LMFDB - Embedded newform 1089.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1089 = 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1089.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:	$8.69570878012$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+2.00000 q^{2} +2.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} -8.00000 q^{10} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} -3.00000 q^{19} -8.00000 q^{20} -2.00000 q^{23} +11.0000 q^{25} -4.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -5.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} -4.00000 q^{35} +3.00000 q^{37} -6.00000 q^{38} +2.00000 q^{41} +12.0000 q^{43} -4.00000 q^{46} -2.00000 q^{47} -6.00000 q^{49} +22.0000 q^{50} -4.00000 q^{52} -6.00000 q^{53} -12.0000 q^{58} +10.0000 q^{59} +3.00000 q^{61} -10.0000 q^{62} -8.00000 q^{64} +8.00000 q^{65} -1.00000 q^{67} -8.00000 q^{68} -8.00000 q^{70} -11.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} +11.0000 q^{79} +16.0000 q^{80} +4.00000 q^{82} -6.00000 q^{83} +16.0000 q^{85} +24.0000 q^{86} -12.0000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -4.00000 q^{94} +12.0000 q^{95} +5.00000 q^{97} -12.0000 q^{98} +O(q^{100})

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

)

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

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	0	0
$10$	−8.00000	−2.52982
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−8.00000	−1.78885
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	2.00000	0.377964
$29$	−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	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−8.00000	−1.37199
$35$	−4.00000	−0.676123
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	22.0000	3.11127
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−12.0000	−1.57568
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	8.00000	0.992278
$66$	0	0
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	−8.00000	−0.970143
$69$	0	0
$70$	−8.00000	−0.956183
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	16.0000	1.78885
$81$	0	0
$82$	4.00000	0.441726
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	24.0000	2.58799
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−4.00000	−0.412568
$95$	12.0000	1.23117
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	−12.0000	−1.21218
$99$	0	0
$100$	22.0000	2.20000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−12.0000	−1.11417
$117$	0	0
$118$	20.0000	1.84115
$119$	−4.00000	−0.366679
$120$	0	0
$121$	0	0
$122$	6.00000	0.543214
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−24.0000	−2.14663
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	16.0000	1.40329
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−2.00000	−0.172774
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	24.0000	1.99309
$146$	−22.0000	−1.82073
$147$	0	0
$148$	6.00000	0.493197
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	22.0000	1.75023
$159$	0	0
$160$	32.0000	2.52982
$161$	−2.00000	−0.157622
$162$	0	0
$163$	25.0000	1.95815	0.979076	−	0.203497i	$-0.0652307\pi$
$163$	25.0000	1.95815	0.979076	+	0.203497i	$0.0652307\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	32.0000	2.45429
$171$	0	0
$172$	24.0000	1.82998
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	11.0000	0.831522
$176$	0	0
$177$	0	0
$178$	−24.0000	−1.79888
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−23.0000	−1.70958	−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000	−1.70958	−0.854788	+	0.518977i	$0.826313\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	−4.00000	−0.291730
$189$	0	0
$190$	24.0000	1.74114
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−12.0000	−0.857143
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	0	0
$202$	20.0000	1.40720
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−14.0000	−0.975426
$207$	0	0
$208$	8.00000	0.554700
$209$	0	0
$210$	0	0
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	36.0000	2.46091
$215$	−48.0000	−3.27357
$216$	0	0
$217$	−5.00000	−0.339422
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	−12.0000	−0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	20.0000	1.30189
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	0	0
$244$	6.00000	0.384111
$245$	24.0000	1.53330
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	−48.0000	−3.03579
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	0	0
$254$	−26.0000	−1.63139
$255$	0	0
$256$	16.0000	1.00000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	16.0000	0.992278
$261$	0	0
$262$	12.0000	0.741362
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	−6.00000	−0.367884
$267$	0	0
$268$	−2.00000	−0.122169
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	16.0000	0.970143
$273$	0	0
$274$	−16.0000	−0.966595
$275$	0	0
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	32.0000	1.91923
