LMFDB - Embedded newform 1620.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.83502 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.83502 q^{7} -1.70739 q^{11} +3.70739 q^{13} +1.70739 q^{17} +0.292611 q^{19} -5.83502 q^{23} +1.00000 q^{25} +8.67004 q^{29} +0.292611 q^{31} +3.83502 q^{35} +11.9627 q^{37} -6.96265 q^{41} +3.70739 q^{43} +1.87237 q^{47} +7.70739 q^{49} -11.6700 q^{53} +1.70739 q^{55} +11.6700 q^{59} +14.9627 q^{61} -3.70739 q^{65} +9.54241 q^{67} +15.9627 q^{71} +8.00000 q^{73} +6.54787 q^{77} +2.00000 q^{79} -11.8350 q^{83} -1.70739 q^{85} -3.00000 q^{89} -14.2179 q^{91} -0.292611 q^{95} -3.67004 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7	$ 3 q - 3 q^{5} + 3 q^{7} + 6 q^{13} + 6 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} - 3 q^{35} + 12 q^{37} + 3 q^{41} + 6 q^{43} + 15 q^{47} + 18 q^{49} - 6 q^{53} + 6 q^{59} + 21 q^{61} - 6 q^{65} + 9 q^{67} + 24 q^{71} + 24 q^{73} + 6 q^{77} + 6 q^{79} - 21 q^{83} - 9 q^{89} - 6 q^{95} + 18 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 6 * q^13 + 6 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^29 + 6 * q^31 - 3 * q^35 + 12 * q^37 + 3 * q^41 + 6 * q^43 + 15 * q^47 + 18 * q^49 - 6 * q^53 + 6 * q^59 + 21 * q^61 - 6 * q^65 + 9 * q^67 + 24 * q^71 + 24 * q^73 + 6 * q^77 + 6 * q^79 - 21 * q^83 - 9 * q^89 - 6 * q^95 + 18 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.83502	−1.44950	−0.724751	−	0.689011i	$-0.758045\pi$
$7$	−3.83502	−1.44950	−0.724751	+	0.689011i	$0.758045\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.70739	−0.514797	−0.257399	−	0.966305i	$-0.582865\pi$
$11$	−1.70739	−0.514797	−0.257399	+	0.966305i	$0.582865\pi$
$12$	0	0
$13$	3.70739	1.02824	0.514122	−	0.857717i	$-0.328118\pi$
$13$	3.70739	1.02824	0.514122	+	0.857717i	$0.328118\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.70739	0.414103	0.207051	−	0.978330i	$-0.433613\pi$
$17$	1.70739	0.414103	0.207051	+	0.978330i	$0.433613\pi$
$18$	0	0
$19$	0.292611	0.0671295	0.0335647	−	0.999437i	$-0.489314\pi$
$19$	0.292611	0.0671295	0.0335647	+	0.999437i	$0.489314\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.83502	−1.21669	−0.608343	−	0.793674i	$-0.708165\pi$
$23$	−5.83502	−1.21669	−0.608343	+	0.793674i	$0.708165\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.67004	1.60999	0.804993	−	0.593284i	$-0.202169\pi$
$29$	8.67004	1.60999	0.804993	+	0.593284i	$0.202169\pi$
$30$	0	0
$31$	0.292611	0.0525544	0.0262772	−	0.999655i	$-0.491635\pi$
$31$	0.292611	0.0525544	0.0262772	+	0.999655i	$0.491635\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.83502	0.648237
$36$	0	0
$37$	11.9627	1.96665	0.983324	−	0.181862i	$-0.0582124\pi$
$37$	11.9627	1.96665	0.983324	+	0.181862i	$0.0582124\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.96265	−1.08738	−0.543692	−	0.839285i	$-0.682974\pi$
$41$	−6.96265	−1.08738	−0.543692	+	0.839285i	$0.682974\pi$
$42$	0	0
$43$	3.70739	0.565372	0.282686	−	0.959213i	$-0.408775\pi$
$43$	3.70739	0.565372	0.282686	+	0.959213i	$0.408775\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.87237	0.273113	0.136556	−	0.990632i	$-0.456396\pi$
$47$	1.87237	0.273113	0.136556	+	0.990632i	$0.456396\pi$
$48$	0	0
$49$	7.70739	1.10106
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.6700	−1.60300	−0.801502	−	0.597992i	$-0.795965\pi$
$53$	−11.6700	−1.60300	−0.801502	+	0.597992i	$0.795965\pi$
$54$	0	0
$55$	1.70739	0.230224
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.6700	1.51931	0.759655	−	0.650326i	$-0.225368\pi$
$59$	11.6700	1.51931	0.759655	+	0.650326i	$0.225368\pi$
$60$	0	0
$61$	14.9627	1.91577	0.957886	−	0.287150i	$-0.0927077\pi$
$61$	14.9627	1.91577	0.957886	+	0.287150i	$0.0927077\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.70739	−0.459845
$66$	0	0
$67$	9.54241	1.16579	0.582896	−	0.812547i	$-0.301920\pi$
$67$	9.54241	1.16579	0.582896	+	0.812547i	$0.301920\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.9627	1.89442	0.947209	−	0.320616i	$-0.103890\pi$
$71$	15.9627	1.89442	0.947209	+	0.320616i	$0.103890\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.54787	0.746200
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.8350	−1.29906	−0.649531	−	0.760335i	$-0.725035\pi$
$83$	−11.8350	−1.29906	−0.649531	+	0.760335i	$0.725035\pi$
$84$	0	0
$85$	−1.70739	−0.185192
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−14.2179	−1.49044
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.292611	−0.0300212
$96$	0	0
$97$	−3.67004	−0.372636	−0.186318	−	0.982489i	$-0.559656\pi$
