LMFDB - Embedded newform 1122.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(1,1122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.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:	$8.95921510679$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} +10.0000 q^{89} -2.00000 q^{90} +8.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	−2.00000	−0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−7.00000	−0.707107
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.00000	−0.196116
$105$	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	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	2.00000	0.190693
$111$	10.0000	0.949158
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	1.00000	0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	16.0000	1.34269
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	7.00000	0.577350
$148$	−10.0000	−0.821995
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	1.00000	0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	0	0
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.00000	−0.468521
$165$	−2.00000	−0.155700
$166$	−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
$170$	−2.00000	−0.153393
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.00000	−0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	20.0000	1.47043
$186$	8.00000	0.586588
$187$	−1.00000	−0.0731272
$188$	8.00000	0.583460
$189$	0	0
$190$	8.00000	0.580381
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	2.00000	0.143592
$195$	−4.00000	−0.286446
$196$	−7.00000	−0.500000
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−1.00000	−0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	12.0000	0.838116
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	−16.0000	−1.09630
$214$	−12.0000	−0.820303
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	−2.00000	−0.135147
$220$	2.00000	0.134840
$221$	−2.00000	−0.134535
$222$	10.0000	0.671156
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−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	$0.369623\pi$
$228$	4.00000	0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	−16.0000	−1.04372
$236$	−12.0000	−0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	2.00000	0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	14.0000	0.894427
$246$	6.00000	0.382546
$247$	8.00000	0.509028
$248$	−8.00000	−0.508001
$249$	12.0000	0.760469
$250$	12.0000	0.758947
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.00000	0.0615457
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−10.0000	−0.611990
$268$	4.00000	0.244339
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	2.00000	0.121716
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−6.00000	−0.362473
$275$	1.00000	0.0603023
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	20.0000	1.19952
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−8.00000	−0.476393
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	16.0000	0.949425
$285$	−8.00000	−0.473879
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	4.00000	0.234888
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	7.00000	0.408248
$295$	24.0000	1.39733
$296$	−10.0000	−0.581238
$297$	1.00000	0.0580259
$298$	22.0000	1.27443
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	20.0000	1.14520
$306$	1.00000	0.0571662
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	16.0000	0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	2.00000	0.113228
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−6.00000	−0.336463
$319$	2.00000	0.111979
$320$	−2.00000	−0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	−6.00000	−0.331801
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−2.00000	−0.110096
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−12.0000	−0.658586
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	6.00000	0.325875
$340$	−2.00000	−0.108465
$341$	8.00000	0.433224
$342$	−4.00000	−0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	12.0000	0.637793
$355$	−32.0000	−1.69838
$356$	10.0000	0.529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−4.00000	−0.209370
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	20.0000	1.03975
$371$	0	0
$372$	8.00000	0.414781
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−1.00000	−0.0517088
$375$	−12.0000	−0.619677
$376$	8.00000	0.412568
$377$	4.00000	0.206010
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	8.00000	0.410391
$381$	−8.00000	−0.409852
$382$	8.00000	0.409316
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	4.00000	0.203331
$388$	2.00000	0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	−26.0000	−1.30986
$395$	16.0000	0.805047
$396$	−1.00000	−0.0502519
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−4.00000	−0.199502
$403$	16.0000	0.797017
$404$	6.00000	0.298511
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	10.0000	0.495682
$408$	−1.00000	−0.0495074
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	12.0000	0.592638
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	−2.00000	−0.0980581
$417$	−20.0000	−0.979404
$418$	4.00000	0.195646
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−4.00000	−0.194717
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	−1.00000	−0.0485071
$426$	−16.0000	−0.775203
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−2.00000	−0.0965609
$430$	−8.00000	−0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	6.00000	0.287348
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	2.00000	0.0953463
$441$	−7.00000	−0.333333
$442$	−2.00000	−0.0951303
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	10.0000	0.474579
$445$	−20.0000	−0.948091
$446$	0	0
$447$	−22.0000	−1.04056
