LMFDB - Embedded newform 72.16.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,16,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	72.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:	$102.739323672$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+42010.0 q^{5} +385728. q^{7} +O(q^{10})$

$q+42010.0 q^{5} +385728. q^{7} -2.04443e7 q^{11} +7.52213e7 q^{13} -6.39708e8 q^{17} +2.67006e8 q^{19} -5.71926e8 q^{23} -2.87527e10 q^{25} +6.24379e10 q^{29} -1.24336e9 q^{31} +1.62044e10 q^{35} -4.08248e11 q^{37} +1.37682e12 q^{41} -1.92352e12 q^{43} +4.74341e12 q^{47} -4.59878e12 q^{49} +3.26049e12 q^{53} -8.58865e11 q^{55} -6.73404e12 q^{59} -1.04988e13 q^{61} +3.16005e12 q^{65} +4.21863e13 q^{67} -8.15267e13 q^{71} -3.75504e13 q^{73} -7.88594e12 q^{77} +1.69301e14 q^{79} -3.57834e14 q^{83} -2.68741e13 q^{85} +1.57672e14 q^{89} +2.90150e13 q^{91} +1.12169e13 q^{95} -1.22044e15 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	42010.0	0.240479	0.120240	−	0.992745i	$-0.461634\pi$
$5$	42010.0	0.240479	0.120240	+	0.992745i	$0.461634\pi$
$6$	0	0
$7$	385728.	0.177030	0.0885148	−	0.996075i	$-0.471788\pi$
$7$	385728.	0.177030	0.0885148	+	0.996075i	$0.471788\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.04443e7	−0.316320	−0.158160	−	0.987413i	$-0.550556\pi$
$11$	−2.04443e7	−0.316320	−0.158160	+	0.987413i	$0.550556\pi$
$12$	0	0
$13$	7.52213e7	0.332480	0.166240	−	0.986085i	$-0.446837\pi$
$13$	7.52213e7	0.332480	0.166240	+	0.986085i	$0.446837\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.39708e8	−0.378107	−0.189053	−	0.981967i	$-0.560542\pi$
$17$	−6.39708e8	−0.378107	−0.189053	+	0.981967i	$0.560542\pi$
$18$	0	0
$19$	2.67006e8	0.0685282	0.0342641	−	0.999413i	$-0.489091\pi$
$19$	2.67006e8	0.0685282	0.0342641	+	0.999413i	$0.489091\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.71926e8	−0.0350252	−0.0175126	−	0.999847i	$-0.505575\pi$
$23$	−5.71926e8	−0.0350252	−0.0175126	+	0.999847i	$0.505575\pi$
$24$	0	0
$25$	−2.87527e10	−0.942170
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.24379e10	0.672146	0.336073	−	0.941836i	$-0.390901\pi$
$29$	6.24379e10	0.672146	0.336073	+	0.941836i	$0.390901\pi$
$30$	0	0
$31$	−1.24336e9	−0.00811677	−0.00405838	−	0.999992i	$-0.501292\pi$
$31$	−1.24336e9	−0.00811677	−0.00405838	+	0.999992i	$0.501292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.62044e10	0.0425719
$36$	0	0
$37$	−4.08248e11	−0.706986	−0.353493	−	0.935437i	$-0.615006\pi$
$37$	−4.08248e11	−0.706986	−0.353493	+	0.935437i	$0.615006\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.37682e12	1.10407	0.552035	−	0.833821i	$-0.313851\pi$
$41$	1.37682e12	1.10407	0.552035	+	0.833821i	$0.313851\pi$
$42$	0	0
$43$	−1.92352e12	−1.07915	−0.539577	−	0.841936i	$-0.681416\pi$
$43$	−1.92352e12	−1.07915	−0.539577	+	0.841936i	$0.681416\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.74341e12	1.36570	0.682852	−	0.730556i	$-0.260739\pi$
$47$	4.74341e12	1.36570	0.682852	+	0.730556i	$0.260739\pi$
$48$	0	0
$49$	−4.59878e12	−0.968661
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.26049e12	0.381254	0.190627	−	0.981663i	$-0.438948\pi$
$53$	3.26049e12	0.381254	0.190627	+	0.981663i	$0.438948\pi$
$54$	0	0
$55$	−8.58865e11	−0.0760685
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.73404e12	−0.352278	−0.176139	−	0.984365i	$-0.556361\pi$
$59$	−6.73404e12	−0.352278	−0.176139	+	0.984365i	$0.556361\pi$
$60$	0	0
$61$	−1.04988e13	−0.427724	−0.213862	−	0.976864i	$-0.568604\pi$
$61$	−1.04988e13	−0.427724	−0.213862	+	0.976864i	$0.568604\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.16005e12	0.0799546
$66$	0	0
$67$	4.21863e13	0.850375	0.425188	−	0.905105i	$-0.360208\pi$
$67$	4.21863e13	0.850375	0.425188	+	0.905105i	$0.360208\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.15267e13	−1.06381	−0.531903	−	0.846806i	$-0.678523\pi$
$71$	−8.15267e13	−1.06381	−0.531903	+	0.846806i	$0.678523\pi$
$72$	0	0
$73$	−3.75504e13	−0.397826	−0.198913	−	0.980017i	$-0.563741\pi$
$73$	−3.75504e13	−0.397826	−0.198913	+	0.980017i	$0.563741\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.88594e12	−0.0559981
$78$	0	0
$79$	1.69301e14	0.991876	0.495938	−	0.868358i	$-0.334824\pi$
$79$	1.69301e14	0.991876	0.495938	+	0.868358i	$0.334824\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.57834e14	−1.44742	−0.723711	−	0.690103i	$-0.757565\pi$
$83$	−3.57834e14	−1.44742	−0.723711	+	0.690103i	$0.757565\pi$
$84$	0	0
$85$	−2.68741e13	−0.0909269
