LMFDB - Embedded newform 1089.2.a.j.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 33)
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} +2.00000 q^{5} -4.00000 q^{7} -3.00000 q^{8} +2.00000 q^{10} +2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -2.00000 q^{20} -8.00000 q^{23} -1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} +6.00000 q^{37} -6.00000 q^{40} -2.00000 q^{41} -8.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +12.0000 q^{56} -6.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} +14.0000 q^{73} +6.00000 q^{74} +4.00000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +12.0000 q^{83} -4.00000 q^{85} +6.00000 q^{89} -8.00000 q^{91} +8.00000 q^{92} -8.00000 q^{94} +2.00000 q^{97} +9.00000 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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	4.00000	0.755929
$29$	−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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−8.00000	−1.35225
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	−6.00000	−0.948683
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−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
$56$	12.0000	1.60357
$57$	0	0
$58$	−6.00000	−0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	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
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	8.00000	0.834058
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000	0.368230
$119$	8.00000	0.733359
$120$	0	0
$121$	0	0
$122$	−6.00000	−0.543214
$123$	0	0
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	4.00000	0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	14.0000	1.15865
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	10.0000	0.790569
$161$	32.0000	2.52195
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	24.0000	1.76930
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	−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
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	2.00000	0.140720
$203$	24.0000	1.68447
$204$	0	0
$205$	−4.00000	−0.279372
$206$	8.00000	0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	32.0000	2.17230
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−16.0000	−1.05501
$231$	0	0
$232$	18.0000	1.18176
$233$	30.0000	1.96537	0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000	1.96537	0.982683	+	0.185296i	$0.0593245\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	−4.00000	−0.260378
$237$	0	0
$238$	8.00000	0.518563
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	6.00000	0.384111
$245$	18.0000	1.14998
$246$	0	0
$247$	0	0
$248$	24.0000	1.52400
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	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
$259$	−24.0000	−1.49129
$260$	−4.00000	−0.248069
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	24.0000	1.43427
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−12.0000	−0.704664
$291$	0	0
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−18.0000	−1.04623
$297$	0	0
$298$	−22.0000	−1.27443
$299$	−16.0000	−0.925304
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	0	0
$310$	−16.0000	−0.908739
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	0	0
$320$	14.0000	0.782624
$321$	0	0
$322$	32.0000	1.78329
$323$	0	0
$324$	0	0
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	32.0000	1.76422
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	4.00000	0.216930
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	22.0000	1.15629
$363$	0	0
$364$	8.00000	0.419314
$365$	28.0000	1.46559
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	12.0000	0.623850
$371$	24.0000	1.24602
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	24.0000	1.23771
$377$	−12.0000	−0.618031
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	−2.00000	−0.101535
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	−27.0000	−1.36371
$393$	0	0
$394$	−14.0000	−0.705310
$395$	8.00000	0.402524
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	24.0000	1.19110
$407$	0	0
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−16.0000	−0.787309
$414$	0	0
$415$	24.0000	1.17811
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	18.0000	0.874157
$425$	2.00000	0.0970143
$426$	0	0
$427$	24.0000	1.16144
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	16.0000	0.757622
$447$	0	0
$448$	−28.0000	−1.32288
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	16.0000	0.746004
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	30.0000	1.38972
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−16.0000	−0.738025
$471$	0	0
$472$	−12.0000	−0.552345
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	24.0000	1.09773
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	0	0
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	18.0000	0.814822
$489$	0	0
