LMFDB - Embedded newform 784.2.f.b.783.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("784.783");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.2.f.b.783.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.73205i q^{5} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.73205i q^{5} -2.00000 q^{9} +1.73205i q^{11} +1.73205i q^{15} +5.19615i q^{17} +7.00000 q^{19} +8.66025i q^{23} +2.00000 q^{25} -5.00000 q^{27} -6.00000 q^{29} +5.00000 q^{31} +1.73205i q^{33} -5.00000 q^{37} -6.92820i q^{41} +3.46410i q^{43} -3.46410i q^{45} +3.00000 q^{47} +5.19615i q^{51} -9.00000 q^{53} -3.00000 q^{55} +7.00000 q^{57} +9.00000 q^{59} +8.66025i q^{61} -5.19615i q^{67} +8.66025i q^{69} -3.46410i q^{71} -1.73205i q^{73} +2.00000 q^{75} -5.19615i q^{79} +1.00000 q^{81} +12.0000 q^{83} -9.00000 q^{85} -6.00000 q^{87} -12.1244i q^{89} +5.00000 q^{93} +12.1244i q^{95} -6.92820i q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^9	$ 2 q + 2 q^{3} - 4 q^{9} + 14 q^{19} + 4 q^{25} - 10 q^{27} - 12 q^{29} + 10 q^{31} - 10 q^{37} + 6 q^{47} - 18 q^{53} - 6 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{75} + 2 q^{81} + 24 q^{83} - 18 q^{85} - 12 q^{87} + 10 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 + 14 * q^19 + 4 * q^25 - 10 * q^27 - 12 * q^29 + 10 * q^31 - 10 * q^37 + 6 * q^47 - 18 * q^53 - 6 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^75 + 2 * q^81 + 24 * q^83 - 18 * q^85 - 12 * q^87 + 10 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.73205i	0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$	1.73205i	0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	1.73205i	0.522233i	0.965307	+	0.261116i	$0.0840907\pi$
$11$	1.73205i	0.522233i	−0.965307	+	0.261116i	$0.915909\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	1.73205i	0.447214i
$16$	0	0
$17$	5.19615i	1.26025i	0.776493	+	0.630126i	$0.216997\pi$
$17$	5.19615i	1.26025i	−0.776493	+	0.630126i	$0.783003\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.66025i	1.80579i	0.429863	+	0.902894i	$0.358562\pi$
$23$	8.66025i	1.80579i	−0.429863	+	0.902894i	$0.641438\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	1.73205i	0.301511i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.92820i	− 1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$	− 6.92820i	− 1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$	0	0
$43$	3.46410i	0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$	3.46410i	0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$	0	0
$45$	− 3.46410i	− 0.516398i
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.19615i	0.727607i
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	8.66025i	1.10883i	0.832240	+	0.554416i	$0.187058\pi$
$61$	8.66025i	1.10883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.19615i	− 0.634811i	−0.948290	−	0.317406i	$-0.897188\pi$
$67$	− 5.19615i	− 0.634811i	0.948290	−	0.317406i	$-0.102812\pi$
$68$	0	0
$69$	8.66025i	1.04257i
$70$	0	0
$71$	− 3.46410i	− 0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$	− 3.46410i	− 0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$	0	0
$73$	− 1.73205i	− 0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	− 1.73205i	− 0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 5.19615i	− 0.584613i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	− 5.19615i	− 0.584613i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	− 12.1244i	− 1.28518i	−0.766211	−	0.642590i	$-0.777860\pi$
$89$	− 12.1244i	− 1.28518i	0.766211	−	0.642590i	$-0.222140\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	12.1244i	1.24393i