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−26.0000	−1.21490
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	−16.0000	−0.741982
$466$	−6.00000	−0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−16.0000	−0.738025
$471$	2.00000	0.0921551
$472$	−12.0000	−0.552345
$473$	−4.00000	−0.183920
$474$	8.00000	0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	16.0000	0.731823
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	2.00000	0.0912871
$481$	20.0000	0.911922
$482$	10.0000	0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−10.0000	−0.452679
$489$	4.00000	0.180886
$490$	14.0000	0.632456
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	6.00000	0.270501
$493$	−2.00000	−0.0900755
$494$	8.00000	0.359937
$495$	2.00000	0.0898933
$496$	−8.00000	−0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	20.0000	0.892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	2.00000	0.0885615
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	2.00000	0.0882162
$515$	16.0000	0.705044
$516$	−4.00000	−0.176090
$517$	−8.00000	−0.351840
$518$	0	0
$519$	18.0000	0.790112
$520$	4.00000	0.175412
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	−8.00000	−0.348485
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	−10.0000	−0.432742
$535$	24.0000	1.03761
$536$	4.00000	0.172774
$537$	4.00000	0.172613
$538$	−26.0000	−1.12094
$539$	7.00000	0.301511
$540$	2.00000	0.0860663
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	24.0000	1.03089
$543$	10.0000	0.429141
$544$	1.00000	0.0428746
$545$	−12.0000	−0.514024
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−6.00000	−0.256307
$549$	−10.0000	−0.426790
$550$	1.00000	0.0426401
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	−20.0000	−0.848953
$556$	20.0000	0.848189
$557$	46.0000	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$557$	46.0000	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$558$	−8.00000	−0.338667
$559$	−8.00000	−0.338364
$560$	0	0
$561$	1.00000	0.0422200
$562$	−6.00000	−0.253095
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	−8.00000	−0.336861
$565$	12.0000	0.504844
$566$	−28.0000	−1.17693
$567$	0	0
$568$	16.0000	0.671345
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−8.00000	−0.335083
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	2.00000	0.0836242
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	1.00000	0.0415945
$579$	6.00000	0.249351
$580$	4.00000	0.166091
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	−6.00000	−0.248495
$584$	2.00000	0.0827606
$585$	4.00000	0.165380
$586$	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	$0.696105\pi$
$588$	7.00000	0.288675
$589$	32.0000	1.31854
$590$	24.0000	0.988064
$591$	26.0000	1.06950
$592$	−10.0000	−0.410997
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	22.0000	0.901155
$597$	−16.0000	−0.654836
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	1.00000	0.0408248
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	16.0000	0.651031
$605$	−2.00000	−0.0813116
$606$	−6.00000	−0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	20.0000	0.809776
$611$	−16.0000	−0.647291
$612$	1.00000	0.0404226
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−4.00000	−0.161427
$615$	−12.0000	−0.483887
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	8.00000	0.321807
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	0	0
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−6.00000	−0.239808
$627$	−4.00000	−0.159745
$628$	−2.00000	−0.0798087
$629$	−10.0000	−0.398726
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−8.00000	−0.318223
$633$	4.00000	0.158986
$634$	22.0000	0.873732
$635$	−16.0000	−0.634941
$636$	−6.00000	−0.237915
$637$	14.0000	0.554700
$638$	2.00000	0.0791808
$639$	16.0000	0.632950
$640$	−2.00000	−0.0790569
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	12.0000	0.473602
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	−4.00000	−0.157378
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	1.00000	0.0392837
$649$	12.0000	0.471041
$650$	2.00000	0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	−6.00000	−0.234619
$655$	−24.0000	−0.937758
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−2.00000	−0.0778499
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	12.0000	0.466393
$663$	2.00000	0.0776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−10.0000	−0.387492
$667$	0	0
$668$	0	0
$669$	0	0
$670$	−8.00000	−0.309067
$671$	10.0000	0.386046
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−22.0000	−0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	−12.0000	−0.459841
$682$	8.00000	0.306336
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.00000	−0.152944
$685$	12.0000	0.458496
$686$	0	0
$687$	26.0000	0.991962
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−40.0000	−1.51729
$696$	2.00000	0.0758098
$697$	−6.00000	−0.227266
$698$	−18.0000	−0.681310
$699$	6.00000	0.226941
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	2.00000	0.0754851
$703$	40.0000	1.50863
$704$	−1.00000	−0.0376889
$705$	16.0000	0.602595
$706$	2.00000	0.0752710
$707$	0	0
$708$	12.0000	0.450988
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	−32.0000	−1.20094
$711$	−8.00000	−0.300023
$712$	10.0000	0.374766
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−4.00000	−0.149487
$717$	−16.0000	−0.597531
$718$	−8.00000	−0.298557
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	2.00000	0.0742781
$726$	−1.00000	−0.0371135
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	4.00000	0.147945
$732$	10.0000	0.369611
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−8.00000	−0.295285
$735$	−14.0000	−0.516398
$736$	0	0
$737$	−4.00000	−0.147342
