LMFDB - Embedded newform 9576.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.64575 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.64575 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.00000 q^{13} +0.354249 q^{17} +1.00000 q^{19} +6.00000 q^{23} +8.29150 q^{25} -5.64575 q^{29} +1.29150 q^{31} -3.64575 q^{35} +5.29150 q^{37} +8.58301 q^{41} -2.00000 q^{43} -2.35425 q^{47} +1.00000 q^{49} +8.93725 q^{53} +7.29150 q^{55} -14.5830 q^{59} +2.70850 q^{61} +14.5830 q^{65} +14.5830 q^{67} -0.354249 q^{71} -9.29150 q^{73} -2.00000 q^{77} +11.2915 q^{79} -16.9373 q^{83} -1.29150 q^{85} -6.00000 q^{89} -4.00000 q^{91} -3.64575 q^{95} +16.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{19} + 12 q^{23} + 6 q^{25} - 6 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{41} - 4 q^{43} - 10 q^{47} + 2 q^{49} + 2 q^{53} + 4 q^{55} - 8 q^{59} + 16 q^{61} + 8 q^{65} + 8 q^{67} - 6 q^{71} - 8 q^{73} - 4 q^{77} + 12 q^{79} - 18 q^{83} + 8 q^{85} - 12 q^{89} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^19 + 12 * q^23 + 6 * q^25 - 6 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^41 - 4 * q^43 - 10 * q^47 + 2 * q^49 + 2 * q^53 + 4 * q^55 - 8 * q^59 + 16 * q^61 + 8 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 4 * q^77 + 12 * q^79 - 18 * q^83 + 8 * q^85 - 12 * q^89 - 8 * q^91 - 2 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.354249	0.0859179	0.0429590	−	0.999077i	$-0.486322\pi$
$17$	0.354249	0.0859179	0.0429590	+	0.999077i	$0.486322\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.64575	−1.04839	−0.524195	−	0.851598i	$-0.675634\pi$
$29$	−5.64575	−1.04839	−0.524195	+	0.851598i	$0.675634\pi$
$30$	0	0
$31$	1.29150	0.231961	0.115980	−	0.993252i	$-0.462999\pi$
$31$	1.29150	0.231961	0.115980	+	0.993252i	$0.462999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.64575	−0.616244
$36$	0	0
$37$	5.29150	0.869918	0.434959	−	0.900450i	$-0.356763\pi$
$37$	5.29150	0.869918	0.434959	+	0.900450i	$0.356763\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.58301	1.34044	0.670220	−	0.742162i	$-0.266200\pi$
$41$	8.58301	1.34044	0.670220	+	0.742162i	$0.266200\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.35425	−0.343402	−0.171701	−	0.985149i	$-0.554926\pi$
$47$	−2.35425	−0.343402	−0.171701	+	0.985149i	$0.554926\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.93725	1.22763	0.613813	−	0.789451i	$-0.289635\pi$
$53$	8.93725	1.22763	0.613813	+	0.789451i	$0.289635\pi$
$54$	0	0
$55$	7.29150	0.983186
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.5830	−1.89855	−0.949273	−	0.314454i	$-0.898179\pi$
$59$	−14.5830	−1.89855	−0.949273	+	0.314454i	$0.898179\pi$
$60$	0	0
$61$	2.70850	0.346788	0.173394	−	0.984853i	$-0.444527\pi$
$61$	2.70850	0.346788	0.173394	+	0.984853i	$0.444527\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.5830	1.80880
$66$	0	0
$67$	14.5830	1.78160	0.890799	−	0.454398i	$-0.150146\pi$
$67$	14.5830	1.78160	0.890799	+	0.454398i	$0.150146\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.354249	−0.0420416	−0.0210208	−	0.999779i	$-0.506692\pi$
$71$	−0.354249	−0.0420416	−0.0210208	+	0.999779i	$0.506692\pi$
$72$	0	0
$73$	−9.29150	−1.08749	−0.543744	−	0.839251i	$-0.682994\pi$
$73$	−9.29150	−1.08749	−0.543744	+	0.839251i	$0.682994\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	11.2915	1.27039	0.635197	−	0.772350i	$-0.280919\pi$
$79$	11.2915	1.27039	0.635197	+	0.772350i	$0.280919\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.9373	−1.85911	−0.929553	−	0.368690i	$-0.879807\pi$
$83$	−16.9373	−1.85911	−0.929553	+	0.368690i	$0.879807\pi$
$84$	0	0
$85$	−1.29150	−0.140083
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.64575	−0.374046
$96$	0	0
$97$	16.5830	1.68375	0.841875	−	0.539673i	$-0.181452\pi$
$97$	16.5830	1.68375	0.841875	+	0.539673i	$0.181452\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.35425	−0.433264	−0.216632	−	0.976253i	$-0.569507\pi$