$96$	0	0
$97$	− 6.92820i	− 0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$	− 6.92820i	− 0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$	0	0
$99$	− 3.46410i	− 0.348155i
$100$	0	0
$101$	5.19615i	0.517036i	0.966006	+	0.258518i	$0.0832342\pi$
$101$	5.19615i	0.517036i	−0.966006	+	0.258518i	$0.916766\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.19615i	− 0.502331i	−0.967944	−	0.251166i	$-0.919186\pi$
$107$	− 5.19615i	− 0.502331i	0.967944	−	0.251166i	$-0.0808138\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−15.0000	−1.39876
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	− 6.92820i	− 0.624695i
$124$	0	0
$125$	12.1244i	1.08444i
$126$	0	0
$127$	− 3.46410i	− 0.307389i	−0.988118	−	0.153695i	$-0.950883\pi$
$127$	− 3.46410i	− 0.307389i	0.988118	−	0.153695i	$-0.0491172\pi$
$128$	0	0
$129$	3.46410i	0.304997i
$130$	0	0
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 8.66025i	− 0.745356i
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 10.3923i	− 0.863034i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	− 12.1244i	− 0.986666i	−0.869841	−	0.493333i	$-0.835778\pi$
$151$	− 12.1244i	− 0.986666i	0.869841	−	0.493333i	$-0.164222\pi$
$152$	0	0
$153$	− 10.3923i	− 0.840168i
$154$	0	0
$155$	8.66025i	0.695608i
$156$	0	0
$157$	− 15.5885i	− 1.24409i	−0.782980	−	0.622047i	$-0.786301\pi$
$157$	− 15.5885i	− 1.24409i	0.782980	−	0.622047i	$-0.213699\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.1244i	− 0.949653i	−0.880079	−	0.474826i	$-0.842511\pi$
$163$	− 12.1244i	− 0.949653i	0.880079	−	0.474826i	$-0.157489\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	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
$169$	13.0000	1.00000
$170$	0	0
$171$	−14.0000	−1.07061
$172$	0	0
$173$	8.66025i	0.658427i	0.944256	+	0.329213i	$0.106784\pi$
$173$	8.66025i	0.658427i	−0.944256	+	0.329213i	$0.893216\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	15.5885i	1.16514i	0.812782	+	0.582568i	$0.197952\pi$
$179$	15.5885i	1.16514i	−0.812782	+	0.582568i	$0.802048\pi$
$180$	0	0
$181$	6.92820i	0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$	6.92820i	0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$	0	0
$183$	8.66025i	0.640184i
$184$	0	0
$185$	− 8.66025i	− 0.636715i
$186$	0	0
$187$	−9.00000	−0.658145
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 12.1244i	− 0.877288i	−0.898661	−	0.438644i	$-0.855459\pi$
$191$	− 12.1244i	− 0.877288i	0.898661	−	0.438644i	$-0.144541\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−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
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	− 5.19615i	− 0.366508i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	− 17.3205i	− 1.20386i
$208$	0	0
$209$	12.1244i	0.838659i
$210$	0	0
$211$	− 24.2487i	− 1.66935i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	− 24.2487i	− 1.66935i	0.550743	−	0.834675i	$-0.314345\pi$
$212$	0	0
$213$	− 3.46410i	− 0.237356i
$214$	0	0
$215$	−6.00000	−0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 1.73205i	− 0.117041i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0	0
$229$	− 12.1244i	− 0.801200i	−0.916253	−	0.400600i	$-0.868802\pi$
$229$	− 12.1244i	− 0.801200i	0.916253	−	0.400600i	$-0.131198\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	5.19615i	0.338960i
$236$	0	0
$237$	− 5.19615i	− 0.337526i
$238$	0	0
$239$	− 10.3923i	− 0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$	− 10.3923i	− 0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$	0	0
