LMFDB - Embedded newform 3042.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 234)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{8} -3.00000 q^{10} +1.00000 q^{16} -5.19615 q^{17} +6.92820 q^{19} -3.00000 q^{20} +4.00000 q^{25} +5.19615 q^{29} -6.92820 q^{31} +1.00000 q^{32} -5.19615 q^{34} -1.73205 q^{37} +6.92820 q^{38} -3.00000 q^{40} -9.00000 q^{41} +4.00000 q^{43} -12.0000 q^{47} -7.00000 q^{49} +4.00000 q^{50} +5.19615 q^{53} +5.19615 q^{58} -12.0000 q^{59} -5.00000 q^{61} -6.92820 q^{62} +1.00000 q^{64} -13.8564 q^{67} -5.19615 q^{68} -12.0000 q^{71} +8.66025 q^{73} -1.73205 q^{74} +6.92820 q^{76} +4.00000 q^{79} -3.00000 q^{80} -9.00000 q^{82} -12.0000 q^{83} +15.5885 q^{85} +4.00000 q^{86} +6.00000 q^{89} -12.0000 q^{94} -20.7846 q^{95} -13.8564 q^{97} -7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 6 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 6 q^{5} + 2 q^{8} - 6 q^{10} + 2 q^{16} - 6 q^{20} + 8 q^{25} + 2 q^{32} - 6 q^{40} - 18 q^{41} + 8 q^{43} - 24 q^{47} - 14 q^{49} + 8 q^{50} - 24 q^{59} - 10 q^{61} + 2 q^{64} - 24 q^{71} + 8 q^{79} - 6 q^{80} - 18 q^{82} - 24 q^{83} + 8 q^{86} + 12 q^{89} - 24 q^{94} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^5 + 2 * q^8 - 6 * q^10 + 2 * q^16 - 6 * q^20 + 8 * q^25 + 2 * q^32 - 6 * q^40 - 18 * q^41 + 8 * q^43 - 24 * q^47 - 14 * q^49 + 8 * q^50 - 24 * q^59 - 10 * q^61 + 2 * q^64 - 24 * q^71 + 8 * q^79 - 6 * q^80 - 18 * q^82 - 24 * q^83 + 8 * q^86 + 12 * q^89 - 24 * q^94 - 14 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	−3.00000	−0.670820
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.19615	0.964901	0.482451	−	0.875923i	$-0.339747\pi$
$29$	5.19615	0.964901	0.482451	+	0.875923i	$0.339747\pi$
$30$	0	0
$31$	−6.92820	−1.24434	−0.622171	−	0.782881i	$-0.713749\pi$
$31$	−6.92820	−1.24434	−0.622171	+	0.782881i	$0.713749\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.19615	−0.891133
$35$	0	0
$36$	0	0
$37$	−1.73205	−0.284747	−0.142374	−	0.989813i	$-0.545473\pi$
$37$	−1.73205	−0.284747	−0.142374	+	0.989813i	$0.545473\pi$
$38$	6.92820	1.12390
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	0	0
$52$	0	0
$53$	5.19615	0.713746	0.356873	−	0.934153i	$-0.383843\pi$
$53$	5.19615	0.713746	0.356873	+	0.934153i	$0.383843\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	5.19615	0.682288
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	−6.92820	−0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−13.8564	−1.69283	−0.846415	−	0.532524i	$-0.821244\pi$
$67$	−13.8564	−1.69283	−0.846415	+	0.532524i	$0.821244\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	8.66025	1.01361	0.506803	−	0.862062i	$-0.330827\pi$
$73$	8.66025	1.01361	0.506803	+	0.862062i	$0.330827\pi$
$74$	−1.73205	−0.201347
$75$	0	0
$76$	6.92820	0.794719
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−9.00000	−0.993884
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	15.5885	1.69081
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−20.7846	−2.13246
$96$	0	0
$97$	−13.8564	−1.40690	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564	−1.40690	−0.703452	+	0.710742i	$0.748359\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	4.00000	0.400000
$101$	−5.19615	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$101$	−5.19615	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	5.19615	0.504695
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.5885	1.46644	0.733219	−	0.679992i	$-0.238017\pi$
$113$	15.5885	1.46644	0.733219	+	0.679992i	$0.238017\pi$
$114$	0	0
$115$	0	0
$116$	5.19615	0.482451
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−5.00000	−0.452679
