LMFDB - Embedded newform 7200.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.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:	$57.4922894553$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	7200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60555 q^{7} -2.00000 q^{11} -3.00000 q^{13} +7.21110 q^{17} +3.60555 q^{19} +6.00000 q^{23} -7.21110 q^{29} -3.60555 q^{31} +2.00000 q^{37} +7.21110 q^{41} -3.60555 q^{43} -4.00000 q^{47} +6.00000 q^{49} +7.21110 q^{53} -12.0000 q^{59} -1.00000 q^{61} +10.8167 q^{67} -6.00000 q^{71} +10.0000 q^{73} +7.21110 q^{77} -14.4222 q^{79} +4.00000 q^{83} +10.8167 q^{91} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{11} - 6 q^{13} + 12 q^{23} + 4 q^{37} - 8 q^{47} + 12 q^{49} - 24 q^{59} - 2 q^{61} - 12 q^{71} + 20 q^{73} + 8 q^{83} + 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^11 - 6 * q^13 + 12 * q^23 + 4 * q^37 - 8 * q^47 + 12 * q^49 - 24 * q^59 - 2 * q^61 - 12 * q^71 + 20 * q^73 + 8 * q^83 + 2 * 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	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.60555	−1.36277	−0.681385	−	0.731925i	$-0.738622\pi$
$7$	−3.60555	−1.36277	−0.681385	+	0.731925i	$0.738622\pi$
$8$	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
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.21110	1.74895	0.874475	−	0.485071i	$-0.161206\pi$
$17$	7.21110	1.74895	0.874475	+	0.485071i	$0.161206\pi$
$18$	0	0
$19$	3.60555	0.827170	0.413585	−	0.910465i	$-0.364276\pi$
$19$	3.60555	0.827170	0.413585	+	0.910465i	$0.364276\pi$
$20$	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
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.21110	−1.33907	−0.669534	−	0.742781i	$-0.733506\pi$
$29$	−7.21110	−1.33907	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	−3.60555	−0.647576	−0.323788	−	0.946130i	$-0.604956\pi$
$31$	−3.60555	−0.647576	−0.323788	+	0.946130i	$0.604956\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.21110	1.12619	0.563093	−	0.826394i	$-0.309611\pi$
$41$	7.21110	1.12619	0.563093	+	0.826394i	$0.309611\pi$
$42$	0	0
$43$	−3.60555	−0.549841	−0.274921	−	0.961467i	$-0.588652\pi$
$43$	−3.60555	−0.549841	−0.274921	+	0.961467i	$0.588652\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.21110	0.990521	0.495261	−	0.868744i	$-0.335073\pi$
$53$	7.21110	0.990521	0.495261	+	0.868744i	$0.335073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8167	1.32146	0.660732	−	0.750622i	$-0.270246\pi$
$67$	10.8167	1.32146	0.660732	+	0.750622i	$0.270246\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.21110	0.821781
$78$	0	0
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	10.8167	1.13389
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.21110	0.717532	0.358766	−	0.933428i	$-0.383198\pi$
$101$	7.21110	0.717532	0.358766	+	0.933428i	$0.383198\pi$
$102$	0	0
$103$	14.4222	1.42106	0.710531	−	0.703666i	$-0.248455\pi$
$103$	14.4222	1.42106	0.710531	+	0.703666i	$0.248455\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.0000	−2.38342
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	−13.0000	−1.12724
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.4222	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$137$	−14.4222	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$138$	0	0
$139$	14.4222	1.22328	0.611638	−	0.791138i	$-0.290511\pi$
$139$	14.4222	1.22328	0.611638	+	0.791138i	$0.290511\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4222	1.18151	0.590757	−	0.806850i	$-0.298829\pi$