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	48.0000	2.81866
$291$	0	0
$292$	−22.0000	−1.28745
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−40.0000	−2.32889
$296$	0	0
$297$	0	0
$298$	32.0000	1.85371
$299$	4.00000	0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	−32.0000	−1.84139
$303$	0	0
$304$	12.0000	0.688247
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	0	0
$310$	40.0000	2.27185
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	22.0000	1.23760
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	0	0
$320$	32.0000	1.78885
$321$	0	0
$322$	−4.00000	−0.222911
$323$	12.0000	0.667698
$324$	0	0
$325$	−22.0000	−1.22034
$326$	50.0000	2.76924
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	36.0000	1.96983
$335$	4.00000	0.218543
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	32.0000	1.73544
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	−48.0000	−2.58050
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	22.0000	1.17595
$351$	0	0
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	0	0
$356$	−24.0000	−1.27200
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−46.0000	−2.41771
$363$	0	0
$364$	−4.00000	−0.209657
$365$	44.0000	2.30307
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	−24.0000	−1.24770
$371$	−6.00000	−0.311504
$372$	0	0
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	10.0000	0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	16.0000	0.806068
$395$	−44.0000	−2.21388
$396$	0	0
$397$	31.0000	1.55585	0.777923	−	0.628360i	$-0.216273\pi$
$397$	31.0000	1.55585	0.777923	+	0.628360i	$0.216273\pi$
$398$	−42.0000	−2.10527
$399$	0	0
$400$	−44.0000	−2.20000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	20.0000	0.995037
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	−16.0000	−0.790184
$411$	0	0
$412$	−14.0000	−0.689730
$413$	10.0000	0.492068
$414$	0	0
$415$	24.0000	1.17811
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−42.0000	−2.04453
$423$	0	0
$424$	0	0
$425$	−44.0000	−2.13431
$426$	0	0
$427$	3.00000	0.145180
$428$	36.0000	1.74013
$429$	0	0
$430$	−96.0000	−4.62953
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	2.00000	0.0957826
$437$	6.00000	0.287019
$438$	0	0
$439$	37.0000	1.76591	0.882957	−	0.469454i	$-0.155549\pi$
$439$	37.0000	1.76591	0.882957	+	0.469454i	$0.155549\pi$
$440$	0	0
$441$	0	0
$442$	16.0000	0.761042
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	48.0000	2.27542
$446$	−34.0000	−1.60995
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	0	0
$452$	−12.0000	−0.564433
$453$	0	0
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−36.0000	−1.68217
$459$	0	0
$460$	16.0000	0.746004
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	−36.0000	−1.66767
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−1.00000	−0.0461757
$470$	16.0000	0.738025
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−33.0000	−1.51414
$476$	−8.00000	−0.366679
$477$	0	0
$478$	−12.0000	−0.548867
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−28.0000	−1.27537
$483$	0	0
$484$	0	0
$485$	−20.0000	−0.908153
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	0	0
$489$	0	0
$490$	48.0000	2.16842
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	12.0000	0.539906
$495$	0	0
$496$	20.0000	0.898027
$497$	0	0
$498$	0	0
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	−48.0000	−2.14663
$501$	0	0
$502$	4.00000	0.178529
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	0	0
$508$	−26.0000	−1.15356
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	32.0000	1.41421
$513$	0	0
$514$	28.0000	1.23503
$515$	28.0000	1.23383
$516$	0	0
$517$	0	0
$518$	6.00000	0.263625
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	20.0000	0.872041
$527$	20.0000	0.871214
$528$	0	0
$529$	−19.0000	−0.826087
$530$	48.0000	2.08499
$531$	0	0
$532$	−6.00000	−0.260133
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−72.0000	−3.11283
$536$	0	0
$537$	0	0
$538$	28.0000	1.20717
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	32.0000	1.37199
$545$	−4.00000	−0.171341
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−16.0000	−0.683486
$549$	0	0
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	11.0000	0.467768
$554$	−22.0000	−0.934690
$555$	0	0
$556$	32.0000	1.35710
$557$	8.00000	0.338971	0.169485	−	0.985533i	$-0.445789\pi$
$557$	8.00000	0.338971	0.169485	+	0.985533i	$0.445789\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	16.0000	0.676123
$561$	0	0
$562$	24.0000	1.01238
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	−22.0000	−0.924729
$567$	0	0
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	0	0
$573$	0	0
$574$	4.00000	0.166957
$575$	−22.0000	−0.917463
$576$	0	0
$577$	15.0000	0.624458	0.312229	−	0.950007i	$-0.398924\pi$
$577$	15.0000	0.624458	0.312229	+	0.950007i	$0.398924\pi$
$578$	−2.00000	−0.0831890
$579$	0	0
$580$	48.0000	1.99309
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−24.0000	−0.991431
