LMFDB - Embedded newform 3420.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.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:	$27.3088374913$
Analytic rank:	$0$
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:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.60555 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.60555 q^{7} +4.60555 q^{11} +4.60555 q^{13} -2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -2.60555 q^{29} +4.00000 q^{31} -2.60555 q^{35} +3.39445 q^{37} -6.60555 q^{41} +10.6056 q^{43} -6.00000 q^{47} -0.211103 q^{49} +4.60555 q^{55} -5.21110 q^{59} -7.21110 q^{61} +4.60555 q^{65} +4.00000 q^{67} +9.21110 q^{71} +6.00000 q^{73} -12.0000 q^{77} +8.00000 q^{79} +11.2111 q^{83} -2.00000 q^{85} -6.60555 q^{89} -12.0000 q^{91} -1.00000 q^{95} +16.6056 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^35 + 14 * q^37 - 6 * q^41 + 14 * q^43 - 12 * q^47 + 14 * q^49 + 2 * q^55 + 4 * q^59 + 2 * q^65 + 8 * q^67 + 4 * q^71 + 12 * q^73 - 24 * q^77 + 16 * q^79 + 8 * q^83 - 4 * q^85 - 6 * q^89 - 24 * q^91 - 2 * q^95 + 26 * 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
$7$	−2.60555	−0.984806	−0.492403	−	0.870367i	$-0.663881\pi$
$7$	−2.60555	−0.984806	−0.492403	+	0.870367i	$0.663881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.60555	1.38863	0.694313	−	0.719673i	$-0.255708\pi$
$11$	4.60555	1.38863	0.694313	+	0.719673i	$0.255708\pi$
$12$	0	0
$13$	4.60555	1.27735	0.638675	−	0.769477i	$-0.279483\pi$
$13$	4.60555	1.27735	0.638675	+	0.769477i	$0.279483\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.60555	−0.483839	−0.241919	−	0.970296i	$-0.577777\pi$
$29$	−2.60555	−0.483839	−0.241919	+	0.970296i	$0.577777\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.60555	−0.440419
$36$	0	0
$37$	3.39445	0.558044	0.279022	−	0.960285i	$-0.409990\pi$
$37$	3.39445	0.558044	0.279022	+	0.960285i	$0.409990\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.60555	−1.03161	−0.515807	−	0.856705i	$-0.672508\pi$
$41$	−6.60555	−1.03161	−0.515807	+	0.856705i	$0.672508\pi$
$42$	0	0
$43$	10.6056	1.61733	0.808666	−	0.588268i	$-0.200190\pi$
$43$	10.6056	1.61733	0.808666	+	0.588268i	$0.200190\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	4.60555	0.621012
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	−7.21110	−0.923287	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	−7.21110	−0.923287	−0.461644	+	0.887066i	$0.652740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.60555	0.571248
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.21110	1.09316	0.546578	−	0.837408i	$-0.315930\pi$
$71$	9.21110	1.09316	0.546578	+	0.837408i	$0.315930\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2111	1.23058	0.615289	−	0.788301i	$-0.289039\pi$
$83$	11.2111	1.23058	0.615289	+	0.788301i	$0.289039\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.60555	−0.700187	−0.350094	−	0.936715i	$-0.613850\pi$
$89$	−6.60555	−0.700187	−0.350094	+	0.936715i	$0.613850\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	16.6056	1.68604	0.843019	−	0.537884i	$-0.180776\pi$
$97$	16.6056	1.68604	0.843019	+	0.537884i	$0.180776\pi$
$98$	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
$103$	18.4222	1.81519	0.907597	−	0.419843i	$-0.137915\pi$
$103$	18.4222	1.81519	0.907597	+	0.419843i	$0.137915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.4222	−1.78094	−0.890471	−	0.455040i	$-0.849625\pi$
$107$	−18.4222	−1.78094	−0.890471	+	0.455040i	$0.849625\pi$
$108$	0	0
$109$	−11.2111	−1.07383	−0.536914	−	0.843637i	$-0.680410\pi$
$109$	−11.2111	−1.07383	−0.536914	+	0.843637i	$0.680410\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.21110	0.113931	0.0569655	−	0.998376i	$-0.481858\pi$