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.57672e14	0.377859	0.188930	−	0.981991i	$-0.439498\pi$
$89$	1.57672e14	0.377859	0.188930	+	0.981991i	$0.439498\pi$
$90$	0	0
$91$	2.90150e13	0.0588588
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.12169e13	0.0164796
$96$	0	0
$97$	−1.22044e15	−1.53366	−0.766831	−	0.641850i	$-0.778167\pi$
$97$	−1.22044e15	−1.53366	−0.766831	+	0.641850i	$0.778167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.11557e14	0.846007	0.423003	−	0.906128i	$-0.360976\pi$
$101$	9.11557e14	0.846007	0.423003	+	0.906128i	$0.360976\pi$
$102$	0	0
$103$	−1.33377e15	−1.06857	−0.534283	−	0.845306i	$-0.679418\pi$
$103$	−1.33377e15	−1.06857	−0.534283	+	0.845306i	$0.679418\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.19234e15	−0.717832	−0.358916	−	0.933370i	$-0.616854\pi$
$107$	−1.19234e15	−0.717832	−0.358916	+	0.933370i	$0.616854\pi$
$108$	0	0
$109$	−3.01769e15	−1.58116	−0.790579	−	0.612360i	$-0.790220\pi$
$109$	−3.01769e15	−1.58116	−0.790579	+	0.612360i	$0.790220\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.89307e14	0.115683	0.0578417	−	0.998326i	$-0.481578\pi$
$113$	2.89307e14	0.115683	0.0578417	+	0.998326i	$0.481578\pi$
$114$	0	0
$115$	−2.40266e13	−0.00842284
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.46753e14	−0.0669361
$120$	0	0
$121$	−3.75928e15	−0.899941
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.48995e15	−0.467052
$126$	0	0
$127$	−5.31096e15	−0.884393	−0.442197	−	0.896918i	$-0.645801\pi$
$127$	−5.31096e15	−0.884393	−0.442197	+	0.896918i	$0.645801\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.39810e15	−0.448438	−0.224219	−	0.974539i	$-0.571983\pi$
$131$	−3.39810e15	−0.448438	−0.224219	+	0.974539i	$0.571983\pi$
$132$	0	0
$133$	1.02992e14	0.0121315
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.79395e16	−1.69202	−0.846010	−	0.533167i	$-0.821002\pi$
$137$	−1.79395e16	−1.69202	−0.846010	+	0.533167i	$0.821002\pi$
$138$	0	0
$139$	−6.25586e15	−0.529268	−0.264634	−	0.964349i	$-0.585251\pi$
$139$	−6.25586e15	−0.529268	−0.264634	+	0.964349i	$0.585251\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.53785e15	−0.105170
$144$	0	0
$145$	2.62302e15	0.161637
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.68134e16	−1.34727	−0.673636	−	0.739063i	$-0.735269\pi$
$149$	−2.68134e16	−1.34727	−0.673636	+	0.739063i	$0.735269\pi$
$150$	0	0
$151$	−2.29535e16	−1.04357	−0.521786	−	0.853076i	$-0.674734\pi$
$151$	−2.29535e16	−1.04357	−0.521786	+	0.853076i	$0.674734\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.22334e13	−0.00195191
$156$	0	0
$157$	−9.26184e15	−0.314376	−0.157188	−	0.987569i	$-0.550243\pi$
$157$	−9.26184e15	−0.314376	−0.157188	+	0.987569i	$0.550243\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.20608e14	−0.00620050
$162$	0	0
$163$	−5.24269e16	−1.34322	−0.671610	−	0.740905i	$-0.734397\pi$
$163$	−5.24269e16	−1.34322	−0.671610	+	0.740905i	$0.734397\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.87173e16	0.827050	0.413525	−	0.910493i	$-0.364297\pi$
$167$	3.87173e16	0.827050	0.413525	+	0.910493i	$0.364297\pi$
$168$	0	0
$169$	−4.55276e16	−0.889457
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.29920e15	0.119655	0.0598273	−	0.998209i	$-0.480945\pi$
$173$	7.29920e15	0.119655	0.0598273	+	0.998209i	$0.480945\pi$
$174$	0	0
$175$	−1.10907e16	−0.166792
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.46685e16	0.693968	0.346984	−	0.937871i	$-0.387206\pi$
$179$	5.46685e16	0.693968	0.346984	+	0.937871i	$0.387206\pi$
$180$	0	0
$181$	−2.89939e16	−0.338624	−0.169312	−	0.985563i	$-0.554154\pi$
$181$	−2.89939e16	−0.338624	−0.169312	+	0.985563i	$0.554154\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.71505e16	−0.170016
$186$	0	0
$187$	1.30784e16	0.119603
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.56191e15	−0.0746096	−0.0373048	−	0.999304i	$-0.511877\pi$
$191$	−9.56191e15	−0.0746096	−0.0373048	+	0.999304i	$0.511877\pi$
$192$	0	0
$193$	1.83986e14	0.00132771	0.000663856	−	1.00000i	$-0.499789\pi$
$193$	1.83986e14	0.00132771	0.000663856		1.00000i	$0.499789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.22633e16	−0.385244	−0.192622	−	0.981273i	$-0.561699\pi$
$197$	−6.22633e16	−0.385244	−0.192622	+	0.981273i	$0.561699\pi$
$198$	0	0
$199$	2.19808e17	1.26080	0.630400	−	0.776271i	$-0.282891\pi$
$199$	2.19808e17	1.26080	0.630400	+	0.776271i	$0.282891\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.40841e16	0.118990
$204$	0	0
$205$	5.78400e16	0.265506
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.45876e15	−0.0216769
