LMFDB - Embedded newform 3042.2.b.m.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.m.1351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{8} +3.00000 q^{10} +1.00000 q^{16} +5.19615 q^{17} -6.92820i q^{19} -3.00000i q^{20} -4.00000 q^{25} +5.19615 q^{29} +6.92820i q^{31} -1.00000i q^{32} -5.19615i q^{34} -1.73205i q^{37} -6.92820 q^{38} -3.00000 q^{40} +9.00000i q^{41} -4.00000 q^{43} -12.0000i q^{47} +7.00000 q^{49} +4.00000i q^{50} +5.19615 q^{53} -5.19615i q^{58} -12.0000i q^{59} -5.00000 q^{61} +6.92820 q^{62} -1.00000 q^{64} +13.8564i q^{67} -5.19615 q^{68} +12.0000i q^{71} +8.66025i q^{73} -1.73205 q^{74} +6.92820i q^{76} +4.00000 q^{79} +3.00000i q^{80} +9.00000 q^{82} +12.0000i q^{83} +15.5885i q^{85} +4.00000i q^{86} +6.00000i q^{89} -12.0000 q^{94} +20.7846 q^{95} +13.8564i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 12 q^{10} + 4 q^{16} - 16 q^{25} - 12 q^{40} - 16 q^{43} + 28 q^{49} - 20 q^{61} - 4 q^{64} + 16 q^{79} + 36 q^{82} - 48 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^10 + 4 * q^16 - 16 * q^25 - 12 * q^40 - 16 * q^43 + 28 * q^49 - 20 * q^61 - 4 * q^64 + 16 * q^79 + 36 * q^82 - 48 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	3.00000	0.948683
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	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	$-0.283003\pi$
$17$	5.19615	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	− 3.00000i	− 0.670820i
$21$	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
$31$	6.92820i	1.24434i	0.782881	+	0.622171i	$0.213749\pi$
$31$	6.92820i	1.24434i	−0.782881	+	0.622171i	$0.786251\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 5.19615i	− 0.891133i
$35$	0	0
$36$	0	0
$37$	− 1.73205i	− 0.284747i	−0.989813	−	0.142374i	$-0.954527\pi$
$37$	− 1.73205i	− 0.284747i	0.989813	−	0.142374i	$-0.0454735\pi$
$38$	−6.92820	−1.12390
$39$	0	0
$40$	−3.00000	−0.474342
$41$	9.00000i	1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$	9.00000i	1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	4.00000i	0.565685i
$51$	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
$58$	− 5.19615i	− 0.682288i
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	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
$67$	13.8564i	1.69283i	0.532524	+	0.846415i	$0.321244\pi$
$67$	13.8564i	1.69283i	−0.532524	+	0.846415i	$0.678756\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	0	0
$73$	8.66025i	1.01361i	0.862062	+	0.506803i	$0.169173\pi$
$73$	8.66025i	1.01361i	−0.862062	+	0.506803i	$0.830827\pi$
$74$	−1.73205	−0.201347
$75$	0	0
$76$	6.92820i	0.794719i
$77$	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	$0.427756\pi$
$80$	3.00000i	0.335410i
$81$	0	0
$82$	9.00000	0.993884
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	15.5885i	1.69081i
$86$	4.00000i	0.431331i
$87$	0	0
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	20.7846	2.13246
$96$	0	0
$97$	13.8564i	1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$	13.8564i	1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	4.00000	0.400000
$101$	5.19615	0.517036	0.258518	−	0.966006i	$-0.416766\pi$
$101$	5.19615	0.517036	0.258518	+	0.966006i	$0.416766\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	− 5.19615i	− 0.504695i
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 6.92820i	− 0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$	− 6.92820i	− 0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$	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
$122$	5.00000i	0.452679i
$123$	0	0
$124$	− 6.92820i	− 0.622171i
$125$	3.00000i	0.268328i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000i	0.0883883i
