LMFDB - Embedded newform 1089.2.a.i.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 121)
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+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} -1.00000 q^{10} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -5.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{23} -4.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} +9.00000 q^{29} -2.00000 q^{31} +5.00000 q^{32} -5.00000 q^{34} -2.00000 q^{35} -3.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -5.00000 q^{41} -2.00000 q^{46} -2.00000 q^{47} -3.00000 q^{49} -4.00000 q^{50} +1.00000 q^{52} -9.00000 q^{53} -6.00000 q^{56} +9.00000 q^{58} -8.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +7.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} +5.00000 q^{68} -2.00000 q^{70} -12.0000 q^{71} +2.00000 q^{73} -3.00000 q^{74} +6.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} -5.00000 q^{82} +6.00000 q^{83} +5.00000 q^{85} +9.00000 q^{89} -2.00000 q^{91} +2.00000 q^{92} -2.00000 q^{94} +6.00000 q^{95} -13.0000 q^{97} -3.00000 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

)

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	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
$25$	−4.00000	−0.800000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−2.00000	−0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−5.00000	−0.857493
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−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
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	−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
$55$	0	0
$56$	−6.00000	−0.801784
$57$	0	0
$58$	9.00000	1.18176
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	5.00000	0.606339
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	6.00000	0.688247
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−5.00000	−0.552158
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	2.00000	0.208514
$93$	0	0
$94$	−2.00000	−0.206284
$95$	6.00000	0.615587
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	4.00000	0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	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
$111$	0	0
$112$	−2.00000	−0.188982
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	−9.00000	−0.835629
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−10.0000	−0.916698
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	0	0
$124$	2.00000	0.179605
$125$	9.00000	0.804984
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	1.00000	0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	2.00000	0.172774
$135$	0	0
$136$	15.0000	1.28624
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	2.00000	0.165521
$147$	0	0
$148$	3.00000	0.246598
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	18.0000	1.45999
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	6.00000	0.465690
$167$	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
$169$	−12.0000	−0.923077
$170$	5.00000	0.383482
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	0	0
$177$	0	0
$178$	9.00000	0.674579
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	6.00000	0.442326
$185$	3.00000	0.220564
$186$	0	0
$187$	0	0
$188$	2.00000	0.145865
$189$	0	0
$190$	6.00000	0.435286
$191$	−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	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	3.00000	0.214286
$197$	−11.0000	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$197$	−11.0000	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	−10.0000	−0.703598
$203$	18.0000	1.26335
$204$	0	0
$205$	5.00000	0.349215
$206$	8.00000	0.557386
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	11.0000	0.745014
$219$	0	0
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	9.00000	0.598671
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−27.0000	−1.77264
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	8.00000	0.520756
$237$	0	0
$238$	−10.0000	−0.648204
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	6.00000	0.384111
$245$	3.00000	0.191663
$246$	0	0
$247$	6.00000	0.381771
$248$	6.00000	0.381000
$249$	0	0
$250$	9.00000	0.569210
$251$	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
$254$	16.0000	1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	0	0
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−12.0000	−0.735767
$267$	0	0
$268$	−2.00000	−0.122169
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	10.0000	0.604122
$275$	0	0
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	2.00000	0.119952
$279$	0	0
$280$	6.00000	0.358569