$210$	0	0
$211$	3.51558e17	1.29981	0.649903	−	0.760017i	$-0.274809\pi$
$211$	3.51558e17	1.29981	0.649903	+	0.760017i	$0.274809\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.08071e16	−0.259514
$216$	0	0
$217$	−4.79598e14	−0.00143691
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.81196e16	−0.125713
$222$	0	0
$223$	8.04389e17	1.96417	0.982086	−	0.188432i	$-0.0603404\pi$
$223$	8.04389e17	1.96417	0.982086	+	0.188432i	$0.0603404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.02068e17	0.218120	0.109060	−	0.994035i	$-0.465216\pi$
$227$	1.02068e17	0.218120	0.109060	+	0.994035i	$0.465216\pi$
$228$	0	0
$229$	2.83060e17	0.566385	0.283192	−	0.959063i	$-0.408606\pi$
$229$	2.83060e17	0.566385	0.283192	+	0.959063i	$0.408606\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.74151e17	0.481748	0.240874	−	0.970556i	$-0.422566\pi$
$233$	2.74151e17	0.481748	0.240874	+	0.970556i	$0.422566\pi$
$234$	0	0
$235$	1.99271e17	0.328424
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.17707e17	1.18744	0.593722	−	0.804670i	$-0.297658\pi$
$239$	8.17707e17	1.18744	0.593722	+	0.804670i	$0.297658\pi$
$240$	0	0
$241$	−4.77148e17	−0.650916	−0.325458	−	0.945556i	$-0.605519\pi$
$241$	−4.77148e17	−0.650916	−0.325458	+	0.945556i	$0.605519\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.93195e17	−0.232943
$246$	0	0
$247$	2.00846e16	0.0227843
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.51217e18	1.52072	0.760358	−	0.649504i	$-0.225023\pi$
$251$	1.51217e18	1.52072	0.760358	+	0.649504i	$0.225023\pi$
$252$	0	0
$253$	1.16926e16	0.0110792
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.46827e17	−0.207919	−0.103960	−	0.994582i	$-0.533151\pi$
$257$	−2.46827e17	−0.207919	−0.103960	+	0.994582i	$0.533151\pi$
$258$	0	0
$259$	−1.57473e17	−0.125157
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.95561e18	1.38552	0.692760	−	0.721168i	$-0.256394\pi$
$263$	1.95561e18	1.38552	0.692760	+	0.721168i	$0.256394\pi$
$264$	0	0
$265$	1.36973e17	0.0916836
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.86530e18	−1.71407	−0.857033	−	0.515262i	$-0.827695\pi$
$269$	−2.86530e18	−1.71407	−0.857033	+	0.515262i	$0.827695\pi$
$270$	0	0
$271$	−7.93544e17	−0.449057	−0.224528	−	0.974468i	$-0.572084\pi$
$271$	−7.93544e17	−0.449057	−0.224528	+	0.974468i	$0.572084\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.87830e17	0.298027
$276$	0	0
$277$	1.80981e18	0.869029	0.434514	−	0.900665i	$-0.356920\pi$
$277$	1.80981e18	0.869029	0.434514	+	0.900665i	$0.356920\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.34507e18	−0.580025	−0.290012	−	0.957023i	$-0.593659\pi$
$281$	−1.34507e18	−0.580025	−0.290012	+	0.957023i	$0.593659\pi$
$282$	0	0
$283$	9.79462e17	0.400488	0.200244	−	0.979746i	$-0.435827\pi$
$283$	9.79462e17	0.400488	0.200244	+	0.979746i	$0.435827\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.31076e17	0.195453
$288$	0	0
$289$	−2.45320e18	−0.857035
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.18129e18	−1.63279	−0.816396	−	0.577492i	$-0.804031\pi$
$293$	−5.18129e18	−1.63279	−0.816396	+	0.577492i	$0.804031\pi$
$294$	0	0
$295$	−2.82897e17	−0.0847156
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.30210e16	−0.0116452
$300$	0	0
$301$	−7.41956e17	−0.191042
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.41053e17	−0.102859
$306$	0	0
$307$	1.46204e18	0.324655	0.162328	−	0.986737i	$-0.448100\pi$
$307$	1.46204e18	0.324655	0.162328	+	0.986737i	$0.448100\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.44852e18	0.291892	0.145946	−	0.989293i	$-0.453377\pi$
$311$	1.44852e18	0.291892	0.145946	+	0.989293i	$0.453377\pi$
$312$	0	0
$313$	−6.13267e18	−1.17779	−0.588894	−	0.808210i	$-0.700436\pi$
$313$	−6.13267e18	−1.17779	−0.588894	+	0.808210i	$0.700436\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.42558e18	−1.29654	−0.648270	−	0.761410i	$-0.724507\pi$
$317$	−7.42558e18	−1.29654	−0.648270	+	0.761410i	$0.724507\pi$
$318$	0	0
$319$	−1.27650e18	−0.212613
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.70806e17	−0.0259110
$324$	0	0
$325$	−2.16282e18	−0.313253
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.82967e18	0.241770
$330$	0	0
$331$	−5.55453e18	−0.701354	−0.350677	−	0.936496i	$-0.614048\pi$
$331$	−5.55453e18	−0.701354	−0.350677	+	0.936496i	$0.614048\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.77225e18	0.204498
$336$	0	0
$337$	−1.48547e19	−1.63923	−0.819613	−	0.572918i	$-0.805811\pi$