$241$	12.1244i	0.780998i	0.920603	+	0.390499i	$0.127698\pi$
$241$	12.1244i	0.780998i	−0.920603	+	0.390499i	$0.872302\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−15.0000	−0.943042
$254$	0	0
$255$	−9.00000	−0.563602
$256$	0	0
$257$	− 19.0526i	− 1.18847i	−0.804293	−	0.594233i	$-0.797456\pi$
$257$	− 19.0526i	− 1.18847i	0.804293	−	0.594233i	$-0.202544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	8.66025i	0.534014i	0.963695	+	0.267007i	$0.0860347\pi$
$263$	8.66025i	0.534014i	−0.963695	+	0.267007i	$0.913965\pi$
$264$	0	0
$265$	− 15.5885i	− 0.957591i
$266$	0	0
$267$	− 12.1244i	− 0.741999i
$268$	0	0
$269$	− 1.73205i	− 0.105605i	−0.998605	−	0.0528025i	$-0.983185\pi$
$269$	− 1.73205i	− 0.105605i	0.998605	−	0.0528025i	$-0.0168154\pi$
$270$	0	0
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.46410i	0.208893i
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	0	0
$285$	12.1244i	0.718185i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.0000	−0.588235
$290$	0	0
$291$	− 6.92820i	− 0.406138i
$292$	0	0
$293$	20.7846i	1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$	20.7846i	1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$	0	0
$295$	15.5885i	0.907595i
$296$	0	0
$297$	− 8.66025i	− 0.502519i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.19615i	0.298511i
$304$	0	0
$305$	−15.0000	−0.858898
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	0	0
$313$	29.4449i	1.66432i	0.554534	+	0.832161i	$0.312897\pi$
$313$	29.4449i	1.66432i	−0.554534	+	0.832161i	$0.687103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	− 10.3923i	− 0.581857i
$320$	0	0
$321$	− 5.19615i	− 0.290021i
$322$	0	0
$323$	36.3731i	2.02385i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.0000	0.608301
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.4449i	1.61844i	0.587508	+	0.809218i	$0.300109\pi$
$331$	29.4449i	1.61844i	−0.587508	+	0.809218i	$0.699891\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	9.00000	0.491723
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	8.66025i	0.468979i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15.0000	−0.807573
$346$	0	0
$347$	8.66025i	0.464907i	0.972608	+	0.232453i	$0.0746753\pi$
$347$	8.66025i	0.464907i	−0.972608	+	0.232453i	$0.925325\pi$
$348$	0	0
$349$	− 27.7128i	− 1.48343i	−0.670714	−	0.741716i	$-0.734012\pi$
$349$	− 27.7128i	− 1.48343i	0.670714	−	0.741716i	$-0.265988\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.0526i	1.01407i	0.861927	+	0.507033i	$0.169258\pi$
$353$	19.0526i	1.01407i	−0.861927	+	0.507033i	$0.830742\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.5885i	0.822727i	0.911471	+	0.411364i	$0.134947\pi$
$359$	15.5885i	0.822727i	−0.911471	+	0.411364i	$0.865053\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	8.00000	0.419891
$364$	0	0
$365$	3.00000	0.157027
$366$	0	0
$367$	1.00000	0.0521996	0.0260998	−	0.999659i	$-0.491691\pi$
$367$	1.00000	0.0521996	0.0260998	+	0.999659i	$0.491691\pi$
$368$	0	0
$369$	13.8564i	0.721336i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	0	0
$375$	12.1244i	0.626099i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.3923i	0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$	10.3923i	0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$	0	0
$381$	− 3.46410i	− 0.177471i
$382$	0	0
$383$	27.0000	1.37964	0.689818	−	0.723983i	$-0.257691\pi$
$383$	27.0000	1.37964	0.689818	+	0.723983i	$0.257691\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 6.92820i	− 0.352180i
$388$	0	0
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	0	0