$113$	1.21110	0.113931	0.0569655	+	0.998376i	$0.481858\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.21110	0.477701
$120$	0	0
$121$	10.2111	0.928282
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.4222	1.63471	0.817353	−	0.576137i	$-0.195441\pi$
$127$	18.4222	1.63471	0.817353	+	0.576137i	$0.195441\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3944	1.34502	0.672510	−	0.740088i	$-0.265216\pi$
$131$	15.3944	1.34502	0.672510	+	0.740088i	$0.265216\pi$
$132$	0	0
$133$	2.60555	0.225930
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.4222	1.06130	0.530650	−	0.847591i	$-0.321948\pi$
$137$	12.4222	1.06130	0.530650	+	0.847591i	$0.321948\pi$
$138$	0	0
$139$	9.21110	0.781276	0.390638	−	0.920544i	$-0.372255\pi$
$139$	9.21110	0.781276	0.390638	+	0.920544i	$0.372255\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.2111	1.77376
$144$	0	0
$145$	−2.60555	−0.216379
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.4222	−1.34536	−0.672680	−	0.739934i	$-0.734857\pi$
$149$	−16.4222	−1.34536	−0.672680	+	0.739934i	$0.734857\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	15.2111	1.21398	0.606989	−	0.794710i	$-0.292377\pi$
$157$	15.2111	1.21398	0.606989	+	0.794710i	$0.292377\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.21110	−0.410692
$162$	0	0
$163$	−17.0278	−1.33372	−0.666858	−	0.745184i	$-0.732361\pi$
$163$	−17.0278	−1.33372	−0.666858	+	0.745184i	$0.732361\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.78890	0.525341	0.262670	−	0.964886i	$-0.415397\pi$
$167$	6.78890	0.525341	0.262670	+	0.964886i	$0.415397\pi$
$168$	0	0
$169$	8.21110	0.631623
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.21110	0.700307	0.350154	−	0.936692i	$-0.386129\pi$
$173$	9.21110	0.700307	0.350154	+	0.936692i	$0.386129\pi$
$174$	0	0
$175$	−2.60555	−0.196961
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.2111	1.58539	0.792696	−	0.609617i	$-0.208677\pi$
$179$	21.2111	1.58539	0.792696	+	0.609617i	$0.208677\pi$
$180$	0	0
$181$	−0.788897	−0.0586383	−0.0293191	−	0.999570i	$-0.509334\pi$
$181$	−0.788897	−0.0586383	−0.0293191	+	0.999570i	$0.509334\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.39445	0.249565
$186$	0	0
$187$	−9.21110	−0.673583
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.81665	0.420878	0.210439	−	0.977607i	$-0.432511\pi$
$191$	5.81665	0.420878	0.210439	+	0.977607i	$0.432511\pi$
$192$	0	0
$193$	−0.605551	−0.0435885	−0.0217943	−	0.999762i	$-0.506938\pi$
$193$	−0.605551	−0.0435885	−0.0217943	+	0.999762i	$0.506938\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	17.2111	1.22006	0.610031	−	0.792377i	$-0.291157\pi$
$199$	17.2111	1.22006	0.610031	+	0.792377i	$0.291157\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.78890	0.476487
$204$	0	0
$205$	−6.60555	−0.461352
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.60555	−0.318573
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.6056	0.723293
$216$	0	0
$217$	−10.4222	−0.707505
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.21110	−0.619606
$222$	0	0
$223$	−10.4222	−0.697922	−0.348961	−	0.937137i	$-0.613466\pi$
$223$	−10.4222	−0.697922	−0.348961	+	0.937137i	$0.613466\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.2111	0.876852	0.438426	−	0.898767i	$-0.355536\pi$
$227$	13.2111	0.876852	0.438426	+	0.898767i	$0.355536\pi$
$228$	0	0
$229$	−3.21110	−0.212196	−0.106098	−	0.994356i	$-0.533836\pi$
$229$	−3.21110	−0.212196	−0.106098	+	0.994356i	$0.533836\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.4222	1.07585	0.537927	−	0.842991i	$-0.319208\pi$
$233$	16.4222	1.07585	0.537927	+	0.842991i	$0.319208\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.8167	−1.66994	−0.834970	−	0.550295i	$-0.814515\pi$