$337$	−1.48547e19	−1.63923	−0.819613	+	0.572918i	$0.805811\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.54196e16	0.00256750
$342$	0	0
$343$	−3.60514e18	−0.348511
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00836e18	0.177980	0.0889899	−	0.996033i	$-0.471636\pi$
$347$	2.00836e18	0.177980	0.0889899	+	0.996033i	$0.471636\pi$
$348$	0	0
$349$	−5.33006e18	−0.452420	−0.226210	−	0.974079i	$-0.572634\pi$
$349$	−5.33006e18	−0.452420	−0.226210	+	0.974079i	$0.572634\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.52521e19	1.18855	0.594276	−	0.804261i	$-0.297439\pi$
$353$	1.52521e19	1.18855	0.594276	+	0.804261i	$0.297439\pi$
$354$	0	0
$355$	−3.42494e18	−0.255823
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.47244e18	−0.650509	−0.325255	−	0.945627i	$-0.605450\pi$
$359$	−9.47244e18	−0.650509	−0.325255	+	0.945627i	$0.605450\pi$
$360$	0	0
$361$	−1.51098e19	−0.995304
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.57749e18	−0.0956690
$366$	0	0
$367$	1.74332e19	1.01480	0.507401	−	0.861710i	$-0.330606\pi$
$367$	1.74332e19	1.01480	0.507401	+	0.861710i	$0.330606\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.25766e18	0.0674932
$372$	0	0
$373$	2.62033e19	1.35064	0.675320	−	0.737525i	$-0.264006\pi$
$373$	2.62033e19	1.35064	0.675320	+	0.737525i	$0.264006\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.69666e18	0.223475
$378$	0	0
$379$	2.19755e19	1.00495	0.502475	−	0.864592i	$-0.332423\pi$
$379$	2.19755e19	1.00495	0.502475	+	0.864592i	$0.332423\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.73716e19	−0.734260	−0.367130	−	0.930170i	$-0.619660\pi$
$383$	−1.73716e19	−0.734260	−0.367130	+	0.930170i	$0.619660\pi$
$384$	0	0
$385$	−3.31288e17	−0.0134664
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.01170e19	−1.13289	−0.566447	−	0.824098i	$-0.691682\pi$
$389$	−3.01170e19	−1.13289	−0.566447	+	0.824098i	$0.691682\pi$
$390$	0	0
$391$	3.65865e17	0.0132433
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.11235e18	0.238526
$396$	0	0
$397$	5.76562e19	1.86173	0.930866	−	0.365362i	$-0.119055\pi$
$397$	5.76562e19	1.86173	0.930866	+	0.365362i	$0.119055\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.30863e19	0.990982	0.495491	−	0.868613i	$-0.334988\pi$
$401$	3.30863e19	0.990982	0.495491	+	0.868613i	$0.334988\pi$
$402$	0	0
$403$	−9.35269e16	−0.00269866
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.34634e18	0.223634
$408$	0	0
$409$	2.00590e19	0.518066	0.259033	−	0.965869i	$-0.416596\pi$
$409$	2.00590e19	0.518066	0.259033	+	0.965869i	$0.416596\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.59751e18	−0.0623636
$414$	0	0
$415$	−1.50326e19	−0.348075
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.27792e18	0.156820	0.0784101	−	0.996921i	$-0.475016\pi$
$419$	7.27792e18	0.156820	0.0784101	+	0.996921i	$0.475016\pi$
$420$	0	0
$421$	5.40680e19	1.12415	0.562076	−	0.827086i	$-0.310003\pi$
$421$	5.40680e19	1.12415	0.562076	+	0.827086i	$0.310003\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.83933e19	0.356241
$426$	0	0
$427$	−4.04966e18	−0.0757198
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.53838e19	0.616915	0.308458	−	0.951238i	$-0.400187\pi$
$431$	3.53838e19	0.616915	0.308458	+	0.951238i	$0.400187\pi$
$432$	0	0
$433$	4.71960e18	0.0794778	0.0397389	−	0.999210i	$-0.487347\pi$
$433$	4.71960e18	0.0794778	0.0397389	+	0.999210i	$0.487347\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.52708e17	−0.00240022
$438$	0	0
$439$	−1.56344e19	−0.237464	−0.118732	−	0.992926i	$-0.537883\pi$
$439$	−1.56344e19	−0.237464	−0.118732	+	0.992926i	$0.537883\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.76193e18	0.0959494	0.0479747	−	0.998849i	$-0.484723\pi$
$443$	6.76193e18	0.0959494	0.0479747	+	0.998849i	$0.484723\pi$
$444$	0	0
$445$	6.62381e18	0.0908673
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.21681e20	1.56090	0.780452	−	0.625216i	$-0.214989\pi$
$449$	1.21681e20	1.56090	0.780452	+	0.625216i	$0.214989\pi$
$450$	0	0
$451$	−2.81480e19	−0.349240
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.21892e18	0.0141543
$456$	0	0
$457$	1.43486e19	0.161227	0.0806137	−	0.996745i	$-0.474312\pi$
$457$	1.43486e19	0.161227	0.0806137	+	0.996745i	$0.474312\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.72277e19	0.918116	0.459058	−	0.888406i	$-0.348187\pi$
$461$	8.72277e19	0.918116	0.459058	+	0.888406i	$0.348187\pi$
$462$	0	0
$463$	−4.05405e19	−0.413077	−0.206539	−	0.978438i	$-0.566220\pi$