$391$	−45.0000	−2.27575
$392$	0	0
$393$	−21.0000	−1.05931
$394$	0	0
$395$	9.00000	0.452839
$396$	0	0
$397$	− 12.1244i	− 0.608504i	−0.952592	−	0.304252i	$-0.901594\pi$
$397$	− 12.1244i	− 0.608504i	0.952592	−	0.304252i	$-0.0984065\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.73205i	0.0860663i
$406$	0	0
$407$	− 8.66025i	− 0.429273i
$408$	0	0
$409$	5.19615i	0.256933i	0.991714	+	0.128467i	$0.0410055\pi$
$409$	5.19615i	0.256933i	−0.991714	+	0.128467i	$0.958994\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0	0
$413$	0	0
$414$	0	0
$415$	20.7846i	1.02028i
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	10.3923i	0.504101i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.4449i	1.41831i	0.705053	+	0.709155i	$0.250923\pi$
$431$	29.4449i	1.41831i	−0.705053	+	0.709155i	$0.749077\pi$
$432$	0	0
$433$	34.6410i	1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$	34.6410i	1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$	0	0
$435$	− 10.3923i	− 0.498273i
$436$	0	0
$437$	60.6218i	2.89993i
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.5167i	1.06980i	0.844916	+	0.534899i	$0.179651\pi$
$443$	22.5167i	1.06980i	−0.844916	+	0.534899i	$0.820349\pi$
$444$	0	0
$445$	21.0000	0.995495
$446$	0	0
$447$	−9.00000	−0.425685
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	− 12.1244i	− 0.569652i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	− 25.9808i	− 1.21268i
$460$	0	0
$461$	− 27.7128i	− 1.29071i	−0.763881	−	0.645357i	$-0.776709\pi$
$461$	− 27.7128i	− 1.29071i	0.763881	−	0.645357i	$-0.223291\pi$
$462$	0	0
$463$	− 3.46410i	− 0.160990i	−0.996755	−	0.0804952i	$-0.974350\pi$
$463$	− 3.46410i	− 0.160990i	0.996755	−	0.0804952i	$-0.0256502\pi$
$464$	0	0
$465$	8.66025i	0.401610i
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 15.5885i	− 0.718278i
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	14.0000	0.642364
$476$	0	0
$477$	18.0000	0.824163
$478$	0	0
$479$	9.00000	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$479$	9.00000	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	8.66025i	0.392434i	0.980561	+	0.196217i	$0.0628656\pi$
$487$	8.66025i	0.392434i	−0.980561	+	0.196217i	$0.937134\pi$
$488$	0	0
$489$	− 12.1244i	− 0.548282i
$490$	0	0
$491$	− 17.3205i	− 0.781664i	−0.920462	−	0.390832i	$-0.872187\pi$
$491$	− 17.3205i	− 0.781664i	0.920462	−	0.390832i	$-0.127813\pi$
$492$	0	0
$493$	− 31.1769i	− 1.40414i
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.1244i	− 0.542761i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	− 12.1244i	− 0.542761i	0.962472	−	0.271380i	$-0.0874801\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	15.5885i	0.690946i	0.938429	+	0.345473i	$0.112282\pi$
$509$	15.5885i	0.690946i	−0.938429	+	0.345473i	$0.887718\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−35.0000	−1.54529
$514$	0	0
$515$	− 1.73205i	− 0.0763233i
$516$	0	0
$517$	5.19615i	0.228527i
$518$	0	0
$519$	8.66025i	0.380143i
$520$	0	0
$521$	− 29.4449i	− 1.29000i	−0.764181	−	0.645001i	$-0.776857\pi$
$521$	− 29.4449i	− 1.29000i	0.764181	−	0.645001i	$-0.223143\pi$
$522$	0	0
$523$	23.0000	1.00572	0.502860	−	0.864368i	$-0.332281\pi$
$523$	23.0000	1.00572	0.502860	+	0.864368i	$0.332281\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.9808i	1.13174i
$528$	0	0
$529$	−52.0000	−2.26087
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	15.5885i	0.672692i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	6.92820i	0.297318i
$544$	0	0
$545$	19.0526i	0.816122i
$546$	0	0
$547$	24.2487i	1.03680i	0.855138	+	0.518400i	$0.173472\pi$