$490$	18.0000	0.813157
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	−11.0000	−0.486136
$513$	0	0
$514$	14.0000	0.617514
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	−24.0000	−1.05450
$519$	0	0
$520$	−12.0000	−0.526235
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	16.0000	0.696971
$528$	0	0
$529$	41.0000	1.78261
$530$	−12.0000	−0.521247
$531$	0	0
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−24.0000	−1.03761
$536$	12.0000	0.518321
$537$	0	0
$538$	2.00000	0.0862261
$539$	0	0
$540$	0	0
$541$	−46.0000	−1.97769	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000	−1.97769	−0.988847	+	0.148933i	$0.952416\pi$
$542$	−20.0000	−0.859074
$543$	0	0
$544$	−10.0000	−0.428746
$545$	4.00000	0.171341
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	26.0000	1.10463
$555$	0	0
$556$	−8.00000	−0.339276
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	0	0
$560$	8.00000	0.338062
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	8.00000	0.333914
$575$	8.00000	0.333623
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	12.0000	0.498273
$581$	−48.0000	−1.99138
$582$	0	0
$583$	0	0
$584$	−42.0000	−1.73797
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	−6.00000	−0.246598
$593$	38.0000	1.56047	0.780236	−	0.625485i	$-0.215099\pi$
$593$	38.0000	1.56047	0.780236	+	0.625485i	$0.215099\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	22.0000	0.901155
$597$	0	0
$598$	−16.0000	−0.654289
$599$	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
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	0	0
$609$	0	0
$610$	−12.0000	−0.485866
$611$	−16.0000	−0.647291
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	24.0000	0.962312
$623$	−24.0000	−0.961540
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−14.0000	−0.558661
$629$	−12.0000	−0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−22.0000	−0.873732
$635$	8.00000	0.317470
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	0	0
$640$	−6.00000	−0.237171
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	−32.0000	−1.26098
$645$	0	0
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	2.00000	0.0780869
$657$	0	0
$658$	32.0000	1.24749
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−36.0000	−1.39707
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	0	0
$670$	−8.00000	−0.309067
$671$	0	0
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	12.0000	0.460179
$681$	0	0
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	−8.00000	−0.305441
$687$	0	0
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	4.00000	0.151838
$695$	16.0000	0.606915
$696$	0	0
$697$	4.00000	0.151511
$698$	−6.00000	−0.227103
$699$	0	0
$700$	−4.00000	−0.151186
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−8.00000	−0.300871
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	−18.0000	−0.674579
$713$	64.0000	2.39682
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−8.00000	−0.298557
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	−19.0000	−0.707107
$723$	0	0
$724$	−22.0000	−0.817624
$725$	6.00000	0.222834
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	24.0000	0.889499
$729$	0	0
$730$	28.0000	1.03633
$731$	0	0
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−40.0000	−1.47442
$737$	0	0
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	24.0000	0.881068
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	−44.0000	−1.61204
$746$	2.00000	0.0732252
$747$	0	0
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−40.0000	−1.45575
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	8.00000	0.289430
$765$	0	0
$766$	16.0000	0.578103
$767$	8.00000	0.288863
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−6.00000	−0.215387
$777$	0	0
$778$	18.0000	0.645331
$779$	0	0
$780$	0	0
$781$	0	0
$782$	16.0000	0.572159
$783$	0	0
$784$	−9.00000	−0.321429
$785$	28.0000	0.999363
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	14.0000	0.498729
$789$	0	0
$790$	8.00000	0.284627
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−12.0000	−0.426132
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	0	0
$797$	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
$799$	16.0000	0.566039
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−26.0000	−0.918092
$803$	0	0
$804$	0	0
$805$	64.0000	2.25570
$806$	−16.0000	−0.563576
$807$	0	0
$808$	−6.00000	−0.211079
$809$	54.0000	1.89854	0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000	1.89854	0.949269	+	0.314464i	$0.101825\pi$
$810$	0	0
$811$	56.0000	1.96643	0.983213	−	0.182462i	$-0.0584065\pi$
$811$	56.0000	1.96643	0.983213	+	0.182462i	$0.0584065\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	0	0
$818$	−18.0000	−0.629355
$819$	0	0
$820$	4.00000	0.139686
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	24.0000	0.833052
$831$	0	0
$832$	14.0000	0.485363