$101$	−4.35425	−0.433264	−0.216632	+	0.976253i	$0.569507\pi$
$102$	0	0
$103$	−2.58301	−0.254511	−0.127256	−	0.991870i	$-0.540617\pi$
$103$	−2.58301	−0.254511	−0.127256	+	0.991870i	$0.540617\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.64575	−0.352448	−0.176224	−	0.984350i	$-0.556388\pi$
$107$	−3.64575	−0.352448	−0.176224	+	0.984350i	$0.556388\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.9373	1.59332	0.796661	−	0.604426i	$-0.206597\pi$
$113$	16.9373	1.59332	0.796661	+	0.604426i	$0.206597\pi$
$114$	0	0
$115$	−21.8745	−2.03981
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.354249	0.0324739
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9373	1.47981	0.739907	−	0.672709i	$-0.234869\pi$
$131$	16.9373	1.47981	0.739907	+	0.672709i	$0.234869\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−22.5830	−1.92940	−0.964698	−	0.263358i	$-0.915170\pi$
$137$	−22.5830	−1.92940	−0.964698	+	0.263358i	$0.915170\pi$
$138$	0	0
$139$	−0.708497	−0.0600940	−0.0300470	−	0.999548i	$-0.509566\pi$
$139$	−0.708497	−0.0600940	−0.0300470	+	0.999548i	$0.509566\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	20.5830	1.70933
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.2915	−1.25273	−0.626364	−	0.779530i	$-0.715458\pi$
$149$	−15.2915	−1.25273	−0.626364	+	0.779530i	$0.715458\pi$
$150$	0	0
$151$	19.2915	1.56992	0.784960	−	0.619546i	$-0.212683\pi$
$151$	19.2915	1.56992	0.784960	+	0.619546i	$0.212683\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.70850	−0.378196
$156$	0	0
$157$	11.8745	0.947689	0.473844	−	0.880609i	$-0.342866\pi$
$157$	11.8745	0.947689	0.473844	+	0.880609i	$0.342866\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−12.5830	−0.985577	−0.492789	−	0.870149i	$-0.664022\pi$
$163$	−12.5830	−0.985577	−0.492789	+	0.870149i	$0.664022\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.2915	−1.18329	−0.591646	−	0.806198i	$-0.701522\pi$
$167$	−15.2915	−1.18329	−0.591646	+	0.806198i	$0.701522\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	8.29150	0.626779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−25.5203	−1.90747	−0.953737	−	0.300643i	$-0.902799\pi$
$179$	−25.5203	−1.90747	−0.953737	+	0.300643i	$0.902799\pi$
$180$	0	0
$181$	16.5830	1.23261	0.616303	−	0.787509i	$-0.288630\pi$
$181$	16.5830	1.23261	0.616303	+	0.787509i	$0.288630\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19.2915	−1.41834
$186$	0	0
$187$	−0.708497	−0.0518105
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	−25.2915	−1.82052	−0.910261	−	0.414035i	$-0.864119\pi$
$193$	−25.2915	−1.82052	−0.910261	+	0.414035i	$0.864119\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.41699	0.385945	0.192972	−	0.981204i	$-0.438187\pi$
$197$	5.41699	0.385945	0.192972	+	0.981204i	$0.438187\pi$
$198$	0	0
$199$	1.41699	0.100448	0.0502240	−	0.998738i	$-0.484006\pi$
$199$	1.41699	0.100448	0.0502240	+	0.998738i	$0.484006\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.64575	−0.396254
$204$	0	0
$205$	−31.2915	−2.18549
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−1.41699	−0.0975499	−0.0487750	−	0.998810i	$-0.515532\pi$
$211$	−1.41699	−0.0975499	−0.0487750	+	0.998810i	$0.515532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.29150	0.497276
$216$	0	0
$217$	1.29150	0.0876729
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.41699	−0.0953174
$222$	0	0
$223$	−5.29150	−0.354345	−0.177173	−	0.984180i	$-0.556695\pi$
$223$	−5.29150	−0.354345	−0.177173	+	0.984180i	$0.556695\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.7085	−0.843493	−0.421746	−	0.906714i	$-0.638583\pi$
$227$	−12.7085	−0.843493	−0.421746	+	0.906714i	$0.638583\pi$
$228$	0	0
$229$	−8.58301	−0.567181	−0.283590	−	0.958945i	$-0.591526\pi$
$229$	−8.58301	−0.567181	−0.283590	+	0.958945i	$0.591526\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.2915	1.00178	0.500890	−	0.865511i	$-0.333006\pi$
