LMFDB - Embedded newform 9196.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,-1,0,3,0,-2,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 8x^{4} + 46x^{3} - 2x^{2} - 48x - 21 $ x^7 - x^6 - 14x^5 + 8x^4 + 46x^3 - 2x^2 - 48*x - 21
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.735972$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.735972 q^{3} +1.46540 q^{5} +2.78748 q^{7} -2.45834 q^{9} -5.24805 q^{13} +1.07850 q^{15} -0.661557 q^{17} -1.00000 q^{19} +2.05151 q^{21} +3.27281 q^{23} -2.85259 q^{25} -4.01719 q^{27} -5.31315 q^{29} -4.78340 q^{31} +4.08479 q^{35} +2.24580 q^{37} -3.86242 q^{39} +5.21043 q^{41} +7.42956 q^{43} -3.60247 q^{45} +13.5211 q^{47} +0.770048 q^{49} -0.486888 q^{51} -9.59489 q^{53} -0.735972 q^{57} -5.95454 q^{59} -10.5573 q^{61} -6.85259 q^{63} -7.69051 q^{65} +15.1410 q^{67} +2.40870 q^{69} +14.4865 q^{71} -8.48754 q^{73} -2.09943 q^{75} -12.3540 q^{79} +4.41849 q^{81} -4.59839 q^{83} -0.969449 q^{85} -3.91033 q^{87} -10.0320 q^{89} -14.6288 q^{91} -3.52045 q^{93} -1.46540 q^{95} -18.0923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} + 3 q^{5} - 2 q^{7} + 8 q^{9} + 11 q^{15} - 12 q^{17} - 7 q^{19} - q^{21} - 5 q^{23} + 4 q^{25} - 13 q^{27} + 2 q^{29} + 5 q^{31} - 16 q^{35} + 8 q^{37} - 15 q^{39} - 31 q^{41} - 12 q^{43}+ \cdots - 25 q^{97}+O(q^{100}) $ 7 * q - q^3 + 3 * q^5 - 2 * q^7 + 8 * q^9 + 11 * q^15 - 12 * q^17 - 7 * q^19 - q^21 - 5 * q^23 + 4 * q^25 - 13 * q^27 + 2 * q^29 + 5 * q^31 - 16 * q^35 + 8 * q^37 - 15 * q^39 - 31 * q^41 - 12 * q^43 - 24 * q^45 + 10 * q^47 - 3 * q^49 - 5 * q^51 - 25 * q^53 + q^57 - 7 * q^59 - q^61 - 24 * q^63 - 13 * q^65 - 5 * q^67 + 31 * q^69 + q^71 + 12 * q^73 - 12 * q^75 - 20 * q^79 + 35 * q^81 - 14 * q^83 + 6 * q^85 - 28 * q^87 - 18 * q^89 + 25 * q^91 - 35 * q^93 - 3 * q^95 - 25 * 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	0
$2$	0	0
$3$	0.735972	0.424914	0.212457	−	0.977170i	$-0.431853\pi$
$3$	0.735972	0.424914	0.212457	+	0.977170i	$0.431853\pi$
$4$	0	0
$5$	1.46540	0.655349	0.327674	−	0.944791i	$-0.393735\pi$
$5$	1.46540	0.655349	0.327674	+	0.944791i	$0.393735\pi$
$6$	0	0
$7$	2.78748	1.05357	0.526784	−	0.849999i	$-0.323398\pi$
$7$	2.78748	1.05357	0.526784	+	0.849999i	$0.323398\pi$
$8$	0	0
$9$	−2.45834	−0.819448
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.24805	−1.45555	−0.727773	−	0.685818i	$-0.759445\pi$
$13$	−5.24805	−1.45555	−0.727773	+	0.685818i	$0.759445\pi$
$14$	0	0
$15$	1.07850	0.278467
$16$	0	0
$17$	−0.661557	−0.160451	−0.0802256	−	0.996777i	$-0.525564\pi$
$17$	−0.661557	−0.160451	−0.0802256	+	0.996777i	$0.525564\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.05151	0.447676
$22$	0	0
$23$	3.27281	0.682429	0.341214	−	0.939985i	$-0.389162\pi$
$23$	3.27281	0.682429	0.341214	+	0.939985i	$0.389162\pi$
$24$	0	0
$25$	−2.85259	−0.570518
$26$	0	0
$27$	−4.01719	−0.773109
$28$	0	0
$29$	−5.31315	−0.986628	−0.493314	−	0.869851i	$-0.664215\pi$
$29$	−5.31315	−0.986628	−0.493314	+	0.869851i	$0.664215\pi$
$30$	0	0
$31$	−4.78340	−0.859124	−0.429562	−	0.903037i	$-0.641332\pi$
$31$	−4.78340	−0.859124	−0.429562	+	0.903037i	$0.641332\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.08479	0.690455
$36$	0	0
$37$	2.24580	0.369207	0.184604	−	0.982813i	$-0.440900\pi$
$37$	2.24580	0.369207	0.184604	+	0.982813i	$0.440900\pi$
$38$	0	0
$39$	−3.86242	−0.618482
$40$	0	0
$41$	5.21043	0.813733	0.406867	−	0.913488i	$-0.366621\pi$
$41$	5.21043	0.813733	0.406867	+	0.913488i	$0.366621\pi$
$42$	0	0
$43$	7.42956	1.13300	0.566499	−	0.824063i	$-0.308298\pi$
$43$	7.42956	1.13300	0.566499	+	0.824063i	$0.308298\pi$
$44$	0	0
$45$	−3.60247	−0.537025
$46$	0	0
$47$	13.5211	1.97226	0.986130	−	0.165976i	$-0.0530773\pi$
$47$	13.5211	1.97226	0.986130	+	0.165976i	$0.0530773\pi$
$48$	0	0
$49$	0.770048	0.110007
$50$	0	0
$51$	−0.486888	−0.0681779
$52$	0	0
$53$	−9.59489	−1.31796	−0.658980	−	0.752161i	$-0.729012\pi$
$53$	−9.59489	−1.31796	−0.658980	+	0.752161i	$0.729012\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.735972	−0.0974819
$58$	0	0
$59$	−5.95454	−0.775215	−0.387608	−	0.921825i	$-0.626698\pi$
$59$	−5.95454	−0.775215	−0.387608	+	0.921825i	$0.626698\pi$
$60$	0	0
$61$	−10.5573	−1.35172	−0.675861	−	0.737030i	$-0.736228\pi$
$61$	−10.5573	−1.35172	−0.675861	+	0.737030i	$0.736228\pi$
$62$	0	0
$63$	−6.85259	−0.863345
$64$	0	0
$65$	−7.69051	−0.953891
$66$	0	0
$67$	15.1410	1.84977	0.924884	−	0.380250i	$-0.124162\pi$
$67$	15.1410	1.84977	0.924884	+	0.380250i	$0.124162\pi$
$68$	0	0
$69$	2.40870	0.289973
$70$	0	0
$71$	14.4865	1.71923	0.859616	−	0.510941i	$-0.170703\pi$
