	1.47726
$983$	6.66142	0.212466	0.106233	−	0.994341i	$-0.466121\pi$
$983$	6.66142	0.212466	0.106233	+	0.994341i	$0.466121\pi$
$984$	0	0
$985$	−15.3031	−0.487596
$986$	−35.4640	−1.12941
$987$	0	0
$988$	0	0
$989$	−22.9785	−0.730674
$990$	0	0
$991$	2.95459	0.0938557	0.0469279	−	0.998898i	$-0.485057\pi$
$991$	2.95459	0.0938557	0.0469279	+	0.998898i	$0.485057\pi$
$992$	41.8896	1.33000
$993$	0	0
$994$	58.2929	1.84894
$995$	0.365647	0.0115918
$996$	0	0
$997$	−45.0908	−1.42804	−0.714020	−	0.700125i	$-0.753128\pi$
$997$	−45.0908	−1.42804	−0.714020	+	0.700125i	$0.753128\pi$
$998$	−64.5501	−2.04330
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.be.1.1		4
3.2	odd	2	inner	3249.2.a.be.1.4		4
19.8	odd	6		171.2.f.c.64.1	✓	8
19.12	odd	6		171.2.f.c.163.1	yes	8
19.18	odd	2		3249.2.a.bd.1.4		4
57.8	even	6		171.2.f.c.64.4	yes	8
57.50	even	6		171.2.f.c.163.4	yes	8
57.56	even	2		3249.2.a.bd.1.1		4
76.27	even	6		2736.2.s.bb.577.2		8
76.31	even	6		2736.2.s.bb.1873.2		8
228.107	odd	6		2736.2.s.bb.1873.3		8
228.179	odd	6		2736.2.s.bb.577.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.f.c.64.1	✓	8	19.8	odd	6
171.2.f.c.64.4	yes	8	57.8	even	6
171.2.f.c.163.1	yes	8	19.12	odd	6
171.2.f.c.163.4	yes	8	57.50	even	6
2736.2.s.bb.577.2		8	76.27	even	6
2736.2.s.bb.577.3		8	228.179	odd	6
2736.2.s.bb.1873.2		8	76.31	even	6
2736.2.s.bb.1873.3		8	228.107	odd	6
3249.2.a.bd.1.1		4	57.56	even	2
3249.2.a.bd.1.4		4	19.18	odd	2
3249.2.a.be.1.1		4	1.1	even	1	trivial
3249.2.a.be.1.4		4	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.33441 q^{2} +3.44949 q^{4} +1.04930 q^{5} -3.44949 q^{7} -3.38371 q^{8} -2.44949 q^{10} -5.71812 q^{11} +1.00000 q^{13} +8.05254 q^{14} +1.00000 q^{16} -2.09859 q^{17} +3.61953 q^{20} +13.3485 q^{22} -3.61953 q^{23} -3.89898 q^{25} -2.33441 q^{26} -11.8990 q^{28} -7.23907 q^{29} +9.44949 q^{31} +4.43300 q^{32} +4.89898 q^{34} -3.61953 q^{35} -3.89898 q^{37} -3.55051 q^{40} -9.33766 q^{41} +6.34847 q^{43} -19.7246 q^{44} +8.44949 q^{46} -9.33766 q^{47} +4.89898 q^{49} +9.10183 q^{50} +3.44949 q^{52} +1.04930 q^{53} -6.00000 q^{55} +11.6721 q^{56} +16.8990 q^{58} +7.81671 q^{59} +5.00000 q^{61} -22.0590 q^{62} -12.3485 q^{64} +1.04930 q^{65} -0.348469 q^{67} -7.23907 q^{68} +8.44949 q^{70} +7.23907 q^{71} +5.00000 q^{73} +9.10183 q^{74} +19.7246 q^{77} -0.348469 q^{79} +1.04930 q^{80} +21.7980 q^{82} +11.4362 q^{83} -2.20204 q^{85} -14.8200 q^{86} +19.3485 q^{88} -5.24648 q^{89} -3.44949 q^{91} -12.4855 q^{92} +21.7980 q^{94} -3.10102 q^{97} -11.4362 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{7} + 4 q^{13} + 4 q^{16} + 24 q^{22} + 4 q^{25} - 28 q^{28} + 28 q^{31} + 4 q^{37} - 24 q^{40} - 4 q^{43} + 24 q^{46} + 4 q^{52} - 24 q^{55} + 48 q^{58} + 20 q^{61} - 20 q^{64} + 28 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) \) 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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