$281$	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
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	8.00000	0.470588
$290$	−9.00000	−0.528498
$291$	0	0
$292$	−2.00000	−0.117041
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	9.00000	0.523114
$297$	0	0
$298$	17.0000	0.984784
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	6.00000	0.344124
$305$	6.00000	0.343559
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	0	0
$310$	2.00000	0.113592
$311$	−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
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	0	0
$320$	−7.00000	−0.391312
$321$	0	0
$322$	−4.00000	−0.222911
$323$	30.0000	1.66924
$324$	0	0
$325$	4.00000	0.221880
$326$	−2.00000	−0.110770
$327$	0	0
$328$	15.0000	0.828236
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	12.0000	0.656611
$335$	−2.00000	−0.109272
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−5.00000	−0.271163
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	6.00000	0.322562
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−9.00000	−0.476999
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	1.00000	0.0525588
$363$	0	0
$364$	2.00000	0.104828
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	3.00000	0.155963
$371$	−18.0000	−0.934513
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	5.00000	0.254493
$387$	0	0
$388$	13.0000	0.659975
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	10.0000	0.505722
$392$	9.00000	0.454569
$393$	0	0
$394$	−11.0000	−0.554172
$395$	−10.0000	−0.503155
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	4.00000	0.200000
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	10.0000	0.497519
$405$	0	0
$406$	18.0000	0.893325
$407$	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	$0.326233\pi$
$410$	5.00000	0.246932
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−5.00000	−0.245145
$417$	0	0
$418$	0	0
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	27.0000	1.31124
$425$	20.0000	0.970143
$426$	0	0
$427$	−12.0000	−0.580721
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−11.0000	−0.526804
$437$	12.0000	0.574038
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	0	0
$442$	5.00000	0.237826
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	−20.0000	−0.947027
$447$	0	0
$448$	14.0000	0.661438
$449$	13.0000	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$449$	13.0000	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$450$	0	0
$451$	0	0
$452$	−9.00000	−0.423324
$453$	0	0
$454$	−24.0000	−1.12638
$455$	2.00000	0.0937614
$456$	0	0
$457$	−39.0000	−1.82434	−0.912172	−	0.409809i	$-0.865595\pi$
$457$	−39.0000	−1.82434	−0.912172	+	0.409809i	$0.865595\pi$
$458$	9.00000	0.420542
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−21.0000	−0.972806
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	2.00000	0.0922531
$471$	0	0
$472$	24.0000	1.10469
$473$	0	0
$474$	0	0
$475$	24.0000	1.10120
$476$	10.0000	0.458349
$477$	0	0
$478$	6.00000	0.274434
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	3.00000	0.136788
$482$	−22.0000	−1.00207
$483$	0	0
$484$	0	0
$485$	13.0000	0.590300
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	18.0000	0.814822
$489$	0	0
$490$	3.00000	0.135526
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−45.0000	−2.02670
$494$	6.00000	0.269953
$495$	0	0
$496$	2.00000	0.0898027
$497$	−24.0000	−1.07655
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	2.00000	0.0892644
$503$	−38.0000	−1.69434	−0.847168	−	0.531325i	$-0.821694\pi$
$503$	−38.0000	−1.69434	−0.847168	+	0.531325i	$0.821694\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−19.0000	−0.838054
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	−6.00000	−0.263625
$519$	0	0
$520$	−3.00000	−0.131559
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	−22.0000	−0.959246
$527$	10.0000	0.435607
$528$	0	0
$529$	−19.0000	−0.826087
$530$	9.00000	0.390935
$531$	0	0
$532$	12.0000	0.520266
$533$	5.00000	0.216574
$534$	0	0
$535$	−6.00000	−0.259403
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	−25.0000	−1.07187
$545$	−11.0000	−0.471188
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	0	0
$551$	−54.0000	−2.30048
$552$	0	0
$553$	20.0000	0.850487
$554$	−1.00000	−0.0424859
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	0	0
$560$	2.00000	0.0845154
$561$	0	0
$562$	6.00000	0.253095
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	−28.0000	−1.17693
$567$	0	0
$568$	36.0000	1.51053
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	−10.0000	−0.417392
$575$	8.00000	0.333623
$576$	0	0
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	9.00000	0.373705