$547$	24.2487i	1.03680i	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	− 17.3205i	− 0.739221i
$550$	0	0
$551$	−42.0000	−1.78926
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 8.66025i	− 0.367607i
$556$	0	0
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	0	0
$565$	10.3923i	0.437208i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	0	0
$571$	− 12.1244i	− 0.507388i	−0.967284	−	0.253694i	$-0.918354\pi$
$571$	− 12.1244i	− 0.507388i	0.967284	−	0.253694i	$-0.0816457\pi$
$572$	0	0
$573$	− 12.1244i	− 0.506502i
$574$	0	0
$575$	17.3205i	0.722315i
$576$	0	0
$577$	− 15.5885i	− 0.648956i	−0.945893	−	0.324478i	$-0.894811\pi$
$577$	− 15.5885i	− 0.648956i	0.945893	−	0.324478i	$-0.105189\pi$
$578$	0	0
$579$	−5.00000	−0.207793
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 15.5885i	− 0.645608i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	35.0000	1.44215
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	− 19.0526i	− 0.782395i	−0.920307	−	0.391197i	$-0.872061\pi$
$593$	− 19.0526i	− 0.782395i	0.920307	−	0.391197i	$-0.127939\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	− 32.9090i	− 1.34462i	−0.740268	−	0.672312i	$-0.765301\pi$
$599$	− 32.9090i	− 1.34462i	0.740268	−	0.672312i	$-0.234699\pi$
$600$	0	0
$601$	6.92820i	0.282607i	0.989966	+	0.141304i	$0.0451294\pi$
$601$	6.92820i	0.282607i	−0.989966	+	0.141304i	$0.954871\pi$
$602$	0	0
$603$	10.3923i	0.423207i
$604$	0	0
$605$	13.8564i	0.563343i
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	− 43.3013i	− 1.73762i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	12.1244i	0.484200i
$628$	0	0
$629$	− 25.9808i	− 1.03592i
$630$	0	0
$631$	− 45.0333i	− 1.79275i	−0.443298	−	0.896374i	$-0.646192\pi$
$631$	− 45.0333i	− 1.79275i	0.443298	−	0.896374i	$-0.353808\pi$
$632$	0	0
$633$	− 24.2487i	− 0.963800i
$634$	0	0
$635$	6.00000	0.238103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.92820i	0.274075i
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	0	0
$649$	15.5885i	0.611900i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	− 36.3731i	− 1.42121i
$656$	0	0
$657$	3.46410i	0.135147i
$658$	0	0
$659$	− 24.2487i	− 0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$	− 24.2487i	− 0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$	0	0
$661$	39.8372i	1.54949i	0.632276	+	0.774743i	$0.282121\pi$
$661$	39.8372i	1.54949i	−0.632276	+	0.774743i	$0.717879\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 51.9615i	− 2.01196i
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−15.0000	−0.579069
$672$	0	0
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	0	0
$675$	−10.0000	−0.384900
$676$	0	0
$677$	− 39.8372i	− 1.53107i	−0.643396	−	0.765533i	$-0.722475\pi$
$677$	− 39.8372i	− 1.53107i	0.643396	−	0.765533i	$-0.277525\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	43.3013i	1.65688i	0.560080	+	0.828439i	$0.310770\pi$
$683$	43.3013i	1.65688i	−0.560080	+	0.828439i	$0.689230\pi$
$684$	0	0
$685$	5.19615i	0.198535i
$686$	0	0
$687$	− 12.1244i	− 0.462573i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.0000	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 6.92820i	− 0.262802i
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	0	0
$705$	5.19615i	0.195698i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	10.3923i	0.389742i
$712$	0	0
$713$	43.3013i	1.62165i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.3923i	− 0.388108i
$718$	0	0
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12.1244i	0.450910i
$724$	0	0
$725$	−12.0000	−0.445669