$71$	14.4865	1.71923	0.859616	+	0.510941i	$0.170703\pi$
$72$	0	0
$73$	−8.48754	−0.993392	−0.496696	−	0.867925i	$-0.665454\pi$
$73$	−8.48754	−0.993392	−0.496696	+	0.867925i	$0.665454\pi$
$74$	0	0
$75$	−2.09943	−0.242421
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.3540	−1.38994	−0.694969	−	0.719040i	$-0.744582\pi$
$79$	−12.3540	−1.38994	−0.694969	+	0.719040i	$0.744582\pi$
$80$	0	0
$81$	4.41849	0.490944
$82$	0	0
$83$	−4.59839	−0.504739	−0.252369	−	0.967631i	$-0.581210\pi$
$83$	−4.59839	−0.504739	−0.252369	+	0.967631i	$0.581210\pi$
$84$	0	0
$85$	−0.969449	−0.105152
$86$	0	0
$87$	−3.91033	−0.419232
$88$	0	0
$89$	−10.0320	−1.06339	−0.531693	−	0.846937i	$-0.678444\pi$
$89$	−10.0320	−1.06339	−0.531693	+	0.846937i	$0.678444\pi$
$90$	0	0
$91$	−14.6288	−1.53352
$92$	0	0
$93$	−3.52045	−0.365053
$94$	0	0
$95$	−1.46540	−0.150347
$96$	0	0
$97$	−18.0923	−1.83699	−0.918495	−	0.395432i	$-0.870595\pi$
$97$	−18.0923	−1.83699	−0.918495	+	0.395432i	$0.870595\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0252	1.19655	0.598275	−	0.801291i	$-0.295853\pi$
$101$	12.0252	1.19655	0.598275	+	0.801291i	$0.295853\pi$
$102$	0	0
$103$	−10.9906	−1.08293	−0.541467	−	0.840722i	$-0.682131\pi$
$103$	−10.9906	−1.08293	−0.541467	+	0.840722i	$0.682131\pi$
$104$	0	0
$105$	3.00629	0.293384
$106$	0	0
$107$	−13.7568	−1.32992	−0.664961	−	0.746878i	$-0.731552\pi$
$107$	−13.7568	−1.32992	−0.664961	+	0.746878i	$0.731552\pi$
$108$	0	0
$109$	−12.5274	−1.19991	−0.599954	−	0.800035i	$-0.704814\pi$
$109$	−12.5274	−1.19991	−0.599954	+	0.800035i	$0.704814\pi$
$110$	0	0
$111$	1.65285	0.156881
$112$	0	0
$113$	−1.71448	−0.161285	−0.0806426	−	0.996743i	$-0.525697\pi$
$113$	−1.71448	−0.161285	−0.0806426	+	0.996743i	$0.525697\pi$
$114$	0	0
$115$	4.79600	0.447229
$116$	0	0
$117$	12.9015	1.19274
$118$	0	0
$119$	−1.84408	−0.169046
$120$	0	0
$121$	0	0
$122$	0	0
$123$	3.83474	0.345766
$124$	0	0
$125$	−11.5072	−1.02924
$126$	0	0
$127$	0.0328621	0.00291604	0.00145802	−	0.999999i	$-0.499536\pi$
$127$	0.0328621	0.00291604	0.00145802	+	0.999999i	$0.499536\pi$
$128$	0	0
$129$	5.46795	0.481426
$130$	0	0
$131$	−6.84827	−0.598336	−0.299168	−	0.954200i	$-0.596709\pi$
$131$	−6.84827	−0.598336	−0.299168	+	0.954200i	$0.596709\pi$
$132$	0	0
$133$	−2.78748	−0.241705
$134$	0	0
$135$	−5.88681	−0.506656
$136$	0	0
$137$	−17.1416	−1.46450	−0.732252	−	0.681033i	$-0.761531\pi$
$137$	−17.1416	−1.46450	−0.732252	+	0.681033i	$0.761531\pi$
$138$	0	0
$139$	2.75054	0.233298	0.116649	−	0.993173i	$-0.462785\pi$
$139$	2.75054	0.233298	0.116649	+	0.993173i	$0.462785\pi$
$140$	0	0
$141$	9.95118	0.838040
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.78592	−0.646586
$146$	0	0
$147$	0.566734	0.0467434
$148$	0	0
$149$	5.26314	0.431173	0.215586	−	0.976485i	$-0.430834\pi$
$149$	5.26314	0.431173	0.215586	+	0.976485i	$0.430834\pi$
$150$	0	0
$151$	−13.3693	−1.08797	−0.543987	−	0.839093i	$-0.683086\pi$
$151$	−13.3693	−1.08797	−0.543987	+	0.839093i	$0.683086\pi$
$152$	0	0
$153$	1.62634	0.131481
$154$	0	0
$155$	−7.00962	−0.563026
$156$	0	0
$157$	16.9724	1.35454	0.677272	−	0.735733i	$-0.263162\pi$
$157$	16.9724	1.35454	0.677272	+	0.735733i	$0.263162\pi$
$158$	0	0
$159$	−7.06157	−0.560019
$160$	0	0
$161$	9.12291	0.718986
$162$	0	0
$163$	5.68072	0.444948	0.222474	−	0.974939i	$-0.428587\pi$
$163$	5.68072	0.444948	0.222474	+	0.974939i	$0.428587\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.532239	0.0411859	0.0205930	−	0.999788i	$-0.493445\pi$
$167$	0.532239	0.0411859	0.0205930	+	0.999788i	$0.493445\pi$
$168$	0	0
$169$	14.5420	1.11861
$170$	0	0
$171$	2.45834	0.187994
$172$	0	0
$173$	−12.2751	−0.933256	−0.466628	−	0.884454i	$-0.654531\pi$
$173$	−12.2751	−0.933256	−0.466628	+	0.884454i	$0.654531\pi$
$174$	0	0
$175$	−7.95154	−0.601080
$176$	0	0
$177$	−4.38238	−0.329400
$178$	0	0
$179$	7.42597	0.555043	0.277521	−	0.960719i	$-0.410487\pi$
$179$	7.42597	0.555043	0.277521	+	0.960719i	$0.410487\pi$
$180$	0	0
$181$	0.0594328	0.00441760	0.00220880	−	0.999998i	$-0.499297\pi$
$181$	0.0594328	0.00441760	0.00220880	+	0.999998i	$0.499297\pi$
$182$	0	0
$183$	−7.76986	−0.574365
$184$	0	0
$185$	3.29100	0.241959
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−11.1978	−0.814523
$190$	0	0
$191$	23.1839	1.67753	0.838764	−	0.544496i	$-0.183279\pi$
$191$	23.1839	1.67753	0.838764	+	0.544496i	$0.183279\pi$
$192$	0	0
$193$	−21.9784	−1.58204	−0.791021	−	0.611789i	$-0.790450\pi$
$193$	−21.9784	−1.58204	−0.791021	+	0.611789i	$0.790450\pi$
$194$	0	0
$195$	−5.66000	−0.405321
$196$	0	0
$197$	−25.9670	−1.85007	−0.925036	−	0.379881i	$-0.875965\pi$