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	9.00000	0.371787
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	8.00000	0.329355
$591$	0	0
$592$	3.00000	0.123299
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	0	0
$595$	10.0000	0.409960
$596$	−17.0000	−0.696347
$597$	0	0
$598$	2.00000	0.0817861
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	6.00000	0.242933
$611$	2.00000	0.0809113
$612$	0	0
$613$	−17.0000	−0.686624	−0.343312	−	0.939222i	$-0.611549\pi$
$613$	−17.0000	−0.686624	−0.343312	+	0.939222i	$0.611549\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	−24.0000	−0.962312
$623$	18.0000	0.721155
$624$	0	0
$625$	11.0000	0.440000
$626$	23.0000	0.919265
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	15.0000	0.598089
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	−30.0000	−1.19334
$633$	0	0
$634$	2.00000	0.0794301
$635$	−16.0000	−0.634941
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	0	0
$640$	3.00000	0.118585
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	30.0000	1.18033
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	0	0
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	2.00000	0.0783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	0	0
$656$	5.00000	0.195217
$657$	0	0
$658$	−4.00000	−0.155936
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−18.0000	−0.698535
$665$	12.0000	0.465340
$666$	0	0
$667$	−18.0000	−0.696963
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	12.0000	0.461538
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	−26.0000	−0.997788
$680$	−15.0000	−0.575224
$681$	0	0
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−20.0000	−0.763604
$687$	0	0
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	28.0000	1.06287
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	25.0000	0.946943
$698$	27.0000	1.02197
$699$	0	0
$700$	8.00000	0.302372
$701$	17.0000	0.642081	0.321041	−	0.947065i	$-0.395967\pi$
$701$	17.0000	0.642081	0.321041	+	0.947065i	$0.395967\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	9.00000	0.338719
$707$	−20.0000	−0.752177
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	−27.0000	−1.01187
$713$	4.00000	0.149801
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	−2.00000	−0.0746393
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	17.0000	0.632674
$723$	0	0
$724$	−1.00000	−0.0371647
$725$	−36.0000	−1.33701
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	0	0
$732$	0	0
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	−10.0000	−0.368605
$737$	0	0
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	−3.00000	−0.110282
$741$	0	0
$742$	−18.0000	−0.660801
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	0	0
$745$	−17.0000	−0.622832
$746$	−22.0000	−0.805477
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	−9.00000	−0.327761
$755$	−16.0000	−0.582300
$756$	0	0
$757$	53.0000	1.92632	0.963159	−	0.268933i	$-0.0866710\pi$
$757$	53.0000	1.92632	0.963159	+	0.268933i	$0.0866710\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	22.0000	0.796453
$764$	8.00000	0.289430
$765$	0	0
$766$	−20.0000	−0.722629
$767$	8.00000	0.288863
$768$	0	0
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	0	0
$772$	−5.00000	−0.179954
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	39.0000	1.40002
$777$	0	0
$778$	3.00000	0.107555
$779$	30.0000	1.07486
$780$	0	0
$781$	0	0
$782$	10.0000	0.357599
$783$	0	0
$784$	3.00000	0.107143
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	11.0000	0.391859
$789$	0	0
$790$	−10.0000	−0.355784
$791$	18.0000	0.640006
$792$	0	0
$793$	6.00000	0.213066
$794$	13.0000	0.461353
$795$	0	0
$796$	−24.0000	−0.850657
$797$	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
$799$	10.0000	0.353775
$800$	−20.0000	−0.707107
$801$	0	0
$802$	−23.0000	−0.812158
$803$	0	0
$804$	0	0
$805$	4.00000	0.140981
$806$	2.00000	0.0704470
$807$	0	0
$808$	30.0000	1.05540
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−18.0000	−0.631676
$813$	0	0
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	0	0
$818$	21.0000	0.734248
$819$	0	0
$820$	−5.00000	−0.174608
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	0	0
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	−7.00000	−0.242681
$833$	15.0000	0.519719
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	−2.00000	−0.0690889
$839$	−46.0000	−1.58810	−0.794048	−	0.607855i	$-0.792030\pi$
$839$	−46.0000	−1.58810	−0.794048	+	0.607855i	$0.792030\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	13.0000	0.448010
$843$	0	0
$844$	12.0000	0.413057
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	9.00000	0.309061