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	−80.0000	−3.29355
$591$	0	0
$592$	−12.0000	−0.493197
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	32.0000	1.31077
$597$	0	0
$598$	8.00000	0.327144
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	24.0000	0.978167
$603$	0	0
$604$	−32.0000	−1.30206
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	24.0000	0.973329
$609$	0	0
$610$	−24.0000	−0.971732
$611$	4.00000	0.161823
$612$	0	0
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	−38.0000	−1.53356
$615$	0	0
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	40.0000	1.60644
$621$	0	0
$622$	−48.0000	−1.92462
$623$	−12.0000	−0.480770
$624$	0	0
$625$	41.0000	1.64000
$626$	−20.0000	−0.799361
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	0	0
$634$	40.0000	1.58860
$635$	52.0000	2.06356
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−37.0000	−1.45914	−0.729569	−	0.683907i	$-0.760279\pi$
$643$	−37.0000	−1.45914	−0.729569	+	0.683907i	$0.760279\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	24.0000	0.944267
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	0	0
$649$	0	0
$650$	−44.0000	−1.72582
$651$	0	0
$652$	50.0000	1.95815
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−8.00000	−0.312348
$657$	0	0
$658$	−4.00000	−0.155936
$659$	−46.0000	−1.79191	−0.895953	−	0.444149i	$-0.853506\pi$
$659$	−46.0000	−1.79191	−0.895953	+	0.444149i	$0.853506\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	−22.0000	−0.855054
$663$	0	0
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	12.0000	0.464642
$668$	36.0000	1.39288
$669$	0	0
$670$	8.00000	0.309067
$671$	0	0
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−18.0000	−0.692308
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	5.00000	0.191882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	0	0
$685$	32.0000	1.22266
$686$	−26.0000	−0.992685
$687$	0	0
$688$	−48.0000	−1.82998
$689$	12.0000	0.457164
$690$	0	0
$691$	11.0000	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$691$	11.0000	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$692$	−48.0000	−1.82469
$693$	0	0
$694$	4.00000	0.151838
$695$	−64.0000	−2.42766
$696$	0	0
$697$	−8.00000	−0.303022
$698$	30.0000	1.13552
$699$	0	0
$700$	22.0000	0.831522
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−9.00000	−0.339441
$704$	0	0
$705$	0	0
$706$	24.0000	0.903252
$707$	10.0000	0.376089
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.0000	0.374503
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−8.00000	−0.298557
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−7.00000	−0.260694
$722$	−20.0000	−0.744323
$723$	0	0
$724$	−46.0000	−1.70958
$725$	−66.0000	−2.45118
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	0	0
$730$	88.0000	3.25703
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	16.0000	0.589768
$737$	0	0
$738$	0	0
$739$	41.0000	1.50821	0.754105	−	0.656754i	$-0.228071\pi$
$739$	41.0000	1.50821	0.754105	+	0.656754i	$0.228071\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	−12.0000	−0.440534
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	0	0
$745$	−64.0000	−2.34478
$746$	14.0000	0.512576
$747$	0	0
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	19.0000	0.693320	0.346660	−	0.937991i	$-0.387316\pi$
$751$	19.0000	0.693320	0.346660	+	0.937991i	$0.387316\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	24.0000	0.874028
$755$	64.0000	2.32920
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	0	0
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	0	0
$763$	1.00000	0.0362024
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−52.0000	−1.87884
$767$	−20.0000	−0.722158
$768$	0	0
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	−55.0000	−1.97566
$776$	0	0
$777$	0	0
$778$	−36.0000	−1.29066
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	16.0000	0.572159
$783$	0	0
$784$	24.0000	0.857143
$785$	4.00000	0.142766
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	16.0000	0.569976
$789$	0	0
$790$	−88.0000	−3.13090
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−6.00000	−0.213066
$794$	62.0000	2.20030
$795$	0	0
$796$	−42.0000	−1.48865
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	−88.0000	−3.11127
$801$	0	0
$802$	56.0000	1.97743
$803$	0	0
$804$	0	0
$805$	8.00000	0.281963
$806$	20.0000	0.704470
$807$	0	0
$808$	0	0
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	0	0
$815$	−100.000	−3.50285
$816$	0	0
$817$	−36.0000	−1.25948
$818$	42.0000	1.46850
$819$	0	0
$820$	−16.0000	−0.558744
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	−27.0000	−0.941161	−0.470580	−	0.882357i	$-0.655955\pi$
$823$	−27.0000	−0.941161	−0.470580	+	0.882357i	$0.655955\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	0	0
$829$	−11.0000	−0.382046	−0.191023	−	0.981586i	$-0.561180\pi$
$829$	−11.0000	−0.382046	−0.191023	+	0.981586i	$0.561180\pi$
$830$	48.0000	1.66610