$97$	−3.67004	−0.372636	−0.186318	+	0.982489i	$0.559656\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.329957	0.0328320	0.0164160	−	0.999865i	$-0.494774\pi$
$101$	0.329957	0.0328320	0.0164160	+	0.999865i	$0.494774\pi$
$102$	0	0
$103$	−3.12217	−0.307636	−0.153818	−	0.988099i	$-0.549157\pi$
$103$	−3.12217	−0.307636	−0.153818	+	0.988099i	$0.549157\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.9198	−1.44236	−0.721178	−	0.692750i	$-0.756399\pi$
$107$	−14.9198	−1.44236	−0.721178	+	0.692750i	$0.756399\pi$
$108$	0	0
$109$	6.70739	0.642451	0.321226	−	0.947003i	$-0.395905\pi$
$109$	6.70739	0.642451	0.321226	+	0.947003i	$0.395905\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.67004	0.533393	0.266696	−	0.963781i	$-0.414068\pi$
$113$	5.67004	0.533393	0.266696	+	0.963781i	$0.414068\pi$
$114$	0	0
$115$	5.83502	0.544119
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.54787	−0.600243
$120$	0	0
$121$	−8.08482	−0.734984
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.54241	−0.491809	−0.245905	−	0.969294i	$-0.579085\pi$
$127$	−5.54241	−0.491809	−0.245905	+	0.969294i	$0.579085\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.8296	1.12092	0.560462	−	0.828180i	$-0.310624\pi$
$131$	12.8296	1.12092	0.560462	+	0.828180i	$0.310624\pi$
$132$	0	0
$133$	−1.12217	−0.0973043
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.08482	0.776169	0.388084	−	0.921624i	$-0.373137\pi$
$137$	9.08482	0.776169	0.388084	+	0.921624i	$0.373137\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.32996	−0.529338
$144$	0	0
$145$	−8.67004	−0.720008
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.12217	−0.665394	−0.332697	−	0.943034i	$-0.607959\pi$
$149$	−8.12217	−0.665394	−0.332697	+	0.943034i	$0.607959\pi$
$150$	0	0
$151$	21.0475	1.71282	0.856410	−	0.516297i	$-0.172690\pi$
$151$	21.0475	1.71282	0.856410	+	0.516297i	$0.172690\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.292611	−0.0235031
$156$	0	0
$157$	−0.255264	−0.0203723	−0.0101861	−	0.999948i	$-0.503242\pi$
$157$	−0.255264	−0.0203723	−0.0101861	+	0.999948i	$0.503242\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.3774	1.76359
$162$	0	0
$163$	−11.7074	−0.916994	−0.458497	−	0.888696i	$-0.651612\pi$
$163$	−11.7074	−0.916994	−0.458497	+	0.888696i	$0.651612\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.1276	1.24799	0.623997	−	0.781427i	$-0.285508\pi$
$167$	16.1276	1.24799	0.623997	+	0.781427i	$0.285508\pi$
$168$	0	0
$169$	0.744736	0.0572874
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.58522	−0.652722	−0.326361	−	0.945245i	$-0.605823\pi$
$173$	−8.58522	−0.652722	−0.326361	+	0.945245i	$0.605823\pi$
$174$	0	0
$175$	−3.83502	−0.289900
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.37743	−0.551415	−0.275708	−	0.961242i	$-0.588912\pi$
$179$	−7.37743	−0.551415	−0.275708	+	0.961242i	$0.588912\pi$
$180$	0	0
$181$	17.4996	1.30074	0.650368	−	0.759620i	$-0.274615\pi$
$181$	17.4996	1.30074	0.650368	+	0.759620i	$0.274615\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.9627	−0.879512
$186$	0	0
$187$	−2.91518	−0.213179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.54787	0.473788	0.236894	−	0.971536i	$-0.423871\pi$
$191$	6.54787	0.473788	0.236894	+	0.971536i	$0.423871\pi$
$192$	0	0
$193$	7.12217	0.512665	0.256332	−	0.966589i	$-0.417486\pi$
$193$	7.12217	0.512665	0.256332	+	0.966589i	$0.417486\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.6700	−1.25894	−0.629469	−	0.777025i	$-0.716728\pi$
$197$	−17.6700	−1.25894	−0.629469	+	0.777025i	$0.716728\pi$
$198$	0	0
$199$	11.6327	0.824620	0.412310	−	0.911044i	$-0.364722\pi$
$199$	11.6327	0.824620	0.412310	+	0.911044i	$0.364722\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−33.2498	−2.33368
$204$	0	0
$205$	6.96265	0.486293
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.499600	−0.0345581
$210$	0	0
$211$	5.63270	0.387771	0.193885	−	0.981024i	$-0.437891\pi$
$211$	5.63270	0.387771	0.193885	+	0.981024i	$0.437891\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.70739	−0.252842
$216$	0	0
$217$	−1.12217	−0.0761777
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.32996	0.425799
$222$	0	0
$223$	−8.12763	−0.544266	−0.272133	−	0.962260i	$-0.587729\pi$
$223$	−8.12763	−0.544266	−0.272133	+	0.962260i	$0.587729\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.96265	0.661245	0.330622	−	0.943763i	$-0.392741\pi$
$227$	9.96265	0.661245	0.330622	+	0.943763i	$0.392741\pi$
$228$	0	0
$229$	1.25526	0.0829502	0.0414751	−	0.999140i	$-0.486794\pi$
$229$	1.25526	0.0829502	0.0414751	+	0.999140i	$0.486794\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.96265	0.259602	0.129801	−	0.991540i	$-0.458566\pi$