$233$	15.2915	1.00178	0.500890	+	0.865511i	$0.333006\pi$
$234$	0	0
$235$	8.58301	0.559894
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.70850	−0.175198	−0.0875991	−	0.996156i	$-0.527919\pi$
$239$	−2.70850	−0.175198	−0.0875991	+	0.996156i	$0.527919\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.64575	−0.232919
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.35425	−0.148599	−0.0742994	−	0.997236i	$-0.523672\pi$
$251$	−2.35425	−0.148599	−0.0742994	+	0.997236i	$0.523672\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.2915	−0.829101	−0.414551	−	0.910026i	$-0.636061\pi$
$257$	−13.2915	−0.829101	−0.414551	+	0.910026i	$0.636061\pi$
$258$	0	0
$259$	5.29150	0.328798
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.70850	−0.167013	−0.0835066	−	0.996507i	$-0.526612\pi$
$263$	−2.70850	−0.167013	−0.0835066	+	0.996507i	$0.526612\pi$
$264$	0	0
$265$	−32.5830	−2.00156
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.4575	1.36926	0.684629	−	0.728891i	$-0.259964\pi$
$269$	22.4575	1.36926	0.684629	+	0.728891i	$0.259964\pi$
$270$	0	0
$271$	4.70850	0.286021	0.143010	−	0.989721i	$-0.454322\pi$
$271$	4.70850	0.286021	0.143010	+	0.989721i	$0.454322\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.5830	−0.999993
$276$	0	0
$277$	−29.8745	−1.79499	−0.897493	−	0.441030i	$-0.854613\pi$
$277$	−29.8745	−1.79499	−0.897493	+	0.441030i	$0.854613\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.06275	0.421328	0.210664	−	0.977559i	$-0.432437\pi$
$281$	7.06275	0.421328	0.210664	+	0.977559i	$0.432437\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.58301	0.506639
$288$	0	0
$289$	−16.8745	−0.992618
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.41699	−0.199623	−0.0998115	−	0.995006i	$-0.531824\pi$
$293$	−3.41699	−0.199623	−0.0998115	+	0.995006i	$0.531824\pi$
$294$	0	0
$295$	53.1660	3.09544
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.87451	−0.565413
$306$	0	0
$307$	−23.8745	−1.36259	−0.681295	−	0.732009i	$-0.738583\pi$
$307$	−23.8745	−1.36259	−0.681295	+	0.732009i	$0.738583\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.3542	0.813955	0.406977	−	0.913438i	$-0.366583\pi$
$311$	14.3542	0.813955	0.406977	+	0.913438i	$0.366583\pi$
$312$	0	0
$313$	17.2915	0.977374	0.488687	−	0.872459i	$-0.337476\pi$
$313$	17.2915	0.977374	0.488687	+	0.872459i	$0.337476\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.93725	0.277304	0.138652	−	0.990341i	$-0.455723\pi$
$317$	4.93725	0.277304	0.138652	+	0.990341i	$0.455723\pi$
$318$	0	0
$319$	11.2915	0.632203
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.354249	0.0197109
$324$	0	0
$325$	−33.1660	−1.83972
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.35425	−0.129794
$330$	0	0
$331$	−24.4575	−1.34431	−0.672153	−	0.740412i	$-0.734630\pi$
$331$	−24.4575	−1.34431	−0.672153	+	0.740412i	$0.734630\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−53.1660	−2.90477
$336$	0	0
$337$	12.5830	0.685440	0.342720	−	0.939438i	$-0.388652\pi$
$337$	12.5830	0.685440	0.342720	+	0.939438i	$0.388652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.58301	−0.139878
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.8745	1.06692	0.533460	−	0.845825i	$-0.320892\pi$
$347$	19.8745	1.06692	0.533460	+	0.845825i	$0.320892\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.6458	1.04564	0.522819	−	0.852444i	$-0.324880\pi$
$353$	19.6458	1.04564	0.522819	+	0.852444i	$0.324880\pi$
$354$	0	0
$355$	1.29150	0.0685458
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.5830	1.71967	0.859833	−	0.510576i	$-0.170568\pi$
$359$	32.5830	1.71967	0.859833	+	0.510576i	$0.170568\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.8745	1.77307
$366$	0	0
$367$	−17.1660	−0.896058	−0.448029	−	0.894019i	$-0.647874\pi$
$367$	−17.1660	−0.896058	−0.448029	+	0.894019i	$0.647874\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.93725	0.463999