$149$	14.4222	1.18151	0.590757	+	0.806850i	$0.298829\pi$
$150$	0	0
$151$	3.60555	0.293416	0.146708	−	0.989180i	$-0.453132\pi$
$151$	3.60555	0.293416	0.146708	+	0.989180i	$0.453132\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−21.6333	−1.70494
$162$	0	0
$163$	−10.8167	−0.847226	−0.423613	−	0.905843i	$-0.639238\pi$
$163$	−10.8167	−0.847226	−0.423613	+	0.905843i	$0.639238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.6333	−1.64475	−0.822375	−	0.568946i	$-0.807351\pi$
$173$	−21.6333	−1.64475	−0.822375	+	0.568946i	$0.807351\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	21.0000	1.56092	0.780459	−	0.625207i	$-0.214986\pi$
$181$	21.0000	1.56092	0.780459	+	0.625207i	$0.214986\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−14.4222	−1.05466
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.4222	−1.02754	−0.513770	−	0.857928i	$-0.671751\pi$
$197$	−14.4222	−1.02754	−0.513770	+	0.857928i	$0.671751\pi$
$198$	0	0
$199$	3.60555	0.255591	0.127795	−	0.991801i	$-0.459210\pi$
$199$	3.60555	0.255591	0.127795	+	0.991801i	$0.459210\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.0000	1.82484
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.21110	−0.498802
$210$	0	0
$211$	−25.2389	−1.73751	−0.868757	−	0.495238i	$-0.835081\pi$
$211$	−25.2389	−1.73751	−0.868757	+	0.495238i	$0.835081\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	13.0000	0.882498
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.6333	−1.45521
$222$	0	0
$223$	−18.0278	−1.20723	−0.603614	−	0.797277i	$-0.706273\pi$
$223$	−18.0278	−1.20723	−0.603614	+	0.797277i	$0.706273\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.21110	0.472415	0.236208	−	0.971703i	$-0.424095\pi$
$233$	7.21110	0.472415	0.236208	+	0.971703i	$0.424095\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	−27.0000	−1.73922	−0.869611	−	0.493737i	$-0.835631\pi$
$241$	−27.0000	−1.73922	−0.869611	+	0.493737i	$0.835631\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.8167	−0.688247
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.4222	−0.899632	−0.449816	−	0.893121i	$-0.648511\pi$
$257$	−14.4222	−0.899632	−0.449816	+	0.893121i	$0.648511\pi$
$258$	0	0
$259$	−7.21110	−0.448076
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.0000	1.35658	0.678289	−	0.734795i	$-0.262722\pi$
$263$	22.0000	1.35658	0.678289	+	0.734795i	$0.262722\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.6333	1.31901	0.659503	−	0.751702i	$-0.270767\pi$
$269$	21.6333	1.31901	0.659503	+	0.751702i	$0.270767\pi$
$270$	0	0
$271$	−14.4222	−0.876087	−0.438043	−	0.898954i	$-0.644328\pi$
$271$	−14.4222	−0.876087	−0.438043	+	0.898954i	$0.644328\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.21110	0.430178	0.215089	−	0.976594i	$-0.430996\pi$
$281$	7.21110	0.430178	0.215089	+	0.976594i	$0.430996\pi$
$282$	0	0
$283$	18.0278	1.07164	0.535819	−	0.844333i	$-0.320003\pi$
$283$	18.0278	1.07164	0.535819	+	0.844333i	$0.320003\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.0000	−1.53473
$288$	0	0
$289$	35.0000	2.05882
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.0000	−1.04097
$300$	0	0
$301$	13.0000	0.749308
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.60555	0.205780	0.102890	−	0.994693i	$-0.467191\pi$
$307$	3.60555	0.205780	0.102890	+	0.994693i	$0.467191\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	27.0000	1.52613	0.763065	−	0.646322i	$-0.223694\pi$