$463$	−4.05405e19	−0.413077	−0.206539	+	0.978438i	$0.566220\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.06168e20	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$467$	−1.06168e20	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$468$	0	0
$469$	1.62724e19	0.150542
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.93251e19	0.341359
$474$	0	0
$475$	−7.67716e18	−0.0645652
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.10758e20	−0.874698	−0.437349	−	0.899292i	$-0.644083\pi$
$479$	−1.10758e20	−0.874698	−0.437349	+	0.899292i	$0.644083\pi$
$480$	0	0
$481$	−3.07089e19	−0.235059
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.12708e19	−0.368814
$486$	0	0
$487$	5.30089e18	0.0369728	0.0184864	−	0.999829i	$-0.494115\pi$
$487$	5.30089e18	0.0369728	0.0184864	+	0.999829i	$0.494115\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.00353e20	−1.31427	−0.657134	−	0.753774i	$-0.728232\pi$
$491$	−2.00353e20	−1.31427	−0.657134	+	0.753774i	$0.728232\pi$
$492$	0	0
$493$	−3.99420e19	−0.254143
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.14471e19	−0.188325
$498$	0	0
$499$	1.64569e19	0.0956298	0.0478149	−	0.998856i	$-0.484774\pi$
$499$	1.64569e19	0.0956298	0.0478149	+	0.998856i	$0.484774\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.25757e20	−1.23561	−0.617805	−	0.786331i	$-0.711978\pi$
$503$	−2.25757e20	−1.23561	−0.617805	+	0.786331i	$0.711978\pi$
$504$	0	0
$505$	3.82945e19	0.203447
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.51070e19	0.225871	0.112935	−	0.993602i	$-0.463975\pi$
$509$	4.51070e19	0.225871	0.112935	+	0.993602i	$0.463975\pi$
$510$	0	0
$511$	−1.44843e19	−0.0704271
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.60316e19	−0.256968
$516$	0	0
$517$	−9.69757e19	−0.432000
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.16580e17	−0.00301288	−0.00150644	−	0.999999i	$-0.500480\pi$
$521$	−7.16580e17	−0.00301288	−0.00150644	+	0.999999i	$0.500480\pi$
$522$	0	0
$523$	−6.60807e19	−0.269968	−0.134984	−	0.990848i	$-0.543098\pi$
$523$	−6.60807e19	−0.269968	−0.134984	+	0.990848i	$0.543098\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.95385e17	0.00306901
$528$	0	0
$529$	−2.66308e20	−0.998773
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.03566e20	0.367082
$534$	0	0
$535$	−5.00903e19	−0.172624
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.40187e19	0.306407
$540$	0	0
$541$	−4.34464e20	−1.37713	−0.688563	−	0.725176i	$-0.741758\pi$
$541$	−4.34464e20	−1.37713	−0.688563	+	0.725176i	$0.741758\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.26773e20	−0.380236
$546$	0	0
$547$	−2.57432e20	−0.751204	−0.375602	−	0.926781i	$-0.622564\pi$
$547$	−2.57432e20	−0.751204	−0.375602	+	0.926781i	$0.622564\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.66713e19	0.0460610
$552$	0	0
$553$	6.53043e19	0.175591
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.86159e20	0.728944	0.364472	−	0.931214i	$-0.381249\pi$
$557$	2.86159e20	0.728944	0.364472	+	0.931214i	$0.381249\pi$
$558$	0	0
$559$	−1.44690e20	−0.358797
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.12051e20	0.498456	0.249228	−	0.968445i	$-0.419823\pi$
$563$	2.12051e20	0.498456	0.249228	+	0.968445i	$0.419823\pi$
$564$	0	0
$565$	1.21538e19	0.0278195
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.20532e20	−0.695871	−0.347936	−	0.937518i	$-0.613117\pi$
$569$	−3.20532e20	−0.695871	−0.347936	+	0.937518i	$0.613117\pi$
$570$	0	0
$571$	−1.50635e20	−0.318534	−0.159267	−	0.987236i	$-0.550913\pi$
$571$	−1.50635e20	−0.318534	−0.159267	+	0.987236i	$0.550913\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.64444e19	0.0329997
$576$	0	0
$577$	2.21986e20	0.434018	0.217009	−	0.976170i	$-0.430370\pi$
$577$	2.21986e20	0.434018	0.217009	+	0.976170i	$0.430370\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.38026e20	−0.256237
$582$	0	0
$583$	−6.66585e19	−0.120598
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.75683e20	1.50509	0.752543	−	0.658544i	$-0.228827\pi$
$587$	8.75683e20	1.50509	0.752543	+	0.658544i	$0.228827\pi$
$588$	0	0
$589$	−3.31984e17	−0.000556228 0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.24620e20	−1.15398	−0.576992	−	0.816749i	$-0.695774\pi$
$593$	−7.24620e20	−1.15398	−0.576992	+	0.816749i	$0.695774\pi$
$594$	0	0
$595$	−1.03661e19	−0.0160968
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.56978e20	0.527157	0.263579	−	0.964638i	$-0.415097\pi$
$599$	3.56978e20	0.527157	0.263579	+	0.964638i	$0.415097\pi$
$600$	0	0
$601$	1.13334e21	1.63231	0.816153	−	0.577836i	$-0.196103\pi$