$239$	−25.8167	−1.66994	−0.834970	+	0.550295i	$0.814515\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.211103	−0.0134868
$246$	0	0
$247$	−4.60555	−0.293044
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.18335	0.390289	0.195145	−	0.980774i	$-0.437482\pi$
$251$	6.18335	0.390289	0.195145	+	0.980774i	$0.437482\pi$
$252$	0	0
$253$	9.21110	0.579097
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.78890	−0.173967	−0.0869833	−	0.996210i	$-0.527723\pi$
$257$	−2.78890	−0.173967	−0.0869833	+	0.996210i	$0.527723\pi$
$258$	0	0
$259$	−8.84441	−0.549565
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.21110	−0.444656	−0.222328	−	0.974972i	$-0.571366\pi$
$263$	−7.21110	−0.444656	−0.222328	+	0.974972i	$0.571366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.3944	1.06056	0.530279	−	0.847823i	$-0.322087\pi$
$269$	17.3944	1.06056	0.530279	+	0.847823i	$0.322087\pi$
$270$	0	0
$271$	11.6333	0.706673	0.353337	−	0.935496i	$-0.385047\pi$
$271$	11.6333	0.706673	0.353337	+	0.935496i	$0.385047\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.60555	0.277725
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.2389	−1.08804	−0.544020	−	0.839073i	$-0.683098\pi$
$281$	−18.2389	−1.08804	−0.544020	+	0.839073i	$0.683098\pi$
$282$	0	0
$283$	−18.6056	−1.10599	−0.552993	−	0.833186i	$-0.686514\pi$
$283$	−18.6056	−1.10599	−0.552993	+	0.833186i	$0.686514\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.2111	1.01594
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.6333	−1.14699	−0.573495	−	0.819209i	$-0.694413\pi$
$293$	−19.6333	−1.14699	−0.573495	+	0.819209i	$0.694413\pi$
$294$	0	0
$295$	−5.21110	−0.303402
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.21110	0.532692
$300$	0	0
$301$	−27.6333	−1.59276
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.21110	−0.412907
$306$	0	0
$307$	−11.6333	−0.663948	−0.331974	−	0.943289i	$-0.607715\pi$
$307$	−11.6333	−0.663948	−0.331974	+	0.943289i	$0.607715\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.6056	−0.941614	−0.470807	−	0.882236i	$-0.656037\pi$
$311$	−16.6056	−0.941614	−0.470807	+	0.882236i	$0.656037\pi$
$312$	0	0
$313$	0.788897	0.0445911	0.0222956	−	0.999751i	$-0.492903\pi$
$313$	0.788897	0.0445911	0.0222956	+	0.999751i	$0.492903\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.42221	0.360707	0.180353	−	0.983602i	$-0.442276\pi$
$317$	6.42221	0.360707	0.180353	+	0.983602i	$0.442276\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	4.60555	0.255470
$326$	0	0
$327$	0	0
$328$	0	0
$329$	15.6333	0.861892
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	3.02776	0.164932	0.0824662	−	0.996594i	$-0.473720\pi$
$337$	3.02776	0.164932	0.0824662	+	0.996594i	$0.473720\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.4222	0.997618
$342$	0	0
$343$	18.7889	1.01451
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.6333	−1.16134	−0.580668	−	0.814140i	$-0.697209\pi$
$347$	−21.6333	−1.16134	−0.580668	+	0.814140i	$0.697209\pi$
$348$	0	0
$349$	34.8444	1.86518	0.932589	−	0.360939i	$-0.117544\pi$
$349$	34.8444	1.86518	0.932589	+	0.360939i	$0.117544\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.422205	−0.0224717	−0.0112359	−	0.999937i	$-0.503577\pi$
$353$	−0.422205	−0.0224717	−0.0112359	+	0.999937i	$0.503577\pi$
$354$	0	0
$355$	9.21110	0.488875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.8167	0.729215	0.364608	−	0.931161i	$-0.381203\pi$
$359$	13.8167	0.729215	0.364608	+	0.931161i	$0.381203\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	25.0278	1.30644	0.653219	−	0.757169i	$-0.273418\pi$
$367$	25.0278	1.30644	0.653219	+	0.757169i	$0.273418\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.6056	0.859803	0.429901	−	0.902876i	$-0.358548\pi$