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.5830	1.16308
$378$	0	0
$379$	7.29150	0.374539	0.187270	−	0.982309i	$-0.440036\pi$
$379$	7.29150	0.374539	0.187270	+	0.982309i	$0.440036\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.12549	0.108608	0.0543038	−	0.998524i	$-0.482706\pi$
$383$	2.12549	0.108608	0.0543038	+	0.998524i	$0.482706\pi$
$384$	0	0
$385$	7.29150	0.371609
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.8745	1.10908	0.554541	−	0.832157i	$-0.312894\pi$
$389$	21.8745	1.10908	0.554541	+	0.832157i	$0.312894\pi$
$390$	0	0
$391$	2.12549	0.107491
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−41.1660	−2.07129
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.6458	−0.681436	−0.340718	−	0.940165i	$-0.610670\pi$
$401$	−13.6458	−0.681436	−0.340718	+	0.940165i	$0.610670\pi$
$402$	0	0
$403$	−5.16601	−0.257337
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.5830	−0.524580
$408$	0	0
$409$	−13.1660	−0.651017	−0.325509	−	0.945539i	$-0.605536\pi$
$409$	−13.1660	−0.651017	−0.325509	+	0.945539i	$0.605536\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.5830	−0.717583
$414$	0	0
$415$	61.7490	3.03114
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.06275	−0.345038	−0.172519	−	0.985006i	$-0.555191\pi$
$419$	−7.06275	−0.345038	−0.172519	+	0.985006i	$0.555191\pi$
$420$	0	0
$421$	19.8745	0.968624	0.484312	−	0.874895i	$-0.339070\pi$
$421$	19.8745	0.968624	0.484312	+	0.874895i	$0.339070\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.93725	0.142478
$426$	0	0
$427$	2.70850	0.131073
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.0627	−0.629210	−0.314605	−	0.949223i	$-0.601872\pi$
$431$	−13.0627	−0.629210	−0.314605	+	0.949223i	$0.601872\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−35.8745	−1.71220	−0.856098	−	0.516813i	$-0.827118\pi$
$439$	−35.8745	−1.71220	−0.856098	+	0.516813i	$0.827118\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.2915	−1.01159	−0.505795	−	0.862654i	$-0.668801\pi$
$443$	−21.2915	−1.01159	−0.505795	+	0.862654i	$0.668801\pi$
$444$	0	0
$445$	21.8745	1.03695
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.64575	−0.455211	−0.227606	−	0.973753i	$-0.573090\pi$
$449$	−9.64575	−0.455211	−0.227606	+	0.973753i	$0.573090\pi$
$450$	0	0
$451$	−17.1660	−0.808316
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.5830	0.683662
$456$	0	0
$457$	−33.8745	−1.58458	−0.792291	−	0.610143i	$-0.791112\pi$
$457$	−33.8745	−1.58458	−0.792291	+	0.610143i	$0.791112\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.9373	−0.509399	−0.254699	−	0.967020i	$-0.581977\pi$
$461$	−10.9373	−0.509399	−0.254699	+	0.967020i	$0.581977\pi$
$462$	0	0
$463$	−29.1660	−1.35546	−0.677730	−	0.735311i	$-0.737036\pi$
$463$	−29.1660	−1.35546	−0.677730	+	0.735311i	$0.737036\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.22876	−0.380781	−0.190391	−	0.981708i	$-0.560975\pi$
$467$	−8.22876	−0.380781	−0.190391	+	0.981708i	$0.560975\pi$
$468$	0	0
$469$	14.5830	0.673381
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	8.29150	0.380440
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.2288	−0.924275	−0.462138	−	0.886808i	$-0.652917\pi$
$479$	−20.2288	−0.924275	−0.462138	+	0.886808i	$0.652917\pi$
$480$	0	0
$481$	−21.1660	−0.965087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−60.4575	−2.74523
$486$	0	0
$487$	35.7490	1.61994	0.809971	−	0.586470i	$-0.199483\pi$
$487$	35.7490	1.61994	0.809971	+	0.586470i	$0.199483\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.29150	−0.419320	−0.209660	−	0.977774i	$-0.567236\pi$
$491$	−9.29150	−0.419320	−0.209660	+	0.977774i	$0.567236\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.354249	−0.0158902
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.5203	1.04872	0.524358	−	0.851498i	$-0.324305\pi$
$503$	23.5203	1.04872	0.524358	+	0.851498i	$0.324305\pi$
$504$	0	0
$505$	15.8745	0.706406