$831$	0	0
$832$	16.0000	0.554700
$833$	24.0000	0.831551
$834$	0	0
$835$	−72.0000	−2.49166
$836$	0	0
$837$	0	0
$838$	−52.0000	−1.79631
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−4.00000	−0.137849
$843$	0	0
$844$	−42.0000	−1.44570
$845$	36.0000	1.23844
$846$	0	0
$847$	0	0
$848$	24.0000	0.824163
$849$	0	0
$850$	−88.0000	−3.01838
$851$	−6.00000	−0.205677
$852$	0	0
$853$	11.0000	0.376633	0.188316	−	0.982108i	$-0.439697\pi$
$853$	11.0000	0.376633	0.188316	+	0.982108i	$0.439697\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	0	0
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	0	0
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	−96.0000	−3.27357
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	96.0000	3.26410
$866$	−34.0000	−1.15537
$867$	0	0
$868$	−10.0000	−0.339422
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	0	0
$874$	12.0000	0.405906
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	74.0000	2.49738
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	49.0000	1.64898	0.824491	−	0.565876i	$-0.191462\pi$
$883$	49.0000	1.64898	0.824491	+	0.565876i	$0.191462\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	−8.00000	−0.268765
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	96.0000	3.21793
$891$	0	0
$892$	−34.0000	−1.13840
$893$	6.00000	0.200782
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	−40.0000	−1.33482
$899$	30.0000	1.00056
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	92.0000	3.05818
$906$	0	0
$907$	33.0000	1.09575	0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000	1.09575	0.547874	+	0.836561i	$0.315438\pi$
$908$	0	0
$909$	0	0
$910$	16.0000	0.530395
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	0	0
$914$	36.0000	1.19077
$915$	0	0
$916$	−36.0000	−1.18947
$917$	6.00000	0.198137
$918$	0	0
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	0	0
$922$	12.0000	0.395199
$923$	0	0
$924$	0	0
$925$	33.0000	1.08503
$926$	32.0000	1.05159
$927$	0	0
$928$	48.0000	1.57568
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	−36.0000	−1.17922
$933$	0	0
$934$	−48.0000	−1.57061
$935$	0	0
$936$	0	0
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	−2.00000	−0.0653023
$939$	0	0
$940$	16.0000	0.521862
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	−40.0000	−1.30189
$945$	0	0
$946$	0	0
$947$	−54.0000	−1.75476	−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000	−1.75476	−0.877382	+	0.479792i	$0.840712\pi$
$948$	0	0
$949$	22.0000	0.714150
$950$	−66.0000	−2.14132
$951$	0	0
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	32.0000	1.03550
$956$	−12.0000	−0.388108
$957$	0	0
$958$	44.0000	1.42158
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−28.0000	−0.901819
$965$	20.0000	0.643823
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	0	0
$970$	−40.0000	−1.28432
$971$	−2.00000	−0.0641831	−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000	−0.0641831	−0.0320915	+	0.999485i	$0.510217\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	−80.0000	−2.56337
$975$	0	0
$976$	−12.0000	−0.384111
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	48.0000	1.53330
$981$	0	0
$982$	28.0000	0.893516
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	48.0000	1.52863
$987$	0	0
$988$	12.0000	0.381771
$989$	−24.0000	−0.763156
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	40.0000	1.27000
$993$	0	0
$994$	0	0
$995$	84.0000	2.66298
$996$	0	0
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	46.0000	1.45610
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.k.1.1		1
3.2	odd	2		363.2.a.a.1.1	✓	1
11.10	odd	2		1089.2.a.a.1.1		1
12.11	even	2		5808.2.a.bh.1.1		1
15.14	odd	2		9075.2.a.t.1.1		1
33.2	even	10		363.2.e.d.202.1		4
33.5	odd	10		363.2.e.i.124.1		4
33.8	even	10		363.2.e.d.130.1		4
33.14	odd	10		363.2.e.i.130.1		4
33.17	even	10		363.2.e.d.124.1		4
33.20	odd	10		363.2.e.i.202.1		4
33.26	odd	10		363.2.e.i.148.1		4
33.29	even	10		363.2.e.d.148.1		4
33.32	even	2		363.2.a.c.1.1	yes	1
132.131	odd	2		5808.2.a.bi.1.1		1
165.164	even	2		9075.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.a.1.1	✓	1	3.2	odd	2
363.2.a.c.1.1	yes	1	33.32	even	2
363.2.e.d.124.1		4	33.17	even	10
363.2.e.d.130.1		4	33.8	even	10
363.2.e.d.148.1		4	33.29	even	10
363.2.e.d.202.1		4	33.2	even	10
363.2.e.i.124.1		4	33.5	odd	10
363.2.e.i.130.1		4	33.14	odd	10
363.2.e.i.148.1		4	33.26	odd	10
363.2.e.i.202.1		4	33.20	odd	10
1089.2.a.a.1.1		1	11.10	odd	2
1089.2.a.k.1.1		1	1.1	even	1	trivial
5808.2.a.bh.1.1		1	12.11	even	2
5808.2.a.bi.1.1		1	132.131	odd	2
9075.2.a.b.1.1		1	165.164	even	2
9075.2.a.t.1.1		1	15.14	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 $\F_{q}$

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Fields

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1089.2.a.k.1.1

Level	$1089$
Weight	$2$
Character	1089.1
Self dual	yes
Analytic conductor	$8.696$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$