$197$	−25.9670	−1.85007	−0.925036	+	0.379881i	$0.875965\pi$
$198$	0	0
$199$	−12.7068	−0.900761	−0.450381	−	0.892837i	$-0.648712\pi$
$199$	−12.7068	−0.900761	−0.450381	+	0.892837i	$0.648712\pi$
$200$	0	0
$201$	11.1434	0.785992
$202$	0	0
$203$	−14.8103	−1.03948
$204$	0	0
$205$	7.63540	0.533279
$206$	0	0
$207$	−8.04571	−0.559215
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.70905	0.461870	0.230935	−	0.972969i	$-0.425822\pi$
$211$	6.70905	0.461870	0.230935	+	0.972969i	$0.425822\pi$
$212$	0	0
$213$	10.6617	0.730525
$214$	0	0
$215$	10.8873	0.742509
$216$	0	0
$217$	−13.3336	−0.905146
$218$	0	0
$219$	−6.24660	−0.422106
$220$	0	0
$221$	3.47188	0.233544
$222$	0	0
$223$	16.3216	1.09297	0.546486	−	0.837468i	$-0.315965\pi$
$223$	16.3216	1.09297	0.546486	+	0.837468i	$0.315965\pi$
$224$	0	0
$225$	7.01265	0.467510
$226$	0	0
$227$	4.68256	0.310792	0.155396	−	0.987852i	$-0.450335\pi$
$227$	4.68256	0.310792	0.155396	+	0.987852i	$0.450335\pi$
$228$	0	0
$229$	−13.1864	−0.871384	−0.435692	−	0.900096i	$-0.643496\pi$
$229$	−13.1864	−0.871384	−0.435692	+	0.900096i	$0.643496\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.27867	0.607866	0.303933	−	0.952693i	$-0.401700\pi$
$233$	9.27867	0.607866	0.303933	+	0.952693i	$0.401700\pi$
$234$	0	0
$235$	19.8139	1.29252
$236$	0	0
$237$	−9.09223	−0.590604
$238$	0	0
$239$	−26.9043	−1.74029	−0.870147	−	0.492792i	$-0.835976\pi$
$239$	−26.9043	−1.74029	−0.870147	+	0.492792i	$0.835976\pi$
$240$	0	0
$241$	3.42981	0.220933	0.110467	−	0.993880i	$-0.464765\pi$
$241$	3.42981	0.220933	0.110467	+	0.993880i	$0.464765\pi$
$242$	0	0
$243$	15.3035	0.981717
$244$	0	0
$245$	1.12843	0.0720929
$246$	0	0
$247$	5.24805	0.333925
$248$	0	0
$249$	−3.38429	−0.214470
$250$	0	0
$251$	6.96954	0.439914	0.219957	−	0.975510i	$-0.429408\pi$
$251$	6.96954	0.439914	0.219957	+	0.975510i	$0.429408\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.713488	−0.0446803
$256$	0	0
$257$	−9.71600	−0.606068	−0.303034	−	0.952980i	$-0.598000\pi$
$257$	−9.71600	−0.606068	−0.303034	+	0.952980i	$0.598000\pi$
$258$	0	0
$259$	6.26012	0.388985
$260$	0	0
$261$	13.0616	0.808491
$262$	0	0
$263$	−20.2544	−1.24894	−0.624469	−	0.781050i	$-0.714685\pi$
$263$	−20.2544	−1.24894	−0.624469	+	0.781050i	$0.714685\pi$
$264$	0	0
$265$	−14.0604	−0.863723
$266$	0	0
$267$	−7.38325	−0.451847
$268$	0	0
$269$	0.577707	0.0352234	0.0176117	−	0.999845i	$-0.494394\pi$
$269$	0.577707	0.0352234	0.0176117	+	0.999845i	$0.494394\pi$
$270$	0	0
$271$	−1.74939	−0.106268	−0.0531340	−	0.998587i	$-0.516921\pi$
$271$	−1.74939	−0.106268	−0.0531340	+	0.998587i	$0.516921\pi$
$272$	0	0
$273$	−10.7664	−0.651613
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.7187	−0.884359	−0.442179	−	0.896927i	$-0.645794\pi$
$277$	−14.7187	−0.884359	−0.442179	+	0.896927i	$0.645794\pi$
$278$	0	0
$279$	11.7592	0.704007
$280$	0	0
$281$	−5.42382	−0.323558	−0.161779	−	0.986827i	$-0.551723\pi$
$281$	−5.42382	−0.323558	−0.161779	+	0.986827i	$0.551723\pi$
$282$	0	0
$283$	20.6442	1.22717	0.613587	−	0.789627i	$-0.289726\pi$
$283$	20.6442	1.22717	0.613587	+	0.789627i	$0.289726\pi$
$284$	0	0
$285$	−1.07850	−0.0638847
$286$	0	0
$287$	14.5240	0.857324
$288$	0	0
$289$	−16.5623	−0.974255
$290$	0	0
$291$	−13.3154	−0.780563
$292$	0	0
$293$	10.6149	0.620130	0.310065	−	0.950715i	$-0.399649\pi$
$293$	10.6149	0.620130	0.310065	+	0.950715i	$0.399649\pi$
$294$	0	0
$295$	−8.72581	−0.508036
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.1759	−0.993307
$300$	0	0
$301$	20.7098	1.19369
$302$	0	0
$303$	8.85020	0.508431
$304$	0	0
$305$	−15.4707	−0.885849
$306$	0	0
$307$	−6.67347	−0.380875	−0.190438	−	0.981699i	$-0.560991\pi$
$307$	−6.67347	−0.380875	−0.190438	+	0.981699i	$0.560991\pi$
$308$	0	0
$309$	−8.08877	−0.460154
$310$	0	0
$311$	−10.6450	−0.603625	−0.301813	−	0.953367i	$-0.597592\pi$
$311$	−10.6450	−0.603625	−0.301813	+	0.953367i	$0.597592\pi$
$312$	0	0
$313$	−11.7192	−0.662411	−0.331205	−	0.943559i	$-0.607455\pi$
$313$	−11.7192	−0.662411	−0.331205	+	0.943559i	$0.607455\pi$
$314$	0	0
$315$	−10.0418	−0.565792
$316$	0	0
$317$	12.7134	0.714055	0.357027	−	0.934094i	$-0.383790\pi$
$317$	12.7134	0.714055	0.357027	+	0.934094i	$0.383790\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−10.1246	−0.565102
$322$	0	0
$323$	0.661557	0.0368100
$324$	0	0
$325$	14.9705	0.830415
$326$	0	0
$327$	−9.21982	−0.509857
$328$	0	0
$329$	37.6899	2.07791
$330$	0	0
$331$	6.90103	0.379315	0.189657	−	0.981850i	$-0.439262\pi$
$331$	6.90103	0.379315	0.189657	+	0.981850i	$0.439262\pi$
$332$	0	0
$333$	−5.52095	−0.302546
$334$	0	0
$335$	22.1877	1.21224
