LMFDB - Embedded newform 5292.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, 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("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 756)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.28799 q^{5} +O(q^{10})$	$q+4.28799 q^{5} -3.81089 q^{11} -3.28799 q^{13} +0.810892 q^{17} -7.09888 q^{19} -6.47710 q^{23} +13.3869 q^{25} +3.81089 q^{29} -3.28799 q^{31} -5.76509 q^{37} -2.09888 q^{41} -8.76509 q^{43} -3.33379 q^{47} +9.86398 q^{53} -16.3411 q^{55} -3.47710 q^{59} -5.95420 q^{61} -14.0989 q^{65} -3.52290 q^{67} +6.05308 q^{71} +10.3869 q^{73} -5.14468 q^{79} +1.71201 q^{83} +3.47710 q^{85} -12.5302 q^{89} -30.4400 q^{95} -1.04580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5}+O(q^{10}) $ 3 * q + q^5	$ 3 q + q^{5} - 5 q^{11} + 2 q^{13} - 4 q^{17} - 3 q^{19} - 14 q^{23} + 10 q^{25} + 5 q^{29} + 2 q^{31} + 12 q^{41} - 9 q^{43} - 9 q^{47} - 6 q^{53} - 8 q^{55} - 5 q^{59} - 7 q^{61} - 24 q^{65} - 16 q^{67} - 11 q^{71} + q^{73} - 8 q^{79} + 17 q^{83} + 5 q^{85} - 3 q^{89} - 32 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q + q^5 - 5 * q^11 + 2 * q^13 - 4 * q^17 - 3 * q^19 - 14 * q^23 + 10 * q^25 + 5 * q^29 + 2 * q^31 + 12 * q^41 - 9 * q^43 - 9 * q^47 - 6 * q^53 - 8 * q^55 - 5 * q^59 - 7 * q^61 - 24 * q^65 - 16 * q^67 - 11 * q^71 + q^73 - 8 * q^79 + 17 * q^83 + 5 * q^85 - 3 * q^89 - 32 * q^95 - 14 * 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
$5$	4.28799	1.91765	0.958824	−	0.284000i	$-0.0916617\pi$
$5$	4.28799	1.91765	0.958824	+	0.284000i	$0.0916617\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.81089	−1.14903	−0.574514	−	0.818495i	$-0.694809\pi$
$11$	−3.81089	−1.14903	−0.574514	+	0.818495i	$0.694809\pi$
$12$	0	0
$13$	−3.28799	−0.911925	−0.455962	−	0.889999i	$-0.650705\pi$
$13$	−3.28799	−0.911925	−0.455962	+	0.889999i	$0.650705\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.810892	0.196670	0.0983351	−	0.995153i	$-0.468648\pi$
$17$	0.810892	0.196670	0.0983351	+	0.995153i	$0.468648\pi$
$18$	0	0
$19$	−7.09888	−1.62860	−0.814298	−	0.580447i	$-0.802878\pi$
$19$	−7.09888	−1.62860	−0.814298	+	0.580447i	$0.802878\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.47710	−1.35057	−0.675284	−	0.737557i	$-0.735979\pi$
$23$	−6.47710	−1.35057	−0.675284	+	0.737557i	$0.735979\pi$
$24$	0	0
$25$	13.3869	2.67738
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.81089	0.707665	0.353832	−	0.935309i	$-0.384878\pi$
$29$	3.81089	0.707665	0.353832	+	0.935309i	$0.384878\pi$
$30$	0	0
$31$	−3.28799	−0.590541	−0.295270	−	0.955414i	$-0.595410\pi$
$31$	−3.28799	−0.590541	−0.295270	+	0.955414i	$0.595410\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.76509	−0.947775	−0.473888	−	0.880585i	$-0.657150\pi$
$37$	−5.76509	−0.947775	−0.473888	+	0.880585i	$0.657150\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.09888	−0.327791	−0.163895	−	0.986478i	$-0.552406\pi$
$41$	−2.09888	−0.327791	−0.163895	+	0.986478i	$0.552406\pi$
$42$	0	0
$43$	−8.76509	−1.33666	−0.668332	−	0.743863i	$-0.732991\pi$
$43$	−8.76509	−1.33666	−0.668332	+	0.743863i	$0.732991\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.33379	−0.486284	−0.243142	−	0.969991i	$-0.578178\pi$
$47$	−3.33379	−0.486284	−0.243142	+	0.969991i	$0.578178\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.86398	1.35492	0.677461	−	0.735559i	$-0.263080\pi$
$53$	9.86398	1.35492	0.677461	+	0.735559i	$0.263080\pi$
$54$	0	0
$55$	−16.3411	−2.20343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.47710	−0.452680	−0.226340	−	0.974048i	$-0.572676\pi$
$59$	−3.47710	−0.452680	−0.226340	+	0.974048i	$0.572676\pi$
$60$	0	0
$61$	−5.95420	−0.762357	−0.381179	−	0.924501i	$-0.624482\pi$
$61$	−5.95420	−0.762357	−0.381179	+	0.924501i	$0.624482\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.0989	−1.74875
$66$	0	0
$67$	−3.52290	−0.430391	−0.215195	−	0.976571i	$-0.569039\pi$
$67$	−3.52290	−0.430391	−0.215195	+	0.976571i	$0.569039\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.05308	0.718369	0.359184	−	0.933267i	$-0.383055\pi$
$71$	6.05308	0.718369	0.359184	+	0.933267i	$0.383055\pi$
$72$	0	0
$73$	10.3869	1.21569	0.607846	−	0.794055i	$-0.292034\pi$
$73$	10.3869	1.21569	0.607846	+	0.794055i	$0.292034\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.14468	−0.578822	−0.289411	−	0.957205i	$-0.593459\pi$