$373$	16.6056	0.859803	0.429901	+	0.902876i	$0.358548\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	26.4222	1.35722	0.678609	−	0.734500i	$-0.262583\pi$
$379$	26.4222	1.35722	0.678609	+	0.734500i	$0.262583\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−36.4222	−1.84668	−0.923340	−	0.383984i	$-0.874552\pi$
$389$	−36.4222	−1.84668	−0.923340	+	0.383984i	$0.874552\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	−7.57779	−0.380319	−0.190159	−	0.981753i	$-0.560900\pi$
$397$	−7.57779	−0.380319	−0.190159	+	0.981753i	$0.560900\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.3944	−1.26814	−0.634069	−	0.773276i	$-0.718617\pi$
$401$	−25.3944	−1.26814	−0.634069	+	0.773276i	$0.718617\pi$
$402$	0	0
$403$	18.4222	0.917675
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.6333	0.774914
$408$	0	0
$409$	−12.7889	−0.632370	−0.316185	−	0.948698i	$-0.602402\pi$
$409$	−12.7889	−0.632370	−0.316185	+	0.948698i	$0.602402\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.5778	0.668120
$414$	0	0
$415$	11.2111	0.550331
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.02776	0.343328	0.171664	−	0.985156i	$-0.445086\pi$
$419$	7.02776	0.343328	0.171664	+	0.985156i	$0.445086\pi$
$420$	0	0
$421$	−8.42221	−0.410473	−0.205237	−	0.978712i	$-0.565796\pi$
$421$	−8.42221	−0.410473	−0.205237	+	0.978712i	$0.565796\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	18.7889	0.909258
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.21110	0.443683	0.221842	−	0.975083i	$-0.428793\pi$
$431$	9.21110	0.443683	0.221842	+	0.975083i	$0.428793\pi$
$432$	0	0
$433$	−16.6056	−0.798012	−0.399006	−	0.916948i	$-0.630645\pi$
$433$	−16.6056	−0.798012	−0.399006	+	0.916948i	$0.630645\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	34.4222	1.64288	0.821441	−	0.570293i	$-0.193170\pi$
$439$	34.4222	1.64288	0.821441	+	0.570293i	$0.193170\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.8444	−0.895325	−0.447662	−	0.894203i	$-0.647743\pi$
$443$	−18.8444	−0.895325	−0.447662	+	0.894203i	$0.647743\pi$
$444$	0	0
$445$	−6.60555	−0.313133
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.2389	0.860745	0.430372	−	0.902651i	$-0.358382\pi$
$449$	18.2389	0.860745	0.430372	+	0.902651i	$0.358382\pi$
$450$	0	0
$451$	−30.4222	−1.43253
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	32.0555	1.49949	0.749747	−	0.661725i	$-0.230175\pi$
$457$	32.0555	1.49949	0.749747	+	0.661725i	$0.230175\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.8444	1.06397	0.531985	−	0.846754i	$-0.321446\pi$
$461$	22.8444	1.06397	0.531985	+	0.846754i	$0.321446\pi$
$462$	0	0
$463$	33.0278	1.53493	0.767465	−	0.641091i	$-0.221518\pi$
$463$	33.0278	1.53493	0.767465	+	0.641091i	$0.221518\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.8444	−1.05711	−0.528557	−	0.848898i	$-0.677267\pi$
$467$	−22.8444	−1.05711	−0.528557	+	0.848898i	$0.677267\pi$
$468$	0	0
$469$	−10.4222	−0.481253
$470$	0	0
$471$	0	0
$472$	0	0
$473$	48.8444	2.24587
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.6056	−1.30702	−0.653510	−	0.756917i	$-0.726704\pi$
$479$	−28.6056	−1.30702	−0.653510	+	0.756917i	$0.726704\pi$
$480$	0	0
$481$	15.6333	0.712817
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.6056	0.754019
$486$	0	0
$487$	0.366692	0.0166164	0.00830821	−	0.999965i	$-0.497355\pi$
$487$	0.366692	0.0166164	0.00830821	+	0.999965i	$0.497355\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.2389	0.552332	0.276166	−	0.961110i	$-0.410936\pi$
$491$	12.2389	0.552332	0.276166	+	0.961110i	$0.410936\pi$
$492$	0	0
$493$	5.21110	0.234696
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−27.6333	−1.23704	−0.618518	−	0.785770i	$-0.712267\pi$