$601$	1.13334e21	1.63231	0.816153	+	0.577836i	$0.196103\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.57927e20	−0.216417
$606$	0	0
$607$	3.41458e20	0.456480	0.228240	−	0.973605i	$-0.426703\pi$
$607$	3.41458e20	0.456480	0.228240	+	0.973605i	$0.426703\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.56806e20	0.454070
$612$	0	0
$613$	1.61060e20	0.200002	0.100001	−	0.994987i	$-0.468115\pi$
$613$	1.61060e20	0.200002	0.100001	+	0.994987i	$0.468115\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.04065e21	−1.23074	−0.615371	−	0.788238i	$-0.710994\pi$
$617$	−1.04065e21	−1.23074	−0.615371	+	0.788238i	$0.710994\pi$
$618$	0	0
$619$	9.65882e20	1.11492	0.557461	−	0.830203i	$-0.311776\pi$
$619$	9.65882e20	1.11492	0.557461	+	0.830203i	$0.311776\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.08186e19	0.0668922
$624$	0	0
$625$	7.72861e20	0.829854
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.61159e20	0.267316
$630$	0	0
$631$	1.06482e21	1.06428	0.532140	−	0.846656i	$-0.321388\pi$
$631$	1.06482e21	1.06428	0.532140	+	0.846656i	$0.321388\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.23114e20	−0.212678
$636$	0	0
$637$	−3.45926e20	−0.322060
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.06674e20	−0.272422	−0.136211	−	0.990680i	$-0.543492\pi$
$641$	−3.06674e20	−0.272422	−0.136211	+	0.990680i	$0.543492\pi$
$642$	0	0
$643$	1.68814e21	1.46496	0.732482	−	0.680786i	$-0.238362\pi$
$643$	1.68814e21	1.46496	0.732482	+	0.680786i	$0.238362\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.09251e21	0.904992	0.452496	−	0.891767i	$-0.350534\pi$
$647$	1.09251e21	0.904992	0.452496	+	0.891767i	$0.350534\pi$
$648$	0	0
$649$	1.37673e20	0.111433
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.76377e20	0.677395	0.338698	−	0.940895i	$-0.390014\pi$
$653$	8.76377e20	0.677395	0.338698	+	0.940895i	$0.390014\pi$
$654$	0	0
$655$	−1.42754e20	−0.107840
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.51631e20	0.398115	0.199057	−	0.979988i	$-0.436212\pi$
$659$	5.51631e20	0.398115	0.199057	+	0.979988i	$0.436212\pi$
$660$	0	0
$661$	−1.66285e21	−1.17312	−0.586559	−	0.809906i	$-0.699518\pi$
$661$	−1.66285e21	−1.17312	−0.586559	+	0.809906i	$0.699518\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.32669e18	0.00291738
$666$	0	0
$667$	−3.57098e19	−0.0235420
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.14640e20	0.135298
$672$	0	0
$673$	−4.08044e20	−0.251532	−0.125766	−	0.992060i	$-0.540139\pi$
$673$	−4.08044e20	−0.251532	−0.125766	+	0.992060i	$0.540139\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.63767e20	0.450346	0.225173	−	0.974319i	$-0.427705\pi$
$677$	7.63767e20	0.450346	0.225173	+	0.974319i	$0.427705\pi$
$678$	0	0
$679$	−4.70759e20	−0.271503
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.77928e20	0.539699	0.269850	−	0.962902i	$-0.413026\pi$
$683$	9.77928e20	0.539699	0.269850	+	0.962902i	$0.413026\pi$
$684$	0	0
$685$	−7.53638e20	−0.406896
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.45258e20	0.126759
$690$	0	0
$691$	1.15976e21	0.586519	0.293259	−	0.956033i	$-0.405260\pi$
$691$	1.15976e21	0.586519	0.293259	+	0.956033i	$0.405260\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.62809e20	−0.127278
$696$	0	0
$697$	−8.80760e20	−0.417457
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.06877e20	−0.0939348	−0.0469674	−	0.998896i	$-0.514956\pi$
$701$	−2.06877e20	−0.0939348	−0.0469674	+	0.998896i	$0.514956\pi$
$702$	0	0
$703$	−1.09005e20	−0.0484485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.51613e20	0.149768
$708$	0	0
$709$	−1.94204e21	−0.809865	−0.404932	−	0.914347i	$-0.632705\pi$
$709$	−1.94204e21	−0.809865	−0.404932	+	0.914347i	$0.632705\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.11108e17	0.000284292 0
$714$	0	0
$715$	−6.46049e19	−0.0252913
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.13366e21	−0.801048	−0.400524	−	0.916286i	$-0.631172\pi$
$719$	−2.13366e21	−0.801048	−0.400524	+	0.916286i	$0.631172\pi$
$720$	0	0
$721$	−5.14472e20	−0.189168
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.79526e21	−0.633275
$726$	0	0
$727$	−1.56209e21	−0.539757	−0.269878	−	0.962894i	$-0.586984\pi$
$727$	−1.56209e21	−0.539757	−0.269878	+	0.962894i	$0.586984\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.23049e21	0.408036
$732$	0	0
$733$	2.13800e21	0.694589	0.347295	−	0.937756i	$-0.387100\pi$
$733$	2.13800e21	0.694589	0.347295	+	0.937756i	$0.387100\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.62470e20	−0.268991