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	20.0000	0.735215
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	8.00000	0.293294
$745$	−44.0000	−1.61204
$746$	−26.0000	−0.951928
$747$	−12.0000	−0.439057
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	8.00000	0.291730
$753$	−20.0000	−0.728841
$754$	4.00000	0.145671
$755$	−32.0000	−1.16460
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	8.00000	0.290191
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	8.00000	0.289430
$765$	−2.00000	−0.0723102
$766$	−24.0000	−0.867155
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−6.00000	−0.215945
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	24.0000	0.859889
$780$	−4.00000	−0.143223
$781$	−16.0000	−0.572525
$782$	0	0
$783$	2.00000	0.0714742
$784$	−7.00000	−0.250000
$785$	4.00000	0.142766
$786$	−12.0000	−0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−26.0000	−0.926212
$789$	−24.0000	−0.854423
$790$	16.0000	0.569254
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	20.0000	0.710221
$794$	14.0000	0.496841
$795$	12.0000	0.425596
$796$	16.0000	0.567105
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	10.0000	0.353112
$803$	−2.00000	−0.0705785
$804$	−4.00000	−0.141069
$805$	0	0
$806$	16.0000	0.563576
$807$	26.0000	0.915243
$808$	6.00000	0.211079
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	−2.00000	−0.0702728
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	10.0000	0.350500
$815$	8.00000	0.280228
$816$	−1.00000	−0.0350070
$817$	−16.0000	−0.559769
$818$	−38.0000	−1.32864
$819$	0	0
$820$	12.0000	0.419058
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	6.00000	0.209274
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	−8.00000	−0.278693
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	24.0000	0.833052
$831$	2.00000	0.0693792
$832$	−2.00000	−0.0693375
$833$	−7.00000	−0.242536
$834$	−20.0000	−0.692543
$835$	0	0
$836$	4.00000	0.138343
$837$	8.00000	0.276520
$838$	28.0000	0.967244
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	6.00000	0.206651
$844$	−4.00000	−0.137686
$845$	18.0000	0.619219
$846$	8.00000	0.275046
$847$	0	0
$848$	6.00000	0.206041
$849$	28.0000	0.960958
$850$	−1.00000	−0.0342997
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	−2.00000	−0.0682789
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	24.0000	0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	36.0000	1.22404
$866$	−14.0000	−0.475739
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	8.00000	0.271381
$870$	−4.00000	−0.135613
$871$	−8.00000	−0.271070
$872$	6.00000	0.203186
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	−16.0000	−0.539974
$879$	−6.00000	−0.202375
$880$	2.00000	0.0674200
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−7.00000	−0.235702
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−2.00000	−0.0672673
$885$	−24.0000	−0.806751
$886$	4.00000	0.134383
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	10.0000	0.335578
$889$	0	0
$890$	−20.0000	−0.670402
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−32.0000	−1.07084
$894$	−22.0000	−0.735790
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	10.0000	0.333704
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	6.00000	0.199889
$902$	6.00000	0.199778
$903$	0	0
$904$	−6.00000	−0.199557
$905$	20.0000	0.664822
$906$	−16.0000	−0.531564
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	4.00000	0.132453
$913$	12.0000	0.397142
$914$	−22.0000	−0.727695
$915$	−20.0000	−0.661180
$916$	−26.0000	−0.859064
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	14.0000	0.461065
$923$	−32.0000	−1.05329
$924$	0	0
$925$	10.0000	0.328798
$926$	16.0000	0.525793
$927$	−8.00000	−0.262754
$928$	−2.00000	−0.0656532
$929$	10.0000	0.328089	0.164045	−	0.986453i	$-0.447546\pi$
$929$	10.0000	0.328089	0.164045	+	0.986453i	$0.447546\pi$
$930$	−16.0000	−0.524661
$931$	28.0000	0.917663
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−36.0000	−1.17796
$935$	2.00000	0.0654070
$936$	−2.00000	−0.0653720
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	−16.0000	−0.521862
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	2.00000	0.0651635
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	8.00000	0.259828
$949$	−4.00000	−0.129845
$950$	4.00000	0.129777
$951$	−22.0000	−0.713399
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000	0.194257
$955$	−16.0000	−0.517748
$956$	16.0000	0.517477
$957$	−2.00000	−0.0646508
$958$	−24.0000	−0.775405
$959$	0	0
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	20.0000	0.644826
$963$	−12.0000	−0.386695
$964$	10.0000	0.322078
$965$	12.0000	0.386294
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	1.00000	0.0321412
$969$	4.00000	0.128499
$970$	−4.00000	−0.128432
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	32.0000	1.02535
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	4.00000	0.127906
$979$	−10.0000	−0.319601
$980$	14.0000	0.447214
$981$	6.00000	0.191565
$982$	−20.0000	−0.638226
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	6.00000	0.191273
$985$	52.0000	1.65686
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	2.00000	0.0635642
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−8.00000	−0.254000
$993$	−12.0000	−0.380808
$994$	0	0
$995$	−32.0000	−1.01447
$996$	12.0000	0.380235
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	44.0000	1.39280
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1122.2.a.e.1.1	✓	1
3.2	odd	2		3366.2.a.k.1.1		1
4.3	odd	2		8976.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1122.2.a.e.1.1	✓	1	1.1	even	1	trivial
3366.2.a.k.1.1		1	3.2	odd	2
8976.2.a.w.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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