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.2915	−0.943729	−0.471865	−	0.881671i	$-0.656419\pi$
$509$	−21.2915	−0.943729	−0.471865	+	0.881671i	$0.656419\pi$
$510$	0	0
$511$	−9.29150	−0.411032
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.41699	0.414962
$516$	0	0
$517$	4.70850	0.207079
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.8745	0.695475	0.347737	−	0.937592i	$-0.386950\pi$
$521$	15.8745	0.695475	0.347737	+	0.937592i	$0.386950\pi$
$522$	0	0
$523$	23.8745	1.04396	0.521980	−	0.852958i	$-0.325194\pi$
$523$	23.8745	1.04396	0.521980	+	0.852958i	$0.325194\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.457513	0.0199296
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−34.3320	−1.48708
$534$	0	0
$535$	13.2915	0.574642
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	15.1660	0.652038	0.326019	−	0.945363i	$-0.394293\pi$
$541$	15.1660	0.652038	0.326019	+	0.945363i	$0.394293\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−36.4575	−1.56167
$546$	0	0
$547$	15.2915	0.653817	0.326909	−	0.945056i	$-0.393993\pi$
$547$	15.2915	0.653817	0.326909	+	0.945056i	$0.393993\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.64575	−0.240517
$552$	0	0
$553$	11.2915	0.480164
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.87451	0.418396	0.209198	−	0.977873i	$-0.432915\pi$
$557$	9.87451	0.418396	0.209198	+	0.977873i	$0.432915\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.8745	0.921901	0.460950	−	0.887426i	$-0.347509\pi$
$563$	21.8745	0.921901	0.460950	+	0.887426i	$0.347509\pi$
$564$	0	0
$565$	−61.7490	−2.59780
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.3542	−0.769450	−0.384725	−	0.923031i	$-0.625704\pi$
$569$	−18.3542	−0.769450	−0.384725	+	0.923031i	$0.625704\pi$
$570$	0	0
$571$	39.7490	1.66344	0.831722	−	0.555192i	$-0.187355\pi$
$571$	39.7490	1.66344	0.831722	+	0.555192i	$0.187355\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	49.7490	2.07468
$576$	0	0
$577$	−41.0405	−1.70854	−0.854270	−	0.519830i	$-0.825995\pi$
$577$	−41.0405	−1.70854	−0.854270	+	0.519830i	$0.825995\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.9373	−0.702676
$582$	0	0
$583$	−17.8745	−0.740286
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.9373	−0.699075	−0.349538	−	0.936922i	$-0.613661\pi$
$587$	−16.9373	−0.699075	−0.349538	+	0.936922i	$0.613661\pi$
$588$	0	0
$589$	1.29150	0.0532154
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.6458	−1.29954	−0.649768	−	0.760133i	$-0.725134\pi$
$593$	−31.6458	−1.29954	−0.649768	+	0.760133i	$0.725134\pi$
$594$	0	0
$595$	−1.29150	−0.0529464
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.5203	−1.53304	−0.766518	−	0.642223i	$-0.778012\pi$
$599$	−37.5203	−1.53304	−0.766518	+	0.642223i	$0.778012\pi$
$600$	0	0
$601$	−44.3320	−1.80834	−0.904170	−	0.427172i	$-0.859510\pi$
$601$	−44.3320	−1.80834	−0.904170	+	0.427172i	$0.859510\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.5203	1.03755
$606$	0	0
$607$	1.41699	0.0575140	0.0287570	−	0.999586i	$-0.490845\pi$
$607$	1.41699	0.0575140	0.0287570	+	0.999586i	$0.490845\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.41699	0.380971
$612$	0	0
$613$	−31.2915	−1.26385	−0.631926	−	0.775029i	$-0.717735\pi$
$613$	−31.2915	−1.26385	−0.631926	+	0.775029i	$0.717735\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.16601	0.207976	0.103988	−	0.994579i	$-0.466840\pi$
$617$	5.16601	0.207976	0.103988	+	0.994579i	$0.466840\pi$
$618$	0	0
$619$	−37.8745	−1.52231	−0.761153	−	0.648573i	$-0.775366\pi$
$619$	−37.8745	−1.52231	−0.761153	+	0.648573i	$0.775366\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	2.29150	0.0916601
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.87451	0.0747415
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.5830	0.578709
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.35425	−0.250978	−0.125489	−	0.992095i	$-0.540050\pi$
$641$	−6.35425	−0.250978	−0.125489	+	0.992095i	$0.540050\pi$