$79$	−5.14468	−0.578822	−0.289411	+	0.957205i	$0.593459\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.71201	0.187917	0.0939586	−	0.995576i	$-0.470048\pi$
$83$	1.71201	0.187917	0.0939586	+	0.995576i	$0.470048\pi$
$84$	0	0
$85$	3.47710	0.377144
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.5302	−1.32820	−0.664098	−	0.747645i	$-0.731184\pi$
$89$	−12.5302	−1.32820	−0.664098	+	0.747645i	$0.731184\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−30.4400	−3.12307
$96$	0	0
$97$	−1.04580	−0.106185	−0.0530925	−	0.998590i	$-0.516908\pi$
$97$	−1.04580	−0.106185	−0.0530925	+	0.998590i	$0.516908\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.9098	−1.78209	−0.891045	−	0.453916i	$-0.850027\pi$
$101$	−17.9098	−1.78209	−0.891045	+	0.453916i	$0.850027\pi$
$102$	0	0
$103$	7.00000	0.689730	0.344865	−	0.938652i	$-0.387925\pi$
$103$	7.00000	0.689730	0.344865	+	0.938652i	$0.387925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.90112	−0.667156	−0.333578	−	0.942722i	$-0.608256\pi$
$107$	−6.90112	−0.667156	−0.333578	+	0.942722i	$0.608256\pi$
$108$	0	0
$109$	−5.62178	−0.538469	−0.269235	−	0.963075i	$-0.586771\pi$
$109$	−5.62178	−0.538469	−0.269235	+	0.963075i	$0.586771\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.1520	−1.89574	−0.947869	−	0.318661i	$-0.896767\pi$
$113$	−20.1520	−1.89574	−0.947869	+	0.318661i	$0.896767\pi$
$114$	0	0
$115$	−27.7738	−2.58992
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.52290	0.320264
$122$	0	0
$123$	0	0
$124$	0	0
$125$	35.9629	3.21662
$126$	0	0
$127$	−2.14468	−0.190310	−0.0951550	−	0.995462i	$-0.530335\pi$
$127$	−2.14468	−0.190310	−0.0951550	+	0.995462i	$0.530335\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.38688	0.295913	0.147956	−	0.988994i	$-0.452730\pi$
$131$	3.38688	0.295913	0.147956	+	0.988994i	$0.452730\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9542	−1.10675	−0.553376	−	0.832932i	$-0.686661\pi$
$137$	−12.9542	−1.10675	−0.553376	+	0.832932i	$0.686661\pi$
$138$	0	0
$139$	17.2880	1.46635	0.733174	−	0.680041i	$-0.238038\pi$
$139$	17.2880	1.46635	0.733174	+	0.680041i	$0.238038\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.5302	1.04783
$144$	0	0
$145$	16.3411	1.35705
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.56870	−0.128513	−0.0642565	−	0.997933i	$-0.520468\pi$
$149$	−1.56870	−0.128513	−0.0642565	+	0.997933i	$0.520468\pi$
$150$	0	0
$151$	−1.38688	−0.112862	−0.0564312	−	0.998406i	$-0.517972\pi$
$151$	−1.38688	−0.112862	−0.0564312	+	0.998406i	$0.517972\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.0989	−1.13245
$156$	0	0
$157$	16.8640	1.34589	0.672946	−	0.739692i	$-0.265029\pi$
$157$	16.8640	1.34589	0.672946	+	0.739692i	$0.265029\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.2967	1.35478	0.677389	−	0.735625i	$-0.263111\pi$
$163$	17.2967	1.35478	0.677389	+	0.735625i	$0.263111\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.9556	1.07991	0.539957	−	0.841692i	$-0.318440\pi$
$167$	13.9556	1.07991	0.539957	+	0.841692i	$0.318440\pi$
$168$	0	0
$169$	−2.18911	−0.168393
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.14331	−0.695153	−0.347576	−	0.937652i	$-0.612995\pi$
$173$	−9.14331	−0.695153	−0.347576	+	0.937652i	$0.612995\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.0617	1.27525	0.637627	−	0.770345i	$-0.279916\pi$
$179$	17.0617	1.27525	0.637627	+	0.770345i	$0.279916\pi$
$180$	0	0
$181$	16.4400	1.22197	0.610986	−	0.791641i	$-0.290773\pi$
$181$	16.4400	1.22197	0.610986	+	0.791641i	$0.290773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.7207	−1.81750
$186$	0	0
$187$	−3.09022	−0.225980
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.96286	0.214385	0.107193	−	0.994238i	$-0.465814\pi$
$191$	2.96286	0.214385	0.107193	+	0.994238i	$0.465814\pi$
$192$	0	0
$193$	−17.4313	−1.25473	−0.627366	−	0.778724i	$-0.715867\pi$
$193$	−17.4313	−1.25473	−0.627366	+	0.778724i	$0.715867\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.2880	0.732989	0.366495	−	0.930420i	$-0.380558\pi$
$197$	10.2880	0.732989	0.366495	+	0.930420i	$0.380558\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.0531	1.87130
$210$	0	0
$211$	16.4327	1.13127	0.565636	−	0.824655i	$-0.308631\pi$
$211$	16.4327	1.13127	0.565636	+	0.824655i	$0.308631\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−37.5846	−2.56325