$336$	0	0
$337$	−23.0296	−1.25450	−0.627251	−	0.778817i	$-0.715820\pi$
$337$	−23.0296	−1.25450	−0.627251	+	0.778817i	$0.715820\pi$
$338$	0	0
$339$	−1.26181	−0.0685323
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−17.3659	−0.937669
$344$	0	0
$345$	3.52972	0.190034
$346$	0	0
$347$	2.11823	0.113712	0.0568562	−	0.998382i	$-0.481892\pi$
$347$	2.11823	0.113712	0.0568562	+	0.998382i	$0.481892\pi$
$348$	0	0
$349$	−6.49706	−0.347780	−0.173890	−	0.984765i	$-0.555634\pi$
$349$	−6.49706	−0.347780	−0.173890	+	0.984765i	$0.555634\pi$
$350$	0	0
$351$	21.0824	1.12530
$352$	0	0
$353$	−14.6837	−0.781532	−0.390766	−	0.920490i	$-0.627790\pi$
$353$	−14.6837	−0.781532	−0.390766	+	0.920490i	$0.627790\pi$
$354$	0	0
$355$	21.2286	1.12670
$356$	0	0
$357$	−1.35719	−0.0718301
$358$	0	0
$359$	1.70057	0.0897526	0.0448763	−	0.998993i	$-0.485711\pi$
$359$	1.70057	0.0897526	0.0448763	+	0.998993i	$0.485711\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.4377	−0.651018
$366$	0	0
$367$	25.8187	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$367$	25.8187	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$368$	0	0
$369$	−12.8090	−0.666812
$370$	0	0
$371$	−26.7456	−1.38856
$372$	0	0
$373$	−32.8818	−1.70256	−0.851278	−	0.524715i	$-0.824172\pi$
$373$	−32.8818	−1.70256	−0.851278	+	0.524715i	$0.824172\pi$
$374$	0	0
$375$	−8.46900	−0.437337
$376$	0	0
$377$	27.8837	1.43608
$378$	0	0
$379$	3.52086	0.180854	0.0904272	−	0.995903i	$-0.471177\pi$
$379$	3.52086	0.180854	0.0904272	+	0.995903i	$0.471177\pi$
$380$	0	0
$381$	0.0241856	0.00123907
$382$	0	0
$383$	−11.1844	−0.571497	−0.285749	−	0.958305i	$-0.592242\pi$
$383$	−11.1844	−0.571497	−0.285749	+	0.958305i	$0.592242\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.2644	−0.928433
$388$	0	0
$389$	9.91397	0.502658	0.251329	−	0.967902i	$-0.419132\pi$
$389$	9.91397	0.502658	0.251329	+	0.967902i	$0.419132\pi$
$390$	0	0
$391$	−2.16515	−0.109497
$392$	0	0
$393$	−5.04013	−0.254241
$394$	0	0
$395$	−18.1037	−0.910895
$396$	0	0
$397$	0.530021	0.0266010	0.0133005	−	0.999912i	$-0.495766\pi$
$397$	0.530021	0.0266010	0.0133005	+	0.999912i	$0.495766\pi$
$398$	0	0
$399$	−2.05151	−0.102704
$400$	0	0
$401$	1.16125	0.0579901	0.0289951	−	0.999580i	$-0.490769\pi$
$401$	1.16125	0.0579901	0.0289951	+	0.999580i	$0.490769\pi$
$402$	0	0
$403$	25.1035	1.25049
$404$	0	0
$405$	6.47488	0.321740
$406$	0	0
$407$	0	0
$408$	0	0
$409$	20.4592	1.01164	0.505822	−	0.862638i	$-0.331189\pi$
$409$	20.4592	1.01164	0.505822	+	0.862638i	$0.331189\pi$
$410$	0	0
$411$	−12.6157	−0.622288
$412$	0	0
$413$	−16.5982	−0.816742
$414$	0	0
$415$	−6.73850	−0.330780
$416$	0	0
$417$	2.02432	0.0991315
$418$	0	0
$419$	33.7923	1.65086	0.825431	−	0.564503i	$-0.190932\pi$
$419$	33.7923	1.65086	0.825431	+	0.564503i	$0.190932\pi$
$420$	0	0
$421$	35.1454	1.71288	0.856441	−	0.516246i	$-0.172671\pi$
$421$	35.1454	1.71288	0.856441	+	0.516246i	$0.172671\pi$
$422$	0	0
$423$	−33.2396	−1.61616
$424$	0	0
$425$	1.88715	0.0915402
$426$	0	0
$427$	−29.4282	−1.42413
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.4993	1.61360	0.806802	−	0.590822i	$-0.201196\pi$
$431$	33.4993	1.61360	0.806802	+	0.590822i	$0.201196\pi$
$432$	0	0
$433$	30.7901	1.47968	0.739840	−	0.672783i	$-0.234901\pi$
$433$	30.7901	1.47968	0.739840	+	0.672783i	$0.234901\pi$
$434$	0	0
$435$	−5.73022	−0.274743
$436$	0	0
$437$	−3.27281	−0.156560
$438$	0	0
$439$	−16.9703	−0.809949	−0.404975	−	0.914328i	$-0.632720\pi$
$439$	−16.9703	−0.809949	−0.404975	+	0.914328i	$0.632720\pi$
$440$	0	0
$441$	−1.89304	−0.0901449
$442$	0	0
$443$	33.6051	1.59663	0.798313	−	0.602243i	$-0.205726\pi$
$443$	33.6051	1.59663	0.798313	+	0.602243i	$0.205726\pi$
$444$	0	0
$445$	−14.7009	−0.696889
$446$	0	0
$447$	3.87352	0.183211
$448$	0	0
$449$	−18.9523	−0.894413	−0.447207	−	0.894431i	$-0.647581\pi$
$449$	−18.9523	−0.894413	−0.447207	+	0.894431i	$0.647581\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−9.83940	−0.462296
$454$	0	0
$455$	−21.4372	−1.00499
$456$	0	0
$457$	42.0582	1.96740	0.983699	−	0.179824i	$-0.0575528\pi$
$457$	42.0582	1.96740	0.983699	+	0.179824i	$0.0575528\pi$
$458$	0	0
$459$	2.65760	0.124046
$460$	0	0
$461$	33.4540	1.55811	0.779054	−	0.626956i	$-0.215700\pi$
$461$	33.4540	1.55811	0.779054	+	0.626956i	$0.215700\pi$
$462$	0	0
$463$	29.0238	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$463$	29.0238	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$464$	0	0
$465$	−5.15888	−0.239237
$466$	0	0
$467$	6.71452	0.310711	0.155356	−	0.987859i	$-0.450348\pi$
$467$	6.71452	0.310711	0.155356	+	0.987859i	$0.450348\pi$
$468$	0	0