$123$	0	0
$124$	−6.92820	−0.622171
$125$	3.00000	0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	20.7846	1.81596	0.907980	−	0.419014i	$-0.137624\pi$
$131$	20.7846	1.81596	0.907980	+	0.419014i	$0.137624\pi$
$132$	0	0
$133$	0	0
$134$	−13.8564	−1.19701
$135$	0	0
$136$	−5.19615	−0.445566
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−15.5885	−1.29455
$146$	8.66025	0.716728
$147$	0	0
$148$	−1.73205	−0.142374
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	6.92820	0.561951
$153$	0	0
$154$	0	0
$155$	20.7846	1.66946
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−3.00000	−0.237171
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	0	0
$170$	15.5885	1.19558
$171$	0	0
$172$	4.00000	0.304997
$173$	−20.7846	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.19615	0.382029
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−20.7846	−1.50787
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	−5.19615	−0.374027	−0.187014	−	0.982357i	$-0.559881\pi$
$193$	−5.19615	−0.374027	−0.187014	+	0.982357i	$0.559881\pi$
$194$	−13.8564	−0.994832
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−5.19615	−0.365600
$203$	0	0
$204$	0	0
$205$	27.0000	1.88576
$206$	4.00000	0.278693
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	5.19615	0.356873
$213$	0	0
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	0	0
$218$	6.92820	0.469237
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.7846	1.39184	0.695920	−	0.718119i	$-0.254997\pi$
$223$	20.7846	1.39184	0.695920	+	0.718119i	$0.254997\pi$
$224$	0	0
$225$	0	0
$226$	15.5885	1.03693
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	20.7846	1.37349	0.686743	−	0.726900i	$-0.259040\pi$
$229$	20.7846	1.37349	0.686743	+	0.726900i	$0.259040\pi$
$230$	0	0
$231$	0	0
$232$	5.19615	0.341144
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	36.0000	2.34838
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−8.66025	−0.557856	−0.278928	−	0.960312i	$-0.589979\pi$
$241$	−8.66025	−0.557856	−0.278928	+	0.960312i	$0.589979\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−5.00000	−0.320092
$245$	21.0000	1.34164
$246$	0	0
$247$	0	0
$248$	−6.92820	−0.439941
$249$	0	0
$250$	3.00000	0.189737
$251$	20.7846	1.31191	0.655956	−	0.754799i	$-0.272265\pi$
$251$	20.7846	1.31191	0.655956	+	0.754799i	$0.272265\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.5885	−0.972381	−0.486191	−	0.873853i	$-0.661614\pi$
$257$	−15.5885	−0.972381	−0.486191	+	0.873853i	$0.661614\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	20.7846	1.28408
$263$	−20.7846	−1.28163	−0.640817	−	0.767694i	$-0.721404\pi$
$263$	−20.7846	−1.28163	−0.640817	+	0.767694i	$0.721404\pi$
$264$	0	0
$265$	−15.5885	−0.957591
$266$	0	0
$267$	0	0
$268$	−13.8564	−0.846415
$269$	20.7846	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$269$	20.7846	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$270$	0	0
$271$	−20.7846	−1.26258	−0.631288	−	0.775549i	$-0.717473\pi$
$271$	−20.7846	−1.26258	−0.631288	+	0.775549i	$0.717473\pi$
$272$	−5.19615	−0.315063
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.0000	0.588235
$290$	−15.5885	−0.915386
$291$	0	0
$292$	8.66025	0.506803
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	−1.73205	−0.100673
$297$	0	0
$298$	3.00000	0.173785
$299$	0	0
$300$	0	0
$301$	0	0
$302$	13.8564	0.797347
$303$	0	0
$304$	6.92820	0.397360
$305$	15.0000	0.858898
$306$	0	0
$307$	−13.8564	−0.790827	−0.395413	−	0.918503i	$-0.629399\pi$
$307$	−13.8564	−0.790827	−0.395413	+	0.918503i	$0.629399\pi$
$308$	0	0
$309$	0	0
$310$	20.7846	1.18049