$642$	0	0
$643$	21.1660	0.834706	0.417353	−	0.908744i	$-0.362958\pi$
$643$	21.1660	0.834706	0.417353	+	0.908744i	$0.362958\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.22876	0.166250	0.0831248	−	0.996539i	$-0.473510\pi$
$647$	4.22876	0.166250	0.0831248	+	0.996539i	$0.473510\pi$
$648$	0	0
$649$	29.1660	1.14487
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.1660	−1.45442	−0.727209	−	0.686416i	$-0.759183\pi$
$653$	−37.1660	−1.45442	−0.727209	+	0.686416i	$0.759183\pi$
$654$	0	0
$655$	−61.7490	−2.41273
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.06275	−0.197217	−0.0986083	−	0.995126i	$-0.531439\pi$
$659$	−5.06275	−0.197217	−0.0986083	+	0.995126i	$0.531439\pi$
$660$	0	0
$661$	−34.5830	−1.34512	−0.672562	−	0.740041i	$-0.734806\pi$
$661$	−34.5830	−1.34512	−0.672562	+	0.740041i	$0.734806\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.64575	−0.141376
$666$	0	0
$667$	−33.8745	−1.31163
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5.41699	−0.209121
$672$	0	0
$673$	−22.4575	−0.865674	−0.432837	−	0.901472i	$-0.642487\pi$
$673$	−22.4575	−0.865674	−0.432837	+	0.901472i	$0.642487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.4575	1.17058	0.585289	−	0.810825i	$-0.300981\pi$
$677$	30.4575	1.17058	0.585289	+	0.810825i	$0.300981\pi$
$678$	0	0
$679$	16.5830	0.636397
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.1033	−0.922286	−0.461143	−	0.887326i	$-0.652560\pi$
$683$	−24.1033	−0.922286	−0.461143	+	0.887326i	$0.652560\pi$
$684$	0	0
$685$	82.3320	3.14574
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−35.7490	−1.36193
$690$	0	0
$691$	−47.2915	−1.79905	−0.899527	−	0.436866i	$-0.856089\pi$
$691$	−47.2915	−1.79905	−0.899527	+	0.436866i	$0.856089\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.58301	0.0979790
$696$	0	0
$697$	3.04052	0.115168
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.1660	−0.648351	−0.324176	−	0.945997i	$-0.605087\pi$
$701$	−17.1660	−0.648351	−0.324176	+	0.945997i	$0.605087\pi$
$702$	0	0
$703$	5.29150	0.199573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.35425	−0.163758
$708$	0	0
$709$	−43.0405	−1.61642	−0.808210	−	0.588894i	$-0.799564\pi$
$709$	−43.0405	−1.61642	−0.808210	+	0.588894i	$0.799564\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.74902	0.290203
$714$	0	0
$715$	−29.1660	−1.09075
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.9373	−1.52670	−0.763351	−	0.645984i	$-0.776447\pi$
$719$	−40.9373	−1.52670	−0.763351	+	0.645984i	$0.776447\pi$
$720$	0	0
$721$	−2.58301	−0.0961961
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−46.8118	−1.73855
$726$	0	0
$727$	−10.1255	−0.375534	−0.187767	−	0.982214i	$-0.560125\pi$
$727$	−10.1255	−0.375534	−0.187767	+	0.982214i	$0.560125\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.708497	−0.0262047
$732$	0	0
$733$	−19.8745	−0.734082	−0.367041	−	0.930205i	$-0.619629\pi$
$733$	−19.8745	−0.734082	−0.367041	+	0.930205i	$0.619629\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29.1660	−1.07434
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.3542	1.04022	0.520108	−	0.854100i	$-0.325892\pi$
$743$	28.3542	1.04022	0.520108	+	0.854100i	$0.325892\pi$
$744$	0	0
$745$	55.7490	2.04249
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.64575	−0.133213
$750$	0	0
$751$	−8.45751	−0.308619	−0.154310	−	0.988023i	$-0.549315\pi$
$751$	−8.45751	−0.308619	−0.154310	+	0.988023i	$0.549315\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−70.3320	−2.55964
$756$	0	0
$757$	32.3320	1.17513	0.587564	−	0.809178i	$-0.300087\pi$
$757$	32.3320	1.17513	0.587564	+	0.809178i	$0.300087\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.0627	−0.473524	−0.236762	−	0.971568i	$-0.576086\pi$
$761$	−13.0627	−0.473524	−0.236762	+	0.971568i	$0.576086\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58.3320	2.10625
$768$	0	0
$769$	30.4575	1.09833	0.549163	−	0.835715i	$-0.314947\pi$