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	4.00000	0.138178
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−26.0000	−0.896019
$843$	0	0
$844$	0	0
$845$	−18.0000	−0.619219
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	2.00000	0.0685994
$851$	−48.0000	−1.64542
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	24.0000	0.821263
$855$	0	0
$856$	36.0000	1.23045
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	34.0000	1.15537
$867$	0	0
$868$	−32.0000	−1.08615
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−6.00000	−0.203186
$873$	0	0
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−28.0000	−0.940678
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	12.0000	0.402241
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	12.0000	0.400892
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	48.0000	1.60089
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	44.0000	1.46261
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	−16.0000	−0.530395
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	48.0000	1.58510
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	48.0000	1.58251
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	16.0000	0.525793
$927$	0	0
$928$	−30.0000	−0.984798
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	−30.0000	−0.982683
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	16.0000	0.521862
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	0	0
$952$	−24.0000	−0.777844
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	−24.0000	−0.776215
$957$	0	0
$958$	8.00000	0.258468
$959$	8.00000	0.258333
$960$	0	0
$961$	33.0000	1.06452
$962$	12.0000	0.386896
$963$	0	0
$964$	10.0000	0.322078
$965$	28.0000	0.901352
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	0	0
$970$	4.00000	0.128432
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	−16.0000	−0.512673
$975$	0	0
$976$	6.00000	0.192055
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	−18.0000	−0.574989
$981$	0	0
$982$	4.00000	0.127645
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−28.0000	−0.892154
$986$	12.0000	0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−40.0000	−1.27000
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−4.00000	−0.126618
$999$	0	0

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up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.j.1.1		1
3.2	odd	2		363.2.a.b.1.1		1
11.10	odd	2		99.2.a.b.1.1		1
12.11	even	2		5808.2.a.t.1.1		1
15.14	odd	2		9075.2.a.q.1.1		1
33.2	even	10		363.2.e.e.202.1		4
33.5	odd	10		363.2.e.g.124.1		4
33.8	even	10		363.2.e.e.130.1		4
33.14	odd	10		363.2.e.g.130.1		4
33.17	even	10		363.2.e.e.124.1		4
33.20	odd	10		363.2.e.g.202.1		4
33.26	odd	10		363.2.e.g.148.1		4
33.29	even	10		363.2.e.e.148.1		4
33.32	even	2		33.2.a.a.1.1	✓	1
44.43	even	2		1584.2.a.o.1.1		1
55.32	even	4		2475.2.c.d.199.1		2
55.43	even	4		2475.2.c.d.199.2		2
55.54	odd	2		2475.2.a.g.1.1		1
77.76	even	2		4851.2.a.b.1.1		1
88.21	odd	2		6336.2.a.x.1.1		1
88.43	even	2		6336.2.a.n.1.1		1
99.32	even	6		891.2.e.e.298.1		2
99.43	odd	6		891.2.e.g.595.1		2
99.65	even	6		891.2.e.e.595.1		2
99.76	odd	6		891.2.e.g.298.1		2
132.131	odd	2		528.2.a.g.1.1		1
165.32	odd	4		825.2.c.a.199.2		2
165.98	odd	4		825.2.c.a.199.1		2
165.164	even	2		825.2.a.a.1.1		1
231.230	odd	2		1617.2.a.j.1.1		1
264.131	odd	2		2112.2.a.j.1.1		1
264.197	even	2		2112.2.a.bb.1.1		1
429.428	even	2		5577.2.a.a.1.1		1
561.560	even	2		9537.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	33.32	even	2
99.2.a.b.1.1		1	11.10	odd	2
363.2.a.b.1.1		1	3.2	odd	2
363.2.e.e.124.1		4	33.17	even	10
363.2.e.e.130.1		4	33.8	even	10
363.2.e.e.148.1		4	33.29	even	10
363.2.e.e.202.1		4	33.2	even	10
363.2.e.g.124.1		4	33.5	odd	10
363.2.e.g.130.1		4	33.14	odd	10
363.2.e.g.148.1		4	33.26	odd	10
363.2.e.g.202.1		4	33.20	odd	10
528.2.a.g.1.1		1	132.131	odd	2
825.2.a.a.1.1		1	165.164	even	2
825.2.c.a.199.1		2	165.98	odd	4
825.2.c.a.199.2		2	165.32	odd	4
891.2.e.e.298.1		2	99.32	even	6
891.2.e.e.595.1		2	99.65	even	6
891.2.e.g.298.1		2	99.76	odd	6
891.2.e.g.595.1		2	99.43	odd	6
1089.2.a.j.1.1		1	1.1	even	1	trivial
1584.2.a.o.1.1		1	44.43	even	2
1617.2.a.j.1.1		1	231.230	odd	2
2112.2.a.j.1.1		1	264.131	odd	2
2112.2.a.bb.1.1		1	264.197	even	2
2475.2.a.g.1.1		1	55.54	odd	2
2475.2.c.d.199.1		2	55.32	even	4
2475.2.c.d.199.2		2	55.43	even	4
4851.2.a.b.1.1		1	77.76	even	2
5577.2.a.a.1.1		1	429.428	even	2
5808.2.a.t.1.1		1	12.11	even	2
6336.2.a.n.1.1		1	88.43	even	2
6336.2.a.x.1.1		1	88.21	odd	2
9075.2.a.q.1.1		1	15.14	odd	2
9537.2.a.m.1.1		1	561.560	even	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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Newspace parameters

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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.j.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	$1$