$311$	20.7846	1.17859	0.589294	−	0.807919i	$-0.299406\pi$
$311$	20.7846	1.17859	0.589294	+	0.807919i	$0.299406\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	4.00000	0.225018
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	0	0
$320$	−3.00000	−0.167705
$321$	0	0
$322$	0	0
$323$	−36.0000	−2.00309
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	13.8564	0.761617	0.380808	−	0.924654i	$-0.375646\pi$
$331$	13.8564	0.761617	0.380808	+	0.924654i	$0.375646\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	12.0000	0.656611
$335$	41.5692	2.27117
$336$	0	0
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	0	0
$339$	0	0
$340$	15.5885	0.845403
$341$	0	0
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−20.7846	−1.11739
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−6.92820	−0.370858	−0.185429	−	0.982658i	$-0.559368\pi$
$349$	−6.92820	−0.370858	−0.185429	+	0.982658i	$0.559368\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.0000	0.798369	0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000	0.798369	0.399185	+	0.916871i	$0.369293\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	6.00000	0.317999
$357$	0	0
$358$	−20.7846	−1.09850
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	1.00000	0.0525588
$363$	0	0
$364$	0	0
$365$	−25.9808	−1.35990
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	0	0
$370$	5.19615	0.270135
$371$	0	0
$372$	0	0
$373$	−23.0000	−1.19089	−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000	−1.19089	−0.595447	+	0.803394i	$0.703025\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	0	0
$379$	13.8564	0.711756	0.355878	−	0.934532i	$-0.384182\pi$
$379$	13.8564	0.711756	0.355878	+	0.934532i	$0.384182\pi$
$380$	−20.7846	−1.06623
$381$	0	0
$382$	−20.7846	−1.06343
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−5.19615	−0.264477
$387$	0	0
$388$	−13.8564	−0.703452
$389$	−5.19615	−0.263455	−0.131728	−	0.991286i	$-0.542052\pi$
$389$	−5.19615	−0.263455	−0.131728	+	0.991286i	$0.542052\pi$
$390$	0	0
$391$	0	0
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−6.92820	−0.347717	−0.173858	−	0.984771i	$-0.555623\pi$
$397$	−6.92820	−0.347717	−0.173858	+	0.984771i	$0.555623\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	4.00000	0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	0	0
$404$	−5.19615	−0.258518
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−15.5885	−0.770800	−0.385400	−	0.922750i	$-0.625936\pi$
$409$	−15.5885	−0.770800	−0.385400	+	0.922750i	$0.625936\pi$
$410$	27.0000	1.33343
$411$	0	0
$412$	4.00000	0.197066
$413$	0	0
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	15.5885	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$421$	15.5885	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	5.19615	0.252347
$425$	−20.7846	−1.00820
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−12.0000	−0.578691
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	0	0
$436$	6.92820	0.331801
$437$	0	0
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	20.7846	0.984180
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	15.5885	0.733219
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	−12.1244	−0.567153	−0.283577	−	0.958950i	$-0.591521\pi$
$457$	−12.1244	−0.567153	−0.283577	+	0.958950i	$0.591521\pi$
$458$	20.7846	0.971201
$459$	0	0
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	6.92820	0.321981	0.160990	−	0.986956i	$-0.448531\pi$
$463$	6.92820	0.321981	0.160990	+	0.986956i	$0.448531\pi$
$464$	5.19615	0.241225
$465$	0	0
$466$	0	0
$467$	−41.5692	−1.92359	−0.961797	−	0.273764i	$-0.911731\pi$
$467$	−41.5692	−1.92359	−0.961797	+	0.273764i	$0.911731\pi$