$313$	27.0000	1.52613	0.763065	+	0.646322i	$0.223694\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.4222	−0.810032	−0.405016	−	0.914310i	$-0.632734\pi$
$317$	−14.4222	−0.810032	−0.405016	+	0.914310i	$0.632734\pi$
$318$	0	0
$319$	14.4222	0.807488
$320$	0	0
$321$	0	0
$322$	0	0
$323$	26.0000	1.44668
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.4222	0.795122
$330$	0	0
$331$	28.8444	1.58543	0.792716	−	0.609591i	$-0.208666\pi$
$331$	28.8444	1.58543	0.792716	+	0.609591i	$0.208666\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.21110	0.390503
$342$	0	0
$343$	3.60555	0.194681
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.6333	−1.15142	−0.575712	−	0.817652i	$-0.695275\pi$
$353$	−21.6333	−1.15142	−0.575712	+	0.817652i	$0.695275\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.4500	1.69387	0.846937	−	0.531693i	$-0.178444\pi$
$367$	32.4500	1.69387	0.846937	+	0.531693i	$0.178444\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26.0000	−1.34985
$372$	0	0
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.6333	1.11417
$378$	0	0
$379$	−32.4500	−1.66684	−0.833421	−	0.552638i	$-0.813621\pi$
$379$	−32.4500	−1.66684	−0.833421	+	0.552638i	$0.813621\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−36.0555	−1.82809	−0.914044	−	0.405616i	$-0.867057\pi$
$389$	−36.0555	−1.82809	−0.914044	+	0.405616i	$0.867057\pi$
$390$	0	0
$391$	43.2666	2.18809
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.6333	1.08032	0.540158	−	0.841564i	$-0.318365\pi$
$401$	21.6333	1.08032	0.540158	+	0.841564i	$0.318365\pi$
$402$	0	0
$403$	10.8167	0.538816
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	43.2666	2.12901
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.60555	0.174485
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.6333	1.03486
$438$	0	0
$439$	25.2389	1.20459	0.602293	−	0.798275i	$-0.294254\pi$
$439$	25.2389	1.20459	0.602293	+	0.798275i	$0.294254\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.4222	0.680626	0.340313	−	0.940312i	$-0.389467\pi$
$449$	14.4222	0.680626	0.340313	+	0.940312i	$0.389467\pi$
$450$	0	0
$451$	−14.4222	−0.679115
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.6333	1.00756	0.503782	−	0.863831i	$-0.331942\pi$
$461$	21.6333	1.00756	0.503782	+	0.863831i	$0.331942\pi$
$462$	0	0
$463$	−14.4222	−0.670257	−0.335128	−	0.942172i	$-0.608780\pi$
$463$	−14.4222	−0.670257	−0.335128	+	0.942172i	$0.608780\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	−39.0000	−1.80085
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.21110	0.331567
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.60555	0.163383	0.0816916	−	0.996658i	$-0.473968\pi$
$487$	3.60555	0.163383	0.0816916	+	0.996658i	$0.473968\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	0	0
$493$	−52.0000	−2.34196
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.6333	0.970386
$498$	0	0
$499$	−39.6611	−1.77547	−0.887737	−	0.460352i	$-0.847723\pi$
$499$	−39.6611	−1.77547	−0.887737	+	0.460352i	$0.847723\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−43.2666	−1.91776	−0.958880	−	0.283813i	$-0.908400\pi$
$509$	−43.2666	−1.91776	−0.958880	+	0.283813i	$0.908400\pi$
$510$	0	0
$511$	−36.0555	−1.59500
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.6333	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$521$	−21.6333	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$522$	0	0
$523$	18.0278	0.788299	0.394149	−	0.919046i	$-0.371039\pi$
$523$	18.0278	0.788299	0.394149	+	0.919046i	$0.371039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.0000	−1.13258