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.66621	−0.179349
$222$	0	0
$223$	−4.13602	−0.276969	−0.138484	−	0.990365i	$-0.544223\pi$
$223$	−4.13602	−0.276969	−0.138484	+	0.990365i	$0.544223\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.2967	1.41351	0.706754	−	0.707459i	$-0.250159\pi$
$227$	21.2967	1.41351	0.706754	+	0.707459i	$0.250159\pi$
$228$	0	0
$229$	5.85532	0.386930	0.193465	−	0.981107i	$-0.438027\pi$
$229$	5.85532	0.386930	0.193465	+	0.981107i	$0.438027\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.04580	−0.527098	−0.263549	−	0.964646i	$-0.584893\pi$
$233$	−8.04580	−0.527098	−0.263549	+	0.964646i	$0.584893\pi$
$234$	0	0
$235$	−14.2953	−0.932521
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.0989	−0.911981	−0.455991	−	0.889985i	$-0.650715\pi$
$239$	−14.0989	−0.911981	−0.455991	+	0.889985i	$0.650715\pi$
$240$	0	0
$241$	26.7280	1.72170	0.860849	−	0.508860i	$-0.169933\pi$
$241$	26.7280	1.72170	0.860849	+	0.508860i	$0.169933\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	23.3411	1.48516
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.43268	−0.153549	−0.0767746	−	0.997048i	$-0.524462\pi$
$251$	−2.43268	−0.153549	−0.0767746	+	0.997048i	$0.524462\pi$
$252$	0	0
$253$	24.6835	1.55184
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.5302	0.781611	0.390806	−	0.920473i	$-0.372196\pi$
$257$	12.5302	0.781611	0.390806	+	0.920473i	$0.372196\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.90112	−0.425541	−0.212771	−	0.977102i	$-0.568249\pi$
$263$	−6.90112	−0.425541	−0.212771	+	0.977102i	$0.568249\pi$
$264$	0	0
$265$	42.2967	2.59826
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.8109	−1.32983	−0.664917	−	0.746917i	$-0.731533\pi$
$269$	−21.8109	−1.32983	−0.664917	+	0.746917i	$0.731533\pi$
$270$	0	0
$271$	−7.52290	−0.456984	−0.228492	−	0.973546i	$-0.573379\pi$
$271$	−7.52290	−0.456984	−0.228492	+	0.973546i	$0.573379\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−51.0159	−3.07638
$276$	0	0
$277$	9.23491	0.554872	0.277436	−	0.960744i	$-0.410515\pi$
$277$	9.23491	0.554872	0.277436	+	0.960744i	$0.410515\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.6749	−0.815774	−0.407887	−	0.913032i	$-0.633734\pi$
$281$	−13.6749	−0.815774	−0.407887	+	0.913032i	$0.633734\pi$
$282$	0	0
$283$	−2.14331	−0.127406	−0.0637032	−	0.997969i	$-0.520291\pi$
$283$	−2.14331	−0.127406	−0.0637032	+	0.997969i	$0.520291\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.3425	−0.961321
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.29665	−0.543116	−0.271558	−	0.962422i	$-0.587539\pi$
$293$	−9.29665	−0.543116	−0.271558	+	0.962422i	$0.587539\pi$
$294$	0	0
$295$	−14.9098	−0.868081
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.2967	1.23162
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−25.5316	−1.46193
$306$	0	0
$307$	10.3869	0.592810	0.296405	−	0.955062i	$-0.404212\pi$
$307$	10.3869	0.592810	0.296405	+	0.955062i	$0.404212\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.57598	−0.146071	−0.0730353	−	0.997329i	$-0.523269\pi$
$311$	−2.57598	−0.146071	−0.0730353	+	0.997329i	$0.523269\pi$
$312$	0	0
$313$	11.4844	0.649136	0.324568	−	0.945862i	$-0.394781\pi$
$313$	11.4844	0.649136	0.324568	+	0.945862i	$0.394781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.1964	−0.853514	−0.426757	−	0.904366i	$-0.640344\pi$
$317$	−15.1964	−0.853514	−0.426757	+	0.904366i	$0.640344\pi$
$318$	0	0
$319$	−14.5229	−0.813126
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.75643	−0.320296
$324$	0	0
$325$	−44.0159	−2.44157
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.5316	0.853692	0.426846	−	0.904324i	$-0.359625\pi$
$331$	15.5316	0.853692	0.426846	+	0.904324i	$0.359625\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.1062	−0.825338
$336$	0	0
$337$	−3.00866	−0.163892	−0.0819461	−	0.996637i	$-0.526114\pi$
$337$	−3.00866	−0.163892	−0.0819461	+	0.996637i	$0.526114\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.5302	0.678548
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.8654	−1.22748	−0.613738	−	0.789510i	$-0.710335\pi$
$347$	−22.8654	−1.22748	−0.613738	+	0.789510i	$0.710335\pi$
$348$	0	0
$349$	−9.04442	−0.484137	−0.242068	−	0.970259i	$-0.577826\pi$
$349$	−9.04442	−0.484137	−0.242068	+	0.970259i	$0.577826\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.1964	−0.968496	−0.484248	−	0.874931i	$-0.660907\pi$