$468$	0	0
$469$	0	0
$470$	36.0000	1.66056
$471$	0	0
$472$	−12.0000	−0.552345
$473$	0	0
$474$	0	0
$475$	27.7128	1.27155
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	−8.66025	−0.394464
$483$	0	0
$484$	−11.0000	−0.500000
$485$	41.5692	1.88756
$486$	0	0
$487$	−6.92820	−0.313947	−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820	−0.313947	−0.156973	+	0.987603i	$0.550174\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	21.0000	0.948683
$491$	−20.7846	−0.937996	−0.468998	−	0.883199i	$-0.655385\pi$
$491$	−20.7846	−0.937996	−0.468998	+	0.883199i	$0.655385\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	0	0
$495$	0	0
$496$	−6.92820	−0.311086
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	20.7846	0.927663
$503$	−20.7846	−0.926740	−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846	−0.926740	−0.463370	+	0.886165i	$0.653360\pi$
$504$	0	0
$505$	15.5885	0.693677
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.5885	−0.687577
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.5885	−0.682943	−0.341471	−	0.939892i	$-0.610925\pi$
$521$	−15.5885	−0.682943	−0.341471	+	0.939892i	$0.610925\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	20.7846	0.907980
$525$	0	0
$526$	−20.7846	−0.906252
$527$	36.0000	1.56818
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−15.5885	−0.677119
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−13.8564	−0.598506
$537$	0	0
$538$	20.7846	0.896088
$539$	0	0
$540$	0	0
$541$	32.9090	1.41487	0.707433	−	0.706780i	$-0.249853\pi$
$541$	32.9090	1.41487	0.707433	+	0.706780i	$0.249853\pi$
$542$	−20.7846	−0.892775
$543$	0	0
$544$	−5.19615	−0.222783
$545$	−20.7846	−0.890315
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	0	0
$554$	19.0000	0.807233
$555$	0	0
$556$	−16.0000	−0.678551
$557$	−45.0000	−1.90671	−0.953356	−	0.301849i	$-0.902396\pi$
$557$	−45.0000	−1.90671	−0.953356	+	0.301849i	$0.902396\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	3.00000	0.126547
$563$	20.7846	0.875967	0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846	0.875967	0.437983	+	0.898983i	$0.355693\pi$
$564$	0	0
$565$	−46.7654	−1.96743
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−12.0000	−0.503509
$569$	41.5692	1.74267	0.871336	−	0.490687i	$-0.163254\pi$
$569$	41.5692	1.74267	0.871336	+	0.490687i	$0.163254\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.3013	1.80266	0.901328	−	0.433138i	$-0.142594\pi$
$577$	43.3013	1.80266	0.901328	+	0.433138i	$0.142594\pi$
$578$	10.0000	0.415945
$579$	0	0
$580$	−15.5885	−0.647275
$581$	0	0
$582$	0	0
$583$	0	0
$584$	8.66025	0.358364
$585$	0	0
$586$	−9.00000	−0.371787
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−48.0000	−1.97781
$590$	36.0000	1.48210
$591$	0	0
$592$	−1.73205	−0.0711868
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	3.00000	0.122885
$597$	0	0
$598$	0	0
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	0	0
$604$	13.8564	0.563809
$605$	33.0000	1.34164
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	6.92820	0.280976
$609$	0	0
$610$	15.0000	0.607332
$611$	0	0
$612$	0	0
$613$	−39.8372	−1.60901	−0.804504	−	0.593947i	$-0.797569\pi$
$613$	−39.8372	−1.60901	−0.804504	+	0.593947i	$0.797569\pi$
$614$	−13.8564	−0.559199
$615$	0	0
$616$	0	0
$617$	−21.0000	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$617$	−21.0000	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$618$	0	0
$619$	−20.7846	−0.835404	−0.417702	−	0.908584i	$-0.637164\pi$
$619$	−20.7846	−0.835404	−0.417702	+	0.908584i	$0.637164\pi$
$620$	20.7846	0.834730
$621$	0	0
$622$	20.7846	0.833387
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	22.0000	0.879297