$129$	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
$136$	5.19615i	0.445566i
$137$	− 9.00000i	− 0.768922i	−0.923141	−	0.384461i	$-0.874387\pi$
$137$	− 9.00000i	− 0.768922i	0.923141	−	0.384461i	$-0.125613\pi$
$138$	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
$145$	15.5885i	1.29455i
$146$	8.66025	0.716728
$147$	0	0
$148$	1.73205i	0.142374i
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	0	0
$151$	13.8564i	1.12762i	0.825905	+	0.563809i	$0.190665\pi$
$151$	13.8564i	1.12762i	−0.825905	+	0.563809i	$0.809335\pi$
$152$	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	$0.590112\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	3.00000	0.237171
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 9.00000i	− 0.702782i
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	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	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	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	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.19615	0.382029
$186$	0	0
$187$	0	0
$188$	12.0000i	0.875190i
$189$	0	0
$190$	− 20.7846i	− 1.50787i
$191$	−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
$193$	− 5.19615i	− 0.374027i	−0.982357	−	0.187014i	$-0.940119\pi$
$193$	− 5.19615i	− 0.374027i	0.982357	−	0.187014i	$-0.0598809\pi$
$194$	13.8564	0.994832
$195$	0	0
$196$	−7.00000	−0.500000
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	− 4.00000i	− 0.282843i
$201$	0	0
$202$	− 5.19615i	− 0.365600i
$203$	0	0
$204$	0	0
$205$	−27.0000	−1.88576
$206$	4.00000i	0.278693i
$207$	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
$215$	− 12.0000i	− 0.818393i
$216$	0	0
$217$	0	0
$218$	−6.92820	−0.469237
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 20.7846i	− 1.39184i	−0.718119	−	0.695920i	$-0.754997\pi$
$223$	− 20.7846i	− 1.39184i	0.718119	−	0.695920i	$-0.245003\pi$
$224$	0	0
$225$	0	0
$226$	− 15.5885i	− 1.03693i
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	20.7846i	1.37349i	0.726900	+	0.686743i	$0.240960\pi$
$229$	20.7846i	1.37349i	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	5.19615i	0.341144i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	36.0000	2.34838
$236$	12.0000i	0.781133i
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	− 8.66025i	− 0.557856i	−0.960312	−	0.278928i	$-0.910021\pi$
$241$	− 8.66025i	− 0.557856i	0.960312	−	0.278928i	$-0.0899791\pi$
$242$	− 11.0000i	− 0.707107i
$243$	0	0
$244$	5.00000	0.320092
$245$	21.0000i	1.34164i
$246$	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	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	0	0
$254$	8.00000i	0.501965i
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.5885	0.972381	0.486191	−	0.873853i	$-0.338386\pi$
$257$	15.5885	0.972381	0.486191	+	0.873853i	$0.338386\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 20.7846i	− 1.28408i
$263$	−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
$265$	15.5885i	0.957591i
$266$	0	0
$267$	0	0
$268$	− 13.8564i	− 0.846415i
$269$	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
$271$	− 20.7846i	− 1.26258i	−0.775549	−	0.631288i	$-0.782527\pi$
$271$	− 20.7846i	− 1.26258i	0.775549	−	0.631288i	$-0.217473\pi$
$272$	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	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	16.0000i	0.959616i
$279$	0	0
$280$	0	0
$281$	3.00000i	0.178965i	0.995988	+	0.0894825i	$0.0285213\pi$
$281$	3.00000i	0.178965i	−0.995988	+	0.0894825i	$0.971479\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	− 12.0000i	− 0.712069i
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.0000	0.588235