$353$	−18.1964	−0.968496	−0.484248	+	0.874931i	$0.660907\pi$
$354$	0	0
$355$	25.9556	1.37758
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.3869	−1.28709	−0.643545	−	0.765408i	$-0.722537\pi$
$359$	−24.3869	−1.28709	−0.643545	+	0.765408i	$0.722537\pi$
$360$	0	0
$361$	31.3942	1.65232
$362$	0	0
$363$	0	0
$364$	0	0
$365$	44.5388	2.33127
$366$	0	0
$367$	8.86535	0.462768	0.231384	−	0.972863i	$-0.425675\pi$
$367$	8.86535	0.462768	0.231384	+	0.972863i	$0.425675\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.15197	−0.163203	−0.0816014	−	0.996665i	$-0.526003\pi$
$373$	−3.15197	−0.163203	−0.0816014	+	0.996665i	$0.526003\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.5302	−0.645337
$378$	0	0
$379$	−2.28799	−0.117526	−0.0587631	−	0.998272i	$-0.518716\pi$
$379$	−2.28799	−0.117526	−0.0587631	+	0.998272i	$0.518716\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.2509	−1.44355	−0.721776	−	0.692127i	$-0.756674\pi$
$383$	−28.2509	−1.44355	−0.721776	+	0.692127i	$0.756674\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.720669	−0.0365394	−0.0182697	−	0.999833i	$-0.505816\pi$
$389$	−0.720669	−0.0365394	−0.0182697	+	0.999833i	$0.505816\pi$
$390$	0	0
$391$	−5.25223	−0.265617
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.0604	−1.10998
$396$	0	0
$397$	−21.1506	−1.06152	−0.530759	−	0.847523i	$-0.678093\pi$
$397$	−21.1506	−1.06152	−0.530759	+	0.847523i	$0.678093\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.4944	1.42294	0.711472	−	0.702715i	$-0.248029\pi$
$401$	28.4944	1.42294	0.711472	+	0.702715i	$0.248029\pi$
$402$	0	0
$403$	10.8109	0.538529
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.9701	1.08902
$408$	0	0
$409$	10.7637	0.532231	0.266116	−	0.963941i	$-0.414260\pi$
$409$	10.7637	0.532231	0.266116	+	0.963941i	$0.414260\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.34108	0.360359
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.0604	1.66396	0.831979	−	0.554807i	$-0.187208\pi$
$419$	34.0604	1.66396	0.831979	+	0.554807i	$0.187208\pi$
$420$	0	0
$421$	20.5833	1.00317	0.501584	−	0.865109i	$-0.332751\pi$
$421$	20.5833	1.00317	0.501584	+	0.865109i	$0.332751\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10.8553	0.526560
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−23.9629	−1.15425	−0.577125	−	0.816656i	$-0.695826\pi$
$431$	−23.9629	−1.15425	−0.577125	+	0.816656i	$0.695826\pi$
$432$	0	0
$433$	−29.9642	−1.43999	−0.719995	−	0.693980i	$-0.755856\pi$
$433$	−29.9642	−1.43999	−0.719995	+	0.693980i	$0.755856\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.9802	2.19953
$438$	0	0
$439$	−28.8196	−1.37548	−0.687741	−	0.725956i	$-0.741398\pi$
$439$	−28.8196	−1.37548	−0.687741	+	0.725956i	$0.741398\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.0087	−1.52078	−0.760389	−	0.649468i	$-0.774992\pi$
$443$	−32.0087	−1.52078	−0.760389	+	0.649468i	$0.774992\pi$
$444$	0	0
$445$	−53.7293	−2.54701
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.0087	1.51058	0.755291	−	0.655390i	$-0.227496\pi$
$449$	32.0087	1.51058	0.755291	+	0.655390i	$0.227496\pi$
$450$	0	0
$451$	7.99862	0.376640
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.8553	−0.975570	−0.487785	−	0.872964i	$-0.662195\pi$
$457$	−20.8553	−0.975570	−0.487785	+	0.872964i	$0.662195\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	25.2509	1.17605	0.588025	−	0.808843i	$-0.299906\pi$
$461$	25.2509	1.17605	0.588025	+	0.808843i	$0.299906\pi$
$462$	0	0
$463$	−21.1520	−0.983015	−0.491508	−	0.870873i	$-0.663554\pi$
$463$	−21.1520	−0.983015	−0.491508	+	0.870873i	$0.663554\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.9171	0.736554	0.368277	−	0.929716i	$-0.379948\pi$
$467$	15.9171	0.736554	0.368277	+	0.929716i	$0.379948\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	33.4028	1.53586
$474$	0	0
$475$	−95.0319	−4.36036
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.95558	−0.0893526	−0.0446763	−	0.999002i	$-0.514226\pi$
$479$	−1.95558	−0.0893526	−0.0446763	+	0.999002i	$0.514226\pi$
$480$	0	0
$481$	18.9556	0.864300
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.48438	−0.203625
$486$	0	0
$487$	19.9629	0.904604	0.452302	−	0.891865i	$-0.350603\pi$
$487$	19.9629	0.904604	0.452302	+	0.891865i	$0.350603\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.0975	−0.997247	−0.498623	−	0.866819i	$-0.666161\pi$