$233$	3.96265	0.259602	0.129801	+	0.991540i	$0.458566\pi$
$234$	0	0
$235$	−1.87237	−0.122140
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.67004	−0.366765	−0.183382	−	0.983042i	$-0.558705\pi$
$239$	−5.67004	−0.366765	−0.183382	+	0.983042i	$0.558705\pi$
$240$	0	0
$241$	−16.6327	−1.07141	−0.535703	−	0.844406i	$-0.679953\pi$
$241$	−16.6327	−1.07141	−0.535703	+	0.844406i	$0.679953\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.70739	−0.492407
$246$	0	0
$247$	1.08482	0.0690255
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.3774	0.844376	0.422188	−	0.906508i	$-0.361262\pi$
$251$	13.3774	0.844376	0.422188	+	0.906508i	$0.361262\pi$
$252$	0	0
$253$	9.96265	0.626347
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.829557	0.0517464	0.0258732	−	0.999665i	$-0.491763\pi$
$257$	0.829557	0.0517464	0.0258732	+	0.999665i	$0.491763\pi$
$258$	0	0
$259$	−45.8770	−2.85066
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.2179	0.753389	0.376695	−	0.926338i	$-0.377061\pi$
$263$	12.2179	0.753389	0.376695	+	0.926338i	$0.377061\pi$
$264$	0	0
$265$	11.6700	0.716885
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.5369	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$269$	17.5369	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$270$	0	0
$271$	−15.1222	−0.918606	−0.459303	−	0.888280i	$-0.651901\pi$
$271$	−15.1222	−0.918606	−0.459303	+	0.888280i	$0.651901\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.70739	−0.102959
$276$	0	0
$277$	−5.70739	−0.342924	−0.171462	−	0.985191i	$-0.554849\pi$
$277$	−5.70739	−0.342924	−0.171462	+	0.985191i	$0.554849\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.29261	0.435041	0.217520	−	0.976056i	$-0.430203\pi$
$281$	7.29261	0.435041	0.217520	+	0.976056i	$0.430203\pi$
$282$	0	0
$283$	−10.1650	−0.604245	−0.302123	−	0.953269i	$-0.597695\pi$
$283$	−10.1650	−0.604245	−0.302123	+	0.953269i	$0.597695\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.7019	1.57616
$288$	0	0
$289$	−14.0848	−0.828519
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.0475	0.762242	0.381121	−	0.924525i	$-0.375538\pi$
$293$	13.0475	0.762242	0.381121	+	0.924525i	$0.375538\pi$
$294$	0	0
$295$	−11.6700	−0.679456
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.6327	−1.25105
$300$	0	0
$301$	−14.2179	−0.819507
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.9627	−0.856759
$306$	0	0
$307$	1.50506	0.0858986	0.0429493	−	0.999077i	$-0.486325\pi$
$307$	1.50506	0.0858986	0.0429493	+	0.999077i	$0.486325\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.45213	−0.309162	−0.154581	−	0.987980i	$-0.549403\pi$
$311$	−5.45213	−0.309162	−0.154581	+	0.987980i	$0.549403\pi$
$312$	0	0
$313$	−10.5479	−0.596201	−0.298101	−	0.954534i	$-0.596353\pi$
$313$	−10.5479	−0.596201	−0.298101	+	0.954534i	$0.596353\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.3774	1.42534	0.712669	−	0.701500i	$-0.247486\pi$
$317$	25.3774	1.42534	0.712669	+	0.701500i	$0.247486\pi$
$318$	0	0
$319$	−14.8031	−0.828817
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.499600	0.0277985
$324$	0	0
$325$	3.70739	0.205649
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.18057	−0.395878
$330$	0	0
$331$	−31.0848	−1.70858	−0.854288	−	0.519800i	$-0.826007\pi$
$331$	−31.0848	−1.70858	−0.854288	+	0.519800i	$0.826007\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.54241	−0.521358
$336$	0	0
$337$	10.2553	0.558640	0.279320	−	0.960198i	$-0.409891\pi$
$337$	10.2553	0.558640	0.279320	+	0.960198i	$0.409891\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.499600	−0.0270549
$342$	0	0
$343$	−2.71285	−0.146480
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.04748	−0.378328	−0.189164	−	0.981945i	$-0.560578\pi$
$347$	−7.04748	−0.378328	−0.189164	+	0.981945i	$0.560578\pi$
$348$	0	0
$349$	29.1696	1.56142	0.780708	−	0.624897i	$-0.214859\pi$
$349$	29.1696	1.56142	0.780708	+	0.624897i	$0.214859\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−15.9627	−0.847210
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.7922	−1.51959	−0.759797	−	0.650160i	$-0.774702\pi$
$359$	−28.7922	−1.51959	−0.759797	+	0.650160i	$0.774702\pi$
$360$	0	0
$361$	−18.9144	−0.995494
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−11.0475	−0.576674	−0.288337	−	0.957529i	$-0.593102\pi$
$367$	−11.0475	−0.576674	−0.288337	+	0.957529i	$0.593102\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	44.7549	2.32356
$372$	0	0
$373$	−15.6700	−0.811364	−0.405682	−	0.914014i	$-0.632966\pi$