$290$	15.5885	0.915386
$291$	0	0
$292$	− 8.66025i	− 0.506803i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	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
$304$	− 6.92820i	− 0.397360i
$305$	− 15.0000i	− 0.858898i
$306$	0	0
$307$	− 13.8564i	− 0.790827i	−0.918503	−	0.395413i	$-0.870601\pi$
$307$	− 13.8564i	− 0.790827i	0.918503	−	0.395413i	$-0.129399\pi$
$308$	0	0
$309$	0	0
$310$	20.7846i	1.18049i
$311$	−20.7846	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$311$	−20.7846	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	7.00000i	0.395033i
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	0	0
$319$	0	0
$320$	− 3.00000i	− 0.167705i
$321$	0	0
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	− 13.8564i	− 0.761617i	−0.924654	−	0.380808i	$-0.875646\pi$
$331$	− 13.8564i	− 0.761617i	0.924654	−	0.380808i	$-0.124354\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	12.0000	0.656611
$335$	−41.5692	−2.27117
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	0	0
$339$	0	0
$340$	− 15.5885i	− 0.845403i
$341$	0	0
$342$	0	0
$343$	0	0
$344$	− 4.00000i	− 0.215666i
$345$	0	0
$346$	− 20.7846i	− 1.11739i
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	− 6.92820i	− 0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$	− 6.92820i	− 0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.0000i	− 0.798369i	−0.916871	−	0.399185i	$-0.869293\pi$
$353$	− 15.0000i	− 0.798369i	0.916871	−	0.399185i	$-0.130707\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	− 6.00000i	− 0.317999i
$357$	0	0
$358$	− 20.7846i	− 1.09850i
$359$	12.0000i	0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$	12.0000i	0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	1.00000i	0.0525588i
$363$	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
$370$	− 5.19615i	− 0.270135i
$371$	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
$379$	− 13.8564i	− 0.711756i	−0.934532	−	0.355878i	$-0.884182\pi$
$379$	− 13.8564i	− 0.711756i	0.934532	−	0.355878i	$-0.115818\pi$
$380$	−20.7846	−1.06623
$381$	0	0
$382$	20.7846i	1.06343i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−5.19615	−0.264477
$387$	0	0
$388$	− 13.8564i	− 0.703452i
$389$	5.19615	0.263455	0.131728	−	0.991286i	$-0.457948\pi$
$389$	5.19615	0.263455	0.131728	+	0.991286i	$0.457948\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	0	0
$394$	6.00000	0.302276
$395$	12.0000i	0.603786i
$396$	0	0
$397$	− 6.92820i	− 0.347717i	−0.984771	−	0.173858i	$-0.944377\pi$
$397$	− 6.92820i	− 0.347717i	0.984771	−	0.173858i	$-0.0556235\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	−4.00000	−0.200000
$401$	− 3.00000i	− 0.149813i	−0.997191	−	0.0749064i	$-0.976134\pi$
$401$	− 3.00000i	− 0.149813i	0.997191	−	0.0749064i	$-0.0238658\pi$
$402$	0	0
$403$	0	0
$404$	−5.19615	−0.258518
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5885i	0.770800i	0.922750	+	0.385400i	$0.125936\pi$
$409$	15.5885i	0.770800i	−0.922750	+	0.385400i	$0.874064\pi$
$410$	27.0000i	1.33343i
$411$	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
$421$	− 15.5885i	− 0.759735i	−0.925041	−	0.379867i	$-0.875970\pi$
$421$	− 15.5885i	− 0.759735i	0.925041	−	0.379867i	$-0.124030\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	5.19615i	0.252347i
$425$	−20.7846	−1.00820
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−12.0000	−0.578691
$431$	− 12.0000i	− 0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$	− 12.0000i	− 0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	6.92820i	0.331801i
$437$	0	0
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	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