$627$	0	0
$628$	−7.00000	−0.279330
$629$	9.00000	0.358854
$630$	0	0
$631$	−13.8564	−0.551615	−0.275807	−	0.961213i	$-0.588945\pi$
$631$	−13.8564	−0.551615	−0.275807	+	0.961213i	$0.588945\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	9.00000	0.357436
$635$	−24.0000	−0.952411
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−3.00000	−0.118585
$641$	5.19615	0.205236	0.102618	−	0.994721i	$-0.467278\pi$
$641$	5.19615	0.205236	0.102618	+	0.994721i	$0.467278\pi$
$642$	0	0
$643$	34.6410	1.36611	0.683054	−	0.730368i	$-0.260651\pi$
$643$	34.6410	1.36611	0.683054	+	0.730368i	$0.260651\pi$
$644$	0	0
$645$	0	0
$646$	−36.0000	−1.41640
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.7846	−0.813365	−0.406682	−	0.913570i	$-0.633314\pi$
$653$	−20.7846	−0.813365	−0.406682	+	0.913570i	$0.633314\pi$
$654$	0	0
$655$	−62.3538	−2.43637
$656$	−9.00000	−0.351391
$657$	0	0
$658$	0	0
$659$	41.5692	1.61931	0.809653	−	0.586908i	$-0.199655\pi$
$659$	41.5692	1.61931	0.809653	+	0.586908i	$0.199655\pi$
$660$	0	0
$661$	5.19615	0.202107	0.101053	−	0.994881i	$-0.467779\pi$
$661$	5.19615	0.202107	0.101053	+	0.994881i	$0.467779\pi$
$662$	13.8564	0.538545
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	12.0000	0.464294
$669$	0	0
$670$	41.5692	1.60596
$671$	0	0
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	−7.00000	−0.269630
$675$	0	0
$676$	0	0
$677$	20.7846	0.798817	0.399409	−	0.916773i	$-0.369215\pi$
$677$	20.7846	0.798817	0.399409	+	0.916773i	$0.369215\pi$
$678$	0	0
$679$	0	0
$680$	15.5885	0.597790
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	−13.8564	−0.527123	−0.263561	−	0.964643i	$-0.584897\pi$
$691$	−13.8564	−0.527123	−0.263561	+	0.964643i	$0.584897\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	0	0
$695$	48.0000	1.82074
$696$	0	0
$697$	46.7654	1.77136
$698$	−6.92820	−0.262236
$699$	0	0
$700$	0	0
$701$	−20.7846	−0.785024	−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846	−0.785024	−0.392512	+	0.919747i	$0.628394\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	15.0000	0.564532
$707$	0	0
$708$	0	0
$709$	25.9808	0.975728	0.487864	−	0.872920i	$-0.337776\pi$
$709$	25.9808	0.975728	0.487864	+	0.872920i	$0.337776\pi$
$710$	36.0000	1.35106
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.7846	−0.776757
$717$	0	0
$718$	12.0000	0.447836
$719$	20.7846	0.775135	0.387568	−	0.921841i	$-0.373315\pi$
$719$	20.7846	0.775135	0.387568	+	0.921841i	$0.373315\pi$
$720$	0	0
$721$	0	0
$722$	29.0000	1.07927
$723$	0	0
$724$	1.00000	0.0371647
$725$	20.7846	0.771921
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	0	0
$730$	−25.9808	−0.961591
$731$	−20.7846	−0.768747
$732$	0	0
$733$	−8.66025	−0.319874	−0.159937	−	0.987127i	$-0.551129\pi$
$733$	−8.66025	−0.319874	−0.159937	+	0.987127i	$0.551129\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−41.5692	−1.52915	−0.764574	−	0.644536i	$-0.777051\pi$
$739$	−41.5692	−1.52915	−0.764574	+	0.644536i	$0.777051\pi$
$740$	5.19615	0.191014
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	−23.0000	−0.842090
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	−41.5692	−1.51286
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	13.8564	0.503287
$759$	0	0
$760$	−20.7846	−0.753937
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	0	0
$764$	−20.7846	−0.751961
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	13.8564	0.499675	0.249837	−	0.968288i	$-0.419623\pi$
$769$	13.8564	0.499675	0.249837	+	0.968288i	$0.419623\pi$
$770$	0	0
$771$	0	0
$772$	−5.19615	−0.187014
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	−27.7128	−0.995474