$849$	0	0
$850$	20.0000	0.685994
$851$	6.00000	0.205677
$852$	0	0
$853$	−17.0000	−0.582069	−0.291034	−	0.956713i	$-0.593999\pi$
$853$	−17.0000	−0.582069	−0.291034	+	0.956713i	$0.593999\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	0	0
$862$	12.0000	0.408722
$863$	54.0000	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$863$	54.0000	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	19.0000	0.645646
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	−33.0000	−1.11752
$873$	0	0
$874$	12.0000	0.405906
$875$	18.0000	0.608511
$876$	0	0
$877$	27.0000	0.911725	0.455863	−	0.890050i	$-0.349331\pi$
$877$	27.0000	0.911725	0.455863	+	0.890050i	$0.349331\pi$
$878$	−22.0000	−0.742464
$879$	0	0
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−5.00000	−0.168168
$885$	0	0
$886$	20.0000	0.671913
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	−9.00000	−0.301681
$891$	0	0
$892$	20.0000	0.669650
$893$	12.0000	0.401565
$894$	0	0
$895$	24.0000	0.802232
$896$	−6.00000	−0.200446
$897$	0	0
$898$	13.0000	0.433816
$899$	−18.0000	−0.600334
$900$	0	0
$901$	45.0000	1.49917
$902$	0	0
$903$	0	0
$904$	−27.0000	−0.898007
$905$	−1.00000	−0.0332411
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	2.00000	0.0662994
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	−39.0000	−1.29001
$915$	0	0
$916$	−9.00000	−0.297368
$917$	0	0
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	33.0000	1.08680
$923$	12.0000	0.394985
$924$	0	0
$925$	12.0000	0.394558
$926$	−20.0000	−0.657241
$927$	0	0
$928$	45.0000	1.47720
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	21.0000	0.687878
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−2.00000	−0.0652328
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	8.00000	0.260378
$945$	0	0
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	24.0000	0.778663
$951$	0	0
$952$	30.0000	0.972306
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−16.0000	−0.516937
$959$	20.0000	0.645834
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.00000	0.0967239
$963$	0	0
$964$	22.0000	0.708572
$965$	−5.00000	−0.160956
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	0	0
$970$	13.0000	0.417405
$971$	−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
$973$	4.00000	0.128234
$974$	2.00000	0.0640841
$975$	0	0
$976$	6.00000	0.192055
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	0	0
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−2.00000	−0.0638226
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	11.0000	0.350489
$986$	−45.0000	−1.43309
$987$	0	0
$988$	−6.00000	−0.190885
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	−24.0000	−0.761234
$995$	−24.0000	−0.760851
$996$	0	0
$997$	−53.0000	−1.67853	−0.839263	−	0.543725i	$-0.817013\pi$
$997$	−53.0000	−1.67853	−0.839263	+	0.543725i	$0.817013\pi$
$998$	8.00000	0.253236
$999$	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.i.1.1		1
3.2	odd	2		121.2.a.a.1.1	✓	1
11.10	odd	2		1089.2.a.c.1.1		1
12.11	even	2		1936.2.a.a.1.1		1
15.14	odd	2		3025.2.a.e.1.1		1
21.20	even	2		5929.2.a.a.1.1		1
24.5	odd	2		7744.2.a.f.1.1		1
24.11	even	2		7744.2.a.be.1.1		1
33.2	even	10		121.2.c.b.81.1		4
33.5	odd	10		121.2.c.d.3.1		4
33.8	even	10		121.2.c.b.9.1		4
33.14	odd	10		121.2.c.d.9.1		4
33.17	even	10		121.2.c.b.3.1		4
33.20	odd	10		121.2.c.d.81.1		4
33.26	odd	10		121.2.c.d.27.1		4
33.29	even	10		121.2.c.b.27.1		4
33.32	even	2		121.2.a.c.1.1	yes	1
132.131	odd	2		1936.2.a.b.1.1		1
165.164	even	2		3025.2.a.b.1.1		1
231.230	odd	2		5929.2.a.g.1.1		1
264.131	odd	2		7744.2.a.bf.1.1		1
264.197	even	2		7744.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.a.a.1.1	✓	1	3.2	odd	2
121.2.a.c.1.1	yes	1	33.32	even	2
121.2.c.b.3.1		4	33.17	even	10
121.2.c.b.9.1		4	33.8	even	10
121.2.c.b.27.1		4	33.29	even	10
121.2.c.b.81.1		4	33.2	even	10
121.2.c.d.3.1		4	33.5	odd	10
121.2.c.d.9.1		4	33.14	odd	10
121.2.c.d.27.1		4	33.26	odd	10
121.2.c.d.81.1		4	33.20	odd	10
1089.2.a.c.1.1		1	11.10	odd	2
1089.2.a.i.1.1		1	1.1	even	1	trivial
1936.2.a.a.1.1		1	12.11	even	2
1936.2.a.b.1.1		1	132.131	odd	2
3025.2.a.b.1.1		1	165.164	even	2
3025.2.a.e.1.1		1	15.14	odd	2
5929.2.a.a.1.1		1	21.20	even	2
5929.2.a.g.1.1		1	231.230	odd	2
7744.2.a.c.1.1		1	264.197	even	2
7744.2.a.f.1.1		1	24.5	odd	2
7744.2.a.be.1.1		1	24.11	even	2
7744.2.a.bf.1.1		1	264.131	odd	2

Overview	Random
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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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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.i.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	$1$