$373$	−15.6700	−0.811364	−0.405682	+	0.914014i	$0.632966\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.1432	1.65546
$378$	0	0
$379$	−6.58522	−0.338260	−0.169130	−	0.985594i	$-0.554096\pi$
$379$	−6.58522	−0.338260	−0.169130	+	0.985594i	$0.554096\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.2179	−0.624306	−0.312153	−	0.950032i	$-0.601050\pi$
$383$	−12.2179	−0.624306	−0.312153	+	0.950032i	$0.601050\pi$
$384$	0	0
$385$	−6.54787	−0.333711
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7175	1.40533	0.702667	−	0.711519i	$-0.251992\pi$
$389$	27.7175	1.40533	0.702667	+	0.711519i	$0.251992\pi$
$390$	0	0
$391$	−9.96265	−0.503833
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.00000	−0.100631
$396$	0	0
$397$	−26.5105	−1.33053	−0.665263	−	0.746609i	$-0.731680\pi$
$397$	−26.5105	−1.33053	−0.665263	+	0.746609i	$0.731680\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.8405	0.840974	0.420487	−	0.907299i	$-0.361859\pi$
$401$	16.8405	0.840974	0.420487	+	0.907299i	$0.361859\pi$
$402$	0	0
$403$	1.08482	0.0540388
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.4249	−1.01243
$408$	0	0
$409$	15.0957	0.746437	0.373218	−	0.927744i	$-0.378254\pi$
$409$	15.0957	0.746437	0.373218	+	0.927744i	$0.378254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−44.7549	−2.20224
$414$	0	0
$415$	11.8350	0.580958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.36730	−0.115650	−0.0578252	−	0.998327i	$-0.518417\pi$
$419$	−2.36730	−0.115650	−0.0578252	+	0.998327i	$0.518417\pi$
$420$	0	0
$421$	9.92531	0.483730	0.241865	−	0.970310i	$-0.422241\pi$
$421$	9.92531	0.483730	0.241865	+	0.970310i	$0.422241\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.70739	0.0828205
$426$	0	0
$427$	−57.3821	−2.77691
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.9253	−0.670758	−0.335379	−	0.942083i	$-0.608864\pi$
$431$	−13.9253	−0.670758	−0.335379	+	0.942083i	$0.608864\pi$
$432$	0	0
$433$	−2.29261	−0.110176	−0.0550879	−	0.998482i	$-0.517544\pi$
$433$	−2.29261	−0.110176	−0.0550879	+	0.998482i	$0.517544\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.70739	−0.0816755
$438$	0	0
$439$	25.0101	1.19367	0.596834	−	0.802365i	$-0.296425\pi$
$439$	25.0101	1.19367	0.596834	+	0.802365i	$0.296425\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.8724	−0.944165	−0.472082	−	0.881554i	$-0.656498\pi$
$443$	−19.8724	−0.944165	−0.472082	+	0.881554i	$0.656498\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.2553	1.52222	0.761110	−	0.648623i	$-0.224655\pi$
$449$	32.2553	1.52222	0.761110	+	0.648623i	$0.224655\pi$
$450$	0	0
$451$	11.8880	0.559782
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.2179	0.666546
$456$	0	0
$457$	19.8880	0.930320	0.465160	−	0.885227i	$-0.345997\pi$
$457$	19.8880	0.930320	0.465160	+	0.885227i	$0.345997\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.7658	−1.01373	−0.506867	−	0.862024i	$-0.669196\pi$
$461$	−21.7658	−1.01373	−0.506867	+	0.862024i	$0.669196\pi$
$462$	0	0
$463$	17.9627	0.834795	0.417398	−	0.908724i	$-0.362942\pi$
$463$	17.9627	0.834795	0.417398	+	0.908724i	$0.362942\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.0475	0.881412	0.440706	−	0.897651i	$-0.354728\pi$
$467$	19.0475	0.881412	0.440706	+	0.897651i	$0.354728\pi$
$468$	0	0
$469$	−36.5953	−1.68982
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.32996	−0.291052
$474$	0	0
$475$	0.292611	0.0134259
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7549	0.948314	0.474157	−	0.880440i	$-0.342753\pi$
$479$	20.7549	0.948314	0.474157	+	0.880440i	$0.342753\pi$
$480$	0	0
$481$	44.3502	2.02220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.67004	0.166648
$486$	0	0
$487$	4.86690	0.220540	0.110270	−	0.993902i	$-0.464828\pi$
$487$	4.86690	0.220540	0.110270	+	0.993902i	$0.464828\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.2553	1.72644	0.863218	−	0.504831i	$-0.168445\pi$
$491$	38.2553	1.72644	0.863218	+	0.504831i	$0.168445\pi$
$492$	0	0
$493$	14.8031	0.666700
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−61.2171	−2.74596
$498$	0	0
$499$	−26.8405	−1.20155	−0.600773	−	0.799420i	$-0.705140\pi$
$499$	−26.8405	−1.20155	−0.600773	+	0.799420i	$0.705140\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.8825	−0.841929	−0.420964	−	0.907077i	$-0.638308\pi$
$503$	−18.8825	−0.841929	−0.420964	+	0.907077i	$0.638308\pi$
$504$	0	0
$505$	−0.329957	−0.0146829
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.1696	1.47022	0.735109	−	0.677949i	$-0.237131\pi$
$509$	33.1696	1.47022	0.735109	+	0.677949i	$0.237131\pi$
$510$	0	0
$511$	−30.6802	−1.35721