$776$	−13.8564	−0.497416
$777$	0	0
$778$	−5.19615	−0.186291
$779$	−62.3538	−2.23406
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−7.00000	−0.250000
$785$	21.0000	0.749522
$786$	0	0
$787$	−48.4974	−1.72875	−0.864373	−	0.502851i	$-0.832285\pi$
$787$	−48.4974	−1.72875	−0.864373	+	0.502851i	$0.832285\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−12.0000	−0.426941
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−6.92820	−0.245873
$795$	0	0
$796$	8.00000	0.283552
$797$	20.7846	0.736229	0.368114	−	0.929781i	$-0.380004\pi$
$797$	20.7846	0.736229	0.368114	+	0.929781i	$0.380004\pi$
$798$	0	0
$799$	62.3538	2.20592
$800$	4.00000	0.141421
$801$	0	0
$802$	−3.00000	−0.105934
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−5.19615	−0.182800
$809$	15.5885	0.548061	0.274030	−	0.961721i	$-0.411643\pi$
$809$	15.5885	0.548061	0.274030	+	0.961721i	$0.411643\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.7128	0.969549
$818$	−15.5885	−0.545038
$819$	0	0
$820$	27.0000	0.942881
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	0	0
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	36.0000	1.24958
$831$	0	0
$832$	0	0
$833$	36.3731	1.26025
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	0	0
$838$	20.7846	0.717992
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−2.00000	−0.0689655
$842$	15.5885	0.537214
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	5.19615	0.178437
$849$	0	0
$850$	−20.7846	−0.712906
$851$	0	0
$852$	0	0
$853$	−36.3731	−1.24539	−0.622695	−	0.782465i	$-0.713962\pi$
$853$	−36.3731	−1.24539	−0.622695	+	0.782465i	$0.713962\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.9808	0.887486	0.443743	−	0.896154i	$-0.353650\pi$
$857$	25.9808	0.887486	0.443743	+	0.896154i	$0.353650\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	12.0000	0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	62.3538	2.12009
$866$	−19.0000	−0.645646
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	6.92820	0.234619
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.0526	−0.643359	−0.321680	−	0.946849i	$-0.604247\pi$
$877$	−19.0526	−0.643359	−0.321680	+	0.946849i	$0.604247\pi$
$878$	4.00000	0.134993
$879$	0	0
$880$	0	0
$881$	−25.9808	−0.875314	−0.437657	−	0.899142i	$-0.644192\pi$
$881$	−25.9808	−0.875314	−0.437657	+	0.899142i	$0.644192\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.7846	0.697879	0.348939	−	0.937145i	$-0.386542\pi$
$887$	20.7846	0.697879	0.348939	+	0.937145i	$0.386542\pi$
$888$	0	0
$889$	0	0
$890$	−18.0000	−0.603361
$891$	0	0
$892$	20.7846	0.695920
$893$	−83.1384	−2.78212
$894$	0	0
$895$	62.3538	2.08426
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	−36.0000	−1.20067
$900$	0	0
$901$	−27.0000	−0.899500
$902$	0	0
$903$	0	0
$904$	15.5885	0.518464
$905$	−3.00000	−0.0997234
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	−20.7846	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$911$	−20.7846	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$912$	0	0
$913$	0	0
$914$	−12.1244	−0.401038
$915$	0	0
$916$	20.7846	0.686743
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	21.0000	0.691598
$923$	0	0
$924$	0	0
$925$	−6.92820	−0.227798
$926$	6.92820	0.227675
$927$	0	0
$928$	5.19615	0.170572
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	−48.4974	−1.58944
$932$	0	0
$933$	0	0
$934$	−41.5692	−1.36019
$935$	0	0
$936$	0	0
$937$	−53.0000	−1.73143	−0.865717	−	0.500533i	$-0.833137\pi$
$937$	−53.0000	−1.73143	−0.865717	+	0.500533i	$0.833137\pi$
$938$	0	0
$939$	0	0
$940$	36.0000	1.17419
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	0	0
$950$	27.7128	0.899122