$499$	−27.6333	−1.23704	−0.618518	+	0.785770i	$0.712267\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.7889	−0.570229	−0.285114	−	0.958494i	$-0.592032\pi$
$503$	−12.7889	−0.570229	−0.285114	+	0.958494i	$0.592032\pi$
$504$	0	0
$505$	7.21110	0.320890
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.4500	1.57129	0.785646	−	0.618676i	$-0.212331\pi$
$509$	35.4500	1.57129	0.785646	+	0.618676i	$0.212331\pi$
$510$	0	0
$511$	−15.6333	−0.691577
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.4222	0.811779
$516$	0	0
$517$	−27.6333	−1.21531
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.18335	0.183276	0.0916379	−	0.995792i	$-0.470790\pi$
$521$	4.18335	0.183276	0.0916379	+	0.995792i	$0.470790\pi$
$522$	0	0
$523$	−25.2111	−1.10240	−0.551202	−	0.834372i	$-0.685831\pi$
$523$	−25.2111	−1.10240	−0.551202	+	0.834372i	$0.685831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.4222	−1.31773
$534$	0	0
$535$	−18.4222	−0.796461
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.972244	−0.0418775
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.2111	−0.480231
$546$	0	0
$547$	−22.7889	−0.974383	−0.487191	−	0.873295i	$-0.661979\pi$
$547$	−22.7889	−0.974383	−0.487191	+	0.873295i	$0.661979\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.60555	0.111000
$552$	0	0
$553$	−20.8444	−0.886394
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.42221	0.187375	0.0936874	−	0.995602i	$-0.470135\pi$
$557$	4.42221	0.187375	0.0936874	+	0.995602i	$0.470135\pi$
$558$	0	0
$559$	48.8444	2.06590
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.6333	−1.33318	−0.666592	−	0.745422i	$-0.732248\pi$
$563$	−31.6333	−1.33318	−0.666592	+	0.745422i	$0.732248\pi$
$564$	0	0
$565$	1.21110	0.0509515
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.4500	−1.65383	−0.826914	−	0.562328i	$-0.809906\pi$
$569$	−39.4500	−1.65383	−0.826914	+	0.562328i	$0.809906\pi$
$570$	0	0
$571$	7.63331	0.319444	0.159722	−	0.987162i	$-0.448940\pi$
$571$	7.63331	0.319444	0.159722	+	0.987162i	$0.448940\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.00000	0.0834058
$576$	0	0
$577$	−23.2111	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$577$	−23.2111	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−29.2111	−1.21188
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.4222	−0.512719	−0.256360	−	0.966581i	$-0.582523\pi$
$587$	−12.4222	−0.512719	−0.256360	+	0.966581i	$0.582523\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.57779	0.146922	0.0734612	−	0.997298i	$-0.476595\pi$
$593$	3.57779	0.146922	0.0734612	+	0.997298i	$0.476595\pi$
$594$	0	0
$595$	5.21110	0.213634
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.8444	−1.50542	−0.752711	−	0.658351i	$-0.771254\pi$
$599$	−36.8444	−1.50542	−0.752711	+	0.658351i	$0.771254\pi$
$600$	0	0
$601$	4.78890	0.195343	0.0976716	−	0.995219i	$-0.468861\pi$
$601$	4.78890	0.195343	0.0976716	+	0.995219i	$0.468861\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.2111	0.415140
$606$	0	0
$607$	−23.6333	−0.959246	−0.479623	−	0.877475i	$-0.659227\pi$
$607$	−23.6333	−0.959246	−0.479623	+	0.877475i	$0.659227\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27.6333	−1.11792
$612$	0	0
$613$	4.78890	0.193422	0.0967109	−	0.995313i	$-0.469168\pi$
$613$	4.78890	0.193422	0.0967109	+	0.995313i	$0.469168\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	−4.36669	−0.175512	−0.0877561	−	0.996142i	$-0.527970\pi$
$619$	−4.36669	−0.175512	−0.0877561	+	0.996142i	$0.527970\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.2111	0.689548
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.78890	−0.270691
$630$	0	0
$631$	36.8444	1.46675	0.733376	−	0.679823i	$-0.237943\pi$