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.12217	0.137579
$516$	0	0
$517$	−3.19686	−0.140598
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−38.0101	−1.66525	−0.832627	−	0.553834i	$-0.813164\pi$
$521$	−38.0101	−1.66525	−0.832627	+	0.553834i	$0.813164\pi$
$522$	0	0
$523$	−37.4677	−1.63835	−0.819174	−	0.573544i	$-0.805568\pi$
$523$	−37.4677	−1.63835	−0.819174	+	0.573544i	$0.805568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.499600	0.0217629
$528$	0	0
$529$	11.0475	0.480325
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25.8133	−1.11810
$534$	0	0
$535$	14.9198	0.645041
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13.1595	−0.566820
$540$	0	0
$541$	31.7549	1.36525	0.682624	−	0.730770i	$-0.260839\pi$
$541$	31.7549	1.36525	0.682624	+	0.730770i	$0.260839\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.70739	−0.287313
$546$	0	0
$547$	13.5051	0.577435	0.288717	−	0.957414i	$-0.406771\pi$
$547$	13.5051	0.577435	0.288717	+	0.957414i	$0.406771\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.53695	0.108078
$552$	0	0
$553$	−7.67004	−0.326163
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	13.7447	0.581340
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.2125	−1.31545	−0.657724	−	0.753259i	$-0.728481\pi$
$563$	−31.2125	−1.31545	−0.657724	+	0.753259i	$0.728481\pi$
$564$	0	0
$565$	−5.67004	−0.238540
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.8296	−0.537843	−0.268922	−	0.963162i	$-0.586667\pi$
$569$	−12.8296	−0.537843	−0.268922	+	0.963162i	$0.586667\pi$
$570$	0	0
$571$	−15.3401	−0.641963	−0.320981	−	0.947086i	$-0.604013\pi$
$571$	−15.3401	−0.641963	−0.320981	+	0.947086i	$0.604013\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.83502	−0.243337
$576$	0	0
$577$	33.3292	1.38751	0.693755	−	0.720211i	$-0.255955\pi$
$577$	33.3292	1.38751	0.693755	+	0.720211i	$0.255955\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.3876	1.88299
$582$	0	0
$583$	19.9253	0.825222
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.5051	1.71310	0.856549	−	0.516066i	$-0.172604\pi$
$587$	41.5051	1.71310	0.856549	+	0.516066i	$0.172604\pi$
$588$	0	0
$589$	0.0856210	0.00352795
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.0584	0.700505	0.350252	−	0.936655i	$-0.386096\pi$
$593$	17.0584	0.700505	0.350252	+	0.936655i	$0.386096\pi$
$594$	0	0
$595$	6.54787	0.268437
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9253	1.05928	0.529640	−	0.848223i	$-0.322327\pi$
$599$	25.9253	1.05928	0.529640	+	0.848223i	$0.322327\pi$
$600$	0	0
$601$	−11.0957	−0.452605	−0.226303	−	0.974057i	$-0.572664\pi$
$601$	−11.0957	−0.452605	−0.226303	+	0.974057i	$0.572664\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.08482	0.328695
$606$	0	0
$607$	26.3829	1.07085	0.535424	−	0.844583i	$-0.320152\pi$
$607$	26.3829	1.07085	0.535424	+	0.844583i	$0.320152\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.94160	0.280827
$612$	0	0
$613$	2.87783	0.116235	0.0581173	−	0.998310i	$-0.481490\pi$
$613$	2.87783	0.116235	0.0581173	+	0.998310i	$0.481490\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.8405	−0.436422	−0.218211	−	0.975902i	$-0.570022\pi$
$617$	−10.8405	−0.436422	−0.218211	+	0.975902i	$0.570022\pi$
$618$	0	0
$619$	8.87783	0.356830	0.178415	−	0.983955i	$-0.442903\pi$
$619$	8.87783	0.356830	0.178415	+	0.983955i	$0.442903\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.5051	0.460941
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.4249	0.814394
$630$	0	0
$631$	14.3300	0.570467	0.285233	−	0.958458i	$-0.407929\pi$
$631$	14.3300	0.570467	0.285233	+	0.958458i	$0.407929\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.54241	0.219944
$636$	0	0
$637$	28.5743	1.13215
$638$	0	0
$639$	0	0
$640$	0	0
$641$	46.0475	1.81877	0.909383	−	0.415960i	$-0.136554\pi$
$641$	46.0475	1.81877	0.909383	+	0.415960i	$0.136554\pi$
$642$	0	0
$643$	3.54241	0.139699	0.0698495	−	0.997558i	$-0.477748\pi$
$643$	3.54241	0.139699	0.0698495	+	0.997558i	$0.477748\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.1276	0.634043	0.317021	−	0.948418i	$-0.397317\pi$
$647$	16.1276	0.634043	0.317021	+	0.948418i	$0.397317\pi$
$648$	0	0
$649$	−19.9253	−0.782137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.5471	−0.999734	−0.499867	−	0.866102i	$-0.666618\pi$
$653$	−25.5471	−0.999734	−0.499867	+	0.866102i	$0.666618\pi$
$654$	0	0
$655$	−12.8296	−0.501292
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.4996	−1.42182	−0.710911	−	0.703282i	$-0.751717\pi$
$659$	−36.4996	−1.42182	−0.710911	+	0.703282i	$0.751717\pi$
$660$	0	0