$951$	0	0
$952$	0	0
$953$	41.5692	1.34656	0.673280	−	0.739388i	$-0.264885\pi$
$953$	41.5692	1.34656	0.673280	+	0.739388i	$0.264885\pi$
$954$	0	0
$955$	62.3538	2.01772
$956$	0	0
$957$	0	0
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	0	0
$964$	−8.66025	−0.278928
$965$	15.5885	0.501810
$966$	0	0
$967$	6.92820	0.222796	0.111398	−	0.993776i	$-0.464467\pi$
$967$	6.92820	0.222796	0.111398	+	0.993776i	$0.464467\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	41.5692	1.33471
$971$	41.5692	1.33402	0.667010	−	0.745049i	$-0.267574\pi$
$971$	41.5692	1.33402	0.667010	+	0.745049i	$0.267574\pi$
$972$	0	0
$973$	0	0
$974$	−6.92820	−0.221994
$975$	0	0
$976$	−5.00000	−0.160046
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	0	0
$979$	0	0
$980$	21.0000	0.670820
$981$	0	0
$982$	−20.7846	−0.663264
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−27.0000	−0.859855
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	−6.92820	−0.219971
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.u.1.1		2
3.2	odd	2		3042.2.a.t.1.2		2
13.5	odd	4		3042.2.b.m.1351.2		4
13.6	odd	12		234.2.l.b.127.1	✓	4
13.8	odd	4		3042.2.b.m.1351.4		4
13.11	odd	12		234.2.l.b.199.1	yes	4
13.12	even	2		3042.2.a.t.1.1		2
39.5	even	4		3042.2.b.m.1351.3		4
39.8	even	4		3042.2.b.m.1351.1		4
39.11	even	12		234.2.l.b.199.2	yes	4
39.32	even	12		234.2.l.b.127.2	yes	4
39.38	odd	2	inner	3042.2.a.u.1.2		2
52.11	even	12		1872.2.by.i.433.1		4
52.19	even	12		1872.2.by.i.1297.2		4
156.11	odd	12		1872.2.by.i.433.2		4
156.71	odd	12		1872.2.by.i.1297.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.l.b.127.1	✓	4	13.6	odd	12
234.2.l.b.127.2	yes	4	39.32	even	12
234.2.l.b.199.1	yes	4	13.11	odd	12
234.2.l.b.199.2	yes	4	39.11	even	12
1872.2.by.i.433.1		4	52.11	even	12
1872.2.by.i.433.2		4	156.11	odd	12
1872.2.by.i.1297.1		4	156.71	odd	12
1872.2.by.i.1297.2		4	52.19	even	12
3042.2.a.t.1.1		2	13.12	even	2
3042.2.a.t.1.2		2	3.2	odd	2
3042.2.a.u.1.1		2	1.1	even	1	trivial
3042.2.a.u.1.2		2	39.38	odd	2	inner
3042.2.b.m.1351.1		4	39.8	even	4
3042.2.b.m.1351.2		4	13.5	odd	4
3042.2.b.m.1351.3		4	39.5	even	4
3042.2.b.m.1351.4		4	13.8	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{8} -3.00000 q^{10} +1.00000 q^{16} -5.19615 q^{17} +6.92820 q^{19} -3.00000 q^{20} +4.00000 q^{25} +5.19615 q^{29} -6.92820 q^{31} +1.00000 q^{32} -5.19615 q^{34} -1.73205 q^{37} +6.92820 q^{38} -3.00000 q^{40} -9.00000 q^{41} +4.00000 q^{43} -12.0000 q^{47} -7.00000 q^{49} +4.00000 q^{50} +5.19615 q^{53} +5.19615 q^{58} -12.0000 q^{59} -5.00000 q^{61} -6.92820 q^{62} +1.00000 q^{64} -13.8564 q^{67} -5.19615 q^{68} -12.0000 q^{71} +8.66025 q^{73} -1.73205 q^{74} +6.92820 q^{76} +4.00000 q^{79} -3.00000 q^{80} -9.00000 q^{82} -12.0000 q^{83} +15.5885 q^{85} +4.00000 q^{86} +6.00000 q^{89} -12.0000 q^{94} -20.7846 q^{95} -13.8564 q^{97} -7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 6 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 6 q^{5} + 2 q^{8} - 6 q^{10} + 2 q^{16} - 6 q^{20} + 8 q^{25} + 2 q^{32} - 6 q^{40} - 18 q^{41} + 8 q^{43} - 24 q^{47} - 14 q^{49} + 8 q^{50} - 24 q^{59} - 10 q^{61} + 2 q^{64} - 24 q^{71} + 8 q^{79} - 6 q^{80} - 18 q^{82} - 24 q^{83} + 8 q^{86} + 12 q^{89} - 24 q^{94} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 6 * q^5 + 2 * q^8 - 6 * q^10 + 2 * q^16 - 6 * q^20 + 8 * q^25 + 2 * q^32 - 6 * q^40 - 18 * q^41 + 8 * q^43 - 24 * q^47 - 14 * q^49 + 8 * q^50 - 24 * q^59 - 10 * q^61 + 2 * q^64 - 24 * q^71 + 8 * q^79 - 6 * q^80 - 18 * q^82 - 24 * q^83 + 8 * q^86 + 12 * q^89 - 24 * q^94 - 14 * q^98

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