$449$	30.0000i	1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$	30.0000i	1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$	0	0
$451$	0	0
$452$	−15.5885	−0.733219
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	12.1244i	0.567153i	0.958950	+	0.283577i	$0.0915211\pi$
$457$	12.1244i	0.567153i	−0.958950	+	0.283577i	$0.908479\pi$
$458$	20.7846	0.971201
$459$	0	0
$460$	0	0
$461$	− 21.0000i	− 0.978068i	−0.872265	−	0.489034i	$-0.837349\pi$
$461$	− 21.0000i	− 0.978068i	0.872265	−	0.489034i	$-0.162651\pi$
$462$	0	0
$463$	6.92820i	0.321981i	0.986956	+	0.160990i	$0.0514688\pi$
$463$	6.92820i	0.321981i	−0.986956	+	0.160990i	$0.948531\pi$
$464$	5.19615	0.241225
$465$	0	0
$466$	0	0
$467$	41.5692	1.92359	0.961797	−	0.273764i	$-0.0882686\pi$
$467$	41.5692	1.92359	0.961797	+	0.273764i	$0.0882686\pi$
$468$	0	0
$469$	0	0
$470$	− 36.0000i	− 1.66056i
$471$	0	0
$472$	12.0000	0.552345
$473$	0	0
$474$	0	0
$475$	27.7128i	1.27155i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	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
$487$	6.92820i	0.313947i	0.987603	+	0.156973i	$0.0501737\pi$
$487$	6.92820i	0.313947i	−0.987603	+	0.156973i	$0.949826\pi$
$488$	− 5.00000i	− 0.226339i
$489$	0	0
$490$	21.0000	0.948683
$491$	20.7846	0.937996	0.468998	−	0.883199i	$-0.344615\pi$
$491$	20.7846	0.937996	0.468998	+	0.883199i	$0.344615\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	0	0
$496$	6.92820i	0.311086i
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 3.00000i	− 0.134164i
$501$	0	0
$502$	20.7846i	0.927663i
$503$	−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
$505$	15.5885i	0.693677i
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 15.5885i	− 0.687577i
$515$	− 12.0000i	− 0.528783i
$516$	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
$526$	20.7846i	0.906252i
$527$	36.0000i	1.56818i
$528$	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
$538$	− 20.7846i	− 0.896088i
$539$	0	0
$540$	0	0
$541$	32.9090i	1.41487i	0.706780	+	0.707433i	$0.250147\pi$
$541$	32.9090i	1.41487i	−0.706780	+	0.707433i	$0.749853\pi$
$542$	−20.7846	−0.892775
$543$	0	0
$544$	− 5.19615i	− 0.222783i
$545$	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	$0.527253\pi$
$548$	9.00000i	0.384461i
$549$	0	0
$550$	0	0
$551$	− 36.0000i	− 1.53365i
$552$	0	0
$553$	0	0
$554$	19.0000i	0.807233i
$555$	0	0
$556$	16.0000	0.678551
$557$	− 45.0000i	− 1.90671i	−0.301849	−	0.953356i	$-0.597604\pi$
$557$	− 45.0000i	− 1.90671i	0.301849	−	0.953356i	$-0.402396\pi$
$558$	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	$-0.644307\pi$
$563$	−20.7846	−0.875967	−0.437983	+	0.898983i	$0.644307\pi$
$564$	0	0
$565$	46.7654i	1.96743i
$566$	− 16.0000i	− 0.672530i
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−41.5692	−1.74267	−0.871336	−	0.490687i	$-0.836746\pi$
$569$	−41.5692	−1.74267	−0.871336	+	0.490687i	$0.836746\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 43.3013i	− 1.80266i	−0.433138	−	0.901328i	$-0.642594\pi$
$577$	− 43.3013i	− 1.80266i	0.433138	−	0.901328i	$-0.357406\pi$
$578$	− 10.0000i	− 0.415945i
$579$	0	0
$580$	− 15.5885i	− 0.647275i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−8.66025	−0.358364
$585$	0	0
$586$	−9.00000	−0.371787
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	− 36.0000i	− 1.48210i
$591$	0	0
$592$	− 1.73205i	− 0.0711868i
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	0	0
$596$	3.00000i	0.122885i
$597$	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
$604$	− 13.8564i	− 0.563809i
$605$	33.0000i	1.34164i
$606$	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
$613$	39.8372i	1.60901i	0.593947	+	0.804504i	$0.297569\pi$