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.6333	−0.937043
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.4222	−0.616649	−0.308324	−	0.951281i	$-0.599768\pi$
$547$	−14.4222	−0.616649	−0.308324	+	0.951281i	$0.599768\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−26.0000	−1.10764
$552$	0	0
$553$	52.0000	2.21126
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.8444	1.22218	0.611088	−	0.791562i	$-0.290732\pi$
$557$	28.8444	1.22218	0.611088	+	0.791562i	$0.290732\pi$
$558$	0	0
$559$	10.8167	0.457496
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.6333	−0.906915	−0.453458	−	0.891278i	$-0.649810\pi$
$569$	−21.6333	−0.906915	−0.453458	+	0.891278i	$0.649810\pi$
$570$	0	0
$571$	−25.2389	−1.05621	−0.528107	−	0.849178i	$-0.677098\pi$
$571$	−25.2389	−1.05621	−0.528107	+	0.849178i	$0.677098\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.0000	−1.70685	−0.853426	−	0.521214i	$-0.825479\pi$
$577$	−41.0000	−1.70685	−0.853426	+	0.521214i	$0.825479\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.4222	−0.598334
$582$	0	0
$583$	−14.4222	−0.597307
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	0	0
$589$	−13.0000	−0.535656
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.6333	0.888373	0.444187	−	0.895934i	$-0.353493\pi$
$593$	21.6333	0.888373	0.444187	+	0.895934i	$0.353493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−5.00000	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.4222	−0.585379	−0.292690	−	0.956208i	$-0.594550\pi$
$607$	−14.4222	−0.585379	−0.292690	+	0.956208i	$0.594550\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.6333	0.870924	0.435462	−	0.900207i	$-0.356585\pi$
$617$	21.6333	0.870924	0.435462	+	0.900207i	$0.356585\pi$
$618$	0	0
$619$	46.8722	1.88395	0.941976	−	0.335681i	$-0.108966\pi$
$619$	46.8722	1.88395	0.941976	+	0.335681i	$0.108966\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.4222	0.575051
$630$	0	0
$631$	10.8167	0.430604	0.215302	−	0.976547i	$-0.430926\pi$
$631$	10.8167	0.430604	0.215302	+	0.976547i	$0.430926\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.2666	1.70893	0.854464	−	0.519510i	$-0.173886\pi$
$641$	43.2666	1.70893	0.854464	+	0.519510i	$0.173886\pi$
$642$	0	0
$643$	−43.2666	−1.70627	−0.853134	−	0.521691i	$-0.825301\pi$
$643$	−43.2666	−1.70627	−0.853134	+	0.521691i	$0.825301\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−43.2666	−1.67529
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.21110	0.277145	0.138573	−	0.990352i	$-0.455749\pi$
$677$	7.21110	0.277145	0.138573	+	0.990352i	$0.455749\pi$
$678$	0	0
$679$	−3.60555	−0.138368
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−21.6333	−0.824163
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	52.0000	1.96964
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.6333	0.817079	0.408539	−	0.912741i	$-0.366038\pi$
$701$	21.6333	0.817079	0.408539	+	0.912741i	$0.366038\pi$
$702$	0	0
$703$	7.21110	0.271972
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.0000	−0.977831
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.6333	−0.810174
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	0	0
$721$	−52.0000	−1.93658
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.60555	−0.133722	−0.0668612	−	0.997762i	$-0.521298\pi$
$727$	−3.60555	−0.133722	−0.0668612	+	0.997762i	$0.521298\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−26.0000	−0.961645
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.6333	−0.796873