$661$	−7.68097	−0.298755	−0.149378	−	0.988780i	$-0.547727\pi$
$661$	−7.68097	−0.298755	−0.149378	+	0.988780i	$0.547727\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.12217	0.0435158
$666$	0	0
$667$	−50.5899	−1.95885
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−25.5471	−0.986234
$672$	0	0
$673$	10.0373	0.386911	0.193456	−	0.981109i	$-0.438030\pi$
$673$	10.0373	0.386911	0.193456	+	0.981109i	$0.438030\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.4521	−1.13194	−0.565969	−	0.824427i	$-0.691498\pi$
$677$	−29.4521	−1.13194	−0.565969	+	0.824427i	$0.691498\pi$
$678$	0	0
$679$	14.0747	0.540137
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.2179	−0.926673	−0.463336	−	0.886182i	$-0.653348\pi$
$683$	−24.2179	−0.926673	−0.463336	+	0.886182i	$0.653348\pi$
$684$	0	0
$685$	−9.08482	−0.347113
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.2654	−1.64828
$690$	0	0
$691$	0.244336	0.00929499	0.00464749	−	0.999989i	$-0.498521\pi$
$691$	0.244336	0.00929499	0.00464749	+	0.999989i	$0.498521\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−11.8880	−0.450289
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.79221	−0.294308	−0.147154	−	0.989114i	$-0.547011\pi$
$701$	−7.79221	−0.294308	−0.147154	+	0.989114i	$0.547011\pi$
$702$	0	0
$703$	3.50040	0.132020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.26539	−0.0475900
$708$	0	0
$709$	−24.8397	−0.932874	−0.466437	−	0.884554i	$-0.654463\pi$
$709$	−24.8397	−0.932874	−0.466437	+	0.884554i	$0.654463\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.70739	−0.0639422
$714$	0	0
$715$	6.32996	0.236727
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.0101	1.30566	0.652829	−	0.757506i	$-0.273582\pi$
$719$	35.0101	1.30566	0.652829	+	0.757506i	$0.273582\pi$
$720$	0	0
$721$	11.9736	0.445919
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.67004	0.321997
$726$	0	0
$727$	22.3082	0.827365	0.413683	−	0.910421i	$-0.364242\pi$
$727$	22.3082	0.827365	0.413683	+	0.910421i	$0.364242\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.32996	0.234122
$732$	0	0
$733$	−16.1696	−0.597239	−0.298620	−	0.954372i	$-0.596526\pi$
$733$	−16.1696	−0.597239	−0.298620	+	0.954372i	$0.596526\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.2926	−0.600146
$738$	0	0
$739$	38.9354	1.43226	0.716132	−	0.697965i	$-0.245911\pi$
$739$	38.9354	1.43226	0.716132	+	0.697965i	$0.245911\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.33542	−0.195738	−0.0978688	−	0.995199i	$-0.531203\pi$
$743$	−5.33542	−0.195738	−0.0978688	+	0.995199i	$0.531203\pi$
$744$	0	0
$745$	8.12217	0.297573
$746$	0	0
$747$	0	0
$748$	0	0
$749$	57.2179	2.09070
$750$	0	0
$751$	34.2070	1.24823	0.624115	−	0.781332i	$-0.285460\pi$
$751$	34.2070	1.24823	0.624115	+	0.781332i	$0.285460\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21.0475	−0.765996
$756$	0	0
$757$	0.952525	0.0346201	0.0173101	−	0.999850i	$-0.494490\pi$
$757$	0.952525	0.0346201	0.0173101	+	0.999850i	$0.494490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.5953	0.601581	0.300790	−	0.953690i	$-0.402750\pi$
$761$	16.5953	0.601581	0.300790	+	0.953690i	$0.402750\pi$
$762$	0	0
$763$	−25.7230	−0.931234
$764$	0	0
$765$	0	0
$766$	0	0
$767$	43.2654	1.56222
$768$	0	0
$769$	6.59535	0.237834	0.118917	−	0.992904i	$-0.462058\pi$
$769$	6.59535	0.237834	0.118917	+	0.992904i	$0.462058\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.8296	0.677252	0.338626	−	0.940921i	$-0.390038\pi$
$773$	18.8296	0.677252	0.338626	+	0.940921i	$0.390038\pi$
$774$	0	0
$775$	0.292611	0.0105109
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.03735	−0.0729955
$780$	0	0
$781$	−27.2545	−0.975241
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.255264	0.00911077
$786$	0	0
$787$	7.12217	0.253878	0.126939	−	0.991911i	$-0.459485\pi$
$787$	7.12217	0.253878	0.126939	+	0.991911i	$0.459485\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−21.7447	−0.773154
$792$	0	0
$793$	55.4724	1.96988
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−48.7175	−1.72566	−0.862832	−	0.505492i	$-0.831311\pi$
$797$	−48.7175	−1.72566	−0.862832	+	0.505492i	$0.831311\pi$
$798$	0	0
$799$	3.19686	0.113097
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.6591	−0.482020
$804$	0	0
$805$	−22.3774	−0.788701
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.1595	0.673613	0.336806	−	0.941574i	$-0.390653\pi$
$809$	19.1595	0.673613	0.336806	+	0.941574i	$0.390653\pi$
$810$	0	0
$811$	13.3401	0.468434	0.234217	−	0.972184i	$-0.424747\pi$
$811$	13.3401	0.468434	0.234217	+	0.972184i	$0.424747\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.7074	0.410092