$769$	30.4575	1.09833	0.549163	+	0.835715i	$0.314947\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.58301	0.308709	0.154355	−	0.988016i	$-0.450670\pi$
$773$	8.58301	0.308709	0.154355	+	0.988016i	$0.450670\pi$
$774$	0	0
$775$	10.7085	0.384661
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.58301	0.307518
$780$	0	0
$781$	0.708497	0.0253520
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−43.2915	−1.54514
$786$	0	0
$787$	34.4575	1.22828	0.614139	−	0.789198i	$-0.289503\pi$
$787$	34.4575	1.22828	0.614139	+	0.789198i	$0.289503\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.9373	0.602219
$792$	0	0
$793$	−10.8340	−0.384726
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.83399	0.171229	0.0856143	−	0.996328i	$-0.472715\pi$
$797$	4.83399	0.171229	0.0856143	+	0.996328i	$0.472715\pi$
$798$	0	0
$799$	−0.833990	−0.0295044
$800$	0	0
$801$	0	0
$802$	0	0
$803$	18.5830	0.655780
$804$	0	0
$805$	−21.8745	−0.770975
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.7490	1.39750	0.698750	−	0.715365i	$-0.253740\pi$
$809$	39.7490	1.39750	0.698750	+	0.715365i	$0.253740\pi$
$810$	0	0
$811$	50.3320	1.76740	0.883698	−	0.468057i	$-0.155046\pi$
$811$	50.3320	1.76740	0.883698	+	0.468057i	$0.155046\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	45.8745	1.60691
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.29150	−0.114874	−0.0574371	−	0.998349i	$-0.518293\pi$
$821$	−3.29150	−0.114874	−0.0574371	+	0.998349i	$0.518293\pi$
$822$	0	0
$823$	−4.58301	−0.159754	−0.0798768	−	0.996805i	$-0.525453\pi$
$823$	−4.58301	−0.159754	−0.0798768	+	0.996805i	$0.525453\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.2288	1.32934	0.664672	−	0.747135i	$-0.268571\pi$
$827$	38.2288	1.32934	0.664672	+	0.747135i	$0.268571\pi$
$828$	0	0
$829$	−7.16601	−0.248886	−0.124443	−	0.992227i	$-0.539714\pi$
$829$	−7.16601	−0.248886	−0.124443	+	0.992227i	$0.539714\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.354249	0.0122740
$834$	0	0
$835$	55.7490	1.92927
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.41699	0.325111	0.162555	−	0.986699i	$-0.448026\pi$
$839$	9.41699	0.325111	0.162555	+	0.986699i	$0.448026\pi$
$840$	0	0
$841$	2.87451	0.0991210
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.9373	−0.376253
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.7490	1.08834
$852$	0	0
$853$	−31.4170	−1.07570	−0.537849	−	0.843041i	$-0.680763\pi$
$853$	−31.4170	−1.07570	−0.537849	+	0.843041i	$0.680763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.7490	−1.69939	−0.849697	−	0.527271i	$-0.823215\pi$
$857$	−49.7490	−1.69939	−0.849697	+	0.527271i	$0.823215\pi$
$858$	0	0
$859$	−17.4170	−0.594260	−0.297130	−	0.954837i	$-0.596030\pi$
$859$	−17.4170	−0.594260	−0.297130	+	0.954837i	$0.596030\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.2288	0.348191	0.174095	−	0.984729i	$-0.444300\pi$
$863$	10.2288	0.348191	0.174095	+	0.984729i	$0.444300\pi$
$864$	0	0
$865$	51.0405	1.73543
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.5830	−0.766076
$870$	0	0
$871$	−58.3320	−1.97651
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	12.5830	0.424898	0.212449	−	0.977172i	$-0.431856\pi$
$877$	12.5830	0.424898	0.212449	+	0.977172i	$0.431856\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.4797	0.487835	0.243917	−	0.969796i	$-0.421567\pi$
$881$	14.4797	0.487835	0.243917	+	0.969796i	$0.421567\pi$
$882$	0	0
$883$	41.1660	1.38535	0.692673	−	0.721252i	$-0.256433\pi$
$883$	41.1660	1.38535	0.692673	+	0.721252i	$0.256433\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.12549	−0.205674	−0.102837	−	0.994698i	$-0.532792\pi$
$887$	−6.12549	−0.205674	−0.102837	+	0.994698i	$0.532792\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.35425	−0.0787819
$894$	0	0
$895$	93.0405	3.11000
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.29150	−0.243185
$900$	0	0
$901$	3.16601	0.105475