$613$	39.8372i	1.60901i	−0.593947	+	0.804504i	$0.702431\pi$
$614$	−13.8564	−0.559199
$615$	0	0
$616$	0	0
$617$	21.0000i	0.845428i	0.906263	+	0.422714i	$0.138923\pi$
$617$	21.0000i	0.845428i	−0.906263	+	0.422714i	$0.861077\pi$
$618$	0	0
$619$	− 20.7846i	− 0.835404i	−0.908584	−	0.417702i	$-0.862836\pi$
$619$	− 20.7846i	− 0.835404i	0.908584	−	0.417702i	$-0.137164\pi$
$620$	20.7846	0.834730
$621$	0	0
$622$	20.7846i	0.833387i
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	− 22.0000i	− 0.879297i
$627$	0	0
$628$	7.00000	0.279330
$629$	− 9.00000i	− 0.358854i
$630$	0	0
$631$	− 13.8564i	− 0.551615i	−0.961213	−	0.275807i	$-0.911055\pi$
$631$	− 13.8564i	− 0.551615i	0.961213	−	0.275807i	$-0.0889452\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	−9.00000	−0.357436
$635$	− 24.0000i	− 0.952411i
$636$	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	$-0.532722\pi$
$641$	−5.19615	−0.205236	−0.102618	+	0.994721i	$0.532722\pi$
$642$	0	0
$643$	− 34.6410i	− 1.36611i	−0.730368	−	0.683054i	$-0.760651\pi$
$643$	− 34.6410i	− 1.36611i	0.730368	−	0.683054i	$-0.239349\pi$
$644$	0	0
$645$	0	0
$646$	−36.0000	−1.41640
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	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
$655$	62.3538i	2.43637i
$656$	9.00000i	0.351391i
$657$	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
$661$	5.19615i	0.202107i	0.994881	+	0.101053i	$0.0322213\pi$
$661$	5.19615i	0.202107i	−0.994881	+	0.101053i	$0.967779\pi$
$662$	−13.8564	−0.538545
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	41.5692i	1.60596i
$671$	0	0
$672$	0	0
$673$	5.00000	0.192736	0.0963679	−	0.995346i	$-0.469277\pi$
$673$	5.00000	0.192736	0.0963679	+	0.995346i	$0.469277\pi$
$674$	− 7.00000i	− 0.269630i
$675$	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
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	13.8564i	0.527123i	0.964643	+	0.263561i	$0.0848971\pi$
$691$	13.8564i	0.527123i	−0.964643	+	0.263561i	$0.915103\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	0	0
$695$	− 48.0000i	− 1.82074i
$696$	0	0
$697$	46.7654i	1.77136i
$698$	−6.92820	−0.262236
$699$	0	0
$700$	0	0
$701$	20.7846	0.785024	0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846	0.785024	0.392512	+	0.919747i	$0.371606\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	0	0
$706$	−15.0000	−0.564532
$707$	0	0
$708$	0	0
$709$	25.9808i	0.975728i	0.872920	+	0.487864i	$0.162224\pi$
$709$	25.9808i	0.975728i	−0.872920	+	0.487864i	$0.837776\pi$
$710$	36.0000i	1.35106i
$711$	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	$-0.626685\pi$
$719$	−20.7846	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$720$	0	0
$721$	0	0
$722$	29.0000i	1.07927i
$723$	0	0
$724$	1.00000	0.0371647
$725$	−20.7846	−0.771921
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	0	0
$730$	25.9808i	0.961591i
$731$	−20.7846	−0.768747
$732$	0	0
$733$	8.66025i	0.319874i	0.987127	+	0.159937i	$0.0511291\pi$
$733$	8.66025i	0.319874i	−0.987127	+	0.159937i	$0.948871\pi$
$734$	− 32.0000i	− 1.18114i
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 41.5692i	− 1.52915i	−0.644536	−	0.764574i	$-0.722949\pi$
$739$	− 41.5692i	− 1.52915i	0.644536	−	0.764574i	$-0.277051\pi$
$740$	−5.19615	−0.191014
$741$	0	0
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	23.0000i	0.842090i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	− 12.0000i	− 0.437595i
$753$	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
$760$	20.7846i	0.753937i
$761$	− 6.00000i	− 0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$	− 6.00000i	− 0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$	0	0
$763$	0	0
$764$	20.7846	0.751961