$816$	0	0
$817$	1.08482	0.0379531
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.62257	−0.0566280	−0.0283140	−	0.999599i	$-0.509014\pi$
$821$	−1.62257	−0.0566280	−0.0283140	+	0.999599i	$0.509014\pi$
$822$	0	0
$823$	26.0529	0.908148	0.454074	−	0.890964i	$-0.349970\pi$
$823$	26.0529	0.908148	0.454074	+	0.890964i	$0.349970\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.4786	1.30326	0.651630	−	0.758537i	$-0.274085\pi$
$827$	37.4786	1.30326	0.651630	+	0.758537i	$0.274085\pi$
$828$	0	0
$829$	32.9627	1.14484	0.572420	−	0.819960i	$-0.306005\pi$
$829$	32.9627	1.14484	0.572420	+	0.819960i	$0.306005\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13.1595	0.455950
$834$	0	0
$835$	−16.1276	−0.558120
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.8778	−0.444592	−0.222296	−	0.974979i	$-0.571355\pi$
$839$	−12.8778	−0.444592	−0.222296	+	0.974979i	$0.571355\pi$
$840$	0	0
$841$	46.1696	1.59206
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.744736	−0.0256197
$846$	0	0
$847$	31.0055	1.06536
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−69.8023	−2.39279
$852$	0	0
$853$	−44.1323	−1.51106	−0.755531	−	0.655113i	$-0.772621\pi$
$853$	−44.1323	−1.51106	−0.755531	+	0.655113i	$0.772621\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.6327	−0.534003	−0.267001	−	0.963696i	$-0.586033\pi$
$857$	−15.6327	−0.534003	−0.267001	+	0.963696i	$0.586033\pi$
$858$	0	0
$859$	7.88796	0.269134	0.134567	−	0.990905i	$-0.457036\pi$
$859$	7.88796	0.269134	0.134567	+	0.990905i	$0.457036\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.53148	0.324455	0.162228	−	0.986753i	$-0.448132\pi$
$863$	9.53148	0.324455	0.162228	+	0.986753i	$0.448132\pi$
$864$	0	0
$865$	8.58522	0.291906
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.41478	−0.115838
$870$	0	0
$871$	35.3774	1.19872
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.83502	0.129647
$876$	0	0
$877$	37.9517	1.28154	0.640769	−	0.767733i	$-0.278615\pi$
$877$	37.9517	1.28154	0.640769	+	0.767733i	$0.278615\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.2070	0.781863	0.390932	−	0.920420i	$-0.372153\pi$
$881$	23.2070	0.781863	0.390932	+	0.920420i	$0.372153\pi$
$882$	0	0
$883$	30.9572	1.04179	0.520896	−	0.853620i	$-0.325598\pi$
$883$	30.9572	1.04179	0.520896	+	0.853620i	$0.325598\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.8770	1.27179	0.635893	−	0.771777i	$-0.280632\pi$
$887$	37.8770	1.27179	0.635893	+	0.771777i	$0.280632\pi$
$888$	0	0
$889$	21.2553	0.712879
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.547875	0.0183339
$894$	0	0
$895$	7.37743	0.246600
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.53695	0.0846119
$900$	0	0
$901$	−19.9253	−0.663808
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.4996	−0.581706
$906$	0	0
$907$	32.3829	1.07526	0.537628	−	0.843182i	$-0.319320\pi$
$907$	32.3829	1.07526	0.537628	+	0.843182i	$0.319320\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−45.2545	−1.49935	−0.749674	−	0.661808i	$-0.769790\pi$
$911$	−45.2545	−1.49935	−0.749674	+	0.661808i	$0.769790\pi$
$912$	0	0
$913$	20.2070	0.668754
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−49.2016	−1.62478
$918$	0	0
$919$	21.9253	0.723249	0.361625	−	0.932324i	$-0.382222\pi$
$919$	21.9253	0.723249	0.361625	+	0.932324i	$0.382222\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	59.1798	1.94793
$924$	0	0
$925$	11.9627	0.393330
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.2654	−0.435223	−0.217612	−	0.976035i	$-0.569827\pi$
$929$	−13.2654	−0.435223	−0.217612	+	0.976035i	$0.569827\pi$
$930$	0	0
$931$	2.25526	0.0739133
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.91518	0.0953365
$936$	0	0
$937$	−40.4996	−1.32306	−0.661532	−	0.749917i	$-0.730093\pi$
$937$	−40.4996	−1.32306	−0.661532	+	0.749917i	$0.730093\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.67004	0.0870409	0.0435205	−	0.999053i	$-0.486143\pi$
$941$	2.67004	0.0870409	0.0435205	+	0.999053i	$0.486143\pi$
$942$	0	0
$943$	40.6272	1.32300
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.8825	1.19852	0.599260	−	0.800554i	$-0.295462\pi$
$947$	36.8825	1.19852	0.599260	+	0.800554i	$0.295462\pi$
$948$	0	0
$949$	29.6591	0.962776
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.7175	0.411961	0.205980	−	0.978556i	$-0.433962\pi$
$953$	12.7175	0.411961	0.205980	+	0.978556i	$0.433962\pi$
$954$	0	0
$955$	−6.54787	−0.211884
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.8405	−1.12506
$960$	0	0
$961$	−30.9144	−0.997238