$491$	−22.0975	−0.997247	−0.498623	+	0.866819i	$0.666161\pi$
$492$	0	0
$493$	3.09022	0.139177
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−33.1148	−1.48242	−0.741212	−	0.671271i	$-0.765749\pi$
$499$	−33.1148	−1.48242	−0.741212	+	0.671271i	$0.765749\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.76509	−0.212465	−0.106232	−	0.994341i	$-0.533879\pi$
$503$	−4.76509	−0.212465	−0.106232	+	0.994341i	$0.533879\pi$
$504$	0	0
$505$	−76.7970	−3.41742
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.7651	−0.477154	−0.238577	−	0.971124i	$-0.576681\pi$
$509$	−10.7651	−0.477154	−0.238577	+	0.971124i	$0.576681\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	30.0159	1.32266
$516$	0	0
$517$	12.7047	0.558753
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.23491	−0.0541023	−0.0270512	−	0.999634i	$-0.508612\pi$
$521$	−1.23491	−0.0541023	−0.0270512	+	0.999634i	$0.508612\pi$
$522$	0	0
$523$	8.56870	0.374683	0.187342	−	0.982295i	$-0.440013\pi$
$523$	8.56870	0.374683	0.187342	+	0.982295i	$0.440013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.66621	−0.116142
$528$	0	0
$529$	18.9528	0.824036
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.90112	0.298920
$534$	0	0
$535$	−29.5919	−1.27937
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.0087	−0.645273	−0.322636	−	0.946523i	$-0.604569\pi$
$541$	−15.0087	−0.645273	−0.322636	+	0.946523i	$0.604569\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.1062	−1.03259
$546$	0	0
$547$	12.2509	0.523809	0.261904	−	0.965094i	$-0.415650\pi$
$547$	12.2509	0.523809	0.261904	+	0.965094i	$0.415650\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.0531	−1.15250
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.7810	−1.09238	−0.546189	−	0.837662i	$-0.683922\pi$
$557$	−25.7810	−1.09238	−0.546189	+	0.837662i	$0.683922\pi$
$558$	0	0
$559$	28.8196	1.21894
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.6835	−1.04029	−0.520143	−	0.854079i	$-0.674121\pi$
$563$	−24.6835	−1.04029	−0.520143	+	0.854079i	$0.674121\pi$
$564$	0	0
$565$	−86.4115	−3.63536
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.2335	0.512856	0.256428	−	0.966563i	$-0.417454\pi$
$569$	12.2335	0.512856	0.256428	+	0.966563i	$0.417454\pi$
$570$	0	0
$571$	35.3126	1.47779	0.738893	−	0.673823i	$-0.235349\pi$
$571$	35.3126	1.47779	0.738893	+	0.673823i	$0.235349\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−86.7081	−3.61598
$576$	0	0
$577$	−9.04442	−0.376524	−0.188262	−	0.982119i	$-0.560285\pi$
$577$	−9.04442	−0.376524	−0.188262	+	0.982119i	$0.560285\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−37.5906	−1.55684
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.8196	−1.47843	−0.739216	−	0.673469i	$-0.764804\pi$
$587$	−35.8196	−1.47843	−0.739216	+	0.673469i	$0.764804\pi$
$588$	0	0
$589$	23.3411	0.961752
$590$	0	0
$591$	0	0
$592$	0	0
$593$	44.0617	1.80940	0.904700	−	0.426050i	$-0.140095\pi$
$593$	44.0617	1.80940	0.904700	+	0.426050i	$0.140095\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0631	1.22835	0.614173	−	0.789171i	$-0.289490\pi$
$599$	30.0631	1.22835	0.614173	+	0.789171i	$0.289490\pi$
$600$	0	0
$601$	−29.1964	−1.19095	−0.595473	−	0.803375i	$-0.703035\pi$
$601$	−29.1964	−1.19095	−0.595473	+	0.803375i	$0.703035\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.1062	0.614153
$606$	0	0
$607$	16.1433	0.655237	0.327618	−	0.944810i	$-0.393754\pi$
$607$	16.1433	0.655237	0.327618	+	0.944810i	$0.393754\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.9615	0.443454
$612$	0	0
$613$	47.1162	1.90301	0.951503	−	0.307640i	$-0.0995392\pi$
$613$	47.1162	1.90301	0.951503	+	0.307640i	$0.0995392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0146	1.53041	0.765204	−	0.643787i	$-0.222638\pi$
$617$	38.0146	1.53041	0.765204	+	0.643787i	$0.222638\pi$
$618$	0	0
$619$	−12.1818	−0.489629	−0.244814	−	0.969570i	$-0.578727\pi$
$619$	−12.1818	−0.489629	−0.244814	+	0.969570i	$0.578727\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	87.2741	3.49096
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.67487	−0.186399
$630$	0	0
$631$	−10.8640	−0.432488	−0.216244	−	0.976339i	$-0.569381\pi$
$631$	−10.8640	−0.432488	−0.216244	+	0.976339i	$0.569381\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.19639	−0.364948