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 13.8564i	− 0.499675i	−0.968288	−	0.249837i	$-0.919623\pi$
$769$	− 13.8564i	− 0.499675i	0.968288	−	0.249837i	$-0.0803772\pi$
$770$	0	0
$771$	0	0
$772$	5.19615i	0.187014i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	− 27.7128i	− 0.995474i
$776$	−13.8564	−0.497416
$777$	0	0
$778$	− 5.19615i	− 0.186291i
$779$	62.3538	2.23406
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	− 21.0000i	− 0.749522i
$786$	0	0
$787$	− 48.4974i	− 1.72875i	−0.502851	−	0.864373i	$-0.667715\pi$
$787$	− 48.4974i	− 1.72875i	0.502851	−	0.864373i	$-0.332285\pi$
$788$	− 6.00000i	− 0.213741i
$789$	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	$-0.619996\pi$
$797$	−20.7846	−0.736229	−0.368114	+	0.929781i	$0.619996\pi$
$798$	0	0
$799$	− 62.3538i	− 2.20592i
$800$	4.00000i	0.141421i
$801$	0	0
$802$	−3.00000	−0.105934
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	5.19615i	0.182800i
$809$	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
$811$	6.92820i	0.243282i	0.992574	+	0.121641i	$0.0388157\pi$
$811$	6.92820i	0.243282i	−0.992574	+	0.121641i	$0.961184\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.7128i	0.969549i
$818$	15.5885	0.545038
$819$	0	0
$820$	27.0000	0.942881
$821$	− 54.0000i	− 1.88461i	−0.334751	−	0.942306i	$-0.608652\pi$
$821$	− 54.0000i	− 1.88461i	0.334751	−	0.942306i	$-0.391348\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	− 4.00000i	− 0.139347i
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	36.0000i	1.24958i
$831$	0	0
$832$	0	0
$833$	36.3731	1.26025
$834$	0	0
$835$	−36.0000	−1.24583
$836$	0	0
$837$	0	0
$838$	− 20.7846i	− 0.717992i
$839$	48.0000i	1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$	48.0000i	1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$	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
$850$	20.7846i	0.712906i
$851$	0	0
$852$	0	0
$853$	− 36.3731i	− 1.24539i	−0.782465	−	0.622695i	$-0.786038\pi$
$853$	− 36.3731i	− 1.24539i	0.782465	−	0.622695i	$-0.213962\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.9808	−0.887486	−0.443743	−	0.896154i	$-0.646350\pi$
$857$	−25.9808	−0.887486	−0.443743	+	0.896154i	$0.646350\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	12.0000i	0.409197i
$861$	0	0
$862$	−12.0000	−0.408722
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	62.3538i	2.12009i
$866$	− 19.0000i	− 0.645646i
$867$	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
$877$	19.0526i	0.643359i	0.946849	+	0.321680i	$0.104247\pi$
$877$	19.0526i	0.643359i	−0.946849	+	0.321680i	$0.895753\pi$
$878$	4.00000i	0.134993i
$879$	0	0
$880$	0	0
$881$	25.9808	0.875314	0.437657	−	0.899142i	$-0.355808\pi$
$881$	25.9808	0.875314	0.437657	+	0.899142i	$0.355808\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	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
$890$	18.0000i	0.603361i
$891$	0	0
$892$	20.7846i	0.695920i
$893$	−83.1384	−2.78212
$894$	0	0
$895$	62.3538i	2.08426i
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	36.0000i	1.20067i
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	0	0
$904$	15.5885i	0.518464i
$905$	− 3.00000i	− 0.0997234i
$906$	0	0
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	− 12.0000i	− 0.398234i
$909$	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
$916$	− 20.7846i	− 0.686743i
$917$	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
$925$	6.92820i	0.227798i
$926$	6.92820	0.227675
$927$	0	0
$928$	− 5.19615i	− 0.170572i
$929$	− 27.0000i	− 0.885841i	−0.896561	−	0.442921i	$-0.853942\pi$
$929$	− 27.0000i	− 0.885841i	0.896561	−	0.442921i	$-0.146058\pi$
$930$	0	0
$931$	− 48.4974i	− 1.58944i