$469$	42.2052	1.94886
$470$	0	0
$471$	12.4912	0.575564
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.85259	0.130886
$476$	0	0
$477$	23.5875	1.08000
$478$	0	0
$479$	31.8500	1.45526	0.727632	−	0.685968i	$-0.240621\pi$
$479$	31.8500	1.45526	0.727632	+	0.685968i	$0.240621\pi$
$480$	0	0
$481$	−11.7861	−0.537398
$482$	0	0
$483$	6.71421	0.305507
$484$	0	0
$485$	−26.5125	−1.20387
$486$	0	0
$487$	−24.1279	−1.09334	−0.546669	−	0.837349i	$-0.684104\pi$
$487$	−24.1279	−1.09334	−0.546669	+	0.837349i	$0.684104\pi$
$488$	0	0
$489$	4.18085	0.189065
$490$	0	0
$491$	24.9575	1.12631	0.563157	−	0.826350i	$-0.309586\pi$
$491$	24.9575	1.12631	0.563157	+	0.826350i	$0.309586\pi$
$492$	0	0
$493$	3.51496	0.158306
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.3808	1.81133
$498$	0	0
$499$	10.9224	0.488952	0.244476	−	0.969655i	$-0.421384\pi$
$499$	10.9224	0.488952	0.244476	+	0.969655i	$0.421384\pi$
$500$	0	0
$501$	0.391713	0.0175005
$502$	0	0
$503$	11.5773	0.516204	0.258102	−	0.966118i	$-0.416903\pi$
$503$	11.5773	0.516204	0.258102	+	0.966118i	$0.416903\pi$
$504$	0	0
$505$	17.6218	0.784158
$506$	0	0
$507$	10.7025	0.475315
$508$	0	0
$509$	32.9551	1.46071	0.730355	−	0.683068i	$-0.239355\pi$
$509$	32.9551	1.46071	0.730355	+	0.683068i	$0.239355\pi$
$510$	0	0
$511$	−23.6589	−1.04661
$512$	0	0
$513$	4.01719	0.177363
$514$	0	0
$515$	−16.1057	−0.709700
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−9.03410	−0.396553
$520$	0	0
$521$	−2.58112	−0.113081	−0.0565405	−	0.998400i	$-0.518007\pi$
$521$	−2.58112	−0.113081	−0.0565405	+	0.998400i	$0.518007\pi$
$522$	0	0
$523$	−4.43555	−0.193953	−0.0969765	−	0.995287i	$-0.530917\pi$
$523$	−4.43555	−0.193953	−0.0969765	+	0.995287i	$0.530917\pi$
$524$	0	0
$525$	−5.85211	−0.255407
$526$	0	0
$527$	3.16449	0.137847
$528$	0	0
$529$	−12.2887	−0.534291
$530$	0	0
$531$	14.6383	0.635249
$532$	0	0
$533$	−27.3446	−1.18443
$534$	0	0
$535$	−20.1593	−0.871563
$536$	0	0
$537$	5.46531	0.235845
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.0115	−0.688390	−0.344195	−	0.938898i	$-0.611848\pi$
$541$	−16.0115	−0.688390	−0.344195	+	0.938898i	$0.611848\pi$
$542$	0	0
$543$	0.0437409	0.00187710
$544$	0	0
$545$	−18.3577	−0.786358
$546$	0	0
$547$	−0.976078	−0.0417341	−0.0208670	−	0.999782i	$-0.506643\pi$
$547$	−0.976078	−0.0417341	−0.0208670	+	0.999782i	$0.506643\pi$
$548$	0	0
$549$	25.9534	1.10767
$550$	0	0
$551$	5.31315	0.226348
$552$	0	0
$553$	−34.4366	−1.46440
$554$	0	0
$555$	2.42209	0.102812
$556$	0	0
$557$	12.2238	0.517937	0.258969	−	0.965886i	$-0.416617\pi$
$557$	12.2238	0.517937	0.258969	+	0.965886i	$0.416617\pi$
$558$	0	0
$559$	−38.9907	−1.64913
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.1164	−1.52213	−0.761063	−	0.648678i	$-0.775322\pi$
$563$	−36.1164	−1.52213	−0.761063	+	0.648678i	$0.775322\pi$
$564$	0	0
$565$	−2.51241	−0.105698
$566$	0	0
$567$	12.3165	0.517243
$568$	0	0
$569$	6.78355	0.284381	0.142191	−	0.989839i	$-0.454585\pi$
$569$	6.78355	0.284381	0.142191	+	0.989839i	$0.454585\pi$
$570$	0	0
$571$	−18.0818	−0.756700	−0.378350	−	0.925663i	$-0.623508\pi$
$571$	−18.0818	−0.756700	−0.378350	+	0.925663i	$0.623508\pi$
$572$	0	0
$573$	17.0627	0.712805
$574$	0	0
$575$	−9.33599	−0.389338
$576$	0	0
$577$	20.1259	0.837850	0.418925	−	0.908021i	$-0.362407\pi$
$577$	20.1259	0.837850	0.418925	+	0.908021i	$0.362407\pi$
$578$	0	0
$579$	−16.1755	−0.672232
$580$	0	0
$581$	−12.8179	−0.531777
$582$	0	0
$583$	0	0
$584$	0	0
$585$	18.9059	0.781664
$586$	0	0
$587$	−11.7864	−0.486476	−0.243238	−	0.969967i	$-0.578210\pi$
$587$	−11.7864	−0.486476	−0.243238	+	0.969967i	$0.578210\pi$
$588$	0	0
$589$	4.78340	0.197096
$590$	0	0
$591$	−19.1110	−0.786121
$592$	0	0
$593$	−44.7098	−1.83601	−0.918005	−	0.396569i	$-0.870201\pi$
$593$	−44.7098	−1.83601	−0.918005	+	0.396569i	$0.870201\pi$
$594$	0	0
$595$	−2.70232	−0.110784
$596$	0	0
$597$	−9.35185	−0.382746
$598$	0	0
$599$	−29.8765	−1.22072	−0.610361	−	0.792124i	$-0.708975\pi$
$599$	−29.8765	−1.22072	−0.610361	+	0.792124i	$0.708975\pi$
$600$	0	0
$601$	12.3581	0.504095	0.252048	−	0.967715i	$-0.418896\pi$
$601$	12.3581	0.504095	0.252048	+	0.967715i	$0.418896\pi$
$602$	0	0
$603$	−37.2218	−1.51579
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.08605	0.125259	0.0626295	−	0.998037i	$-0.480051\pi$
$607$	3.08605	0.125259	0.0626295	+	0.998037i	$0.480051\pi$
$608$	0	0
$609$	−10.9000	−0.441689
$610$	0	0
$611$	−70.9595	−2.87071
$612$	0	0
$613$	−15.4192	−0.622776	−0.311388	−	0.950283i	$-0.600794\pi$
$613$	−15.4192	−0.622776	−0.311388	+	0.950283i	$0.600794\pi$
$614$	0	0
$615$	5.61944	0.226598
$616$	0	0