$726$	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
$729$	13.0000	0.481481
$730$	0	0
$731$	−18.0000	−0.665754
$732$	0	0
$733$	− 19.0526i	− 0.703722i	−0.936052	−	0.351861i	$-0.885549\pi$
$733$	− 19.0526i	− 0.703722i	0.936052	−	0.351861i	$-0.114451\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.00000	0.331519
$738$	0	0
$739$	1.73205i	0.0637145i	0.999492	+	0.0318573i	$0.0101422\pi$
$739$	1.73205i	0.0637145i	−0.999492	+	0.0318573i	$0.989858\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.1051i	1.39794i	0.715150	+	0.698971i	$0.246358\pi$
$743$	38.1051i	1.39794i	−0.715150	+	0.698971i	$0.753642\pi$
$744$	0	0
$745$	− 15.5885i	− 0.571117i
$746$	0	0
$747$	−24.0000	−0.878114
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 39.8372i	− 1.45368i	−0.686807	−	0.726839i	$-0.740988\pi$
$751$	− 39.8372i	− 1.45368i	0.686807	−	0.726839i	$-0.259012\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21.0000	0.764268
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−15.0000	−0.544466
$760$	0	0
$761$	− 5.19615i	− 0.188360i	−0.995555	−	0.0941802i	$-0.969977\pi$
$761$	− 5.19615i	− 0.188360i	0.995555	−	0.0941802i	$-0.0300230\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	18.0000	0.650791
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.92820i	0.249837i	0.992167	+	0.124919i	$0.0398670\pi$
$769$	6.92820i	0.249837i	−0.992167	+	0.124919i	$0.960133\pi$
$770$	0	0
$771$	− 19.0526i	− 0.686161i
$772$	0	0
$773$	− 8.66025i	− 0.311488i	−0.987797	−	0.155744i	$-0.950223\pi$
$773$	− 8.66025i	− 0.311488i	0.987797	−	0.155744i	$-0.0497774\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 48.4974i	− 1.73760i
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	30.0000	1.07211
$784$	0	0
$785$	27.0000	0.963671
$786$	0	0
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	0	0
$789$	8.66025i	0.308313i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 15.5885i	− 0.552866i
$796$	0	0
$797$	− 13.8564i	− 0.490819i	−0.969419	−	0.245410i	$-0.921078\pi$
$797$	− 13.8564i	− 0.490819i	0.969419	−	0.245410i	$-0.0789224\pi$
$798$	0	0
$799$	15.5885i	0.551480i
$800$	0	0
$801$	24.2487i	0.856786i
$802$	0	0
$803$	3.00000	0.105868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.73205i	− 0.0609711i
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	−1.00000	−0.0350715
$814$	0	0
$815$	21.0000	0.735598
$816$	0	0
$817$	24.2487i	0.848355i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−57.0000	−1.98931	−0.994657	−	0.103236i	$-0.967080\pi$
$821$	−57.0000	−1.98931	−0.994657	+	0.103236i	$0.967080\pi$
$822$	0	0
$823$	22.5167i	0.784881i	0.919777	+	0.392441i	$0.128369\pi$
$823$	22.5167i	0.784881i	−0.919777	+	0.392441i	$0.871631\pi$
$824$	0	0
$825$	3.46410i	0.120605i
$826$	0	0
$827$	− 38.1051i	− 1.32504i	−0.749042	−	0.662522i	$-0.769486\pi$
$827$	− 38.1051i	− 1.32504i	0.749042	−	0.662522i	$-0.230514\pi$
$828$	0	0
$829$	− 8.66025i	− 0.300783i	−0.988627	−	0.150392i	$-0.951947\pi$
$829$	− 8.66025i	− 0.300783i	0.988627	−	0.150392i	$-0.0480534\pi$
$830$	0	0
$831$	−17.0000	−0.589723
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.7846i	0.719281i
$836$	0	0
$837$	−25.0000	−0.864126
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	22.5167i	0.774597i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11.0000	−0.377519
$850$	0	0
$851$	− 43.3013i	− 1.48435i
$852$	0	0
$853$	13.8564i	0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$	13.8564i	0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$	0	0
$855$	− 24.2487i	− 0.829288i
$856$	0	0