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.4575	−2.00968
$906$	0	0
$907$	−25.1660	−0.835624	−0.417812	−	0.908534i	$-0.637203\pi$
$907$	−25.1660	−0.835624	−0.417812	+	0.908534i	$0.637203\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	18.9373	0.627419	0.313710	−	0.949519i	$-0.398428\pi$
$911$	18.9373	0.627419	0.313710	+	0.949519i	$0.398428\pi$
$912$	0	0
$913$	33.8745	1.12108
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.9373	0.559317
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.41699	0.0466410
$924$	0	0
$925$	43.8745	1.44258
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.6458	−0.907028	−0.453514	−	0.891249i	$-0.649830\pi$
$929$	−27.6458	−0.907028	−0.453514	+	0.891249i	$0.649830\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.58301	0.0844733
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.29150	−0.172498	−0.0862490	−	0.996274i	$-0.527488\pi$
$941$	−5.29150	−0.172498	−0.0862490	+	0.996274i	$0.527488\pi$
$942$	0	0
$943$	51.4980	1.67701
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	0	0
$949$	37.1660	1.20646
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.5203	0.632323	0.316162	−	0.948705i	$-0.397606\pi$
$953$	19.5203	0.632323	0.316162	+	0.948705i	$0.397606\pi$
$954$	0	0
$955$	21.8745	0.707842
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.5830	−0.729243
$960$	0	0
$961$	−29.3320	−0.946194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	92.2065	2.96823
$966$	0	0
$967$	0.833990	0.0268193	0.0134096	−	0.999910i	$-0.495731\pi$
$967$	0.833990	0.0268193	0.0134096	+	0.999910i	$0.495731\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.41699	0.173840	0.0869198	−	0.996215i	$-0.472298\pi$
$971$	5.41699	0.173840	0.0869198	+	0.996215i	$0.472298\pi$
$972$	0	0
$973$	−0.708497	−0.0227134
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.39477	−0.300565	−0.150283	−	0.988643i	$-0.548018\pi$
$977$	−9.39477	−0.300565	−0.150283	+	0.988643i	$0.548018\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.5830	1.35819	0.679093	−	0.734052i	$-0.262373\pi$
$983$	42.5830	1.35819	0.679093	+	0.734052i	$0.262373\pi$
$984$	0	0
$985$	−19.7490	−0.629256
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.16601	−0.163774
$996$	0	0
$997$	−23.8745	−0.756113	−0.378057	−	0.925782i	$-0.623408\pi$
$997$	−23.8745	−0.756113	−0.378057	+	0.925782i	$0.623408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bh.1.1	✓	2
3.2	odd	2		9576.2.a.bu.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bh.1.1	✓	2	1.1	even	1	trivial
9576.2.a.bu.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q-3.64575 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.64575 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.00000 q^{13} +0.354249 q^{17} +1.00000 q^{19} +6.00000 q^{23} +8.29150 q^{25} -5.64575 q^{29} +1.29150 q^{31} -3.64575 q^{35} +5.29150 q^{37} +8.58301 q^{41} -2.00000 q^{43} -2.35425 q^{47} +1.00000 q^{49} +8.93725 q^{53} +7.29150 q^{55} -14.5830 q^{59} +2.70850 q^{61} +14.5830 q^{65} +14.5830 q^{67} -0.354249 q^{71} -9.29150 q^{73} -2.00000 q^{77} +11.2915 q^{79} -16.9373 q^{83} -1.29150 q^{85} -6.00000 q^{89} -4.00000 q^{91} -3.64575 q^{95} +16.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{19} + 12 q^{23} + 6 q^{25} - 6 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{41} - 4 q^{43} - 10 q^{47} + 2 q^{49} + 2 q^{53} + 4 q^{55} - 8 q^{59} + 16 q^{61} + 8 q^{65} + 8 q^{67} - 6 q^{71} - 8 q^{73} - 4 q^{77} + 12 q^{79} - 18 q^{83} + 8 q^{85} - 12 q^{89} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^19 + 12 * q^23 + 6 * q^25 - 6 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^41 - 4 * q^43 - 10 * q^47 + 2 * q^49 + 2 * q^53 + 4 * q^55 - 8 * q^59 + 16 * q^61 + 8 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 4 * q^77 + 12 * q^79 - 18 * q^83 + 8 * q^85 - 12 * q^89 - 8 * q^91 - 2 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more