$631$	36.8444	1.46675	0.733376	+	0.679823i	$0.237943\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.4222	0.731063
$636$	0	0
$637$	−0.972244	−0.0385217
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.4500	−1.40019	−0.700095	−	0.714050i	$-0.746859\pi$
$641$	−35.4500	−1.40019	−0.700095	+	0.714050i	$0.746859\pi$
$642$	0	0
$643$	5.39445	0.212736	0.106368	−	0.994327i	$-0.466078\pi$
$643$	5.39445	0.212736	0.106368	+	0.994327i	$0.466078\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.36669	0.0930443	0.0465221	−	0.998917i	$-0.485186\pi$
$647$	2.36669	0.0930443	0.0465221	+	0.998917i	$0.485186\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	15.3944	0.601511
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−31.2111	−1.21397	−0.606986	−	0.794713i	$-0.707621\pi$
$661$	−31.2111	−1.21397	−0.606986	+	0.794713i	$0.707621\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.60555	0.101039
$666$	0	0
$667$	−5.21110	−0.201775
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.2111	−1.28210
$672$	0	0
$673$	36.6056	1.41104	0.705520	−	0.708690i	$-0.250713\pi$
$673$	36.6056	1.41104	0.705520	+	0.708690i	$0.250713\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.4222	−0.400558	−0.200279	−	0.979739i	$-0.564185\pi$
$677$	−10.4222	−0.400558	−0.200279	+	0.979739i	$0.564185\pi$
$678$	0	0
$679$	−43.2666	−1.66042
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	12.4222	0.474628
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−42.0555	−1.59987	−0.799934	−	0.600089i	$-0.795132\pi$
$691$	−42.0555	−1.59987	−0.799934	+	0.600089i	$0.795132\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.21110	0.349397
$696$	0	0
$697$	13.2111	0.500406
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−3.39445	−0.128024
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.7889	−0.706629
$708$	0	0
$709$	−25.6333	−0.962679	−0.481340	−	0.876534i	$-0.659850\pi$
$709$	−25.6333	−0.962679	−0.481340	+	0.876534i	$0.659850\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	21.2111	0.793250
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.2389	−0.605607	−0.302804	−	0.953053i	$-0.597923\pi$
$719$	−16.2389	−0.605607	−0.302804	+	0.953053i	$0.597923\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.60555	−0.0967677
$726$	0	0
$727$	−17.3944	−0.645124	−0.322562	−	0.946548i	$-0.604544\pi$
$727$	−17.3944	−0.645124	−0.322562	+	0.946548i	$0.604544\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.2111	−0.784521
$732$	0	0
$733$	−24.4222	−0.902055	−0.451027	−	0.892510i	$-0.648942\pi$
$733$	−24.4222	−0.902055	−0.451027	+	0.892510i	$0.648942\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.4222	0.678591
$738$	0	0
$739$	38.4222	1.41338	0.706692	−	0.707521i	$-0.250187\pi$
$739$	38.4222	1.41338	0.706692	+	0.707521i	$0.250187\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	−16.4222	−0.601663
$746$	0	0
$747$	0	0
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	40.8444	1.49043	0.745217	−	0.666822i	$-0.232346\pi$
$751$	40.8444	1.49043	0.745217	+	0.666822i	$0.232346\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−36.0555	−1.31046	−0.655230	−	0.755429i	$-0.727428\pi$
$757$	−36.0555	−1.31046	−0.655230	+	0.755429i	$0.727428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	29.2111	1.05751
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−20.4222	−0.736444	−0.368222	−	0.929738i	$-0.620033\pi$
$769$	−20.4222	−0.736444	−0.368222	+	0.929738i	$0.620033\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.2666	1.55619	0.778096	−	0.628145i	$-0.216186\pi$
$773$	43.2666	1.55619	0.778096	+	0.628145i	$0.216186\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.60555	0.236668
$780$	0	0
$781$	42.4222	1.51799