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.4400	−1.20231	−0.601153	−	0.799134i	$-0.705292\pi$
$641$	−30.4400	−1.20231	−0.601153	+	0.799134i	$0.705292\pi$
$642$	0	0
$643$	37.8640	1.49321	0.746605	−	0.665268i	$-0.231683\pi$
$643$	37.8640	1.49321	0.746605	+	0.665268i	$0.231683\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.1964	1.42303	0.711513	−	0.702672i	$-0.248010\pi$
$647$	36.1964	1.42303	0.711513	+	0.702672i	$0.248010\pi$
$648$	0	0
$649$	13.2509	0.520141
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.0975	0.864742	0.432371	−	0.901696i	$-0.357677\pi$
$653$	22.0975	0.864742	0.432371	+	0.901696i	$0.357677\pi$
$654$	0	0
$655$	14.5229	0.567457
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−44.0677	−1.71663	−0.858316	−	0.513121i	$-0.828489\pi$
$659$	−44.0677	−1.71663	−0.858316	+	0.513121i	$0.828489\pi$
$660$	0	0
$661$	−5.90702	−0.229757	−0.114878	−	0.993380i	$-0.536648\pi$
$661$	−5.90702	−0.229757	−0.114878	+	0.993380i	$0.536648\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.6835	−0.955750
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.6908	0.875969
$672$	0	0
$673$	−2.19777	−0.0847178	−0.0423589	−	0.999102i	$-0.513487\pi$
$673$	−2.19777	−0.0847178	−0.0423589	+	0.999102i	$0.513487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.0631	−1.15542	−0.577710	−	0.816242i	$-0.696053\pi$
$677$	−30.0631	−1.15542	−0.577710	+	0.816242i	$0.696053\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.29528	−0.317410	−0.158705	−	0.987326i	$-0.550732\pi$
$683$	−8.29528	−0.317410	−0.158705	+	0.987326i	$0.550732\pi$
$684$	0	0
$685$	−55.5475	−2.12236
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.4327	−1.23559
$690$	0	0
$691$	1.19639	0.0455129	0.0227564	−	0.999741i	$-0.492756\pi$
$691$	1.19639	0.0455129	0.0227564	+	0.999741i	$0.492756\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	74.1308	2.81194
$696$	0	0
$697$	−1.70197	−0.0644667
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.8813	−1.20414	−0.602070	−	0.798443i	$-0.705657\pi$
$701$	−31.8813	−1.20414	−0.602070	+	0.798443i	$0.705657\pi$
$702$	0	0
$703$	40.9257	1.54354
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.4327	−0.804921	−0.402461	−	0.915437i	$-0.631845\pi$
$709$	−21.4327	−0.804921	−0.402461	+	0.915437i	$0.631845\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.2967	0.797566
$714$	0	0
$715$	53.7293	2.00936
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.2609	1.61336	0.806680	−	0.590989i	$-0.201262\pi$
$719$	43.2609	1.61336	0.806680	+	0.590989i	$0.201262\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	51.0159	1.89468
$726$	0	0
$727$	−12.1818	−0.451799	−0.225899	−	0.974151i	$-0.572532\pi$
$727$	−12.1818	−0.451799	−0.225899	+	0.974151i	$0.572532\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.10755	−0.262882
$732$	0	0
$733$	−13.4486	−0.496736	−0.248368	−	0.968666i	$-0.579894\pi$
$733$	−13.4486	−0.496736	−0.248368	+	0.968666i	$0.579894\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.4254	0.494531
$738$	0	0
$739$	−9.87264	−0.363171	−0.181585	−	0.983375i	$-0.558123\pi$
$739$	−9.87264	−0.363171	−0.181585	+	0.983375i	$0.558123\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.5687	−0.827965	−0.413983	−	0.910285i	$-0.635863\pi$
$743$	−22.5687	−0.827965	−0.413983	+	0.910285i	$0.635863\pi$
$744$	0	0
$745$	−6.72658	−0.246443
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.575984	−0.0210180	−0.0105090	−	0.999945i	$-0.503345\pi$
$751$	−0.575984	−0.0210180	−0.0105090	+	0.999945i	$0.503345\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.94692	−0.216430
$756$	0	0
$757$	22.8196	0.829391	0.414695	−	0.909960i	$-0.363888\pi$
$757$	22.8196	0.829391	0.414695	+	0.909960i	$0.363888\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.7810	−0.717062	−0.358531	−	0.933518i	$-0.616722\pi$
$761$	−19.7810	−0.717062	−0.358531	+	0.933518i	$0.616722\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.4327	0.412810
$768$	0	0
$769$	28.7207	1.03569	0.517847	−	0.855473i	$-0.326734\pi$
$769$	28.7207	1.03569	0.517847	+	0.855473i	$0.326734\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.20505	−0.295115	−0.147558	−	0.989053i	$-0.547141\pi$
$773$	−8.20505	−0.295115	−0.147558	+	0.989053i	$0.547141\pi$
$774$	0	0
$775$	−44.0159	−1.58110
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.8997	0.533839
$780$	0	0