$617$	−34.0019	−1.36886	−0.684432	−	0.729077i	$-0.739950\pi$
$617$	−34.0019	−1.36886	−0.684432	+	0.729077i	$0.739950\pi$
$618$	0	0
$619$	−3.06520	−0.123201	−0.0616005	−	0.998101i	$-0.519620\pi$
$619$	−3.06520	−0.123201	−0.0616005	+	0.998101i	$0.519620\pi$
$620$	0	0
$621$	−13.1475	−0.527592
$622$	0	0
$623$	−27.9639	−1.12035
$624$	0	0
$625$	−2.59980	−0.103992
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.48572	−0.0592397
$630$	0	0
$631$	−4.56632	−0.181782	−0.0908912	−	0.995861i	$-0.528972\pi$
$631$	−4.56632	−0.181782	−0.0908912	+	0.995861i	$0.528972\pi$
$632$	0	0
$633$	4.93767	0.196255
$634$	0	0
$635$	0.0481563	0.00191103
$636$	0	0
$637$	−4.04125	−0.160120
$638$	0	0
$639$	−35.6128	−1.40882
$640$	0	0
$641$	24.6689	0.974364	0.487182	−	0.873301i	$-0.338025\pi$
$641$	24.6689	0.974364	0.487182	+	0.873301i	$0.338025\pi$
$642$	0	0
$643$	12.8861	0.508178	0.254089	−	0.967181i	$-0.418224\pi$
$643$	12.8861	0.508178	0.254089	+	0.967181i	$0.418224\pi$
$644$	0	0
$645$	8.01276	0.315502
$646$	0	0
$647$	−9.04643	−0.355652	−0.177826	−	0.984062i	$-0.556906\pi$
$647$	−9.04643	−0.355652	−0.177826	+	0.984062i	$0.556906\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−9.81318	−0.384609
$652$	0	0
$653$	14.4810	0.566686	0.283343	−	0.959019i	$-0.408557\pi$
$653$	14.4810	0.566686	0.283343	+	0.959019i	$0.408557\pi$
$654$	0	0
$655$	−10.0355	−0.392119
$656$	0	0
$657$	20.8653	0.814033
$658$	0	0
$659$	−25.8823	−1.00823	−0.504115	−	0.863636i	$-0.668181\pi$
$659$	−25.8823	−1.00823	−0.504115	+	0.863636i	$0.668181\pi$
$660$	0	0
$661$	28.0516	1.09108	0.545541	−	0.838084i	$-0.316324\pi$
$661$	28.0516	1.09108	0.545541	+	0.838084i	$0.316324\pi$
$662$	0	0
$663$	2.55521	0.0992361
$664$	0	0
$665$	−4.08479	−0.158401
$666$	0	0
$667$	−17.3890	−0.673303
$668$	0	0
$669$	12.0122	0.464419
$670$	0	0
$671$	0	0
$672$	0	0
$673$	48.7740	1.88010	0.940050	−	0.341036i	$-0.110778\pi$
$673$	48.7740	1.88010	0.940050	+	0.341036i	$0.110778\pi$
$674$	0	0
$675$	11.4594	0.441072
$676$	0	0
$677$	−46.2533	−1.77766	−0.888828	−	0.458240i	$-0.848480\pi$
$677$	−46.2533	−1.77766	−0.888828	+	0.458240i	$0.848480\pi$
$678$	0	0
$679$	−50.4318	−1.93540
$680$	0	0
$681$	3.44623	0.132060
$682$	0	0
$683$	7.80364	0.298598	0.149299	−	0.988792i	$-0.452298\pi$
$683$	7.80364	0.298598	0.149299	+	0.988792i	$0.452298\pi$
$684$	0	0
$685$	−25.1194	−0.959762
$686$	0	0
$687$	−9.70485	−0.370263
$688$	0	0
$689$	50.3544	1.91835
$690$	0	0
$691$	10.7800	0.410089	0.205045	−	0.978753i	$-0.434266\pi$
$691$	10.7800	0.410089	0.205045	+	0.978753i	$0.434266\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.03066	0.152892
$696$	0	0
$697$	−3.44700	−0.130564
$698$	0	0
$699$	6.82884	0.258290
$700$	0	0
$701$	−41.2228	−1.55696	−0.778481	−	0.627668i	$-0.784010\pi$
$701$	−41.2228	−1.55696	−0.778481	+	0.627668i	$0.784010\pi$
$702$	0	0
$703$	−2.24580	−0.0847019
$704$	0	0
$705$	14.5825	0.549209
$706$	0	0
$707$	33.5200	1.26065
$708$	0	0
$709$	−35.3192	−1.32644	−0.663219	−	0.748425i	$-0.730810\pi$
$709$	−35.3192	−1.32644	−0.663219	+	0.748425i	$0.730810\pi$
$710$	0	0
$711$	30.3705	1.13898
$712$	0	0
$713$	−15.6552	−0.586291
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.8008	−0.739475
$718$	0	0
$719$	−45.6912	−1.70399	−0.851997	−	0.523546i	$-0.824609\pi$
$719$	−45.6912	−1.70399	−0.851997	+	0.523546i	$0.824609\pi$
$720$	0	0
$721$	−30.6360	−1.14095
$722$	0	0
$723$	2.52424	0.0938777
$724$	0	0
$725$	15.1562	0.562889
$726$	0	0
$727$	−50.2094	−1.86216	−0.931082	−	0.364810i	$-0.881134\pi$
$727$	−50.2094	−1.86216	−0.931082	+	0.364810i	$0.881134\pi$
$728$	0	0
$729$	−1.99256	−0.0737985
$730$	0	0
$731$	−4.91508	−0.181791
$732$	0	0
$733$	12.7694	0.471649	0.235824	−	0.971796i	$-0.424221\pi$
$733$	12.7694	0.471649	0.235824	+	0.971796i	$0.424221\pi$
$734$	0	0
$735$	0.830495	0.0306333
$736$	0	0
$737$	0	0
$738$	0	0
$739$	43.5923	1.60357	0.801783	−	0.597615i	$-0.203885\pi$
$739$	43.5923	1.60357	0.801783	+	0.597615i	$0.203885\pi$
$740$	0	0
$741$	3.86242	0.141889
$742$	0	0
$743$	−26.7661	−0.981955	−0.490977	−	0.871172i	$-0.663360\pi$
$743$	−26.7661	−0.981955	−0.490977	+	0.871172i	$0.663360\pi$
$744$	0	0
$745$	7.71262	0.282569
$746$	0	0
$747$	11.3044	0.413607
$748$	0	0
$749$	−38.3469	−1.40116
$750$	0	0
$751$	−11.0819	−0.404386	−0.202193	−	0.979346i	$-0.564807\pi$
$751$	−11.0819	−0.404386	−0.202193	+	0.979346i	$0.564807\pi$
$752$	0	0
$753$	5.12939	0.186925
$754$	0	0
$755$	−19.5914	−0.713003
$756$	0	0
$757$	−9.37562	−0.340763	−0.170381	−	0.985378i	$-0.554500\pi$
$757$	−9.37562	−0.340763	−0.170381	+	0.985378i	$0.554500\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.04634	0.255430	0.127715	−	0.991811i	$-0.459236\pi$