$932$	0	0
$933$	0	0
$934$	− 41.5692i	− 1.36019i
$935$	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
$941$	18.0000i	0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$	18.0000i	0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	0	0
$943$	0	0
$944$	− 12.0000i	− 0.390567i
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	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	$-0.735115\pi$
$953$	−41.5692	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$954$	0	0
$955$	− 62.3538i	− 2.01772i
$956$	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
$964$	8.66025i	0.278928i
$965$	15.5885	0.501810
$966$	0	0
$967$	− 6.92820i	− 0.222796i	−0.993776	−	0.111398i	$-0.964467\pi$
$967$	− 6.92820i	− 0.222796i	0.993776	−	0.111398i	$-0.0355328\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	41.5692i	1.33471i
$971$	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
$977$	3.00000i	0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$	3.00000i	0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$	0	0
$979$	0	0
$980$	− 21.0000i	− 0.670820i
$981$	0	0
$982$	− 20.7846i	− 0.663264i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	− 27.0000i	− 0.859855i
$987$	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
$995$	− 24.0000i	− 0.760851i
$996$	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.b.m.1351.2		4
3.2	odd	2	inner	3042.2.b.m.1351.3		4
13.5	odd	4		3042.2.a.t.1.1		2
13.8	odd	4		3042.2.a.u.1.1		2
13.9	even	3		234.2.l.b.127.1	✓	4
13.10	even	6		234.2.l.b.199.1	yes	4
13.12	even	2	inner	3042.2.b.m.1351.4		4
39.5	even	4		3042.2.a.u.1.2		2
39.8	even	4		3042.2.a.t.1.2		2
39.23	odd	6		234.2.l.b.199.2	yes	4
39.35	odd	6		234.2.l.b.127.2	yes	4
39.38	odd	2	inner	3042.2.b.m.1351.1		4
52.23	odd	6		1872.2.by.i.433.1		4
52.35	odd	6		1872.2.by.i.1297.2		4
156.23	even	6		1872.2.by.i.433.2		4
156.35	even	6		1872.2.by.i.1297.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.l.b.127.1	✓	4	13.9	even	3
234.2.l.b.127.2	yes	4	39.35	odd	6
234.2.l.b.199.1	yes	4	13.10	even	6
234.2.l.b.199.2	yes	4	39.23	odd	6
1872.2.by.i.433.1		4	52.23	odd	6
1872.2.by.i.433.2		4	156.23	even	6
1872.2.by.i.1297.1		4	156.35	even	6
1872.2.by.i.1297.2		4	52.35	odd	6
3042.2.a.t.1.1		2	13.5	odd	4
3042.2.a.t.1.2		2	39.8	even	4
3042.2.a.u.1.1		2	13.8	odd	4
3042.2.a.u.1.2		2	39.5	even	4
3042.2.b.m.1351.1		4	39.38	odd	2	inner
3042.2.b.m.1351.2		4	1.1	even	1	trivial
3042.2.b.m.1351.3		4	3.2	odd	2	inner
3042.2.b.m.1351.4		4	13.12	even	2	inner

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{8} +3.00000 q^{10} +1.00000 q^{16} +5.19615 q^{17} -6.92820i q^{19} -3.00000i q^{20} -4.00000 q^{25} +5.19615 q^{29} +6.92820i q^{31} -1.00000i q^{32} -5.19615i q^{34} -1.73205i q^{37} -6.92820 q^{38} -3.00000 q^{40} +9.00000i q^{41} -4.00000 q^{43} -12.0000i q^{47} +7.00000 q^{49} +4.00000i q^{50} +5.19615 q^{53} -5.19615i q^{58} -12.0000i q^{59} -5.00000 q^{61} +6.92820 q^{62} -1.00000 q^{64} +13.8564i q^{67} -5.19615 q^{68} +12.0000i q^{71} +8.66025i q^{73} -1.73205 q^{74} +6.92820i q^{76} +4.00000 q^{79} +3.00000i q^{80} +9.00000 q^{82} +12.0000i q^{83} +15.5885i q^{85} +4.00000i q^{86} +6.00000i q^{89} -12.0000 q^{94} +20.7846 q^{95} +13.8564i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 12 q^{10} + 4 q^{16} - 16 q^{25} - 12 q^{40} - 16 q^{43} + 28 q^{49} - 20 q^{61} - 4 q^{64} + 16 q^{79} + 36 q^{82} - 48 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^10 + 4 * q^16 - 16 * q^25 - 12 * q^40 - 16 * q^43 + 28 * q^49 - 20 * q^61 - 4 * q^64 + 16 * q^79 + 36 * q^82 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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