$781$	−23.0677	−0.825425
$782$	0	0
$783$	0	0
$784$	0	0
$785$	72.3126	2.58095
$786$	0	0
$787$	52.6364	1.87628	0.938142	−	0.346252i	$-0.112546\pi$
$787$	52.6364	1.87628	0.938142	+	0.346252i	$0.112546\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	19.5774	0.695212
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.5329	0.621049	0.310524	−	0.950565i	$-0.399495\pi$
$797$	17.5329	0.621049	0.310524	+	0.950565i	$0.399495\pi$
$798$	0	0
$799$	−2.70335	−0.0956375
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−39.5833	−1.39686
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.15059	−0.251401	−0.125701	−	0.992068i	$-0.540118\pi$
$809$	−7.15059	−0.251401	−0.125701	+	0.992068i	$0.540118\pi$
$810$	0	0
$811$	31.6835	1.11256	0.556280	−	0.830995i	$-0.312228\pi$
$811$	31.6835	1.11256	0.556280	+	0.830995i	$0.312228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	74.1679	2.59799
$816$	0	0
$817$	62.2224	2.17689
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.0690	−0.525913	−0.262956	−	0.964808i	$-0.584698\pi$
$821$	−15.0690	−0.525913	−0.262956	+	0.964808i	$0.584698\pi$
$822$	0	0
$823$	−20.9556	−0.730465	−0.365233	−	0.930916i	$-0.619011\pi$
$823$	−20.9556	−0.730465	−0.365233	+	0.930916i	$0.619011\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.7897	1.27930	0.639652	−	0.768665i	$-0.279079\pi$
$827$	36.7897	1.27930	0.639652	+	0.768665i	$0.279079\pi$
$828$	0	0
$829$	−46.3053	−1.60825	−0.804125	−	0.594460i	$-0.797366\pi$
$829$	−46.3053	−1.60825	−0.804125	+	0.594460i	$0.797366\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	59.8414	2.07090
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.6377	0.885113	0.442556	−	0.896741i	$-0.354072\pi$
$839$	25.6377	0.885113	0.442556	+	0.896741i	$0.354072\pi$
$840$	0	0
$841$	−14.4771	−0.499210
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.38688	−0.322918
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	37.3411	1.28004
$852$	0	0
$853$	5.43130	0.185964	0.0929821	−	0.995668i	$-0.470360\pi$
$853$	5.43130	0.185964	0.0929821	+	0.995668i	$0.470360\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.23491	−0.247140	−0.123570	−	0.992336i	$-0.539434\pi$
$857$	−7.23491	−0.247140	−0.123570	+	0.992336i	$0.539434\pi$
$858$	0	0
$859$	5.43130	0.185314	0.0926568	−	0.995698i	$-0.470464\pi$
$859$	5.43130	0.185314	0.0926568	+	0.995698i	$0.470464\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.7120	0.364641	0.182320	−	0.983239i	$-0.441639\pi$
$863$	10.7120	0.364641	0.182320	+	0.983239i	$0.441639\pi$
$864$	0	0
$865$	−39.2064	−1.33306
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.6058	0.665083
$870$	0	0
$871$	11.5833	0.392484
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.8168	−0.466560	−0.233280	−	0.972410i	$-0.574946\pi$
$877$	−13.8168	−0.466560	−0.233280	+	0.972410i	$0.574946\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.6662	−0.797335	−0.398667	−	0.917096i	$-0.630527\pi$
$881$	−23.6662	−0.797335	−0.398667	+	0.917096i	$0.630527\pi$
$882$	0	0
$883$	18.9615	0.638105	0.319052	−	0.947737i	$-0.396635\pi$
$883$	18.9615	0.638105	0.319052	+	0.947737i	$0.396635\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.3955	−0.684815	−0.342408	−	0.939552i	$-0.611242\pi$
$887$	−20.3955	−0.684815	−0.342408	+	0.939552i	$0.611242\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	23.6662	0.791959
$894$	0	0
$895$	73.1606	2.44549
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.5302	−0.417905
$900$	0	0
$901$	7.99862	0.266473
$902$	0	0
$903$	0	0
$904$	0	0
$905$	70.4944	2.34331
$906$	0	0
$907$	40.0146	1.32866	0.664331	−	0.747439i	$-0.268717\pi$
$907$	40.0146	1.32866	0.664331	+	0.747439i	$0.268717\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.8268	0.424972	0.212486	−	0.977164i	$-0.431844\pi$
$911$	12.8268	0.424972	0.212486	+	0.977164i	$0.431844\pi$
$912$	0	0
$913$	−6.52428	−0.215922
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−43.9542	−1.44992	−0.724958	−	0.688793i	$-0.758141\pi$
$919$	−43.9542	−1.44992	−0.724958	+	0.688793i	$0.758141\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−19.9025	−0.655099
$924$	0	0
$925$	−77.1766	−2.53755
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.2349	1.32006	0.660032	−	0.751237i	$-0.270543\pi$
$929$	40.2349	1.32006	0.660032	+	0.751237i	$0.270543\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.2509	−0.433349