$738$	0	0
$739$	−2.44319e20	−0.0746661	−0.0373331	−	0.999303i	$-0.511886\pi$
$739$	−2.44319e20	−0.0746661	−0.0373331	+	0.999303i	$0.511886\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.17781e21	−1.51960	−0.759802	−	0.650155i	$-0.774704\pi$
$743$	−5.17781e21	−1.51960	−0.759802	+	0.650155i	$0.774704\pi$
$744$	0	0
$745$	−1.12643e21	−0.323991
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.59920e20	−0.127078
$750$	0	0
$751$	−3.35351e21	−0.908238	−0.454119	−	0.890941i	$-0.650046\pi$
$751$	−3.35351e21	−0.908238	−0.454119	+	0.890941i	$0.650046\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.64278e20	−0.250957
$756$	0	0
$757$	1.79273e21	0.457399	0.228699	−	0.973497i	$-0.426553\pi$
$757$	1.79273e21	0.457399	0.228699	+	0.973497i	$0.426553\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.56278e20	−0.185480	−0.0927399	−	0.995690i	$-0.529563\pi$
$761$	−7.56278e20	−0.185480	−0.0927399	+	0.995690i	$0.529563\pi$
$762$	0	0
$763$	−1.16401e21	−0.279912
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.06543e20	−0.117125
$768$	0	0
$769$	−1.62929e20	−0.0369447	−0.0184723	−	0.999829i	$-0.505880\pi$
$769$	−1.62929e20	−0.0369447	−0.0184723	+	0.999829i	$0.505880\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.72838e21	−0.595058	−0.297529	−	0.954713i	$-0.596162\pi$
$773$	−2.72838e21	−0.595058	−0.297529	+	0.954713i	$0.596162\pi$
$774$	0	0
$775$	3.57499e19	0.00764737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.67618e20	0.0756600
$780$	0	0
$781$	1.66676e21	0.336503
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.89090e20	−0.0756010
$786$	0	0
$787$	−5.57208e21	−1.06220	−0.531100	−	0.847309i	$-0.678221\pi$
$787$	−5.57208e21	−1.06220	−0.531100	+	0.847309i	$0.678221\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.11594e20	0.0204794
$792$	0	0
$793$	−7.89730e20	−0.142210
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.16563e21	−0.895749	−0.447874	−	0.894097i	$-0.647819\pi$
$797$	−5.16563e21	−0.895749	−0.447874	+	0.894097i	$0.647819\pi$
$798$	0	0
$799$	−3.03440e21	−0.516383
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.67693e20	0.125841
$804$	0	0
$805$	−9.26773e18	−0.00149109
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.79086e21	1.36275	0.681377	−	0.731933i	$-0.261381\pi$
$809$	8.79086e21	1.36275	0.681377	+	0.731933i	$0.261381\pi$
$810$	0	0
$811$	−7.33836e21	−1.11672	−0.558358	−	0.829600i	$-0.688568\pi$
$811$	−7.33836e21	−1.11672	−0.558358	+	0.829600i	$0.688568\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.20245e21	−0.323017
$816$	0	0
$817$	−5.13592e20	−0.0739526
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.11179e22	1.54329	0.771647	−	0.636051i	$-0.219433\pi$
$821$	1.11179e22	1.54329	0.771647	+	0.636051i	$0.219433\pi$
$822$	0	0
$823$	−9.34802e20	−0.127415	−0.0637075	−	0.997969i	$-0.520292\pi$
$823$	−9.34802e20	−0.127415	−0.0637075	+	0.997969i	$0.520292\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.53535e21	0.858969	0.429485	−	0.903074i	$-0.358695\pi$
$827$	6.53535e21	0.858969	0.429485	+	0.903074i	$0.358695\pi$
$828$	0	0
$829$	3.20337e21	0.413474	0.206737	−	0.978397i	$-0.433716\pi$
$829$	3.20337e21	0.413474	0.206737	+	0.978397i	$0.433716\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.94187e21	0.366257
$834$	0	0
$835$	1.62651e21	0.198888
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.81725e21	−0.568309	−0.284155	−	0.958778i	$-0.591713\pi$
$839$	−4.81725e21	−0.568309	−0.284155	+	0.958778i	$0.591713\pi$
$840$	0	0
$841$	−4.73070e21	−0.548220
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.91262e21	−0.213896
$846$	0	0
$847$	−1.45006e21	−0.159316
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.33487e20	0.0247623
$852$	0	0
$853$	1.04745e22	1.09148	0.545742	−	0.837953i	$-0.316248\pi$
$853$	1.04745e22	1.09148	0.545742	+	0.837953i	$0.316248\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.58463e22	1.59430	0.797152	−	0.603779i	$-0.206339\pi$
$857$	1.58463e22	1.59430	0.797152	+	0.603779i	$0.206339\pi$
$858$	0	0
$859$	1.46055e22	1.44400	0.722001	−	0.691892i	$-0.243223\pi$
$859$	1.46055e22	1.44400	0.722001	+	0.691892i	$0.243223\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.35151e22	−1.29044	−0.645218	−	0.763998i	$-0.723234\pi$
$863$	−1.35151e22	−1.29044	−0.645218	+	0.763998i	$0.723234\pi$
$864$	0	0
$865$	3.06639e20	0.0287745
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.46125e21	−0.313751
$870$	0	0
$871$	3.17331e21	0.282733
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.60442e20	−0.0826819