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.2111	0.542908
$786$	0	0
$787$	−19.6333	−0.699852	−0.349926	−	0.936777i	$-0.613793\pi$
$787$	−19.6333	−0.699852	−0.349926	+	0.936777i	$0.613793\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.15559	−0.112200
$792$	0	0
$793$	−33.2111	−1.17936
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.2111	−0.609649	−0.304824	−	0.952409i	$-0.598598\pi$
$797$	−17.2111	−0.609649	−0.304824	+	0.952409i	$0.598598\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.6333	0.975158
$804$	0	0
$805$	−5.21110	−0.183667
$806$	0	0
$807$	0	0
$808$	0	0
$809$	48.4222	1.70243	0.851217	−	0.524814i	$-0.175865\pi$
$809$	48.4222	1.70243	0.851217	+	0.524814i	$0.175865\pi$
$810$	0	0
$811$	−40.8444	−1.43424	−0.717121	−	0.696949i	$-0.754540\pi$
$811$	−40.8444	−1.43424	−0.717121	+	0.696949i	$0.754540\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.0278	−0.596456
$816$	0	0
$817$	−10.6056	−0.371041
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.2111	−1.50808	−0.754039	−	0.656830i	$-0.771897\pi$
$821$	−43.2111	−1.50808	−0.754039	+	0.656830i	$0.771897\pi$
$822$	0	0
$823$	−47.8167	−1.66678	−0.833392	−	0.552683i	$-0.813604\pi$
$823$	−47.8167	−1.66678	−0.833392	+	0.552683i	$0.813604\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.6333	0.960904	0.480452	−	0.877021i	$-0.340473\pi$
$827$	27.6333	0.960904	0.480452	+	0.877021i	$0.340473\pi$
$828$	0	0
$829$	15.2111	0.528303	0.264152	−	0.964481i	$-0.414908\pi$
$829$	15.2111	0.528303	0.264152	+	0.964481i	$0.414908\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.422205	0.0146285
$834$	0	0
$835$	6.78890	0.234939
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.6333	−0.954008	−0.477004	−	0.878901i	$-0.658277\pi$
$839$	−27.6333	−0.954008	−0.477004	+	0.878901i	$0.658277\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.21110	0.282471
$846$	0	0
$847$	−26.6056	−0.914178
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.78890	0.232720
$852$	0	0
$853$	29.6333	1.01463	0.507313	−	0.861762i	$-0.330639\pi$
$853$	29.6333	1.01463	0.507313	+	0.861762i	$0.330639\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.42221	−0.219378	−0.109689	−	0.993966i	$-0.534986\pi$
$857$	−6.42221	−0.219378	−0.109689	+	0.993966i	$0.534986\pi$
$858$	0	0
$859$	8.84441	0.301767	0.150884	−	0.988552i	$-0.451788\pi$
$859$	8.84441	0.301767	0.150884	+	0.988552i	$0.451788\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.42221	−0.218614	−0.109307	−	0.994008i	$-0.534863\pi$
$863$	−6.42221	−0.218614	−0.109307	+	0.994008i	$0.534863\pi$
$864$	0	0
$865$	9.21110	0.313187
$866$	0	0
$867$	0	0
$868$	0	0
$869$	36.8444	1.24986
$870$	0	0
$871$	18.4222	0.624213
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.60555	−0.0880837
$876$	0	0
$877$	27.0278	0.912662	0.456331	−	0.889810i	$-0.349163\pi$
$877$	27.0278	0.912662	0.456331	+	0.889810i	$0.349163\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−59.2111	−1.99487	−0.997436	−	0.0715590i	$-0.977203\pi$
$881$	−59.2111	−1.99487	−0.997436	+	0.0715590i	$0.977203\pi$
$882$	0	0
$883$	27.8167	0.936105	0.468052	−	0.883701i	$-0.344956\pi$
$883$	27.8167	0.936105	0.468052	+	0.883701i	$0.344956\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.8444	0.699887	0.349943	−	0.936771i	$-0.386201\pi$
$887$	20.8444	0.699887	0.349943	+	0.936771i	$0.386201\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	21.2111	0.709009
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.4222	−0.347600
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.788897	−0.0262238
$906$	0	0
$907$	30.0555	0.997977	0.498988	−	0.866609i	$-0.333705\pi$
$907$	30.0555	0.997977	0.498988	+	0.866609i	$0.333705\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10.4222	−0.345303	−0.172652	−	0.984983i	$-0.555233\pi$