$738$	0	0
$739$	28.8444	1.06106	0.530529	−	0.847667i	$-0.321993\pi$
$739$	28.8444	1.06106	0.530529	+	0.847667i	$0.321993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	64.8999	2.37139
$750$	0	0
$751$	28.8444	1.05255	0.526274	−	0.850315i	$-0.323589\pi$
$751$	28.8444	1.05255	0.526274	+	0.850315i	$0.323589\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.8444	−1.04561	−0.522805	−	0.852453i	$-0.675114\pi$
$761$	−28.8444	−1.04561	−0.522805	+	0.852453i	$0.675114\pi$
$762$	0	0
$763$	39.6611	1.43583
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	15.0000	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.21110	−0.259365	−0.129683	−	0.991556i	$-0.541396\pi$
$773$	−7.21110	−0.259365	−0.129683	+	0.991556i	$0.541396\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.0000	0.931547
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.0278	0.642620	0.321310	−	0.946974i	$-0.395877\pi$
$787$	18.0278	0.642620	0.321310	+	0.946974i	$0.395877\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.00000	0.106533
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−43.2666	−1.53258	−0.766291	−	0.642494i	$-0.777900\pi$
$797$	−43.2666	−1.53258	−0.766291	+	0.642494i	$0.777900\pi$
$798$	0	0
$799$	−28.8444	−1.02044
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.6333	−0.760587	−0.380293	−	0.924866i	$-0.624177\pi$
$809$	−21.6333	−0.760587	−0.380293	+	0.924866i	$0.624177\pi$
$810$	0	0
$811$	18.0278	0.633040	0.316520	−	0.948586i	$-0.397486\pi$
$811$	18.0278	0.633040	0.316520	+	0.948586i	$0.397486\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−13.0000	−0.454812
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.8444	−1.00668	−0.503338	−	0.864089i	$-0.667895\pi$
$821$	−28.8444	−1.00668	−0.503338	+	0.864089i	$0.667895\pi$
$822$	0	0
$823$	10.8167	0.377045	0.188522	−	0.982069i	$-0.439630\pi$
$823$	10.8167	0.377045	0.188522	+	0.982069i	$0.439630\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	43.2666	1.49910
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.2389	0.867217
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	−17.0000	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−43.2666	−1.47796	−0.738980	−	0.673728i	$-0.764692\pi$
$857$	−43.2666	−1.47796	−0.738980	+	0.673728i	$0.764692\pi$
$858$	0	0
$859$	−28.8444	−0.984159	−0.492079	−	0.870550i	$-0.663763\pi$
$859$	−28.8444	−0.984159	−0.492079	+	0.870550i	$0.663763\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28.8444	0.978480
$870$	0	0
$871$	−32.4500	−1.09952
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−53.0000	−1.78968	−0.894841	−	0.446384i	$-0.852711\pi$
$877$	−53.0000	−1.78968	−0.894841	+	0.446384i	$0.852711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.4222	−0.485896	−0.242948	−	0.970039i	$-0.578115\pi$
$881$	−14.4222	−0.485896	−0.242948	+	0.970039i	$0.578115\pi$
$882$	0	0
$883$	−3.60555	−0.121336	−0.0606682	−	0.998158i	$-0.519323\pi$
$883$	−3.60555	−0.121336	−0.0606682	+	0.998158i	$0.519323\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14.4222	−0.482621
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.0000	0.867149
$900$	0	0
$901$	52.0000	1.73237
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	57.6888	1.91553	0.957763	−	0.287559i	$-0.0928437\pi$
$907$	57.6888	1.91553	0.957763	+	0.287559i	$0.0928437\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.21110	−0.238132
$918$	0	0
$919$	−39.6611	−1.30830	−0.654149	−	0.756366i	$-0.726973\pi$