$876$	0	0
$877$	1.34760e22	1.14042	0.570210	−	0.821499i	$-0.306862\pi$
$877$	1.34760e22	1.14042	0.570210	+	0.821499i	$0.306862\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.16284e22	0.951041	0.475521	−	0.879705i	$-0.342260\pi$
$881$	1.16284e22	0.951041	0.475521	+	0.879705i	$0.342260\pi$
$882$	0	0
$883$	2.19163e22	1.76223	0.881114	−	0.472904i	$-0.156794\pi$
$883$	2.19163e22	1.76223	0.881114	+	0.472904i	$0.156794\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.99328e22	−1.54932	−0.774660	−	0.632378i	$-0.782079\pi$
$887$	−1.99328e22	−1.54932	−0.774660	+	0.632378i	$0.782079\pi$
$888$	0	0
$889$	−2.04859e21	−0.156564
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.26652e21	0.0935894
$894$	0	0
$895$	2.29662e21	0.166885
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.76326e19	−0.00545565
$900$	0	0
$901$	−2.08576e21	−0.144155
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.21803e21	−0.0814320
$906$	0	0
$907$	4.38465e21	0.288323	0.144162	−	0.989554i	$-0.453951\pi$
$907$	4.38465e21	0.288323	0.144162	+	0.989554i	$0.453951\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.07093e22	0.681354	0.340677	−	0.940180i	$-0.389344\pi$
$911$	1.07093e22	0.681354	0.340677	+	0.940180i	$0.389344\pi$
$912$	0	0
$913$	7.31566e21	0.457849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.31074e21	−0.0793867
$918$	0	0
$919$	−1.35100e21	−0.0804987	−0.0402494	−	0.999190i	$-0.512815\pi$
$919$	−1.35100e21	−0.0804987	−0.0402494	+	0.999190i	$0.512815\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.13254e21	−0.353694
$924$	0	0
$925$	1.17382e22	0.666101
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.57185e21	0.415991	0.207996	−	0.978130i	$-0.433306\pi$
$929$	7.57185e21	0.415991	0.207996	+	0.978130i	$0.433306\pi$
$930$	0	0
$931$	−1.22790e21	−0.0663806
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.49423e20	0.0287620
$936$	0	0
$937$	−1.90115e19	−0.000979422 0	−0.000489711	−	1.00000i	$-0.500156\pi$
$937$	−1.90115e19	−0.000979422 0	−0.000489711		1.00000i	$0.500156\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−7.15736e20	−0.0357134	−0.0178567	−	0.999841i	$-0.505684\pi$
$941$	−7.15736e20	−0.0357134	−0.0178567	+	0.999841i	$0.505684\pi$
$942$	0	0
$943$	−7.87436e20	−0.0386703
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.54930e22	1.21282	0.606409	−	0.795153i	$-0.292610\pi$
$947$	2.54930e22	1.21282	0.606409	+	0.795153i	$0.292610\pi$
$948$	0	0
$949$	−2.82459e21	−0.132269
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.36539e21	−0.379568	−0.189784	−	0.981826i	$-0.560779\pi$
$953$	−8.36539e21	−0.379568	−0.189784	+	0.981826i	$0.560779\pi$
$954$	0	0
$955$	−4.01696e20	−0.0179421
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.91976e21	−0.299538
$960$	0	0
$961$	−2.34637e22	−0.999934
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.72924e18	0.000319287 0
$966$	0	0
$967$	−2.12635e22	−0.864839	−0.432420	−	0.901673i	$-0.642340\pi$
$967$	−2.12635e22	−0.864839	−0.432420	+	0.901673i	$0.642340\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.78832e22	1.88816	0.944080	−	0.329716i	$-0.106953\pi$
$971$	4.78832e22	1.88816	0.944080	+	0.329716i	$0.106953\pi$
$972$	0	0
$973$	−2.41306e21	−0.0936961
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.90417e22	−0.716962	−0.358481	−	0.933537i	$-0.616705\pi$
$977$	−1.90417e22	−0.716962	−0.358481	+	0.933537i	$0.616705\pi$
$978$	0	0
$979$	−3.22350e21	−0.119525
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.52918e22	−0.549928	−0.274964	−	0.961455i	$-0.588666\pi$
$983$	−1.52918e22	−0.549928	−0.274964	+	0.961455i	$0.588666\pi$
$984$	0	0
$985$	−2.61568e21	−0.0926431
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.10011e21	0.0377976
$990$	0	0
$991$	−4.70053e22	−1.59072	−0.795362	−	0.606135i	$-0.792719\pi$
$991$	−4.70053e22	−1.59072	−0.795362	+	0.606135i	$0.792719\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.23414e21	0.303196
$996$	0	0
$997$	−2.91083e22	−0.941463	−0.470731	−	0.882277i	$-0.656010\pi$
$997$	−2.91083e22	−0.941463	−0.470731	+	0.882277i	$0.656010\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.16.a.b.1.1		1
3.2	odd	2		24.16.a.a.1.1	✓	1
4.3	odd	2		144.16.a.i.1.1		1
12.11	even	2		48.16.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.16.a.a.1.1	✓	1	3.2	odd	2
48.16.a.e.1.1		1	12.11	even	2
72.16.a.b.1.1		1	1.1	even	1	trivial
144.16.a.i.1.1		1	4.3	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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