$761$	7.04634	0.255430	0.127715	+	0.991811i	$0.459236\pi$
$762$	0	0
$763$	−34.9199	−1.26418
$764$	0	0
$765$	2.38324	0.0861662
$766$	0	0
$767$	31.2497	1.12836
$768$	0	0
$769$	42.3274	1.52637	0.763183	−	0.646183i	$-0.223636\pi$
$769$	42.3274	1.52637	0.763183	+	0.646183i	$0.223636\pi$
$770$	0	0
$771$	−7.15071	−0.257526
$772$	0	0
$773$	13.5595	0.487701	0.243850	−	0.969813i	$-0.421589\pi$
$773$	13.5595	0.487701	0.243850	+	0.969813i	$0.421589\pi$
$774$	0	0
$775$	13.6451	0.490145
$776$	0	0
$777$	4.60727	0.165285
$778$	0	0
$779$	−5.21043	−0.186683
$780$	0	0
$781$	0	0
$782$	0	0
$783$	21.3439	0.762771
$784$	0	0
$785$	24.8714	0.887699
$786$	0	0
$787$	19.7436	0.703784	0.351892	−	0.936041i	$-0.385538\pi$
$787$	19.7436	0.703784	0.351892	+	0.936041i	$0.385538\pi$
$788$	0	0
$789$	−14.9067	−0.530691
$790$	0	0
$791$	−4.77909	−0.169925
$792$	0	0
$793$	55.4051	1.96749
$794$	0	0
$795$	−10.3481	−0.367008
$796$	0	0
$797$	−20.7217	−0.733999	−0.367000	−	0.930221i	$-0.619615\pi$
$797$	−20.7217	−0.733999	−0.367000	+	0.930221i	$0.619615\pi$
$798$	0	0
$799$	−8.94500	−0.316451
$800$	0	0
$801$	24.6620	0.871390
$802$	0	0
$803$	0	0
$804$	0	0
$805$	13.3688	0.471187
$806$	0	0
$807$	0.425176	0.0149669
$808$	0	0
$809$	−20.1883	−0.709783	−0.354891	−	0.934908i	$-0.615482\pi$
$809$	−20.1883	−0.709783	−0.354891	+	0.934908i	$0.615482\pi$
$810$	0	0
$811$	−2.03212	−0.0713573	−0.0356786	−	0.999363i	$-0.511359\pi$
$811$	−2.03212	−0.0713573	−0.0356786	+	0.999363i	$0.511359\pi$
$812$	0	0
$813$	−1.28750	−0.0451547
$814$	0	0
$815$	8.32455	0.291596
$816$	0	0
$817$	−7.42956	−0.259927
$818$	0	0
$819$	35.9627	1.25664
$820$	0	0
$821$	−32.2030	−1.12389	−0.561946	−	0.827174i	$-0.689947\pi$
$821$	−32.2030	−1.12389	−0.561946	+	0.827174i	$0.689947\pi$
$822$	0	0
$823$	5.04397	0.175822	0.0879109	−	0.996128i	$-0.471981\pi$
$823$	5.04397	0.175822	0.0879109	+	0.996128i	$0.471981\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.33031	−0.289673	−0.144837	−	0.989456i	$-0.546266\pi$
$827$	−8.33031	−0.289673	−0.144837	+	0.989456i	$0.546266\pi$
$828$	0	0
$829$	−44.2158	−1.53568	−0.767839	−	0.640643i	$-0.778668\pi$
$829$	−44.2158	−1.53568	−0.767839	+	0.640643i	$0.778668\pi$
$830$	0	0
$831$	−10.8325	−0.375776
$832$	0	0
$833$	−0.509431	−0.0176507
$834$	0	0
$835$	0.779946	0.0269912
$836$	0	0
$837$	19.2158	0.664196
$838$	0	0
$839$	10.8395	0.374222	0.187111	−	0.982339i	$-0.440088\pi$
$839$	10.8395	0.374222	0.187111	+	0.982339i	$0.440088\pi$
$840$	0	0
$841$	−0.770397	−0.0265654
$842$	0	0
$843$	−3.99178	−0.137484
$844$	0	0
$845$	21.3099	0.733083
$846$	0	0
$847$	0	0
$848$	0	0
$849$	15.1936	0.521443
$850$	0	0
$851$	7.35008	0.251958
$852$	0	0
$853$	14.2376	0.487485	0.243742	−	0.969840i	$-0.421625\pi$
$853$	14.2376	0.487485	0.243742	+	0.969840i	$0.421625\pi$
$854$	0	0
$855$	3.60247	0.123202
$856$	0	0
$857$	−1.80268	−0.0615783	−0.0307891	−	0.999526i	$-0.509802\pi$
$857$	−1.80268	−0.0615783	−0.0307891	+	0.999526i	$0.509802\pi$
$858$	0	0
$859$	19.8861	0.678506	0.339253	−	0.940695i	$-0.389826\pi$
$859$	19.8861	0.678506	0.339253	+	0.940695i	$0.389826\pi$
$860$	0	0
$861$	10.6893	0.364289
$862$	0	0
$863$	6.66877	0.227008	0.113504	−	0.993538i	$-0.463793\pi$
$863$	6.66877	0.227008	0.113504	+	0.993538i	$0.463793\pi$
$864$	0	0
$865$	−17.9879	−0.611608
$866$	0	0
$867$	−12.1894	−0.413975
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−79.4607	−2.69242
$872$	0	0
$873$	44.4770	1.50532
$874$	0	0
$875$	−32.0762	−1.08437
$876$	0	0
$877$	28.9853	0.978764	0.489382	−	0.872070i	$-0.337222\pi$
$877$	28.9853	0.978764	0.489382	+	0.872070i	$0.337222\pi$
$878$	0	0
$879$	7.81228	0.263502
$880$	0	0
$881$	−34.4951	−1.16217	−0.581084	−	0.813843i	$-0.697371\pi$
$881$	−34.4951	−1.16217	−0.581084	+	0.813843i	$0.697371\pi$
$882$	0	0
$883$	11.9912	0.403538	0.201769	−	0.979433i	$-0.435331\pi$
$883$	11.9912	0.403538	0.201769	+	0.979433i	$0.435331\pi$
$884$	0	0
$885$	−6.42196	−0.215872
$886$	0	0
$887$	14.2750	0.479308	0.239654	−	0.970858i	$-0.422966\pi$
$887$	14.2750	0.479308	0.239654	+	0.970858i	$0.422966\pi$
$888$	0	0
$889$	0.0916025	0.00307225
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.5211	−0.452467
$894$	0	0
$895$	10.8821	0.363747
$896$	0	0
$897$	−12.6410	−0.422070
$898$	0	0
$899$	25.4149	0.847635
$900$	0	0
$901$	6.34757	0.211468
$902$	0	0
$903$	15.2418	0.507215
$904$	0	0
$905$	0.0870931	0.00289507
$906$	0	0
$907$	22.8888	0.760009	0.380005	−	0.924985i	$-0.375922\pi$
$907$	22.8888	0.760009	0.380005	+	0.924985i	$0.375922\pi$
$908$	0	0
$909$	−29.5620	−0.980511
$910$	0	0