$919$	−39.6611	−1.30830	−0.654149	+	0.756366i	$0.726973\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18.0000	0.592477
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.6333	0.709766	0.354883	−	0.934911i	$-0.384521\pi$
$929$	21.6333	0.709766	0.354883	+	0.934911i	$0.384521\pi$
$930$	0	0
$931$	21.6333	0.709003
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.8444	0.940301	0.470150	−	0.882586i	$-0.344200\pi$
$941$	28.8444	0.940301	0.470150	+	0.882586i	$0.344200\pi$
$942$	0	0
$943$	43.2666	1.40895
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	0	0
$949$	−30.0000	−0.973841
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28.8444	−0.934362	−0.467181	−	0.884162i	$-0.654730\pi$
$953$	−28.8444	−0.934362	−0.467181	+	0.884162i	$0.654730\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	52.0000	1.67917
$960$	0	0
$961$	−18.0000	−0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.4222	−0.463787	−0.231893	−	0.972741i	$-0.574492\pi$
$967$	−14.4222	−0.463787	−0.231893	+	0.972741i	$0.574492\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−52.0000	−1.66704
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.2666	−1.38422	−0.692111	−	0.721791i	$-0.743319\pi$
$977$	−43.2666	−1.38422	−0.692111	+	0.721791i	$0.743319\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.6333	−0.687899
$990$	0	0
$991$	−3.60555	−0.114534	−0.0572671	−	0.998359i	$-0.518239\pi$
$991$	−3.60555	−0.114534	−0.0572671	+	0.998359i	$0.518239\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.a.cd.1.1	✓	2
3.2	odd	2		7200.2.a.co.1.1	yes	2
4.3	odd	2		7200.2.a.co.1.2	yes	2
5.2	odd	4		7200.2.f.bd.6049.2		4
5.3	odd	4		7200.2.f.bd.6049.3		4
5.4	even	2		7200.2.a.ce.1.2	yes	2
12.11	even	2	inner	7200.2.a.cd.1.2	yes	2
15.2	even	4		7200.2.f.bk.6049.2		4
15.8	even	4		7200.2.f.bk.6049.3		4
15.14	odd	2		7200.2.a.cp.1.2	yes	2
20.3	even	4		7200.2.f.bk.6049.1		4
20.7	even	4		7200.2.f.bk.6049.4		4
20.19	odd	2		7200.2.a.cp.1.1	yes	2
60.23	odd	4		7200.2.f.bd.6049.1		4
60.47	odd	4		7200.2.f.bd.6049.4		4
60.59	even	2		7200.2.a.ce.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7200.2.a.cd.1.1	✓	2	1.1	even	1	trivial
7200.2.a.cd.1.2	yes	2	12.11	even	2	inner
7200.2.a.ce.1.1	yes	2	60.59	even	2
7200.2.a.ce.1.2	yes	2	5.4	even	2
7200.2.a.co.1.1	yes	2	3.2	odd	2
7200.2.a.co.1.2	yes	2	4.3	odd	2
7200.2.a.cp.1.1	yes	2	20.19	odd	2
7200.2.a.cp.1.2	yes	2	15.14	odd	2
7200.2.f.bd.6049.1		4	60.23	odd	4
7200.2.f.bd.6049.2		4	5.2	odd	4
7200.2.f.bd.6049.3		4	5.3	odd	4
7200.2.f.bd.6049.4		4	60.47	odd	4
7200.2.f.bk.6049.1		4	20.3	even	4
7200.2.f.bk.6049.2		4	15.2	even	4
7200.2.f.bk.6049.3		4	15.8	even	4
7200.2.f.bk.6049.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.60555 q^{7} -2.00000 q^{11} -3.00000 q^{13} +7.21110 q^{17} +3.60555 q^{19} +6.00000 q^{23} -7.21110 q^{29} -3.60555 q^{31} +2.00000 q^{37} +7.21110 q^{41} -3.60555 q^{43} -4.00000 q^{47} +6.00000 q^{49} +7.21110 q^{53} -12.0000 q^{59} -1.00000 q^{61} +10.8167 q^{67} -6.00000 q^{71} +10.0000 q^{73} +7.21110 q^{77} -14.4222 q^{79} +4.00000 q^{83} +10.8167 q^{91} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{11} - 6 q^{13} + 12 q^{23} + 4 q^{37} - 8 q^{47} + 12 q^{49} - 24 q^{59} - 2 q^{61} - 12 q^{71} + 20 q^{73} + 8 q^{83} + 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^11 - 6 * q^13 + 12 * q^23 + 4 * q^37 - 8 * q^47 + 12 * q^49 - 24 * q^59 - 2 * q^61 - 12 * q^71 + 20 * q^73 + 8 * q^83 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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