$857$	− 43.3013i	− 1.47914i	−0.673078	−	0.739572i	$-0.735028\pi$
$857$	− 43.3013i	− 1.47914i	0.673078	−	0.739572i	$-0.264972\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 46.7654i	− 1.59191i	−0.605355	−	0.795956i	$-0.706969\pi$
$863$	− 46.7654i	− 1.59191i	0.605355	−	0.795956i	$-0.293031\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	0	0
$867$	−10.0000	−0.339618
$868$	0	0
$869$	9.00000	0.305304
$870$	0	0
$871$	0	0
$872$	0	0
$873$	13.8564i	0.468968i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.0000	1.58708	0.793539	−	0.608520i	$-0.208236\pi$
$877$	47.0000	1.58708	0.793539	+	0.608520i	$0.208236\pi$
$878$	0	0
$879$	20.7846i	0.701047i
$880$	0	0
$881$	13.8564i	0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$	13.8564i	0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$	0	0
$883$	45.0333i	1.51549i	0.652550	+	0.757746i	$0.273699\pi$
$883$	45.0333i	1.51549i	−0.652550	+	0.757746i	$0.726301\pi$
$884$	0	0
$885$	15.5885i	0.524000i
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.73205i	0.0580259i
$892$	0	0
$893$	21.0000	0.702738
$894$	0	0
$895$	−27.0000	−0.902510
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.0000	−1.00056
$900$	0	0
$901$	− 46.7654i	− 1.55798i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	− 5.19615i	− 0.172535i	−0.996272	−	0.0862677i	$-0.972506\pi$
$907$	− 5.19615i	− 0.172535i	0.996272	−	0.0862677i	$-0.0274940\pi$
$908$	0	0
$909$	− 10.3923i	− 0.344691i
$910$	0	0
$911$	10.3923i	0.344312i	0.985070	+	0.172156i	$0.0550734\pi$
$911$	10.3923i	0.344312i	−0.985070	+	0.172156i	$0.944927\pi$
$912$	0	0
$913$	20.7846i	0.687870i
$914$	0	0
$915$	−15.0000	−0.495885
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.3013i	1.42838i	0.699953	+	0.714189i	$0.253204\pi$
$919$	43.3013i	1.42838i	−0.699953	+	0.714189i	$0.746796\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	2.00000	0.0656886
$928$	0	0
$929$	− 25.9808i	− 0.852401i	−0.904629	−	0.426201i	$-0.859852\pi$
$929$	− 25.9808i	− 0.852401i	0.904629	−	0.426201i	$-0.140148\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	33.0000	1.08037
$934$	0	0
$935$	− 15.5885i	− 0.509797i
$936$	0	0
$937$	− 41.5692i	− 1.35801i	−0.734135	−	0.679004i	$-0.762412\pi$
$937$	− 41.5692i	− 1.35801i	0.734135	−	0.679004i	$-0.237588\pi$
$938$	0	0
$939$	29.4449i	0.960897i
$940$	0	0
$941$	− 29.4449i	− 0.959875i	−0.877303	−	0.479938i	$-0.840659\pi$
$941$	− 29.4449i	− 0.959875i	0.877303	−	0.479938i	$-0.159341\pi$
$942$	0	0
$943$	60.0000	1.95387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.4449i	0.956830i	0.878134	+	0.478415i	$0.158788\pi$
$947$	29.4449i	0.956830i	−0.878134	+	0.478415i	$0.841212\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−21.0000	−0.680972
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	21.0000	0.679544
$956$	0	0
$957$	− 10.3923i	− 0.335936i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	10.3923i	0.334887i
$964$	0	0
$965$	− 8.66025i	− 0.278783i
$966$	0	0
$967$	− 31.1769i	− 1.00258i	−0.865279	−	0.501291i	$-0.832859\pi$
$967$	− 31.1769i	− 1.00258i	0.865279	−	0.501291i	$-0.167141\pi$
$968$	0	0
$969$	36.3731i	1.16847i
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.0000	0.863807	0.431903	−	0.901920i	$-0.357842\pi$
$977$	27.0000	0.863807	0.431903	+	0.901920i	$0.357842\pi$
$978$	0	0
$979$	21.0000	0.671163
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	− 10.3923i	− 0.331126i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.0000	−0.953945
$990$	0	0
$991$	− 5.19615i	− 0.165061i	−0.996589	−	0.0825306i	$-0.973700\pi$