$911$	0.120096	0.00397895	0.00198948	−	0.999998i	$-0.499367\pi$
$911$	0.120096	0.00397895	0.00198948	+	0.999998i	$0.499367\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−11.3860	−0.376409
$916$	0	0
$917$	−19.0894	−0.630388
$918$	0	0
$919$	46.4700	1.53290	0.766452	−	0.642301i	$-0.222020\pi$
$919$	46.4700	1.53290	0.766452	+	0.642301i	$0.222020\pi$
$920$	0	0
$921$	−4.91149	−0.161839
$922$	0	0
$923$	−76.0258	−2.50242
$924$	0	0
$925$	−6.40634	−0.210639
$926$	0	0
$927$	27.0187	0.887409
$928$	0	0
$929$	4.31451	0.141555	0.0707773	−	0.997492i	$-0.477452\pi$
$929$	4.31451	0.141555	0.0707773	+	0.997492i	$0.477452\pi$
$930$	0	0
$931$	−0.770048	−0.0252373
$932$	0	0
$933$	−7.83446	−0.256489
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.23075	0.301555	0.150778	−	0.988568i	$-0.451822\pi$
$937$	9.23075	0.301555	0.150778	+	0.988568i	$0.451822\pi$
$938$	0	0
$939$	−8.62504	−0.281467
$940$	0	0
$941$	57.9211	1.88817	0.944086	−	0.329699i	$-0.106947\pi$
$941$	57.9211	1.88817	0.944086	+	0.329699i	$0.106947\pi$
$942$	0	0
$943$	17.0528	0.555315
$944$	0	0
$945$	−16.4094	−0.533797
$946$	0	0
$947$	−23.9217	−0.777350	−0.388675	−	0.921375i	$-0.627067\pi$
$947$	−23.9217	−0.777350	−0.388675	+	0.921375i	$0.627067\pi$
$948$	0	0
$949$	44.5430	1.44593
$950$	0	0
$951$	9.35670	0.303412
$952$	0	0
$953$	−32.3468	−1.04782	−0.523908	−	0.851775i	$-0.675527\pi$
$953$	−32.3468	−1.04782	−0.523908	+	0.851775i	$0.675527\pi$
$954$	0	0
$955$	33.9738	1.09937
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−47.7818	−1.54296
$960$	0	0
$961$	−8.11910	−0.261906
$962$	0	0
$963$	33.8190	1.08980
$964$	0	0
$965$	−32.2073	−1.03679
$966$	0	0
$967$	−1.02137	−0.0328451	−0.0164225	−	0.999865i	$-0.505228\pi$
$967$	−1.02137	−0.0328451	−0.0164225	+	0.999865i	$0.505228\pi$
$968$	0	0
$969$	0.486888	0.0156411
$970$	0	0
$971$	−50.7514	−1.62869	−0.814345	−	0.580381i	$-0.802904\pi$
$971$	−50.7514	−1.62869	−0.814345	+	0.580381i	$0.802904\pi$
$972$	0	0
$973$	7.66709	0.245795
$974$	0	0
$975$	11.0179	0.352855
$976$	0	0
$977$	−13.1735	−0.421456	−0.210728	−	0.977545i	$-0.567583\pi$
$977$	−13.1735	−0.421456	−0.210728	+	0.977545i	$0.567583\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.7967	0.983262
$982$	0	0
$983$	−9.24612	−0.294905	−0.147453	−	0.989069i	$-0.547107\pi$
$983$	−9.24612	−0.294905	−0.147453	+	0.989069i	$0.547107\pi$
$984$	0	0
$985$	−38.0521	−1.21244
$986$	0	0
$987$	27.7387	0.882933
$988$	0	0
$989$	24.3156	0.773190
$990$	0	0
$991$	10.6286	0.337627	0.168814	−	0.985648i	$-0.446006\pi$
$991$	10.6286	0.337627	0.168814	+	0.985648i	$0.446006\pi$
$992$	0	0
$993$	5.07896	0.161176
$994$	0	0
$995$	−18.6206	−0.590313
$996$	0	0
$997$	−5.06696	−0.160472	−0.0802361	−	0.996776i	$-0.525567\pi$
$997$	−5.06696	−0.160472	−0.0802361	+	0.996776i	$0.525567\pi$
$998$	0	0
$999$	−9.02180	−0.285437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.o.1.4	✓	7
11.10	odd	2		9196.2.a.p.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.o.1.4	✓	7	1.1	even	1	trivial
9196.2.a.p.1.4	yes	7	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+0.735972 q^{3} +1.46540 q^{5} +2.78748 q^{7} -2.45834 q^{9} -5.24805 q^{13} +1.07850 q^{15} -0.661557 q^{17} -1.00000 q^{19} +2.05151 q^{21} +3.27281 q^{23} -2.85259 q^{25} -4.01719 q^{27} -5.31315 q^{29} -4.78340 q^{31} +4.08479 q^{35} +2.24580 q^{37} -3.86242 q^{39} +5.21043 q^{41} +7.42956 q^{43} -3.60247 q^{45} +13.5211 q^{47} +0.770048 q^{49} -0.486888 q^{51} -9.59489 q^{53} -0.735972 q^{57} -5.95454 q^{59} -10.5573 q^{61} -6.85259 q^{63} -7.69051 q^{65} +15.1410 q^{67} +2.40870 q^{69} +14.4865 q^{71} -8.48754 q^{73} -2.09943 q^{75} -12.3540 q^{79} +4.41849 q^{81} -4.59839 q^{83} -0.969449 q^{85} -3.91033 q^{87} -10.0320 q^{89} -14.6288 q^{91} -3.52045 q^{93} -1.46540 q^{95} -18.0923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} + 3 q^{5} - 2 q^{7} + 8 q^{9} + 11 q^{15} - 12 q^{17} - 7 q^{19} - q^{21} - 5 q^{23} + 4 q^{25} - 13 q^{27} + 2 q^{29} + 5 q^{31} - 16 q^{35} + 8 q^{37} - 15 q^{39} - 31 q^{41} - 12 q^{43}+ \cdots - 25 q^{97}+O(q^{100}) \) 7 * q - q^3 + 3 * q^5 - 2 * q^7 + 8 * q^9 + 11 * q^15 - 12 * q^17 - 7 * q^19 - q^21 - 5 * q^23 + 4 * q^25 - 13 * q^27 + 2 * q^29 + 5 * q^31 - 16 * q^35 + 8 * q^37 - 15 * q^39 - 31 * q^41 - 12 * q^43 - 24 * q^45 + 10 * q^47 - 3 * q^49 - 5 * q^51 - 25 * q^53 + q^57 - 7 * q^59 - q^61 - 24 * q^63 - 13 * q^65 - 5 * q^67 + 31 * q^69 + q^71 + 12 * q^73 - 12 * q^75 - 20 * q^79 + 35 * q^81 - 14 * q^83 + 6 * q^85 - 28 * q^87 - 18 * q^89 + 25 * q^91 - 35 * q^93 - 3 * q^95 - 25 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more