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.12217	−0.229271
$966$	0	0
$967$	42.3455	1.36174	0.680871	−	0.732404i	$-0.261602\pi$
$967$	42.3455	1.36174	0.680871	+	0.732404i	$0.261602\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.38836	0.172921	0.0864604	−	0.996255i	$-0.472444\pi$
$971$	5.38836	0.172921	0.0864604	+	0.996255i	$0.472444\pi$
$972$	0	0
$973$	15.3401	0.491781
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.1323	0.900032	0.450016	−	0.893021i	$-0.351418\pi$
$977$	28.1323	0.900032	0.450016	+	0.893021i	$0.351418\pi$
$978$	0	0
$979$	5.12217	0.163705
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.7603	−0.821627	−0.410813	−	0.911719i	$-0.634755\pi$
$983$	−25.7603	−0.821627	−0.410813	+	0.911719i	$0.634755\pi$
$984$	0	0
$985$	17.6700	0.563014
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.6327	−0.687880
$990$	0	0
$991$	23.0848	0.733314	0.366657	−	0.930356i	$-0.380502\pi$
$991$	23.0848	0.733314	0.366657	+	0.930356i	$0.380502\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.6327	−0.368781
$996$	0	0
$997$	−32.8405	−1.04007	−0.520034	−	0.854145i	$-0.674081\pi$
$997$	−32.8405	−1.04007	−0.520034	+	0.854145i	$0.674081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.a.i.1.1		3
3.2	odd	2		1620.2.a.j.1.1		3
4.3	odd	2		6480.2.a.bt.1.3		3
5.2	odd	4		8100.2.d.p.649.2		6
5.3	odd	4		8100.2.d.p.649.5		6
5.4	even	2		8100.2.a.v.1.3		3
9.2	odd	6		540.2.i.b.361.3		6
9.4	even	3		180.2.i.b.61.2	✓	6
9.5	odd	6		540.2.i.b.181.3		6
9.7	even	3		180.2.i.b.121.2	yes	6
12.11	even	2		6480.2.a.bw.1.3		3
15.2	even	4		8100.2.d.o.649.2		6
15.8	even	4		8100.2.d.o.649.5		6
15.14	odd	2		8100.2.a.u.1.3		3
36.7	odd	6		720.2.q.k.481.2		6
36.11	even	6		2160.2.q.i.1441.1		6
36.23	even	6		2160.2.q.i.721.1		6
36.31	odd	6		720.2.q.k.241.2		6
45.2	even	12		2700.2.s.c.1549.2		12
45.4	even	6		900.2.i.c.601.2		6
45.7	odd	12		900.2.s.c.49.6		12
45.13	odd	12		900.2.s.c.349.6		12
45.14	odd	6		2700.2.i.c.1801.1		6
45.22	odd	12		900.2.s.c.349.1		12
45.23	even	12		2700.2.s.c.2449.2		12
45.29	odd	6		2700.2.i.c.901.1		6
45.32	even	12		2700.2.s.c.2449.5		12
45.34	even	6		900.2.i.c.301.2		6
45.38	even	12		2700.2.s.c.1549.5		12
45.43	odd	12		900.2.s.c.49.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.2	✓	6	9.4	even	3
180.2.i.b.121.2	yes	6	9.7	even	3
540.2.i.b.181.3		6	9.5	odd	6
540.2.i.b.361.3		6	9.2	odd	6
720.2.q.k.241.2		6	36.31	odd	6
720.2.q.k.481.2		6	36.7	odd	6
900.2.i.c.301.2		6	45.34	even	6
900.2.i.c.601.2		6	45.4	even	6
900.2.s.c.49.1		12	45.43	odd	12
900.2.s.c.49.6		12	45.7	odd	12
900.2.s.c.349.1		12	45.22	odd	12
900.2.s.c.349.6		12	45.13	odd	12
1620.2.a.i.1.1		3	1.1	even	1	trivial
1620.2.a.j.1.1		3	3.2	odd	2
2160.2.q.i.721.1		6	36.23	even	6
2160.2.q.i.1441.1		6	36.11	even	6
2700.2.i.c.901.1		6	45.29	odd	6
2700.2.i.c.1801.1		6	45.14	odd	6
2700.2.s.c.1549.2		12	45.2	even	12
2700.2.s.c.1549.5		12	45.38	even	12
2700.2.s.c.2449.2		12	45.23	even	12
2700.2.s.c.2449.5		12	45.32	even	12
6480.2.a.bt.1.3		3	4.3	odd	2
6480.2.a.bw.1.3		3	12.11	even	2
8100.2.a.u.1.3		3	15.14	odd	2
8100.2.a.v.1.3		3	5.4	even	2
8100.2.d.o.649.2		6	15.2	even	4
8100.2.d.o.649.5		6	15.8	even	4
8100.2.d.p.649.2		6	5.2	odd	4
8100.2.d.p.649.5		6	5.3	odd	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.83502 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.83502 q^{7} -1.70739 q^{11} +3.70739 q^{13} +1.70739 q^{17} +0.292611 q^{19} -5.83502 q^{23} +1.00000 q^{25} +8.67004 q^{29} +0.292611 q^{31} +3.83502 q^{35} +11.9627 q^{37} -6.96265 q^{41} +3.70739 q^{43} +1.87237 q^{47} +7.70739 q^{49} -11.6700 q^{53} +1.70739 q^{55} +11.6700 q^{59} +14.9627 q^{61} -3.70739 q^{65} +9.54241 q^{67} +15.9627 q^{71} +8.00000 q^{73} +6.54787 q^{77} +2.00000 q^{79} -11.8350 q^{83} -1.70739 q^{85} -3.00000 q^{89} -14.2179 q^{91} -0.292611 q^{95} -3.67004 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7	\( 3 q - 3 q^{5} + 3 q^{7} + 6 q^{13} + 6 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} - 3 q^{35} + 12 q^{37} + 3 q^{41} + 6 q^{43} + 15 q^{47} + 18 q^{49} - 6 q^{53} + 6 q^{59} + 21 q^{61} - 6 q^{65} + 9 q^{67} + 24 q^{71} + 24 q^{73} + 6 q^{77} + 6 q^{79} - 21 q^{83} - 9 q^{89} - 6 q^{95} + 18 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 6 * q^13 + 6 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^29 + 6 * q^31 - 3 * q^35 + 12 * q^37 + 3 * q^41 + 6 * q^43 + 15 * q^47 + 18 * q^49 - 6 * q^53 + 6 * q^59 + 21 * q^61 - 6 * q^65 + 9 * q^67 + 24 * q^71 + 24 * q^73 + 6 * q^77 + 6 * q^79 - 21 * q^83 - 9 * q^89 - 6 * q^95 + 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more