$991$	− 5.19615i	− 0.165061i	0.996589	−	0.0825306i	$-0.0263002\pi$
$992$	0	0
$993$	29.4449i	0.934405i
$994$	0	0
$995$	− 19.0526i	− 0.604007i
$996$	0	0
$997$	− 1.73205i	− 0.0548546i	−0.999624	−	0.0274273i	$-0.991269\pi$
$997$	− 1.73205i	− 0.0548546i	0.999624	−	0.0274273i	$-0.00873148\pi$
$998$	0	0
$999$	25.0000	0.790965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.f.b.783.2		2
3.2	odd	2		7056.2.b.m.1567.1		2
4.3	odd	2		784.2.f.a.783.2		2
7.2	even	3		112.2.p.a.31.1	✓	2
7.3	odd	6		112.2.p.b.47.1	yes	2
7.4	even	3		784.2.p.c.607.1		2
7.5	odd	6		784.2.p.d.31.1		2
7.6	odd	2		784.2.f.a.783.1		2
8.3	odd	2		3136.2.f.b.3135.1		2
8.5	even	2		3136.2.f.a.3135.1		2
12.11	even	2		7056.2.b.b.1567.1		2
21.2	odd	6		1008.2.cs.f.703.1		2
21.17	even	6		1008.2.cs.c.271.1		2
21.20	even	2		7056.2.b.b.1567.2		2
28.3	even	6		112.2.p.a.47.1	yes	2
28.11	odd	6		784.2.p.d.607.1		2
28.19	even	6		784.2.p.c.31.1		2
28.23	odd	6		112.2.p.b.31.1	yes	2
28.27	even	2	inner	784.2.f.b.783.1		2
56.3	even	6		448.2.p.b.383.1		2
56.13	odd	2		3136.2.f.b.3135.2		2
56.27	even	2		3136.2.f.a.3135.2		2
56.37	even	6		448.2.p.b.255.1		2
56.45	odd	6		448.2.p.a.383.1		2
56.51	odd	6		448.2.p.a.255.1		2
84.23	even	6		1008.2.cs.c.703.1		2
84.59	odd	6		1008.2.cs.f.271.1		2
84.83	odd	2		7056.2.b.m.1567.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.p.a.31.1	✓	2	7.2	even	3
112.2.p.a.47.1	yes	2	28.3	even	6
112.2.p.b.31.1	yes	2	28.23	odd	6
112.2.p.b.47.1	yes	2	7.3	odd	6
448.2.p.a.255.1		2	56.51	odd	6
448.2.p.a.383.1		2	56.45	odd	6
448.2.p.b.255.1		2	56.37	even	6
448.2.p.b.383.1		2	56.3	even	6
784.2.f.a.783.1		2	7.6	odd	2
784.2.f.a.783.2		2	4.3	odd	2
784.2.f.b.783.1		2	28.27	even	2	inner
784.2.f.b.783.2		2	1.1	even	1	trivial
784.2.p.c.31.1		2	28.19	even	6
784.2.p.c.607.1		2	7.4	even	3
784.2.p.d.31.1		2	7.5	odd	6
784.2.p.d.607.1		2	28.11	odd	6
1008.2.cs.c.271.1		2	21.17	even	6
1008.2.cs.c.703.1		2	84.23	even	6
1008.2.cs.f.271.1		2	84.59	odd	6
1008.2.cs.f.703.1		2	21.2	odd	6
3136.2.f.a.3135.1		2	8.5	even	2
3136.2.f.a.3135.2		2	56.27	even	2
3136.2.f.b.3135.1		2	8.3	odd	2
3136.2.f.b.3135.2		2	56.13	odd	2
7056.2.b.b.1567.1		2	12.11	even	2
7056.2.b.b.1567.2		2	21.20	even	2
7056.2.b.m.1567.1		2	3.2	odd	2
7056.2.b.m.1567.2		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.73205i q^{5} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.73205i q^{5} -2.00000 q^{9} +1.73205i q^{11} +1.73205i q^{15} +5.19615i q^{17} +7.00000 q^{19} +8.66025i q^{23} +2.00000 q^{25} -5.00000 q^{27} -6.00000 q^{29} +5.00000 q^{31} +1.73205i q^{33} -5.00000 q^{37} -6.92820i q^{41} +3.46410i q^{43} -3.46410i q^{45} +3.00000 q^{47} +5.19615i q^{51} -9.00000 q^{53} -3.00000 q^{55} +7.00000 q^{57} +9.00000 q^{59} +8.66025i q^{61} -5.19615i q^{67} +8.66025i q^{69} -3.46410i q^{71} -1.73205i q^{73} +2.00000 q^{75} -5.19615i q^{79} +1.00000 q^{81} +12.0000 q^{83} -9.00000 q^{85} -6.00000 q^{87} -12.1244i q^{89} +5.00000 q^{93} +12.1244i q^{95} -6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^9	\( 2 q + 2 q^{3} - 4 q^{9} + 14 q^{19} + 4 q^{25} - 10 q^{27} - 12 q^{29} + 10 q^{31} - 10 q^{37} + 6 q^{47} - 18 q^{53} - 6 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{75} + 2 q^{81} + 24 q^{83} - 18 q^{85} - 12 q^{87} + 10 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 + 14 * q^19 + 4 * q^25 - 10 * q^27 - 12 * q^29 + 10 * q^31 - 10 * q^37 + 6 * q^47 - 18 * q^53 - 6 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^75 + 2 * q^81 + 24 * q^83 - 18 * q^85 - 12 * q^87 + 10 * q^93

Introduction

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