$911$	−10.4222	−0.345303	−0.172652	+	0.984983i	$0.555233\pi$
$912$	0	0
$913$	51.6333	1.70881
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.1110	−1.32458
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	42.4222	1.39634
$924$	0	0
$925$	3.39445	0.111609
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.6333	1.49718	0.748590	−	0.663033i	$-0.230731\pi$
$929$	45.6333	1.49718	0.748590	+	0.663033i	$0.230731\pi$
$930$	0	0
$931$	0.211103	0.00691861
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.21110	−0.301235
$936$	0	0
$937$	−19.2111	−0.627599	−0.313800	−	0.949489i	$-0.601602\pi$
$937$	−19.2111	−0.627599	−0.313800	+	0.949489i	$0.601602\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.2389	0.985759	0.492879	−	0.870098i	$-0.335944\pi$
$941$	30.2389	0.985759	0.492879	+	0.870098i	$0.335944\pi$
$942$	0	0
$943$	−13.2111	−0.430213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.21110	0.234329	0.117165	−	0.993113i	$-0.462619\pi$
$947$	7.21110	0.234329	0.117165	+	0.993113i	$0.462619\pi$
$948$	0	0
$949$	27.6333	0.897015
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.21110	−0.168804	−0.0844021	−	0.996432i	$-0.526898\pi$
$953$	−5.21110	−0.168804	−0.0844021	+	0.996432i	$0.526898\pi$
$954$	0	0
$955$	5.81665	0.188222
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.3667	−1.04518
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.605551	−0.0194934
$966$	0	0
$967$	−26.2389	−0.843785	−0.421892	−	0.906646i	$-0.638634\pi$
$967$	−26.2389	−0.843785	−0.421892	+	0.906646i	$0.638634\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.42221	−0.205465	−0.102732	−	0.994709i	$-0.532758\pi$
$977$	−6.42221	−0.205465	−0.102732	+	0.994709i	$0.532758\pi$
$978$	0	0
$979$	−30.4222	−0.972298
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.21110	0.293789	0.146894	−	0.989152i	$-0.453072\pi$
$983$	9.21110	0.293789	0.146894	+	0.989152i	$0.453072\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.2111	0.674474
$990$	0	0
$991$	50.4222	1.60171	0.800857	−	0.598856i	$-0.204378\pi$
$991$	50.4222	1.60171	0.800857	+	0.598856i	$0.204378\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.2111	0.545629
$996$	0	0
$997$	−61.6333	−1.95195	−0.975973	−	0.217891i	$-0.930082\pi$
$997$	−61.6333	−1.95195	−0.975973	+	0.217891i	$0.930082\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.i.1.1		2
3.2	odd	2		1140.2.a.e.1.1	✓	2
12.11	even	2		4560.2.a.bl.1.2		2
15.2	even	4		5700.2.f.n.3649.3		4
15.8	even	4		5700.2.f.n.3649.2		4
15.14	odd	2		5700.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.e.1.1	✓	2	3.2	odd	2
3420.2.a.i.1.1		2	1.1	even	1	trivial
4560.2.a.bl.1.2		2	12.11	even	2
5700.2.a.u.1.2		2	15.14	odd	2
5700.2.f.n.3649.2		4	15.8	even	4
5700.2.f.n.3649.3		4	15.2	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.60555 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.60555 q^{7} +4.60555 q^{11} +4.60555 q^{13} -2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -2.60555 q^{29} +4.00000 q^{31} -2.60555 q^{35} +3.39445 q^{37} -6.60555 q^{41} +10.6056 q^{43} -6.00000 q^{47} -0.211103 q^{49} +4.60555 q^{55} -5.21110 q^{59} -7.21110 q^{61} +4.60555 q^{65} +4.00000 q^{67} +9.21110 q^{71} +6.00000 q^{73} -12.0000 q^{77} +8.00000 q^{79} +11.2111 q^{83} -2.00000 q^{85} -6.60555 q^{89} -12.0000 q^{91} -1.00000 q^{95} +16.6056 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^35 + 14 * q^37 - 6 * q^41 + 14 * q^43 - 12 * q^47 + 14 * q^49 + 2 * q^55 + 4 * q^59 + 2 * q^65 + 8 * q^67 + 4 * q^71 + 12 * q^73 - 24 * q^77 + 16 * q^79 + 8 * q^83 - 4 * q^85 - 6 * q^89 - 24 * q^91 - 2 * q^95 + 26 * q^97

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