$936$	0	0
$937$	3.06175	0.100023	0.0500114	−	0.998749i	$-0.484074\pi$
$937$	3.06175	0.100023	0.0500114	+	0.998749i	$0.484074\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.1891	0.658146	0.329073	−	0.944304i	$-0.393264\pi$
$941$	20.1891	0.658146	0.329073	+	0.944304i	$0.393264\pi$
$942$	0	0
$943$	13.5947	0.442704
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.14468	0.0371973	0.0185986	−	0.999827i	$-0.494080\pi$
$947$	1.14468	0.0371973	0.0185986	+	0.999827i	$0.494080\pi$
$948$	0	0
$949$	−34.1520	−1.10862
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.9701	−1.39194	−0.695970	−	0.718071i	$-0.745025\pi$
$953$	−42.9701	−1.39194	−0.695970	+	0.718071i	$0.745025\pi$
$954$	0	0
$955$	12.7047	0.411115
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.1891	−0.651262
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−74.7453	−2.40614
$966$	0	0
$967$	9.99862	0.321534	0.160767	−	0.986992i	$-0.448603\pi$
$967$	9.99862	0.321534	0.160767	+	0.986992i	$0.448603\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.6291	1.04712	0.523558	−	0.851990i	$-0.324604\pi$
$971$	32.6291	1.04712	0.523558	+	0.851990i	$0.324604\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.4958	−1.32757	−0.663784	−	0.747924i	$-0.731051\pi$
$977$	−41.4958	−1.32757	−0.663784	+	0.747924i	$0.731051\pi$
$978$	0	0
$979$	47.7512	1.52613
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.4327	1.13013	0.565063	−	0.825047i	$-0.308852\pi$
$983$	35.4327	1.13013	0.565063	+	0.825047i	$0.308852\pi$
$984$	0	0
$985$	44.1148	1.40562
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.7724	1.80526
$990$	0	0
$991$	−48.5919	−1.54357	−0.771787	−	0.635881i	$-0.780637\pi$
$991$	−48.5919	−1.54357	−0.771787	+	0.635881i	$0.780637\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.0159	0.951569
$996$	0	0
$997$	−47.0260	−1.48933	−0.744664	−	0.667440i	$-0.767390\pi$
$997$	−47.0260	−1.48933	−0.744664	+	0.667440i	$0.767390\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.w.1.3		3
3.2	odd	2		5292.2.a.v.1.1		3
7.3	odd	6		756.2.k.f.541.3	yes	6
7.5	odd	6		756.2.k.f.109.3	yes	6
7.6	odd	2		5292.2.a.u.1.1		3
21.5	even	6		756.2.k.e.109.1	✓	6
21.17	even	6		756.2.k.e.541.1	yes	6
21.20	even	2		5292.2.a.x.1.3		3
63.5	even	6		2268.2.i.j.865.1		6
63.31	odd	6		2268.2.l.j.541.1		6
63.38	even	6		2268.2.i.j.2053.1		6
63.40	odd	6		2268.2.i.k.865.3		6
63.47	even	6		2268.2.l.k.109.3		6
63.52	odd	6		2268.2.i.k.2053.3		6
63.59	even	6		2268.2.l.k.541.3		6
63.61	odd	6		2268.2.l.j.109.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.e.109.1	✓	6	21.5	even	6
756.2.k.e.541.1	yes	6	21.17	even	6
756.2.k.f.109.3	yes	6	7.5	odd	6
756.2.k.f.541.3	yes	6	7.3	odd	6
2268.2.i.j.865.1		6	63.5	even	6
2268.2.i.j.2053.1		6	63.38	even	6
2268.2.i.k.865.3		6	63.40	odd	6
2268.2.i.k.2053.3		6	63.52	odd	6
2268.2.l.j.109.1		6	63.61	odd	6
2268.2.l.j.541.1		6	63.31	odd	6
2268.2.l.k.109.3		6	63.47	even	6
2268.2.l.k.541.3		6	63.59	even	6
5292.2.a.u.1.1		3	7.6	odd	2
5292.2.a.v.1.1		3	3.2	odd	2
5292.2.a.w.1.3		3	1.1	even	1	trivial
5292.2.a.x.1.3		3	21.20	even	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q+4.28799 q^{5} +O(q^{10})\)	\(q+4.28799 q^{5} -3.81089 q^{11} -3.28799 q^{13} +0.810892 q^{17} -7.09888 q^{19} -6.47710 q^{23} +13.3869 q^{25} +3.81089 q^{29} -3.28799 q^{31} -5.76509 q^{37} -2.09888 q^{41} -8.76509 q^{43} -3.33379 q^{47} +9.86398 q^{53} -16.3411 q^{55} -3.47710 q^{59} -5.95420 q^{61} -14.0989 q^{65} -3.52290 q^{67} +6.05308 q^{71} +10.3869 q^{73} -5.14468 q^{79} +1.71201 q^{83} +3.47710 q^{85} -12.5302 q^{89} -30.4400 q^{95} -1.04580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5}+O(q^{10}) \) 3 * q + q^5	\( 3 q + q^{5} - 5 q^{11} + 2 q^{13} - 4 q^{17} - 3 q^{19} - 14 q^{23} + 10 q^{25} + 5 q^{29} + 2 q^{31} + 12 q^{41} - 9 q^{43} - 9 q^{47} - 6 q^{53} - 8 q^{55} - 5 q^{59} - 7 q^{61} - 24 q^{65} - 16 q^{67} - 11 q^{71} + q^{73} - 8 q^{79} + 17 q^{83} + 5 q^{85} - 3 q^{89} - 32 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q + q^5 - 5 * q^11 + 2 * q^13 - 4 * q^17 - 3 * q^19 - 14 * q^23 + 10 * q^25 + 5 * q^29 + 2 * q^31 + 12 * q^41 - 9 * q^43 - 9 * q^47 - 6 * q^53 - 8 * q^55 - 5 * q^59 - 7 * q^61 - 24 * q^65 - 16 * q^67 - 11 * q^71 + q^73 - 8 * q^79 + 17 * q^83 + 5 * q^85 - 3 * q^89 - 32 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more